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abstract: |
Let $T$ be Takagi’s continuous but nowhere-differentiable function. Using a representation in terms of Rademacher series due to N. Kono \[[*Acta Math. Hungar.*]{} [**49**]{} (1987)\], we give a complete characterization of those points where $T$ has a left-sided, right-sided, or two-sided infinite derivative. This characterization is illustrated by several examples. A consequence of the main result is that the sets of points where $T'(x)=\pm\infty$ have Hausdorff dimension one. As a byproduct of the method of proof, some exact results concerning the modulus of continuity of $T$ are also obtained.
: 26A27 (primary); 26A15 (secondary)
: Takagi’s function, Nowhere-differentiable function, Improper derivative, Modulus of continuity
title: |
The improper infinite derivatives of\
Takagi’s nowhere-differentiable function
---
Introduction
============
Takagi’s function is one of the simplest examples of a nowhere-differentiable continuous function. It was first discovered in 1903 [@Takagi], and is defined by $$T(x)=\sum_{n=1}^\infty \frac{1}{2^n} \phi^{(n)}(x), \qquad 0\leq x\leq 1,
\label{eq:takagi-function}$$ where $\phi^{(1)}:=\phi$ is the “tent map" defined by $$\phi(x):=\begin{cases}
2x, & \mbox{if $0\leq x\leq 1/2$},\\
2-2x, & \mbox{if $1/2\leq x\leq 1$};
\end{cases}$$ and inductively, $\phi^{(n)}:=\phi\circ\phi^{(n-1)}$ for $n\geq 2$. Takagi’s function was rediscovered independently by Van der Waerden, Hildebrandt, De Rham and others, and is known alternatively as Van der Waerden’s function. Although $T$ does not have a finite derivative anywhere, it is known to have an improper infinite derivative at many points. At which points exactly this is the case appeared to be settled in 1936 by Begle and Ayres [@Begle]. Let $O_n$ be the number of zeros, and $I_n=n-O_n$ the number of ones, among the first $n$ binary digits of $x$, and let $D_n=O_n-I_n$. Begle and Ayres claimed that $T'(x)=\infty$ if $D_n\to\infty$, and $T'(x)=-\infty$ if $D_n\to-\infty$. Unfortunately, in their proof they considered only the case $D_n\to\infty$, and only the right-hand derivative, believing the condition for the left-hand derivative to be the same. It is not. In fact, Kruppel [@Kruppel], unaware of Begle and Ayres’ paper, recently published a counterexample to their claim, which we explain in Section \[sec:example\] below.
The main purpose of the present article, then, is to give a complete characterization of those points $x$ at which $T$ has an improper infinite derivative. Guided by Kruppel’s counterexample, we replace the condition of Begle and Ayres by a stronger condition, expressed in terms of the binary expansion of $x$. Since the condition we obtain is somewhat intransparent, we illustrate it with several examples. This is done in Section \[sec:main-result\]. The main result is proved in Section \[sec:proof\], using a representation in terms of Rademacher series due to Kono [@Kono]. In Section \[sec:modulus\] we extend, with little extra effort, another recent result of Kruppel [@Kruppel] concerning the modulus of continuity of $T$.
Kruppel’s counterexample {#sec:example}
========================
The following example, which is essentially Example 7.2 of [@Kruppel], shows that $T'(x)$ may not exist even if $D_n\to\infty$. We present the argument here in a somewhat different (and, we hope, easier to visualize) form. This section may be skipped without loss of continuity. It does, however, lay down the basic idea upon which the proof of necessity in Section \[sec:proof\] is based.
Let $x=\sum_{n=1}^\infty 2^{-a_n}$, where $a_n=4^n$. For this $x$, we certainly have $D_n\to\infty$. A well-known formula for $T(x)$ at dyadic rational points is $$T\left(\frac{k}{2^m}\right)=\frac{1}{2^m}\sum_{j=0}^{k-1}(m-2s_j),
\label{eq:Takagi-dyadic}$$ where $s_j$ is the number of ones in the binary representation of the integer $j$. (See, e.g., [@Kruppel], p. 44.) For given $m$, let $k$ be the integer such that $k/2^m<x<(k+1)/2^m$. Then $$T\left(\frac{k+1}{2^m}\right)-T\left(\frac{k}{2^m}\right)=\frac{1}{2^m}(m-2s_k)=\frac{1}{2^m} D_m,$$ so the secant slopes over the dyadic intervals $[k/2^m,(k+1)/2^m]$ containing $x$ indeed tend to $+\infty$. However, if put $m=a_{n+1}-1$, then $s_k=n$ whereas $s_{k-1}=n+a_{n+1}-a_n-2$ and $s_{k-2}=n+a_{n+1}-a_n-3$. Thus, a simple calculation using yields $$\begin{aligned}
2^m\left[T\left(\frac{k+1}{2^m}\right)-T\left(\frac{k-2}{2^m}\right)\right]
&=3m-2s_k-2s_{k-1}-2s_{k-2}\\
&=4a_n-a_{n+1}-6n+7\to-\infty,\end{aligned}$$ as $n\to\infty$. Since the intervals $[(k-2)/2^m,(k+1)/2^m]$ also contain $x$, it follows that $T$ cannot have an infinite derivative at $x$.
It is easy to imagine how this idea can be extended for sequences $\{a_n\}$ which do not grow quite as fast as $4^n$, by enlarging the intervals even further to the left; that is, we can take the secant slopes over $[(k-j)/2^m,(k+1)/2^m]$ where $j=3,4,\dots$. In fact, we can even let $j$ depend on $m$. It is essentially this realization that lead us to the correct condition for the existence of an improper derivative at a point $x$, as stated in the next section. But, since we wish to consider the left-hand and right-hand derivatives separately, we will use a slightly different approach that does not make use of .
Improper derivatives {#sec:main-result}
====================
Define $$\begin{aligned}
T'_{+}(x)&:=\lim_{h \downarrow 0} \frac{T(x+h)-T(x)}{h}, \\
T'_{-}(x)&:=\lim_{h \uparrow 0} \frac{T(x+h)-T(x)}{h},\end{aligned}$$ provided each limit exists as an extended real number. It has been pointed out by various authors (e.g. [@Begle; @Kruppel]) that if $x$ is a dyadic rational (that is, a point of the form $x=k/2^m$), then $T'_+(x)=+\infty$ and $T'_-(x)=-\infty$. We now treat the non-dyadic case.
\[thm:main\] Let $x\in(0,1)$ be non-dyadic, and write $$x=\sum_{n=1}^\infty 2^{-a_n}, \qquad 1-x=\sum_{n=1}^\infty 2^{-b_n},
\label{eq:expansions}$$ where $\{a_n\}$ and $\{b_n\}$ are strictly increasing sequences of positive integers, determined uniquely by $x$. Then:
(i) $T'_+(x)=+\infty$ if and only if $a_n-2n\to\infty$.
(ii) $T'_-(x)=+\infty$ if and only if $$a_{n+1}-2a_n+2n-\log_2(a_{n+1}-a_n)\to-\infty.
\label{eq:NS-condition+}$$
(iii) $T'_+(x)=-\infty$ if and only if $$b_{n+1}-2b_n+2n-\log_2(b_{n+1}-b_n)\to-\infty.
\label{eq:NS-condition-}$$
(iv) $T'_-(x)=-\infty$ if and only if $b_n-2n\to\infty$.
In the notation of Theorem \[thm:main\], we have:
(i) $T'(x)=+\infty$ if and only if holds;
(ii) $T'(x)=-\infty$ if and only if holds.
The condition implies that $a_n-2n\to\infty$. For: $$\begin{aligned}
a_{n+1}-2a_n+2n-&\log_2(a_{n+1}-a_n)\\
&=a_{n+1}-a_n-\log_2(a_{n+1}-a_n)-(a_n-2n)\\
&\geq -(a_n-2n).\end{aligned}$$ This gives the first statement; the second follows by symmetry.
[The condition $a_n-2n\to\infty$ is equivalent to the condition of Begle and Ayres. First, if $D_k\to\infty$, then $a_n-2n=D_{a_n}\to\infty$. Conversely, suppose $a_n-2n\to\infty$. Then for $a_n\leq k<a_{n+1}$, $$D_k=k-2I_k=k-2n\geq a_n-2n\to\infty.$$ Conditions and , on the other hand, may look a bit mysterious. The examples below aim to provide more insight into their meaning. Since the conditions are quite analogous, we focus on . ]{}
[If the number of consecutive zeros in the binary expansion of $x$ is bounded, say by $M$, then $a_{n+1}-a_n\leq M+1$, and so $$a_{n+1}-2a_n+2n-\log_2(a_{n+1}-a_n)\leq M+1-(a_n-2n).$$ Thus, we obtain Kruppel’s result [@Kruppel Proposition 5.3]: if $D_n\to\infty$ and the number of consecutive $0$’s in the binary expansion of $x$ is bounded, then $T'(x)=+\infty$. Similarly, if $D_n\to-\infty$ and the number of consecutive $1$’s is bounded, then $T'(x)=-\infty$. ]{}
[If $\limsup_{n\to\infty}a_{n+1}/a_n>2$, then fails. To see this, write $a_{n+1}=\lambda_n a_n$. Then, whenever $\lambda_n\geq 2+{{\varepsilon}}$, $$\begin{aligned}
a_{n+1}-2a_n+2n-&\log_2(a_{n+1}-a_n)\\
&=(\lambda_n-2)a_n+2n-\log_2((\lambda_n-1)a_n)\\
&\geq(\lambda_n-2)a_n+2n-\log_2((\lambda_n-2)a_n)-(2/{{\varepsilon}})\\
&\geq 2n-(2/{{\varepsilon}})\to\infty,\end{aligned}$$ where the first inequality follows since, by the mean value theorem, $$\log_2(\lambda_n-1)-\log_2(\lambda_n-2)\leq\frac{1}{(\lambda_n-2)\log 2}<2/{{\varepsilon}}.$$ Thus, even if the $4$ in Kruppel’s counterexample in Section \[sec:example\] is replaced by a smaller number $\gamma>2$, $T$ will not have an improper derivative at $x$. ]{}
[On the other hand, a sufficient condition for to hold is that, for some $0<{{\varepsilon}}\leq 1$, $$\limsup_{n\to\infty}\frac{a_{n+1}}{a_n}=2-{{\varepsilon}}\qquad\mbox{and}\qquad \liminf_{n\to\infty}\frac{a_n}{n}>\frac{2}{{{\varepsilon}}}.
\label{eq:sufficient}$$ (We leave the easy verification to the reader.) Thus, for instance, holds for $a_n=3n$; for any increasing polynomial of degree $2$ or higher; and for any exponential sequence $a_n=\lfloor\alpha^n\rfloor$ with $1<\alpha<2$. As another example, let $a_n$ be the $n$-th prime number; then $a_n/n\log n\to 1$ by the Prime Number Theorem. Thus, $a_n$ satisfies with ${{\varepsilon}}=1$, and hence it satisfies . ]{}
If $\limsup_{n\to\infty}a_{n+1}/a_n=2$, then a finer examination of the asymptotics of the sequence $\{a_n\}$ is necessary, as the next example shows.
[The sequence $a_n=2^n$ does not satisfy ; neither does $a_n=2^n+n$. But $a_n=2^n+(1+{{\varepsilon}})n$ satisfies for any ${{\varepsilon}}>0$: $$\begin{aligned}
a_{n+1}-2a_n+2n-\log_2(a_{n+1}-a_n)
&=(1-{{\varepsilon}})n+1+{{\varepsilon}}-\log_2(2^n+1+{{\varepsilon}})\\
&\leq -{{\varepsilon}}n+1+{{\varepsilon}}\to-\infty.\end{aligned}$$ ]{}
This last example also illustrates that the logarithmic term in can sometimes be of critical importance.
An important subset of $[0,1]$ is formed by the points $x$ whose binary expansion has a density; that is, points $x=\sum_{k=1}^\infty 2^{-k}{{\varepsilon}}_k$ for which the limit $$d_1(x):=\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}{{\varepsilon}}_k
\label{eq:density1}$$ exists. Note that $d_1(x)$ expresses the long-run proportion of $1$’s in the binary expansion of $x$. If it exists, we define $$d_0(x):=1-d_1(x)
\label{eq:density0}$$ to denote the long-run proportion of $0$’s. An immediate consequence of Theorem \[thm:main\] is that $T(x)$ has an infinite derivative at the majority of points $x$ for which $d_1(x)$ exists.
\[cor:density\] Let $x$ be a non-dyadic point and suppose $d_1(x)$ exists. If either
(a) $0<d_1(x)<1/2$, or
(b) $d_1(x)=0$ and $\limsup_{n\to\infty}a_{n+1}/a_n<2$,
then $T'(x)=+\infty$. Similarly, if either
(a) $1/2<d_1(x)<1$, or
(b) $d_1(x)=1$ and $\limsup_{n\to\infty}b_{n+1}/b_n<2$,
then $T'(x)=-\infty$.
By the definition , $n/a_n\to d_1(x)$ and $n/b_n\to d_1(1-x)=d_0(x)$. In particular, if $0<d_1(x)<1$, it follows that $a_{n+1}/a_n\to 1$ and $b_{n+1}/b_n\to 1$. Thus, under the conditions of the corollary, (or its analog for the sequence $\{b_n\}$) is satisfied.
Corollary \[cor:density\] has a remarkable consequence for the Hausdorff dimension of the set of points where $T'(x)=\pm\infty$. (See [@Falconer] for the definition and basic properties of Hausdorff dimension.)
Let $S_\infty=\{x\in[0,1]:T'(x)=\infty\}$, and $S_{-\infty}=\{x\in[0,1]:T'(x)=-\infty\}$. Then $$\dim_H S_\infty=\dim_H S_{-\infty}=1,$$ where $\dim_H$ denotes the Hausdorff dimension.
By Corollary \[cor:density\], $S_\infty$ contains the sets $F(\alpha):=\{x\in[0,1]:d_1(x)=\alpha\}$, for $0<\alpha<1/2$. It is well known that $$\dim_H F(\alpha)=\frac{-\alpha\log(\alpha)-(1-\alpha)\log(1-\alpha)}{\log2}$$ (see [@Falconer Proposition 10.1]). Thus, $$\dim_H S_\infty \geq \dim_H \bigcup_{0<\alpha < 1/2}F(\alpha)
=\sup_{0<\alpha < 1/2} \dim_H F(\alpha) = 1.$$ The dimension of $S_{-\infty}$ follows in the same way.
Corollary \[cor:density\] left out the (binary) [*normal numbers*]{}; that is, those numbers $x$ for which $d_1(x)=1/2$. These numbers form a set of Lebesgue measure one by Borel’s theorem. However, the law of the iterated logarithm implies that for almost all of those, $\limsup_{n\to\infty}D_n=+\infty$ and $\liminf_{n\to\infty}D_n=-\infty$. Hence, at almost all normal numbers, $T$ does not even have a one-sided infinite derivative. Less extremely, for any [*rational*]{} normal number $x$ (such as $x=1/3$), $D_n$ oscillates between finite bounds so that $T'(x)$ does not exist.
Nonetheless, many normal numbers satisfy $T'(x)=\pm\infty$. For instance, if $a_n=2n+\lfloor \sqrt{n}\rfloor$ or $a_n=2n+\lfloor\log n\rfloor$, it is easy to see that is satisfied. On the other hand, surprisingly perhaps, there exist normal numbers for which $T_+'(x)=+\infty$, but $T_-'(x)$ fails to exist. Here we construct one such example.
[Let $a_1=3$, and for $n\geq 1$, define $a_{n+1}$ recursively as follows. If $a_n\leq 2n+\lfloor\sqrt{n}\rfloor$, put $a_{n+1}=2n+3\lfloor\sqrt{n}\rfloor$; otherwise, put $a_{n+1}=a_n+1$. Since $a_n$ always increases by at least $1$ and $2n+\lfloor\sqrt{n}\rfloor$ increases by at most $3$ at each step, it is clear that for every $n$, $a_n\geq 2n+\lfloor\sqrt{n}\rfloor-1$. Hence $a_n-2n\to\infty$. Furthermore, $a_n\leq 2n+3\lfloor\sqrt{n}\rfloor$ for each $n$, and so $a_n/n\to 2$. Finally, it is easy to check that $a_n\leq 2n+\lfloor\sqrt{n}\rfloor$ for infinitely many $n$. Thus, infinitely often, $$\begin{aligned}
a_{n+1}-2a_n+2n-&\log_2(a_{n+1}-a_n)\\
&\geq 2n+3\lfloor\sqrt{n}\rfloor-2(2n+\lfloor\sqrt{n}\rfloor)+2n-\log_2(2\lfloor\sqrt{n}\rfloor+1)\\
&\geq\lfloor\sqrt{n}\rfloor-\frac12\log n-2\to\infty.\end{aligned}$$ ]{}
Proof of the main theorem {#sec:proof}
=========================
To prove Theorem \[thm:main\] we will use an approach by N. Kono [@Kono]. Let $x$ and $h$ be real numbers such that $0\leq x<x+h<1$, and write $$x=\sum_{k=1}^\infty 2^{-k}{{\varepsilon}}_k, \qquad x+h=\sum_{k=1}^\infty 2^{-k}{{\varepsilon}}_k',$$ where ${{\varepsilon}}_k,{{\varepsilon}}_k'\in\{0,1\}$. When $x$ is dyadic rational, there are two binary expansions, but we choose the one which is eventually all zeros.
For $h>0$, let $p:=p(h) \in {\mathrm{I\!N}}$ such that $2^{-p-1}<h\leq 2^{-p}$ and let $$k_0:=\max\{k: {{\varepsilon}}_1={{\varepsilon}}_1',\dots,{{\varepsilon}}_k={{\varepsilon}}_k'\}$$ (or $k_0=0$ if ${{\varepsilon}}_1\neq {{\varepsilon}}_1'$). Clearly $0\leq k_0\leq p$, and we have the implications $$\begin{aligned}
k_0<p \quad&\Rightarrow\quad {{\varepsilon}}_{k_0+1}=0 \mbox{ and } {{\varepsilon}}_{k_0+1}'=1, \label{eq:kono-fact1}\\
k_0+2\leq p \quad&\Rightarrow\quad {{\varepsilon}}_k'=0 \mbox{ and } {{\varepsilon}}_k=1 \quad \mbox{for} \quad k_0+2\leq k\leq p. \label{eq:kono-fact2}\end{aligned}$$
Observe that by the assumption for the expression of $x$, $k_0 \to \infty$ as $h\downarrow 0$. Let $X_n(x):=1-2{{\varepsilon}}_n(x)=(-1)^{{{\varepsilon}}_n(x)}$ denote the $n$-th Rademacher function. For $h>0$, the following representation is a special case of Lemma 3 in [@Kono]: $$T(x+h)-T(x)=\Sigma_1+\Sigma_2+\Sigma_3,$$ where $$\begin{gathered}
\Sigma_1=h\sum_{n=1}^{k_0}X_n(x)=hD_{k_0},\\
\Sigma_2=\left[\sum_{k=p+1}^\infty 2^{-k}(1-{{\varepsilon}}_k-{{\varepsilon}}_k')\right]\sum_{n=k_0+1}^p X_n(x), \\
\Sigma_3=\frac12\sum_{n=p+1}^{\infty}\sum_{k=n+1}^{\infty}\left[X_n(x)X_k(x)-X_n(x+h)X_k(x+h)\right]2^{-k}.\end{gathered}$$ Since $\Sigma_3=O(h)$, it plays no role in determining whether $T'_+(x)=\pm \infty$. In fact, for many points $x$ the behavior of the difference quotient is controlled by $\Sigma_1$ alone, but in some cases, $\Sigma_2$ may be of the same order of magnitude but with the opposite sign. The key to the proof of Theorem \[thm:main\], then, is a careful analysis of this ‘rogue’ term, especially the factor $\sum_{k=p+1}^\infty 2^{-k}(1-{{\varepsilon}}_{k}-{{\varepsilon}}_k')$. Note that the other factor can be written more simply: if $k_0<p$, then $$\sum_{n=k_0+1}^p X_n(x)=-(p-k_0-2),
\label{eq:middle-sum}$$ in view of and .
\[lem:key-inequality\] Assume $k_0<p$. Then $$\sum_{k=p+1}^\infty 2^{-k}(1-{{\varepsilon}}_{k}-{{\varepsilon}}_k')\leq h.$$ Moreover, if $m\geq 0$ such that ${{\varepsilon}}_{p+m+1}=0$, then $$\sum_{k=p+1}^\infty 2^{-k}(1-{{\varepsilon}}_{k}-{{\varepsilon}}_k')\geq -h(1-2^{-m}).$$
If $h<2^{-p}$, then a “$1$" is carried from position $p+1$ to position $p$ in the addition of $x$ and $h$ because of the assumption that $k_0<p$. If $h=2^{-p}$, then ${{\varepsilon}}_k={{\varepsilon}}_k'$ for all $k>p$. In both cases, we have $$\sum_{k=p+1}^\infty 2^{-k}{{\varepsilon}}_k+h=2^{-p}+\sum_{k=p+1}^\infty 2^{-k}{{\varepsilon}}_k',
\label{eq:addition}$$ and so $$h-\sum_{k=p+1}^\infty 2^{-k}(1-{{\varepsilon}}_{k}-{{\varepsilon}}_k')
=2^{-p}+\sum_{k=p+1}^\infty 2^{-k}(2{{\varepsilon}}_k'-1)\geq 0.$$ For the second statement, observe that $h(1-2^{-m})\geq h-2^{-m-p}$ since $h\leq 2^{-p}$. Thus, using we obtain $$\begin{aligned}
h(1-2^{-m})+\sum_{k=p+1}^\infty &2^{-k}(1-{{\varepsilon}}_{k}-{{\varepsilon}}_k')\\
&\geq 2^{-p}+\sum_{k=p+1}^\infty 2^{-k}(1-2{{\varepsilon}}_k)-2^{-m-p}\\
&\geq 2^{-p}+\sum_{k=p+1}^\infty 2^{-k}(-1)+2\cdot 2^{-(p+m+1)}-2^{-m-p}=0,\end{aligned}$$ where the last inequality follows since ${{\varepsilon}}_{p+m+1}=0$.
\[lem:maximize\] Let $c\geq 1$, and define the function $f:\{0,1,2,\dots\}\to{\mathrm{I\!R}}$ by $$f(m)=(1-2^{-m})(c-m).$$ Let $m^*$ be the largest integer $m$ where $f(m)$ attains its maximum. Then $$\log_2 c-2<m^*\leq \log_2 c+1.$$
An easy calculation gives $$f(m+1)-f(m)=2^{-(m+1)}(c+1-m)-1,$$ so $f(m+1)\geq f(m)$ if and only if $2^{m+1}+m\leq c+1$. Thus, $$\begin{gathered}
2^{m^*}+m^*-1\leq c+1, \label{eq:lower-estimate}\\
2^{m^*+1}+m^*>c+1. \label{eq:upper-estimate}\end{gathered}$$ From we obtain $m^*\leq\log_2 c+1$. On the other hand, yields $$2^{m^*+2}>2^{m^*+1}+m^*>c,$$ so that $m^*>\log_2 c-2$.
Since $T(1-x)=T(x)$, it is enough to prove parts (i) and (iii). Statements (ii) and (iv) follow from these by replacing $x$ with $1-x$.
We first prove part (i). Assume $D_n\to\infty$, and let $h>0$. Suppose first that $k_0\leq p-2$. Then $p-k_0-2\geq 0$, so it follows from and Lemma \[lem:key-inequality\] that $$\Sigma_2\geq -h(p-k_0-2).$$ And, since $O_{k_0}=O_p-1$, $$\Sigma_1=h(O_{k_0}-I_{k_0})=h(2O_{k_0}-k_0)=h(2O_p-k_0-2),$$ so that $$\begin{aligned}
T(x+h)-T(x)&=\Sigma_1+\Sigma_2+\Sigma_3\\
&\geq h(2O_p-k_0-2)-h(p-k_0-2)+O(h)\\
&=h(2O_p-p)+O(h)=hD_p+O(h).\end{aligned}$$ If, on the other hand, $k_0\geq p-1$, then $\Sigma_2=O(h)$, and $O_{k_0}\geq O_p-1$. Thus, $$T(x+h)-T(x)=\Sigma_1+O(h)\geq h(D_p-2)+O(h).$$ In both cases, $$\frac{T(x+h)-T(x)}{h}\geq D_p+O(1),$$ and hence, $T_+'(x)=+\infty$.
Conversely, suppose $\liminf_{n\to\infty}D_n<\infty$. Choose a sequence $\{n_k\}$ increasing to $+\infty$ so that $\lim_{k\to\infty}D_{n_k}<\infty$, and let $p_k:=\min\{n\geq n_k:{{\varepsilon}}_n=0\}$. For $h=2^{-p_k}$ and $p=p_k$, we have $k_0=p-1$ and so $\Sigma_2=O(h)$. Hence $T(x+h)-T(x)=\Sigma_1+O(h)=hD_{k_0}+O(h)$. Since $D_{k_0}=D_{p_k}-1\leq D_{n_k}$, it follows that $$\liminf_{h\downarrow 0}\frac{T(x+h)-T(x)}{h}\leq\lim_{k\to\infty}D_{n_k}+O(1)<\infty.$$
Next, we prove statement (iii). Suppose first that holds. Let $h>0$, $2^{-p-1}<h\leq 2^{-p}$, and let $n$ be the integer such that $b_n\leq p<b_{n+1}$. Put $m=b_{n+1}-p-1$; then $p+m+1=b_{n+1}$ and so ${{\varepsilon}}_{p+m+1}=0$, since the $b_n$’s indicate the locations of the zeros in the binary expansion of $x$. Now it follows from that $b_n-2n\to\infty$, or equivalently, $D_k\to-\infty$. If $p-k_0<2$, then $\Sigma_1=hD_{k_0}$ and $\Sigma_2=O(h)$, and so $$\frac{T(x+h)-T(x)}{h}=D_{k_0}+O(1)\to-\infty.$$ Assume then that $p-k_0\geq 2$. By and , $k_0=b_n-1$, and since $O_{b_n}=n$, we have $$\Sigma_1=h(2O_{k_0}-k_0)\leq h(2O_{b_n}-k_0)=h(2n-k_0).$$ As for $\Sigma_2$, Lemma \[lem:key-inequality\] gives $$\Sigma_2\leq h(1-2^{-m})(p-k_0-2)\leq h(1-2^{-m})(p-k_0).$$ Hence, $$\begin{aligned}
\begin{split}
\frac{T(x+h)-T(x)}{h}&\leq 2n-k_0+(1-2^{-m})(p-k_0)+O(1)\\
&=2n-b_n+(1-2^{-m})(b_{n+1}-b_n-m)+O(1).
\end{split}
\label{eq:high-estimate}\end{aligned}$$ For given $n$, let $m_n$ be the largest value of $m$ which maximizes the function $$f_n(m)=(1-2^{-m})(b_{n+1}-b_n-m).$$ By Lemma \[lem:maximize\] we have, for any $m$, $$\begin{aligned}
2n-b_n+f_n(m)&\leq b_{n+1}-2b_n+2n-m_n\\
&\leq b_{n+1}-2b_n+2n-\log_2(b_{n+1}-b_n)+2.\end{aligned}$$ This, in combination with ,, and the already established result for the case $p-k_0<2$, yields $T_+'(x)=-\infty$.
For the converse, assume that fails. Suppose first that $D_n\to-\infty$, or equivalently, $b_n-2n\to\infty$. Replacing the sequence $\{b_n\}$ with a suitable subsequence if necessary, we may assume there exists $M\in{\mathrm{I\!R}}$ such that $$b_{n+1}-2b_n+2n-\log_2(b_{n+1}-b_n)>M \qquad\mbox{for all $n$}.
\label{eq:bounded-below}$$ Fix $n\in{\mathrm{I\!N}}$ temporarily, let $m=m_n$, and let $h=2^{-p}$, where $p=b_{n+1}-m$. By Lemma \[lem:maximize\], $$\begin{aligned}
b_{n+1}-m-b_n&\geq b_{n+1}-b_n-\log_2(b_{n+1}-b_n)-1\\
&>M+(b_n-2n)-1\to\infty.\end{aligned}$$ Thus, for all sufficiently large $n$, $b_n<p<b_{n+1}$. Therefore $k_0=b_n-1$, and $$\Sigma_1=h(2O_{k_0}-k_0)=h(2n-b_n-1).$$ Furthermore, $p-k_0-2=b_{n+1}-b_n-m-1\geq 0$ for $n$ large enough, and $$\begin{aligned}
\sum_{k=p+1}^\infty 2^{-k}(1-{{\varepsilon}}_k-{{\varepsilon}}_k')
&\leq -\sum_{k=p+1}^{b_{n+1}-1}2^{-k}+\sum_{k=b_{n+1}}^\infty 2^{-k}\\
&=-2^{-p}\left(1-2^{-(b_{n+1}-p-2)}\right)=-h\left(1-2^{-(m-2)}\right),\end{aligned}$$ where the inequality follows since ${{\varepsilon}}_k={{\varepsilon}}_k'=1$ for $k=p+1,\dots,b_{n+1}-1$. Hence $$\Sigma_2\geq h\left(1-2^{-(m-2)}\right)(b_{n+1}-b_n-m-1).$$ Putting these results together, we obtain $$\begin{aligned}
\frac{T(x+h)-T(x)}{h}&\geq 2n-b_n-1+\left(1-2^{-(m-2)}\right)(b_{n+1}-b_n-m-1)+O(1)\\
&\geq b_{n+1}-2b_n+2n-m-2^{-(m-2)}(b_{n+1}-b_n)+O(1).\end{aligned}$$ By Lemma \[lem:maximize\], $m\leq\log_2(b_{n+1}-b_n)+1$, and the term $2^{-(m-2)}(b_{n+1}-b_n)$ is bounded. Thus, $$\frac{T(x+h)-T(x)}{h}\geq b_{n+1}-2b_n+2n-\log_2(b_{n+1}-b_n)+O(1),
\label{eq:DQ-lower-bound}$$ which is bounded below, by .
If $\limsup_{n\to\infty}D_n>-\infty$, then we can choose a sequence $\{n_k\}$ increasing to $+\infty$ so that $\lim_{k\to\infty}D_{n_k}>-\infty$. Let $p_k:=\max\{n\leq n_k:{{\varepsilon}}_n=0\}$; then $D_{p_k}\geq D_{n_k}$. For $h=2^{-p_k}$ and $p=p_k$, we have $k_0=p-1$ and so $\Sigma_2=O(h)$. Hence $T(x+h)-T(x)=hD_{k_0}+O(h)$. Since $D_{k_0}=D_{p_k}-1\geq D_{n_k}-1$, it follows that $$\limsup_{h\downarrow 0}\frac{T(x+h)-T(x)}{h}\geq\lim_{k\to\infty}D_{n_k}+O(1)>-\infty,$$ completing the proof.
The modulus of continuity {#sec:modulus}
=========================
In this final section we present some exact results concerning the modulus of continuity of $T$. Let $d_1(x)$ and $d_0(x)$ denote the densities of $1$ and $0$ in the binary expansion of $x$, respectively, provided they exist (see and ). Kruppel [@Kruppel] recently proved that if $x$ is dyadic, then $$\lim_{h \to 0} \frac{T(x+h)-T(x)}{|h| \log_2(1/|h|)}=1,
\label{eq:dyadic-case}$$ while if $x$ is non-dyadic but rational, then $$\lim_{h \to 0} \frac{T(x+h)-T(x)}{h \log_2(1/|h|)}= 1-\frac{2({{\varepsilon}}_{k+1}+{{\varepsilon}}_{k+2}+\cdots +{{\varepsilon}}_{k+m})}{m},
\label{eq:modulus-limit}$$ where ${{\varepsilon}}_{k+1} {{\varepsilon}}_{k+2} \cdots {{\varepsilon}}_{k+m}$ is a repeating part in the binary expansion of $x$. Observe that for an $x$ of the latter type, $d_1(x)=({{\varepsilon}}_{k+1}+{{\varepsilon}}_{k+2}+\cdots +{{\varepsilon}}_{k+m})/m$, so the right hand side of can be written as $d_0(x)-d_1(x)$. Here we will give a simpler proof of , and generalize to arbitrary real numbers. More precisely, we give a complete characterization of the set of points $x$ for which the limit in exists, and show that if it does, it must equal $d_0(x)-d_1(x)$.
A point $x\in[0,1]$ is [*density-regular*]{} if $d_1(x)$ exists and one of the following holds:
(a) $0<d_1(x)<1$; or
(b) $d_1(x)=0$ and $a_{n+1}/a_n\to 1$; or
(c) $d_1(x)=1$ and $b_{n+1}/b_n\to 1$.
Here, $\{a_n\}$ and $\{b_n\}$ are the sequences determined by .
\[lem:right-limit\] Let $x\in[0,1]$ and suppose $d_1(x)$ exists.
\(i) If $d_1(x)<1$, then $$\lim_{h \downarrow 0} \frac{T(x+h)-T(x)}{h\log_2(1/|h|)}=d_{0}(x)-d_{1}(x).
\label{eq:exact-modulus}$$
\(ii) Suppose $d_1(x)=1$. Then $$\lim_{h \downarrow 0} \frac{T(x+h)-T(x)}{h\log_2(1/|h|)} \quad\mbox{exists}$$ if and only if $b_{n+1}/b_n\to 1$, in which case the limit is equal to $-1$.
Assume throughout that $d_1(x)$ exists. If $d_1(x)<1$, then $b_{n+1}/b_n\to 1$ holds automatically (see the proof of Corollary \[cor:density\]). Thus, we can prove the two statements by a single argument.
Suppose first that $b_{n+1}/b_n\to 1$. Then $k_0/p\to 1$ as $h\downarrow 0$. We can write $\Sigma_1=hD_{k_0}=h(O_{k_0}-I_{k_0})$. Since $p \leq \log_2(1/|h|)<p+1$ and $$\lim_{k_0 \to \infty}\frac{O_{k_0}-I_{k_0}}{k_0}=d_0(x)-d_1(x),$$ it follows that $$\lim_{h \downarrow 0} \frac{\Sigma_1}{h\log_2(1/|h|)}=d_{0}(x)-d_{1}(x).$$
Next, by Lemma \[lem:key-inequality\] and , we have $|\Sigma_2|\leq h(p-k_0)$, and hence, $$\frac{|\Sigma_2|}{h\log_2(1/|h|)}\leq \frac{p-k_0}{p}\to 0.$$ Finally, since $\Sigma_3=O(h)$, follows.
Conversely, suppose $d_1(x)=1$ and $b_{n+1}/b_n$ does not tend to $1$; in other words, $\limsup_{n\to\infty}b_n/b_{n+1}<1$. On the one hand, we can choose an increasing index sequence $\{p_n\}$ such that ${{\varepsilon}}_{p_n}=0$ for each $n$; such a sequence exists even if $x$ is dyadic, in view of our convention of choosing the representation ending in all zeros for such points. Put $h_n:=2^{-p_n}$. Then $k_0=p_n-1$, so $\Sigma_2=O(h_n)$ and $$\lim_{n\to\infty}\frac{T(x+h_n)-T(x)}{h_n\log_2(1/|h_n|)}=\lim_{n\to\infty}\frac{\Sigma_1}{h_n\log_2(1/|h_n|)}=-1,$$ as above. On the other hand, we can let $h$ approach $0$ along a sequence $\{h_n\}$ just as in the last part of the proof of Theorem \[thm:main\]. Since $p=b_{n+1}-m_n\sim b_{n+1}$, dividing both sides by $\log_2(1/|h|)$ in gives $$\liminf_{n\to\infty}\frac{T(x+h_n)-T(x)}{h_n\log_2(1/|h_n|)}\geq \liminf_{n\to\infty}\frac{b_{n+1}-2b_n}{b_{n+1}}>-1,$$ since the remaining terms in are of smaller order than $b_{n+1}$ in view of $n/b_n\to 0$. Thus, the limit in does not exist.
\[cor:Kruppel\] If $x$ is dyadic, then holds.
If $x$ is dyadic, then $d_1(x)=d_1(1-x)=0$. Thus, the statement follows by applying Lemma \[lem:right-limit\] first to $x$ and then to $1-x$, since for $h<0$, $$\frac{T(x+h)-T(x)}{|h|\log_2(1/|h|)}
=\frac{T(1-x+|h|)-T(1-x)}{|h|\log_2(1/|h|)},$$ by the symmetry of $T$.
For non-dyadic $x$, we obtain the following result. Before stating it we observe that, if $\lim_{n\to\infty}n/b_n=d$, then $d_0(x)$ exists and is equal to $d$. (This is straightforward to verify.)
\[thm:kiko\] Let $x$ be non-dyadic, and define $a_n$ and $b_n$ as in . The limit $$\lim_{h \to 0} \frac{T(x+h)-T(x)}{h \log_2(1/|h|)}
$$ exists if and only if $x$ is density-regular, in which case the limit is equal to $d_0(x)-d_1(x)$.
If $d_1(x)$ exists, the result follows by applying Lemma \[lem:right-limit\] first to $x$ and then to $1-x$, since $d_1(x)=1-d_1(1-x)$.
Suppose $d_1(x)$ does not exist. For $n\in{\mathrm{I\!N}}$, let $p=b_n$ and $h=2^{-p}$; then $k_0=p-1$, so $\Sigma_2=O(h)$ and $\Sigma_1=h D_{k_0}=h(D_p-1)=h(2n-b_n-1)$. Thus, $$\frac{T(x+h)-T(x)}{h \log_2(1/|h|)}=\frac{2n-b_n-1}{b_n}+o(1)=2\frac{n}{b_n}-1+o(1),$$ which does not have a limit as $n\to\infty$.
[5]{}
and [W. L. Ayres]{}, On Hildebrandt’s example of a function without a finite derivative, [*Amer. Math. Monthly*]{} [**43**]{} (1936), no. 5, 294-296.
, [*Fractal Geometry. Mathematical Foundations and Applications*]{}, 2nd Edition, Wiley (2003)
, On generalized Takagi functions, [*Acta Math. Hungar.*]{} [**49**]{} (1987), 315-324.
, On the extrema and the improper derivatives of Takagi’s continuous nowhere differentiable function, [*Rostock. Math. Kolloq*]{} [**62**]{} (2007), 41-59.
, A simple example of the continuous function without derivative, [*Phys.-Math. Soc. Japan*]{} [**1**]{} (1903), 176-177. [*The Collected Papers of Teiji Takagi*]{}, S. Kuroda, Ed., Iwanami (1973), 5-6.
|
---
abstract: |
We report the discovery of [[*Kepler*]{}-432]{}[b]{}, a giant planet (${\ifmmode{M_{\rm b}}\else $M_{\rm b}$\fi}=
5.41^{+0.32}_{-0.18}~{\ifmmode{M_{\rm Jup}}\else $M_{\rm Jup}$\fi}$, ${\ifmmode{R_{\rm b}}\else $R_{\rm b}$\fi}= 1.145^{+0.036}_{-0.039}~{\ifmmode{R_{\rm Jup}}\else $R_{\rm Jup}$\fi}$) transiting an evolved star ($M_\star = 1.32^{+0.10}_{-0.07}~{\ifmmode{M_{\odot}}\else $M_{\odot}$\fi},
R_\star = 4.06^{+0.12}_{-0.08}~{\ifmmode{R_{\odot}}\else $R_{\odot}$\fi}$) with an orbital period of $P_{\rm b} = 52.501129^{+0.000067}_{-0.000053}$ days. Radial velocities (RVs) reveal that [[*Kepler*]{}-432]{}[b]{} orbits its parent star with an eccentricity of $e=0.5134^{+0.0098}_{-0.0089}$, which we also measure independently with asterodensity profiling (AP; $e=0.507^{+0.039}_{-0.114}$), thereby confirming the validity of AP on this particular evolved star. The well determined planetary properties and unusually large mass also make this planet an important benchmark for theoretical models of super-Jupiter formation. Long-term RV monitoring detected the presence of a non-transiting outer planet ([[*Kepler*]{}-432]{}[c]{}; ${\ifmmode{M_{\rm c}}\else $M_{\rm c}$\fi}\sin{i_{\rm c}} = 2.43^{+0.22}_{-0.24}~{\ifmmode{M_{\rm Jup}}\else $M_{\rm Jup}$\fi}$, $P_{\rm
c} = 406.2^{+3.9}_{-2.5}$ days), and adaptive optics imaging revealed a nearby ($0\farcs87$), faint companion ([[*Kepler*]{}-432]{}[B]{}) that is a physically bound M dwarf. The host star exhibits high S/N asteroseismic oscillations, which enable precise measurements of the stellar mass, radius and age. Analysis of the rotational splitting of the oscillation modes additionally reveals the stellar spin axis to be nearly edge-on, which suggests that the stellar spin is likely well-aligned with the orbit of the transiting planet. Despite its long period, the obliquity of the $52.5$-day orbit may have been shaped by star–planet interaction in a manner similar to hot Jupiter systems, and we present observational and theoretical evidence to support this scenario. Finally, as a short-period outlier among giant planets orbiting giant stars, study of [[*Kepler*]{}-432]{}[b]{} may help explain the distribution of massive planets orbiting giant stars interior to $1$AU.
author:
- 'Samuel N. Quinn, Timothy. R. White, David W. Latham, William J. Chaplin, Rasmus Handberg, Daniel Huber, David M. Kipping, Matthew J. Payne, Chen Jiang, Victor Silva Aguirre, Dennis Stello, David H. Sliski, David R. Ciardi, Lars A. Buchhave, Timothy R. Bedding, Guy R. Davies, Saskia Hekker, Hans Kjeldsen, James S. Kuszlewicz, Mark E. Everett Steve B. Howell, Sarbani Basu, Tiago L. Campante, J[ø]{}rgen Christensen-Dalsgaard, Yvonne P. Elsworth, Christoffer Karoff, Steven D. Kawaler, Mikkel N. Lund, Mia Lundkvist, Gilbert A. Esquerdo, Michael L. Calkins, and Perry Berlind'
title: '[[*Kepler*]{}-432]{}: a Red Giant Interacting with One of its Two Long Period Giant Planets'
---
Introduction
============
The NASA [*Kepler*]{} mission [@borucki:2010], at its heart a statistical endeavor, has provided a rich dataset that enables ensemble studies of planetary populations, from gas giants to Earth-sized planets. Such investigations can yield valuable statistical constraints for theories of planetary formation and subsequent dynamical evolution [e.g., @buchhave:2014; @steffen:2012]. Individual discoveries, however, provide important case studies to explore these processes in detail, especially in parameter space for which populations remain small. Because of its unprecedented photometric sensitivity, duty cycle, and time coverage, companions that are intrinsically rare or otherwise difficult to detect are expected to be found by [ *Kepler*]{}, and detailed study of such discoveries can lead to characterization of poorly understood classes of objects and physical processes.
Planets orbiting red giants are of interest because they trace the planetary population around their progenitors, many of which are massive and can be hard to survey while they reside on the main sequence. Stars more massive than about $1.3~{\ifmmode{M_{\odot}}\else $M_{\odot}$\fi}$ [the so-called Kraft break; @kraft:1967] have negligible convective envelopes, which prevents the generation of the magnetic winds that drive angular momentum loss in smaller stars. Their rapid rotation and high temperatures—resulting in broad spectral features that are sparse in the optical—make precise radial velocities (RVs) extremely difficult with current techniques. However, as they evolve to become giant stars, they cool and spin down, making them ideal targets for precise radial velocity work [see, e.g., @johnson:2011 and references therein]. There is already good evidence that planetary populations around intermediate mass stars are substantially different from those around their low mass counterparts. Higher mass stars seem to harbor more Jupiters than do Sun-like stars [@johnson:2010], and the typical planetary mass correlates with the stellar mass [@lovis:2007; @dollinger:2009; @bowler:2010], but there are not many planets within $1$AU of more massive stars [@johnson:2007; @sato:2008], and their orbits tend to be less eccentric than Jupiters orbiting low mass stars [@jones:2014]. Due to the observational difficulties associated with massive and intermediate-mass main-sequence stars, many of the more massive stars known to host planets have already reached an advanced evolutionary state, and it is not yet clear whether most of the orbital differences can be attributed to mass-dependent formation and migration, or if planetary engulfment and/or tidal evolution as the star swells on the giant branch plays a more important role.
While the number of planets known to orbit evolved stars has become substantial, because of their typically long periods, not many transit, and thus very few are amenable to detailed study. In fact, [*Kepler*]{}-91b [@lillo:2014a; @lillo:2014b; @barclay:2014] and [*Kepler*]{}-56c [@huber:2013b] are the only two massive planets ($M_p>0.5~{\ifmmode{M_{\rm Jup}}\else $M_{\rm Jup}$\fi}$) orbiting giant stars ($\log{g}<3.9$) that are known to transit[^1]. A transit leads to a radius measurement, enabling investigation of interior structure and composition via bulk density constraints and theoretical models [e.g., @fortney:2007], and also opens up the possibility of atmospheric studies [e.g., @charbonneau:2002; @knutson:2008; @berta:2012; @poppenhaeger:2013], which can yield more specific details about planetary structure, weather, or atomic and molecular abundances within the atmosphere of the planet. Such information can provide additional clues about the process of planet formation around hot stars.
In studies of orbital migration of giant planets, the stellar obliquity—the angle between the stellar spin axis and the orbital angular momentum vector—has proven to be a valuable measurement, as it holds clues about the dynamical history of the planetary system [see, e.g., @albrecht:2012]. Assuming that the protoplanetary disk is coplanar with the stellar equator, and thus the rotational and orbital angular momenta start out well-aligned, some migration mechanisms–e.g., Type II migration [@goldreich:1980; @lin:1986]—are expected to preserve low obliquities, while others—for example, Kozai cycles [e.g., @wu:2003; @fabrycky:2007], secular chaos [@wu:2011], or planet-planet scattering [e.g., @rasio:1996; @juric:2008]—may excite large orbital inclinations. Measurements of stellar obliquity can thus potentially distinguish between classes of planetary migration.
The Rossiter–McLaughlin effect [@mclaughlin:1924; @rossiter:1924] has been the main source of (projected) obliquity measurements, although at various times starspot crossings [e.g., @desert:2011; @nutzman:2011; @sanchis:2011], Doppler tomography of planetary transits [e.g., @collier:2010; @johnson:2014], gravity-darkened models of the stellar disk during transit [e.g., @barnes:2011], and the rotational splitting of asteroseismic oscillation modes [e.g., @huber:2013b] have all been used to determine absolute or projected stellar obliquities. Most obliquity measurements to date have been for hot Jupiters orbiting Sun-like stars, but to get a full picture of planetary migration, we must study the dynamical histories of planets across a range of separations and in a variety of environments. Luckily, the diversity of techniques with which we can gather this information allows us to begin investigating spin–orbit misalignment for long period planets and around stars of various masses and evolutionary states. For long-period planets orbiting slowly rotating giant stars, the Rossiter–McLaughlin amplitudes are small because they scale with the stellar rotational velocity and the planet-to-star area ratio, and few transits occur, which limits the opportunities to obtain follow-up measurements or to identify the transit geometry from spot crossings. Fortunately, the detection of asteroseismic modes does not require rapid rotation, and is independent of the planetary properties, so it becomes a valuable tool for long period planets orbiting evolved stars. The high precision, high duty-cycle, long timespan, photometric observations of [*Kepler*]{} are ideal for both identifying long period transiting planets and examining the asteroseismic properties of their host stars.
In this paper, we highlight the discovery of [[*Kepler*]{}-432]{}[b]{} and c, a pair of giant planets in long period orbits ($>50$ days) around an oscillating, intermediate mass red giant. We present the photometric observations and transit light curve analysis of [[*Kepler*]{}-432]{} in and the follow-up imaging and spectroscopy in , followed by the asteroseismic and radial velocity analyses in . False positive scenarios and orbital stability are investigated in , and we discuss the system in the context of planet formation and migration in , paying particular attention to star–planet interactions (SPIs) and orbital evolution during the red giant phase. We provide a summary in .
Photometry {#sec:phot}
==========
Kepler Observations {#sec:kepler}
-------------------
![ Detrended and normalized long cadence [*Kepler*]{} time series for [[*Kepler*]{}-432]{}, spanning $1470.5$ days. The transits, which are clearly visible even in the unfolded data, are indicated by red lines. Solid lines denote full transits, while dotted lines denote that only a partial transit was observed. Three of the $28$ expected transits occurred entirely during data gaps. \[fig:kepts\]](fig1.pdf){width="48.00000%"}
The [*Kepler*]{} mission and its photometric performance are described in @borucki:2010, and the characteristics of the detector on board the spacecraft are described in @koch:2010 and @vancleve:2008. The photometric observations of [[*Kepler*]{}-432]{} span [*Kepler*]{} observation Quarters 0 through 17 (JD 2454953.5 to 2456424.0), a total of $1470.5$ days. [[*Kepler*]{}-432]{}[b]{} was published by the [*Kepler*]{} team as a [*Kepler*]{} Object of Interest (KOI) and planetary candidate [designated KOI-1299; see @batalha:2013], and after also being identified as a promising asteroseismic target, it was observed in short cadence (SC) mode for $8$ quarters. We note that a pair of recent papers have now confirmed the planetary nature of this transiting companion via radial velocity measurements [@ciceri:2015; @ortiz:2015].
The full photometric timeseries, normalized in each quarter, is shown in [Figure \[fig:kepts\]]{}. A transit signature with a period of ${\mathord{\sim}}52.5$ days is apparent in the data, and our investigation of the transits is described in the following section.
Light Curve Analysis {#sec:lc}
--------------------
A transit light curve analysis of [[*Kepler*]{}-432]{}[b]{} was performed previously by @sliski:2014. In that work, the authors first detrended the Simple Aperture Photometry (SAP) [*Kepler*]{} data[^2] for quarters $1$–$17$ using the CoFiAM (Cosine Filtering with Autocorrelation Minimization) algorithm and then regressed the cleaned data with the multimodal nested sampling algorithm MultiNest [@feroz:2009] coupled to a @mandel:2002 planetary transit model. Details on the priors employed and treatment of limb darkening are described in @sliski:2014. The authors compared the light curve derived stellar density, $\rho_{\star,\rm{obs}}$, to that from asteroseismology, $\rho_{\star,\rm{astero}}$, in a procedure dubbed “Asterodensity Profiling” [AP; @kipping:2012; @kipping:2014a] to constrain the planet’s minimum orbital eccentricity as being $e_{\rm min} = 0.488_{-0.051}^{+0.025}$. The minimum eccentricity is most easily retrieved with AP but the proper eccentricity (and argument of periastron, $\omega$) can be estimated by including $e$ and $\omega$ as free parameters in the light curve fit and marginalizing over $\omega$. In order to estimate the proper eccentricity, we were motivated to re-visit the [*Kepler*]{} data, as described below.
![ Folded short cadence [*Kepler*]{} light curve, shown as gray points. For clarity, the binned data (every $100$ points) are overplotted as large dark circles, and the best fit, with parameters reported in [Table \[tab:transit\]]{}, is indicated by the solid red line. \[fig:folded\_transit\]](fig2.pdf){width="48.00000%"}
We first detrended the [*Kepler*]{} SAP data as was done in @sliski:2014, by using the CoFiAM algorithm, which is described in detail in @kipping:2013. CoFiAM acts like a harmonic filter, removing any long term periodicities in the data but protecting those variations occurring on the timescale of the transit or shorter, so as to retain the true light curve shape. The algorithm requires an estimate of the times of transit minimum, orbital period, and full transit duration. Since @sliski:2014 provided refined values for these quantities, we used these updated values to conduct a revised CoFiAM detrending of the [*Kepler*]{} data. As with the previous analysis, the final light curves are optimized for a window within three transit durations of the transit minima.
Due to the effects of stellar granulation on the photometry, we find that the light curve scatter clearly exceeds the typical photometric uncertainties. In order to obtain more realistic parameter uncertainties, we added a “jitter” term in quadrature to the photometric uncertainties to yield a reduced chi-squared of unity for the out-of-transit data. This was done independently for the long- and short-cadence data, although the photometric jitter terms were (as expected) nearly identical at $177.4$ and $175.9\,$ppm for the short- and long-cadence data, respectively.
The $13$ long-cadence transits for which no SC data was available and the $11$ SC transits were stitched together and regressed to a transit model using MultiNest. Our light curve model employs the quadratic limb darkening @mandel:2002 routine with the @kipping:2010 “resampling” prescription for accounting for the smearing of the long-cadence data. The seven basic parameters in our light curve fit were ratio-of-radii, $R_{\rm b}/R_\star$, stellar density, $\rho_{\star}$, impact parameter, $b$, time of transit minimum, $T_0$, orbital period, $P$, and the limb darkening coefficients $q_1$ and $q_2$ described in @kipping:2013a. In addition, we included an 8th parameter for the log of the contaminated light fraction from a blend source, $\log_{10}\beta=\log_{10}(F_{\mathrm{blend}}/F_{\star})$. This was constrained from adaptive optics imaging (AO; see ) to be $\log \beta = -2.647\pm0.042$ with a Gaussian prior, assuming Gaussian uncertainties on the magnitudes measured from AO.
Ordinarily, a transit light curve contains very little information on the orbital eccentricity and thus it is not possible to reach a converged eccentricity solution with photometry alone [@kipping:2008]. However, in cases where the parent star’s mean density is independently constrained, a transit light curve can be used to constrain the orbital eccentricity and argument of periastron [@dawson:2012; @kipping:2014a]. This technique, an example of AP, enables us to include $e$ and $\omega$ as our 9th and 10th transit model parameters.
To enable the use of AP, we impose an informative Gaussian prior on the mean stellar density given by the asteroseismology constraint ($\rho_{\star,\mathrm{astero}} = 27.94^{+0.55}_{-0.58}$kgm$^{-3}$; ). We use the ECCSAMPLES code [@kipping:2014b] to draw samples from an appropriate joint $e$–$\omega$ prior. This code describes the eccentricity distribution as following a Beta distribution (a weakly informative prior) and then accounts for the bias in both $e$ and $\omega$ caused by the fact that the planet is known to transit. For the Beta distribution shape parameters, we use the “short” period calibration ($P<380$d) of @kipping:2013b: $a_{\beta}=0.697$ and $b_{\beta}=3.27$.
The maximum a posteriori folded transit light curve is presented in [Figure \[fig:folded\_transit\]]{}. The transit parameters and associated 68.3% uncertainties, derived solely from this photometric fit, are reported in [Table \[tab:transit\]]{}.
[lr]{} $R_{\rm b}/R_\star$ & $0.02914^{+0.00038}_{-0.00093}$\
$\rho_\star~({\rm kg\,m}^{-3})$ & $27.94^{+0.54}_{-0.58}$\
$b$ & $0.503^{+0.090}_{-0.168}$\
$T_{\rm 0,b}$ (BJD) & $2455949.5374^{+0.0018}_{-0.0016}$\
$P_{\rm b}$ (days) & $52.501134^{+0.000070}_{-0.000107}$\
$q_1$ & $0.309^{+0.047}_{-0.042}$\
$q_2$ & $0.674^{+0.151}_{-0.098}$\
$\log\beta$ & $-2.647 \pm 0.042$\
$e_{\rm b}$ & $0.507^{+0.039}_{-0.114}$\
$\omega_{\rm b}$ (degrees) & $76^{+59}_{-24}$\
\[-2.4ex\]
High Spatial Resolution Imaging {#sec:im}
===============================
Speckle Imaging {#sec:speck}
---------------
Speckle imaging observations of [[*Kepler*]{}-432]{} were performed on UT 2011 June 16 at the $3.5$ m WIYN telescope on Kitt Peak, AZ, using the Differential Speckle Survey Instrument [DSSI; @horch:2010]. DSSI provides simultaneous images in two filters using a dichroic beam splitter and two identical EMCCDs. These images were obtained in the $R$ ($6920$ Å) and $I$ ($8800$ Å) bands. Data reduction and analysis of these images is described in @torres:2011, @horch:2010, and @howell:2011. The reconstructed $R$- and $I$-band images reveal no stellar companions brighter than $\Delta R {\mathord{\sim}}4.5$ magnitudes and $\Delta I {\mathord{\sim}}3.5$ magnitudes, within the annulus from $0\farcs05$ to $2\arcsec$. The contrasts achieved as a function of distance are plotted in [Figure \[fig:sens\]]{}, and represent $5$-$\sigma$ detection thresholds.
Adaptive Optics Imaging {#sec:ao}
-----------------------
![ NIRC2 AO images in $J$-band ($1.260~\micron$) and $\rm{Br}\gamma$ ($2.165~\micron$), $4\arcsec\!\times\!4\arcsec$ in size, with a logarithmic flux scale. A faint companion to the northeast is clearly detected in both images, with separation $0\farcs8730 \pm 0\farcs0014$ and PA $20.86{\ensuremath{^\circ}}\pm 0.07{\ensuremath{^\circ}}$. North is up and east is left. \[fig:ao\]](fig4a.pdf "fig:"){width="23.00000%"}![ NIRC2 AO images in $J$-band ($1.260~\micron$) and $\rm{Br}\gamma$ ($2.165~\micron$), $4\arcsec\!\times\!4\arcsec$ in size, with a logarithmic flux scale. A faint companion to the northeast is clearly detected in both images, with separation $0\farcs8730 \pm 0\farcs0014$ and PA $20.86{\ensuremath{^\circ}}\pm 0.07{\ensuremath{^\circ}}$. North is up and east is left. \[fig:ao\]](fig4b.pdf "fig:"){width="23.00000%"}
Adaptive optics imaging was obtained using the Near InfraRed Camera 2 (NIRC2) mounted on the Keck II $10$ m telescope on Mauna Kea, HI on UT 2014 September 4. Images were obtained in both $J$ ($1.260~\micron$) and Br$\gamma$ ($2.165~\micron$; a good proxy for both $K$ and $K_s$). NIRC2 has a field of view $10\arcsec\!\times\!10\arcsec$, a pixel scale of about $0\farcs01\,{\rm pix}^{-1}$, and a rotator accuracy of $0\fdg02$. The overlap region of the dither pattern of the observations (i.e., the size of the final combined images) is ${\mathord{\sim}}4\arcsec\!\times\!4\arcsec$. In both filters, the FWHM of the stellar PSF was better than $0\farcs05$, and the achieved contrasts were $\Delta K {\mathord{\sim}}9$ ($\Delta J {\mathord{\sim}}8$) beyond $0\farcs5$ (see [Figure \[fig:sens\]]{}).
A visual companion was detected in both images ([Figure \[fig:ao\]]{}) with separation $0\farcs8730 \pm 0\farcs0014$ and PA $20\fdg86 \pm 0\fdg07$ (east of north). Relative to [[*Kepler*]{}-432]{}, we calculate the companion to have magnitudes ${\Delta}J = 5.59 \pm 0.06$ and ${\Delta}K = 5.16 \pm
0.02$, implying $J-K= 0.99 \pm 0.07$. Using the $J-K$ colors, we estimate the magnitude in the [*Kepler*]{} bandpass to be $K_{\rm p}
{\mathord{\sim}}18.8$. The object was not detected in the speckle images because they were taken with less aperture and the expected contrast ratios are larger in $R$ and $I$—using the properties of the companion as derived in the following section, we estimate $\Delta R
{\mathord{\sim}}6.7$ and $\Delta I {\mathord{\sim}}6.6$. These magnitudes are consistent with non-detections in the speckle images, as plotted in [Figure \[fig:sens\]]{}.
Properties of the Visual Companion
----------------------------------
The faint visual companion to [[*Kepler*]{}-432]{} could be a background star or a physically bound main-sequence companion. We argue that it is unlikely to be a background star, and present two pieces of evidence to support this conclusion. We first estimate the background stellar density in the direction of [[*Kepler*]{}-432]{} using the TRILEGAL stellar population synthesis tool [@girardi:2005]: we expect $49,\!000$ sources per deg$^2$ that are brighter than $K_s{\mathord{\sim}}19$ (the detection limit of our observation). This translates to $0.06$ sources (of [*any*]{} brightness and color) expected in our $16~{\rm arcsec}^2$ image, and thus the a priori probability of a chance alignment is low.
Furthermore, since we know the properties of the primary star, we can determine whether there exists a coeval main-sequence star that could adequately produce the observed colors and magnitude differences. Using the asteroseismically derived mass, radius, and age of the primary (see ), we place the primary star on an appropriate Padova PARSEC isochrone [@bressan:2012; @chen:2014]. We then use the observed magnitude differences between the stars to search for an appropriate match to the companion in the isochrone. If the visual companion is actually a background giant, it is unlikely that it would happen to be at the right distance to match both the colors and brightness of a physically bound companion. Therefore, we do not expect a background giant star to lie on the isochrone. However, we [*do*]{} find a close match to the observed colors and magnitudes of the companion (see [Figure \[fig:isochrone\]]{}), further suggesting that it is not a background object, but truly a physical companion, and this allows us to estimate its properties from the isochrone.
![ [[*Kepler*]{}-432]{} (blue circle) placed on a $3.5$ Gyr, ${\rm
[m/H]=-0.07}$ PARSEC isochrone [@bressan:2012], plotted as a solid black line. The upper right (red) axes represent the $J-K$ and apparent $K$ mag corresponding to the $T_{\rm eff}$ and $\log{L/L\odot}$ on the lower left (blue) axes. The visual companion (red triangle) is placed on the plot according to its measured $J-K$ and $K$ mag. It lies very near the isochrone for the system, suggesting that it is indeed coeval with and physically bound to [[*Kepler*]{}-432]{}, rather than a background star.[]{data-label="fig:isochrone"}](fig5.pdf){width="48.00000%"}
We conclude that the companion is most likely a physically bound, coeval M dwarf with a mass of ${\mathord{\sim}}0.52~{\ifmmode{M_{\odot}}\else $M_{\odot}$\fi}$ and an effective temperature of ${\mathord{\sim}}3660$ K. The distance to the system (${\mathord{\sim}}870$ pc; ), implies that the projected separation of the companion is ${\mathord{\sim}}750$ AU. Using this as an estimate of the semi-major axis, the binary orbital period is on the order of $15,\!000$ yr. In reality, the semi-major axis may be smaller (if we observed it near apastron of an eccentric orbit with the major axis in the plane of the sky), or significantly larger (due to projections into the plane of the sky and the unknown orbital phase).
We discuss the possibility of false positives due to this previously undetected companion in .
Spectroscopic Follow-Up {#sec:spec}
=======================
Spectroscopic Observations
--------------------------
We used the Tillinghast Reflector Echelle Spectrograph [TRES; @furesz:2008] mounted on the $1.5$-m Tillinghast Reflector at the Fred L. Whipple Observatory (FLWO) on Mt. Hopkins, AZ to obtain $84$ high resolution spectra of [[*Kepler*]{}-432]{} between UT 2011 March 23 and 2014 June 18. TRES is a temperature-controlled, fiber-fed instrument with a resolving power of $R {\mathord{\sim}}44,\!000$ and a wavelength coverage of ${\mathord{\sim}}3850$–$9100$ Å, spanning $51$ echelle orders. Typical exposure times were $15$-$30$ minutes, and resulted in extracted signal-to-noise ratios (S/Ns) between about $20$ and $45$ per resolution element. The goal of the intial observations was to rule out false positives involving stellar binaries as part of the [*Kepler*]{} Follow-up Observing Program (KFOP, which has evolved into CFOP[^3]), but upon analysis of the first few spectra, it became clear that the planet was massive enough to confirm with an instrument like TRES that has a modest aperture and ${\mathord{\sim}}10~{\ifmmode{\rm m\thinspace s^{-1}}\else m\thinspace s$^{-1}$\fi}$ precision [see, e.g., @quinn:2014]. By the second observing season, an additional velocity trend was observed, which led to an extended campaign of observations.
Precise wavelength calibration of the spectra was established by obtaining ThAr emission-line spectra before and after each spectrum, through the same fiber as the science exposures. Nightly observations of the IAU RV standard star HD 182488 helped us track the achieved instrumental precision and correct for any RV zero point drift. We also shift the absolute velocities from each run so that the median RV of HD 182488 is $-21.508~{\ifmmode{\rm km\thinspace s^{-1}}\else km\thinspace s$^{-1}$\fi}$ [@nidever:2002]. This allows us to report the absolute systemic velocity, $\gamma_{\rm abs}$. We are aware of specific TRES hardware malfunctions (and upgrades) that occurred during the timespan of our data that, in addition to small zero point shifts (typically $<10{\ifmmode{\rm m\thinspace s^{-1}}\else m\thinspace s$^{-1}$\fi}$), caused degradation (or improvement) of RV precision for particular observing runs. For example, the installation of a new dewar lens caused a zero point shift after BJD 2455750, and a second shift (accompanied by significant improvement in precision) occurred when the fiber positioner was fixed in place on BJD 2456013. It will be important to treat these with care so that the radial velocities are accurate and each receives its appropriate weight in our analysis.
We also obtained five spectra with the FIber-fed Echelle Spectrograph [FIES; @frandsen:1999] on the $2.5$-m Nordic Optical Telescope [NOT; @djupvik:2010] at La Palma, Spain during the first observing season (UT 2011 August 4 through 2011 October 7) to confirm the initial RV detection before continuing to monitor the star with TRES. Like TRES, FIES is a temperature-controlled, fiber-fed instrument, and has a resolving power through the medium fiber of ${\rm R}{\mathord{\sim}}46,\!000$, a wavelength coverage of ${\mathord{\sim}}3600$–$7400$ Å, and wavelength calibration determined from ThAr emission-line spectra.
Spectroscopic Reduction and Radial Velocity Determination
---------------------------------------------------------
We will discuss the reduction of spectra from both instruments collectively but only briefly [more details can be found in @buchhave:2010] while detailing the challenges presented by our particular data set. Spectra were optimally extracted, rectified to intensity versus wavelength, and cross-correlated, order by order, using the strongest exposure as a template. We used $21$ orders (spanning $4290$–$6280$ Å), rejecting those plagued by telluric lines, fringing in the red, and low S/N in the blue. For each epoch, the cross-correlation functions (CCFs) from all orders were added and fit with a Gaussian to determine the relative RV for that epoch. Using the summed CCF rather than the mean of RVs from each order naturally weights the orders with high correlation coefficients more strongly. Internal error estimates for each observation were calculated as $\sigma_{\rm int}={\rm rms}(\vec{v})/\sqrt{N}$, where $\vec{v}$ is the RV of each order, $N$ is the number of orders, and rms denotes the root mean squared velocity difference from the mean. These internal errors account for photon noise and the precision with which we can measure the line centers, which in turn depends on the characteristics of the [[*Kepler*]{}-432]{} spectrum (line shapes, number of lines, etc), but do [*not*]{} account for errors introduced by the instrument itself.
The nightly observations of RV standards were used to correct for systematic velocity shifts between runs and to estimate the instrumental precision. The median RV of HD 182488 was calculated for each run, which we applied as shifts to the [[*Kepler*]{}-432]{} velocities, keeping in mind that each shift introduces additional uncertainty. By also applying the run-to-run offsets to the standard star RVs themselves, we were able to evaluate the residual RV noise introduced by the limited instrumental precision (separate from systematic zero point shifts). After correction, the rms of the standard star RVs in each run was consistent with the internal errors. This indicates that the additional uncertainty introduced by run-to-run correction already adequately accounts for the instrumental uncertainty, and we do not need to explicitly include an additional error term to account for it. The final error budget of [[*Kepler*]{}-432]{} RVs was assumed to be the sum by quadrature of all RV error sources—internal errors, run-to-run offset uncertainties, and TRES instrumental precision: $\sigma_{\rm
RV}^2=\sigma_{\rm int}^2 + \sigma_{\rm run}^2 + \sigma_{\rm TRES}^2$, where the final term $\sigma_{\rm TRES}=0$ because it is implicitly incorporated into $\sigma_{\rm run}$. The final radial velocities are listed in [Table \[tab:rv\]]{}. We recognize that stellar jitter or additional undetected planets may also act as noise sources, and we address this during the orbital fitting analysis in .
[rrr|rrr]{} $ 644.00217$ & $235.4$ & $ 65.1$ & $1083.87676$ & $338.7$ & $36.7$\
$ 722.95848$ & $453.1$ & $ 23.0$ & $1091.93504$ & $489.2$ & $44.3$\
$ 727.87487$ & $452.2$ & $ 72.1$ & $1117.90004$ & $211.7$ & $41.6$\
$ 734.86675$ & $544.4$ & $ 68.2$ & $1132.79275$ & $410.9$ & $42.4$\
$ 755.93317$ & $ 47.2$ & $ 18.9$ & $1137.83647$ & $333.7$ & $21.8$\
$ 757.82852$ & $197.1$ & $ 23.7$ & $1175.82536$ & $210.1$ & $62.5$\
$ 758.88922$ & $121.7$ & $ 18.9$ & $1197.74504$ & $394.7$ & $22.3$\
$ 760.95496$ & $193.8$ & $ 25.9$ & $1202.70980$ & $495.3$ & $20.5$\
$ 764.92787$ & $270.3$ & $ 36.7$ & $1227.64276$ & $ 75.4$ & $23.1$\
$ 768.78698$ & $262.1$ & $ 33.9$ & $1233.59012$ & $213.1$ & $32.0$\
$ 768.80782$ & $277.4$ & $ 30.8$ & $1258.59157$ & $564.4$ & $24.3$\
$ 822.70873$ & $197.0$ & $ 54.9$ & $1279.58992$ & $ 96.6$ & $28.9$\
$ 825.63058$ & $265.0$ & $ 53.1$ & $1376.99835$ & $ 29.1$ & $25.1$\
$ 826.64436$ & $310.1$ & $ 53.0$ & $1382.93180$ & $148.7$ & $23.9$\
$ 827.71851$ & $362.0$ & $ 55.3$ & $1389.91148$ & $179.3$ & $26.7$\
$ 828.61567$ & $358.8$ & $ 60.8$ & $1400.93674$ & $384.5$ & $31.0$\
$ 829.72558$ & $358.7$ & $ 53.9$ & $1405.90249$ & $465.6$ & $30.5$\
$ 830.69401$ & $446.6$ & $ 56.2$ & $1409.87680$ & $507.5$ & $28.4$\
$ 837.59392$ & $588.0$ & $ 51.1$ & $1413.96428$ & $646.1$ & $37.2$\
$ 840.65649$ & $603.7$ & $ 51.0$ & $1429.95861$ & $111.4$ & $23.7$\
$ 841.70196$ & $706.7$ & $ 71.9$ & $1436.83822$ & $153.9$ & $22.7$\
$ 842.57863$ & $574.4$ & $ 57.1$ & $1442.84894$ & $191.6$ & $26.9$\
$ 843.59367$ & $437.4$ & $ 43.8$ & $1446.78723$ & $285.9$ & $21.2$\
$ 844.64075$ & $346.9$ & $ 46.7$ & $1503.72052$ & $351.9$ & $35.0$\
$ 845.68516$ & $157.7$ & $ 45.8$ & $1547.69956$ & $189.1$ & $19.5$\
$ 846.62110$ & $ 91.9$ & $ 45.3$ & $1551.63468$ & $217.2$ & $20.0$\
$ 852.61538$ & $ 64.2$ & $ 47.3$ & $1556.67843$ & $292.6$ & $20.9$\
$ 854.62438$ & $ 57.6$ & $ 48.9$ & $1561.63231$ & $370.3$ & $18.2$\
$ 856.71790$ & $132.6$ & $ 56.1$ & $1575.69266$ & $593.5$ & $20.0$\
$ 858.64825$ & $105.2$ & $ 48.9$ & $1581.66560$ & $141.4$ & $18.9$\
$1027.94566$ & $284.4$ & $ 26.4$ & $1586.63377$ & $ 40.5$ & $18.4$\
$1046.89934$ & $594.1$ & $ 20.6$ & $1591.70580$ & $100.0$ & $21.4$\
$1047.87727$ & $702.8$ & $ 38.6$ & $1729.01428$ & $553.2$ & $37.9$\
$1048.86080$ & $693.7$ & $ 18.9$ & $1740.94781$ & $ 98.6$ & $33.0$\
$1049.97884$ & $661.7$ & $ 28.7$ & $1743.99496$ & $ 34.8$ & $29.3$\
$1051.99004$ & $631.1$ & $ 24.6$ & $1799.87657$ & $ 37.0$ & $26.3$\
$1052.93268$ & $583.1$ & $ 30.0$ & $1816.87894$ & $248.9$ & $32.0$\
$1053.90973$ & $461.0$ & $ 23.4$ & $1822.88725$ & $364.1$ & $24.3$\
$1054.88376$ & $383.9$ & $ 29.3$ & $1826.79500$ & $419.8$ & $20.0$\
$1055.88175$ & $282.3$ & $ 37.3$ & $777.59554$ & $-100.9$ & $10.7$\
$1056.88078$ & $239.3$ & $ 26.4$ & $778.59121$ & $ -81.1$ & $10.5$\
$1057.95303$ & $158.7$ & $ 24.6$ & $779.59677$ & $ -51.2$ & $14.1$\
$1058.84781$ & $160.9$ & $ 33.7$ & $782.57231$ & $ 0.0$ & $10.2$\
$1074.90585$ & $267.8$ & $ 31.1$ & $842.41733$ & $ 47.7$ & $10.2$\
$1080.95245$ & $354.5$ & $ 30.6$ & & &\
\[-2.4ex\]
Spectroscopic Classification {#sec:spec_class}
----------------------------
[lcc]{} Asteroseismic & Grid-based Modeling & Frequency Modeling\
\
\[-2ex\] () & $18.59\pm0.04$ & …\
() & $266\pm3$ & …\
(dex) & $3.340\pm0.006$ & $3.345 \pm 0.006$\
$\rho_\star$ (${\rm g~cm}^{-3}$) & $0.02723^{+0.00054}_{-0.00057}$ & $0.02794^{+0.00055}_{-0.00058}$\
$R$ ([$R_{\odot}$]{}) & $4.12^{+0.12}_{-0.08}$ & $4.06^{+0.12}_{-0.08}$\
$M$ ([$M_{\odot}$]{}) & $1.35^{+0.10}_{-0.07}$ & $1.32^{+0.10}_{-0.07}$\
Age (Gyr) & $3.5^{+0.7}_{-0.8}$ & $4.2^{+0.8}_{-1.0}$\
\
\[-2ex\] Spectroscopic & &\
\
\[-2ex\] $T_{\rm eff}$ (K) &\
$\log{g}$ (cgs) &\
$[m/H]$ &\
$v\sin{i_\star}$ ([kms$^{-1}$]{})&\
\
\[-2ex\] Photometric & &\
\
\[-2ex\] $V$ (mag) &\
$K_{\rm p}$ (mag) &\
$J$ (mag) &\
$H$ (mag) &\
$K_{\rm s}$ (mag) &\
\
\[-2ex\] Derived & &\
\
\[-2ex\] $L_\star$ ([$L_{\odot}$]{}) &\
$d$ (pc) &\
$i_\star$ ([$^\circ$]{}) &\
$P_{\rm rot}$ (days) &\
\[-2.4ex\]
We initially determined the spectroscopic stellar properties (effective temperature, $T_{\rm eff}$; surface gravity, $\log{g}$; projected rotational velocity, $v\sin{i}$; and metallicity, \[m/H\]) using Stellar Parameter Classification [SPC; @buchhave:2012], with the goal of providing an accurate temperature for the asteroseismic modeling (see ). SPC cross-correlates an observed spectrum against a grid of synthetic spectra, and uses the correlation peak heights to fit a three-dimensional surface in order to find the best combination of atmospheric parameters ($v\sin{i}$ is fit iteratively since it only weakly correlates with the other parameters). We used the CfA library of synthetic spectra, which are based on Kurucz model atmospheres [@kurucz:1992]. SPC, like other spectroscopic classifications, can be limited by degeneracy between $T_{\rm eff}$, $\log{g}$, and \[m/H\] [see a discussion in @torres:2012], but asteroseismology provides a nearly independent measure of the surface gravity (depending only weakly on the effective temperature and metallicity). This allows one to iterate the two analyses until agreement is reached, generally requiring only $1$ iteration [see, e.g., @huber:2013a]. In our initial analysis, we found $T_{\rm eff}=5072 \pm 55$ K, $\log{g}=3.49 \pm 0.11$, ${\rm
[m/H]}=-0.02 \pm 0.08$, and $v\sin{i}=2.5 \pm 0.5~{\ifmmode{\rm km\thinspace s^{-1}}\else km\thinspace s$^{-1}$\fi}$. After iterating with the asteroseismic analysis and fixing the final asteroseismic gravity, we find similar values: $T_{\rm eff}=4995 \pm
78$ K, $\log{g}=3.345 \pm 0.006$, ${\rm [m/H]}=-0.07 \pm 0.10$, and $v\sin{i}=2.7 \pm 0.5~{\ifmmode{\rm km\thinspace s^{-1}}\else km\thinspace s$^{-1}$\fi}$. We adopt the values from the combined analysis, and these final spectroscopic parameters are listed in [Table \[tab:stellar\]]{}.
Asteroseismology of [[*Kepler*]{}-432]{} {#sec:seis}
========================================
Background
----------
Cool stars exhibit brightness variations due to oscillations driven by near-surface convection [@houdek:1999; @aerts:2010], which are a powerful tool to study their density profiles and evolutionary states. A simple asteroseismic analysis is based on the average separation of modes of equal spherical degree () and the frequency of maximum oscillation power (), using scaling relations to estimate the mean stellar density, surface gravity, radius, and mass [@kjeldsen:1995; @stello:2008; @kallinger:2010; @belkacem:2011]. @huber:2013a presented an asteroseismic analysis of [[*Kepler*]{}-432]{} by measuring and using three quarters of SC data, combined with an SPC analysis [@buchhave:2012] of high-resolution spectra obtained with the FIES and TRES spectrographs. The results showed that [[*Kepler*]{}-432]{} is an evolved star just beginning to ascend the red-giant branch (RGB), with a radius of $R=4.16\pm0.12$[$R_{\odot}$]{} and a mass of $1.35\pm0.10$[$M_{\odot}$]{} ([Table \[tab:stellar\]]{}, [Figure \[fig:isochrone\]]{}).
Compared to average oscillation properties, individual frequencies offer a greatly increased amount of information by probing the interior sound speed profile. In particular, evolved stars oscillate in mixed modes, which occur when pressure modes excited on the surface couple with gravity modes confined to the core [@aizenman:1977]. Mixed modes place tight constraints on fundamental properties such as stellar age, and provide the possibility to probe the core structure and rotation [@bedding:2011; @beck:2012; @mosser:2012a]. Importantly, relative amplitudes of individual oscillation modes that are split by rotation can be used to infer the stellar line-of-sight inclination [@gizon:2003], providing valuable information on the orbital architectures of transiting exoplanet systems [e.g., @chaplin:2013; @huber:2013b; @benomar:2014; @lund:2014; @vaneylen:2014]. In the following section we expand on the initial asteroseismic analysis by @huber:2013a by performing a detailed individual frequency analysis based on all eight quarters (Q9–17) of [*Kepler*]{} short-cadence data.
Frequency Analysis
------------------
The time series was prepared for asteroseismic analysis from the raw [*Kepler*]{} target pixel data using the [*Kepler Asteroseismic Science Operations Center*]{} (KASOC) filter [@handberg:2014]. The KASOC filter removes instrumental and transit signals from the light curve, which may produce spurious peaks in the frequency domain. The power spectrum, shown in [Figure \[fig:powspec\]]{}, was computed using a Lomb–Scargle periodogram [@lomb:1976; @scargle:1982] calibrated to satisfy Parseval’s theorem.
The pattern of oscillation modes in the power spectrum is typical of red giants, with $\ell=0$ modes of consecutive order being approximately equally spaced by $\Delta\nu$, adjacent to $\ell=2$ modes. In addition to the $\ell=0, 2$ pairs, several $\ell=1$ mixed modes are observed in each radial order which, on inspection, are the outer components of rotationally split triplets corresponding to the $m=\pm1$ modes. This indicates that the star is seen equator-on (see ).
The relative $p$- and $g$-mode behavior of each mixed mode depends on the strength of the coupling between the oscillation cavities in the stellar core and envelope. Detecting the $\ell=1$ modes with the greatest $g$-mode character may be challenging because they have low amplitudes, and overlap in frequency with the $\ell=0$ and $\ell=2$ modes. Adding to the possible confusion, mixed $\ell=2$ and $\ell=3$ modes may also be present, although the weaker coupling between the $p$- and $g$-modes results in only the most $p$-like modes having an observable amplitude.
[lccccc]{} 9 & $194.111^{+0.018}_{-0.014}$ & $202.500^{+0.006}_{-0.008}$ & $0.184^{+0.014}_{-0.013}$ & $209.990^{+0.010}_{-0.023}$ &\
& & $204.392^{+0.015}_{-0.010}$ & $0.274^{+0.018}_{-0.022}$ & &\
& & $207.713^{+0.006}_{-0.007}$ & $0.335^{+0.009}_{-0.008}$ & &\
10 & $212.214^{+0.026}_{-0.020}$ & $215.394^{+0.008}_{-0.007}$ & $0.342^{+0.014}_{-0.015}$ & $228.236^{+0.019}_{-0.019}$ &\
& & $219.215^{+0.006}_{-0.008}$ & $0.284^{+0.007}_{-0.008}$ & &\
& & $221.586^{+0.011}_{-0.019}$ & $0.165^{+0.014}_{-0.011}$ & &\
& & $224.590^{+0.016}_{-0.008}$ & $0.325^{+0.017}_{-0.017}$ & &\
& & $228.905^{+0.020}_{-0.013}$ & $0.328^{+0.022}_{-0.017}$ & &\
11 & $230.704^{+0.009}_{-0.008}$ & $233.410^{+0.013}_{-0.017}$ & $0.280^{+0.015}_{-0.018}$ & $247.031^{+0.013}_{-0.016}$ & $253.534^{+0.018}_{-0.034}$\
& & $237.904^{+0.014}_{-0.011}$ & $0.279^{+0.010}_{-0.010}$ & &\
& & $240.439^{+0.011}_{-0.009}$ & $0.169^{+0.008}_{-0.009}$ & &\
& & $244.157^{+0.015}_{-0.009}$ & $0.302^{+0.009}_{-0.019}$ & &\
12 & $249.259^{+0.016}_{-0.009}$ & $254.553^{+0.005}_{-0.006}$ & $0.313^{+0.005}_{-0.005}$ & $265.636^{+0.011}_{-0.043}$ & $272.212^{+0.015}_{-0.006}$\
& & $258.288^{+0.006}_{-0.008}$ & $0.143^{+0.010}_{-0.011}$ & &\
& & $261.490^{+0.003}_{-0.004}$ & $0.298^{+0.005}_{-0.006}$ & &\
13 & $267.757^{+0.017}_{-0.016}$ & $273.216^{+0.007}_{-0.007}$ & $0.301^{+0.007}_{-0.007}$ & $284.389^{+0.013}_{-0.011}$ & $291.410^{+0.008}_{-0.008}$\
& & $277.110^{+0.006}_{-0.008}$ & $0.140^{+0.012}_{-0.014}$ & &\
& & $281.091^{+0.006}_{-0.006}$ & $0.304^{+0.007}_{-0.008}$ & &\
14 & $286.457^{+0.006}_{-0.009}$ & $294.131^{+0.015}_{-0.014}$ & $0.246^{+0.012}_{-0.011}$ & $303.352^{+0.021}_{-0.015}$ &\
& & $297.229^{+0.012}_{-0.017}$ & $0.173^{+0.016}_{-0.014}$ & &\
15 & $305.284^{+0.015}_{-0.014}$ & $315.134^{+0.032}_{-0.011}$ & $0.207^{+0.026}_{-0.026}$ & &\
\[-2.4ex\]
{width="\textwidth"}
The first step to fitting the oscillation modes and extracting their frequencies is to correctly identify the modes present. Fortunately, the $\ell=1$ mixed modes follow a frequency pattern that arises from coupling of the $p$-modes in the envelope, which have approximately equal spacing in frequency ($\Delta\nu$), to $g$-modes in the core, which are approximately equally spaced in period ($\Delta\Pi$). This pattern is well described by the asymptotic relation for mixed modes [@mosser:2012b]. We calculated the asymptotic mixed mode frequencies by fitting this relation to several of the highest-amplitude $\ell=1$ modes. From these calculations, nearby peaks could be associated with $\ell=1$ mixed modes. In this way we have been able to identify both of the $m=\pm1$ components for 21 out of 27 $\ell=1$ mixed modes between $200$ and $320$ .
Following a strategy that has been implemented in the mode fitting of other [*Kepler*]{} stars [e.g., @appourchaux:2012], three teams performed fits to the identified modes. The mode frequencies from each fit were compared to the mean values, and the fitter that differed least overall was selected to provide the frequency solution. This fitter performed a final fit to the power spectrum to include modes that other fitters had detected, but were absent from this fitter’s initial solution.
The final fit was made using a Markov chain Monte Carlo (MCMC) method that performs a global fit to the oscillation spectrum, with the modes modeled as Lorentzian profiles [@handberg:2011]. Each $\ell=1$ triplet was modeled with the frequency splitting and inclination angle as additional parameters to the usual frequency, height, and width that define a single Lorentzian profile. Owing to the differing sensitivity of the $\ell=1$ mixed modes to rotation at different depths within the star, each $\ell=1$ triplet was fitted with an independent frequency splitting, although a common inclination angle was used. We discuss the rotation of the star further in .
The measured mode frequencies are given in [Table \[tab:oscillation\]]{}. The values of the $\ell=1$ modes are presented as the central frequency of the rotationally split mode profile, which corresponds to the value of the $m=0$ component, along with the value of the rotational splitting between the $m=0$ and $m=\pm1$ components, $\nu_s$. Revised values of and can be obtained from the measured mode frequencies and amplitudes. We find $=~18.59\pm0.04$ and $=~266\pm3$ , both of which are in agreement with the values provided by @huber:2013a. Additionally, we measure the underlying $\ell=1$ $g$-mode period spacing, $\Delta\Pi_1$, to be $89.9\pm0.3$s, which is consistent with a red giant branch star with a mass below $\sim1.6\,$[$M_{\odot}$]{} [e.g., @stello:2013].
Host Star Inclination {#sec:inc}
---------------------
The line-of-sight inclination of a rotating star can be determined by measuring the relative heights of rotationally split modes [@gizon:2003]. A star viewed pole-on produces no visible splitting, while stars viewed with an inclination near $i=45^{\circ}$ would produce a frequency triplet. [Figure \[fig:powspec\]]{} shows that all dipole modes observed for [[*Kepler*]{}-432]{} are split into doublets, which we interpret as triplets with the central peak missing, indicating a rotation axis nearly perpendicular to the line of sight (inclination $i=90^{\circ}$).
To measure the inclination of [[*Kepler*]{}-432]{}, we included rotationally split Lorentzian profiles for each of the 21 dipole modes in the global MCMC fit of the power spectrum. [Figure \[fig:incl\]]{} shows the posterior distribution of the stellar inclination. The mode of the posterior distribution and $68.3\%$ highest probability density region is $90.0^{+0.0}_{-3.7}$ deg.
The inclination estimate is based on three important assumptions: the inclination is the same for $p$-dominated and $g$-dominated mixed modes, that there is equipartition of energy between modes with the same $n$ and $\ell$, and that the modes are well-resolved.
To test the first assumption, we performed additional fits to individual $\ell=1$ modes using the Python implementaiton of the nested sampling algorithm MultiNest, pyMultiNest [@feroz:2013; @buchner:2014]. No significant difference was found between the inclination angle for $p$-dominated and $g$-dominated mixed modes. We therefore use the results of our global MCMC fit. @huber:2013b similarly found no difference between the inclination angle of $p$-dominated and $g$-dominated mixed modes in [*Kepler*]{}-56. @beck:2014 have shown that these modes actually have slightly different pulsation cavities. They identified asymmetric rotational splittings between $m=0$ and $m=\pm1$ modes in the red giant KIC5006817, which results from the modes having varying $p$- and $g$-mode characteristics. Besides the effect on the rotational splitting, there is also a small impact on mode heights and lifetimes. @beck:2014 further note that the asymmetries are mirrored about the frequency of the uncoupled $p$-modes. This means that the heights of the $m=\pm1$ components relative to the $m=0$ component will change in opposite directions, so the effect can be mitigated by forcing the $m=\pm1$ components to have the same height in the fit, as well as by performing a global fit to all modes, as we have done.
Unless mode lifetimes are much shorter than the observing baseline, the Lorentzian profiles of the modes will not be well-resolved, and the mode heights will vary. We determined the effect on the measured inclination angle in the manner of @huber:2013b, by investigating the impact on simulated data with similar properties to the frequency spectrum of [[*Kepler*]{}-432]{}. Taking this effect into account in the determination of our measurement uncertainty, we find a final value of $i=90^{+0}_{-8}$ deg. The asteroseismic analysis therefore shows directly that the spin axis of the host star is in the plane of the sky.
![Posterior of the host star line-of-sight inclination derived from the MCMC analysis of the oscillation power spectrum of [[*Kepler*]{}-432]{}. The red point indicates the mode of the distribution, and the blue dashed line indicates the limit of the 68% highest probability density region.[]{data-label="fig:incl"}](fig7.pdf){width="48.00000%"}
Modeling
--------
Two approaches may be used when performing asteroseismic modeling. The first is the so-called [*grid-based method*]{}, which uses evolutionary tracks that cover a wide range of metallicities and masses, and searches for the best fitting model using , , , and as constraints [e.g. @stello:2009; @basu:2010; @gai:2011; @chaplin:2014]. The second is [*detailed frequency modeling*]{}, which uses individual mode frequencies instead of the global asteroseismic parameters to more precisely determine the best-fitting model [e.g., @metcalfe:2010; @jiang:2011]. For comparison, we have modeled [[*Kepler*]{}-432]{} using both approaches.
For the grid-based method we used the Garching Stellar Evolution Code [@weiss:2008]. The detailed parameters of this grid are described by @silvaaguirre:2012 and its coverage has been now extended to stars evolved in the RGB phase. The spectroscopic values of and found in , and our new asteroseismic measurements of and were used as inputs in a Bayesian scheme as described in @silvaaguirre:2014. Note that while \[Fe/H\] is the model input, our spectroscopic analysis yields an estimate of \[m/H\]. We have assumed that the two are equivalent for [[*Kepler*]{}-432]{} (i.e., that the star has a scaled solar composition). If this assumption is invalid, it may introduce a small bias in our results, which are given in [Table \[tab:stellar\]]{}. The values of mass and radius agree well with those from @huber:2013a, who also used the grid-based method. We note that a comparison of results provided by several grid-pipelines by @chaplin:2014 found typical systematic uncertainties of $3.7\%$ in mass, $1.3\%$ in radius, and $12\%$ in age across an ensemble of main-sequence and subgiant stars with spectroscopic constraints on and .
We performed two detailed modeling analyses using separate stellar evolution codes in order to better account for systematic uncertainties. The first analysis modeled the star using the integrated astero extension within MESA [@paxton:2013]. After an initial grid search to determine the approximate location of the the global minimum, we found the best-fitting model using the build-in simplex minimization routine, which automatically adjusted the mass, metallicity, and the mixing length parameter. The theoretical frequencies were calculated using GYRE [@townsend:2013] and were corrected for near-surface effects using the power-law correction of @kjeldsen:2008 for radial modes. The non-radial modes in red giants are mixed with $g$-mode characteristics in the core, so they are less affected by near-surface effects. To account for this, MESA-astero follows @brandao:2011 in scaling the correction term for non-radial modes by $Q_{n,\ell}^{-1}$, where $Q_{n,\ell}$ is the ratio of the inertia of the mode to the inertia of a radial mode at the same frequency.
The second analysis was performed with the Aarhus Stellar Evolution Code [ASTEC; @cd08a], with theoretical frequencies calculated using the Aarhus adiabatic oscillation package [@cd08b]. The best-fitting model was found in a similar manner as the first analysis, although the mixing length parameter was kept fixed at a value of $\alpha=1.8$.
[Figure \[fig:echelle\]]{} shows the best-fitting models compared to the observed frequencies in an échelle diagram. Both analyses found an asteroseismic mass and radius of $M=1.32^{+0.10}_{-0.07}$[$M_{\odot}$]{}and $R=4.06^{+0.12}_{-0.08}$[$R_{\odot}$]{}, but differ in the value of the age, with the best-fitting MESA and ASTEC models having ages of $4.2^{+0.8}_{-1.0}$ and $2.9^{+0.6}_{-0.7}$Gyr, respectively. Uncertainties were estimated by adopting the fractional uncertainties of the grid-based method, thereby accounting for systematic uncertainties in model input physics and treatment of near-surface effects. The consistency between the detailed model fitting results and grid-based results demonstrates the precise stellar characterization that can be provided by asteroseismology. Throughout the remainder of the paper we adopt the results obtained with the MESA code, though we use an age of $3.5$Gyr, which is consistent with both detailed frequency analyses as well as the age from the grid-based modeling.
![Échelle diagram of [[*Kepler*]{}-432]{} showing observed frequencies in white. Modes are identified as $\ell=0$ (circles), $\ell=1$ (triangles), $\ell=2$ (squares) and $\ell=3$ (diamonds). Frequencies of the best-fitting MESA and ASTEC models are indicated by the red and blue open symbols, respectively. For reference, a gray-scale map of the power spectrum is shown in the background. Numbers to the right of the plot indicate the radial order of the $\ell=0$ modes.[]{data-label="fig:echelle"}](fig8.pdf){width="48.00000%"}
Distance and Reddening {#sec:dist}
----------------------
The stellar model best-fit to the derived stellar properties provides color indices that may be compared against measured values as a consistency check, and as a means to determine a photometric distance to the system. Given the physical stellar parameters derived from the asteroseismic and spectroscopic measurements, the PARSEC isochrones [@bressan:2012] predict $J-K_{\rm s}=0.564$, in good agreement with the measured 2MASS colors ($J-K_{\rm s}=0.563 \pm 0.028$). While this indicates the dust extinction along the line of sight is probably low, we attempt to correct for it nonetheless using galactic dust maps. The mean of the reddening values reported by @schlafly:2011 and @schlegel:1998—$E(B-V)=0.079 \pm
0.008$—is indeed low, implying extinction in the infrared of ${\rm
A_J} = 0.070$, ${\rm A_H} = 0.045$, and ${\rm A_{K_s}} =
0.028$. Applying these corrections, the observed 2MASS index becomes $J-K_{\rm s}=0.521$, now slightly inconsistent at the $1$-$\sigma$ level with the PARSEC model colors. We note that if only a small fraction of the dust column lies between us and [[*Kepler*]{}-432]{}, it would lead to a slight over-correction of the magnitudes and distance. The distances derived in the two cases (using $J, H, {\rm
and}~K_s$ magnitudes) are $878 \pm 9$ pc (no extinction) and $859 \pm
13$ pc (entire column of extinction). Because the colors agree more closely [*without*]{} an extinction correction, it is tempting to conclude that only a small fraction of the extinction in the direction of [[*Kepler*]{}-432]{} actually lies between us and the star. However, in the direction to the star (i.e., out of the galactic plane), it seems unlikely that a significant column of absorbers would lie [*beyond*]{} ${\mathord{\sim}}1$ kpc. In reality, the model magnitudes are probably not a perfect match for the star and the appropriate reddening for this star is probably between $0$ and that implied by the full column ($0.079$). The derived distance does not depend strongly on the value we adopt for reddening, and we choose to use the mean of the two distance estimates and slightly inflate the errors: $d = 870 \pm
20$ pc.
{width="20.00000%"}{width="20.00000%"}{width="20.00000%"}{width="20.00000%"}{width="20.00000%"}\
{width="20.00000%"}{width="20.00000%"}{width="20.00000%"}{width="20.00000%"}{width="20.00000%"}
Orbital Solution {#sec:rv}
================
After recognition of the signature of the non-transiting planet in the [[*Kepler*]{}-432]{} RVs (the outer planet was identified both visually and via periodogram analysis), they were fit with two Keplerian orbits using a MCMC algorithm with the Metropolis-Hastings rule [@metropolis:1953; @hastings:1970] and a Gibbs sampler [a review of which can be found in @casella:1992]. Twelve parameters were included in the fit: for each planet, the times of inferior conjunction $T_{\rm 0}$, orbital periods $P$, radial-velocity semi-amplitudes $K$, and the orthogonal quantities $\sqrt{e}\sin{\omega}$ and $\sqrt{e}\cos{\omega}$, where $e$ is orbital eccentricity and $\omega$ is the longitude of periastron; the systemic velocity, $\gamma_{\rm rel}$, in the arbitrary zero point of the TRES relative RV data set; and the FIES RV offset, ${\Delta}RV_{\rm FIES}$. (The absolute systemic velocity, $\gamma_{\rm
abs}$, was calculated based on $\gamma_{\rm rel}$ and the offset between relative and absolute RVs, which are discussed in .) We applied Gaussian priors on $T_{\rm 0,b}$ and $P_{\rm b}$ based on the results of the light curve fitting.
[lrr]{}\[!h\] Inner Planet & &\
$T_{\rm 0,b}$ (BJD) & $2455949.5374^{+0.0018}_{-0.0016}$ & $2455949.5374^{+0.0011}_{-0.0012}$\
$P_{\rm b}$ (days) & $52.501134^{+0.000070}_{-0.000107}$ & $52.501129^{+0.000067}_{-0.000053}$\
$K_{\rm b}$ ([ms$^{-1}$]{}) & $285.9^{+4.1}_{-4.7}$ & $286.8^{+4.7}_{-4.0}$\
$\sqrt{e_{\rm b}}\cos{\omega_{\rm b}}$ & $0.294^{+0.022}_{-0.015}$ & $0.311^{+0.018}_{-0.016}$\
$\sqrt{e_{\rm b}}\sin{\omega_{\rm b}}$ & $0.6482^{+0.0134}_{-0.0094}$ & $0.645^{+0.012}_{-0.011}$\
$e_{\rm b}$ & $0.5121^{+0.0084}_{-0.0107}$ & $0.5134^{+0.0098}_{-0.0089}$\
$\omega_{\rm b}$ (deg) & $65.6^{+1.5}_{-1.8}$ & $64.1^{+1.6}_{-1.5}$\
$i_b$ (deg) & $88.17^{+0.61}_{-0.33}$ & $88.17^{+0.61}_{-0.33}$\
$M_{\rm b}$ ([$M_{\rm Jup}$]{}) & $5.41^{+0.30}_{-0.19}$ & $5.41^{+0.32}_{-0.18}$\
$R_{\rm b}$ ([$R_{\rm Jup}$]{}) & $1.145^{+0.036}_{-0.039}$ & $1.145^{+0.036}_{-0.039}$\
$\rho_{\rm b}~({\rm g\,cm}^{-3})$ & $4.46^{+0.36}_{-0.29}$ & $4.46^{+0.37}_{-0.29}$\
$a_{\rm b}$ (AU) & $0.301^{+0.016}_{-0.011}$ & $0.301^{+0.016}_{-0.011}$\
$\langle F_{\rm b} \rangle$ ($\langle F_\oplus \rangle$) & $118 \pm 10$ & $118 \pm 10$\
Outer Planet & &\
$T_{\rm 0,c}$ (BJD) & $2456134.9^{+3.0}_{-2.1}$ & $2456139.3^{+3.6}_{-2.9}$\
$P_{\rm c}$ (days) & $411.0^{+0.9}_{-3.2}$ & $406.2^{+3.9}_{-2.5}$\
$K_{\rm c}$ ([ms$^{-1}$]{}) & $73^{+25}_{-15}$ & $62.1^{+6.1}_{-5.8}$\
$\sqrt{e_{\rm c}}\cos{\omega_{\rm c}}$ & $0.47^{+0.12}_{-0.14}$ & $0.336^{+0.115}_{-0.076}$\
$\sqrt{e_{\rm c}}\sin{\omega_{\rm c}}$ & $0.648^{+0.072}_{-0.073}$ & $0.602^{+0.048}_{-0.071}$\
$e_{\rm c}$ & $0.64^{+0.14}_{-0.13}$ & $0.498^{+0.029}_{-0.059}$\
$\omega_{\rm c}$ (deg) & $53.2^{+8.7}_{-9.5}$ & $60.8^{+7.0}_{-11.2}$\
$M_{\rm c} \sin{i_{\rm c}}$ ([$M_{\rm Jup}$]{}) & $2.63^{+0.43}_{-0.35}$ & $2.43^{+0.22}_{-0.24}$\
$a_{\rm c} \sin{i_{\rm c}}$ (AU) & $1.188^{+0.062}_{-0.042}$ & $1.178^{+0.063}_{-0.042}$\
$\langle F_{\rm c} \rangle$ ($\langle F_\oplus \rangle$) & $8.5^{+2.1}_{-1.2}$ & $7.7^{+0.7}_{-0.8}$\
Other Parameters & &\
$\gamma_{\rm rel}$ ([ms$^{-1}$]{}) & $313.4^{+5.1}_{-4.5}$ & $306.7^{+2.6}_{-2.7}$\
$\gamma_{\rm abs}$ ([kms$^{-1}$]{}) & $-35.22 \pm 0.19$ & $-35.22 \pm 0.19$\
$\Delta RV_{\rm FIES}$ ([ms$^{-1}$]{}) & $-493.5^{+8.4}_{-10.4}$ & $-490.5^{+8.6}_{-7.7}$\
RV Jitter ([ms$^{-1}$]{}) & $20$ & $20$\
\[-2.4ex\]
We ran a chain with $1.01 \times 10^7$ steps, treating the first $10^5$ realizations as burn-in and thinning the chain by saving every tenth entry, for a final chain length of $10^6$. The marginalized posterior distributions are shown in [Figure \[fig:post\]]{}. It is apparent that while the parameters of the inner planet are very well constrained, there is a high-eccentricity tail of solutions for the outer planet that cannot be ruled out by RVs alone (we investigate this further using N-body simulations in ). Because several of the posteriors are non-Gaussian, we cannot simply adopt the median and central $68.3\%$ confidence interval as our best fit parameters and $1$-$\sigma$ errors as we normally might. Instead, we adopt best fit parameters from the mode of each distribution, which we identify from the peak of the probability density function (PDF). We generate the PDFs using a Gaussian kernel density estimator with bandwidths for each parameter chosen according to Silverman’s rule. We assign errors from the region that encloses $68.3\%$ of the PDF, and for which the bounding values have identical probability densities. That is, we require the $\pm1$-$\sigma$ values to have equal likelihoods. The resulting orbital solution using these parameters has velocity residuals larger than expected from the nominal RV uncertainties. We attribute this to some combination of astrophysical jitter (e.g., stellar activity or additional undetected planets) and imperfect treatment of the various noise sources described in . An analysis of the residuals does not reveal any significant periodicity, but given our measurement precision, we would not expect to detect any additional planets unless they were also massive gas giants, or orbiting at very small separations. To account for the observed velocity residuals, we re-run our MCMC with the inclusion of an additional RV jitter term. Tuning this until $\chi^2$ is equal to the number of degrees of freedom, we find an additional $20~{\ifmmode{\rm m\thinspace s^{-1}}\else m\thinspace s$^{-1}$\fi}$ jitter is required. While part of this jitter may be due to instrumental effects, we do not include separate jitter terms for the TRES and FIES RVs because the FIES data set is not rich enough to reliably determine the observed scatter. We report the best fit orbital and physical planetary parameters in [Table \[tab:planetary\]]{}, which also includes the set of parameters additionally constrained by dynamical stability simulations, as described in the N-body analysis of . The corresponding orbital solution is shown in [Figure \[fig:orbit\]]{}.
False Positive Scenarios {#sec:fp}
========================
An apparent planetary signal (transit or RV) can sometimes be caused by astrophysical false positives. We consider several scenarios in which one of the [[*Kepler*]{}-432]{} planetary signals is caused by something other than a planet, and we run a number of tests to rule these out.
For an object with a deep transit, such as [[*Kepler*]{}-432]{}[b]{}, one may worry that the orbiting object is actually a small star, or that the signal is caused by a blend with an eclipsing binary system. @sliski:2014 noted that there are very few planets orbiting evolved stars with periods shorter than $100$ days ([[*Kepler*]{}-432]{}[b]{} would be somewhat of an outlier), and also found via AP that either the transit signal must be caused by a blend, or it must have significant eccentricity ($e>0.488$). Without additional evidence, this would be cause for concern, but our radial velocity curve demonstrates that the transiting object is indeed orbiting the target star, that its mass is planetary, and, consistent with the prediction of @sliski:2014, its eccentricity is $0.5134$.
If a planet does not transit, as is the case for [[*Kepler*]{}-432]{}[c]{}, determining the authenticity of the planetary signal is less straightforward. An apparent radial-velocity orbit can be induced by a genuine planet, spots rotating on the stellar surface [@queloz:2001], or a blended stellar binary [@mandushev:2005]. Both of these false positive scenarios should manifest themselves in the shapes of the stellar spectral lines. That is, spots with enough contrast with the photosphere to induce apparent RV variations will also deform the line profiles, as should blended binaries bright enough to influence the derived RVs. A standard prescription for characterizing the shape of a line is to measure the relative velocity at its top and bottom; this difference is referred to as a line bisector span [see, e.g., @torres:2005]. To test against the scenarios described, we computed the line bisector spans for the TRES spectra. We do find a possible correlation with the RVs of the outer planet, having a Spearman’s rank correlation value of $-0.21$ and a significance of $94\%$. While this is a potential concern and we cannot conclusively demonstrate the planetary nature of the $407$-day signal, we find it to be the most likely interpretation. In the following paragraphs, we explain why other interpretations are unlikely.
With the discovery of a close stellar companion (see ), it is reasonable to ask how that might affect interpretation of the outer planetary signal. That is, if the stellar companion is itself a binary with a period of ${\mathord{\sim}}400$ days, could it cause the RV signal we observe? The answer is unequivocally no, as the companion is far too faint in the optical compared to the primary star ($\Delta V{\mathord{\sim}}7$) to contribute any significant light to the spectrum, let alone induce a variation of ${\mathord{\sim}}100~{\ifmmode{\rm m\thinspace s^{-1}}\else m\thinspace s$^{-1}$\fi}$ or affect the bisector spans. If there is also a brighter visual companion inside the resolution limit of our high resolution images (${\mathord{\sim}}0\farcs05 \simeq 45$ AU projected separation), it could be a binary with a $400$-day period responsible for the RV and bisector span variations on that timescale. However, such a close physically bound binary may pose problems for formation of the $52.5$-day planet, and the a priori likelihood of a background star bright enough to cause the observed variations within $0.05\arcsec$ is extremely low; a TRILEGAL simulation suggests ${\mathord{\sim}}5 \times 10^{-6}$ background sources should be expected, and only a small fraction of those would be expected to host a binary with the correct systemic velocity.
To rule out spot-induced velocity variation, we examine the [*Kepler*]{} light curve for evidence of spot activity. From the measured $v\sin{i_\star}$, $i_\star$, and $R_\star$, the stellar rotation period is $77 \pm 14$ days. Not only is this inconsistent with the observed outer orbital period ($407$ days), but we detect no significant photometric signal near either of these periods. For [[*Kepler*]{}-432]{} ($v\sin{i_\star}=2.7~{\ifmmode{\rm km\thinspace s^{-1}}\else km\thinspace s$^{-1}$\fi}$), a spot must cover ${\mathord{\sim}}4$%–$5$% of the stellar surface to induce the observed RV amplitude [@saar:1997], and such a spot would have been apparent in the high precision [*Kepler*]{} light curve.
As we have shown in , [[*Kepler*]{}-432]{} exhibits strong oscillations, so one may wonder whether these could induce the observed RV signal for the outer planet. Oscillations on a $400$-day timescale are intrinsically unlikely, as there is no known driving mechanism that could cause them in giant stars; the well-known stochastically driven oscillations are confined to much higher frequencies. Furthermore, if there were such a mechanism, a mode with ${\mathord{\sim}}50~{\ifmmode{\rm m\thinspace s^{-1}}\else m\thinspace s$^{-1}$\fi}$ velocity semi-amplitude should cause a photometric variation of ${\mathord{\sim}}1.3$mmag [see @kjeldsen:1995], which is clearly ruled out by the [*Kepler*]{} data.
Upon examining all of the evidence available, we conclude that both detected orbits are caused by bona fide planets orbiting the primary star.
Orbital Stability Analysis {#sec:nbody}
==========================
Methodology {#sec:nbody_method}
-----------
Following the Keplerian MCMC fitting procedure described in , we wish to understand whether these posterior solutions are dynamically stable—i.e., whether they describe realistic systems which could survive to the $3.5$ Gyr age of the system. When presenting our results in the sections that follow, we use the planet–planet separation as an easily visualized proxy for stability: given the semi-major axes of the two planets—${\mathord{\sim}}0.3$ and ${\mathord{\sim}}1.2$AU—separations greater than ${\mathord{\sim}}10$AU are clear indications that the system has suffered an instability and the planets subsequently scattered. If some solutions do prove to be unstable, it will lead to further constraints on the orbital elements, and thus the planetary masses. We perform integrations of both coplanar and inclined systems. We first explore the less computationally expensive coplanar case to understand the behavior of the system, and then extend the simulations to include inclination in the outer orbit, which has the additional potential to constrain the mutual inclination of the planets.
To understand this stability, we utilize the integration algorithm described in @payne:2013. This algorithm uses a symplectic method in Jacobi coordinates, making it both accurate and rapid for systems with arbitrary planet-to-star mass ratios. It uses calculations of the tangent equations to evaluate the Lyapunov exponents for the system, providing a detailed insight into whether the system is stable (up to the length of the simulation examined) or exhibits chaos (and hence instability).
For the coplanar systems, we take the ensemble of $10^6$ solutions generated in (which form the basis of the reported elements in column 2 of [Table \[tab:planetary\]]{}) and convert these to Cartesian coordinates, assuming that the system is coplanar and edge-on ($90{\ensuremath{^\circ}}$), and hence both planets have their minimum masses. We then evolve the systems forward for a fixed period of time (more detail supplied in below) and examine some critical diagnostics for the system (e.g., the Lyapunov time, and the planet–planet separation) to understand whether the system remains stable, or whether some significant instability has become apparent.
For the inclined systems, we assume that the inner (transiting) planet is edge-on, and hence retains its measured mass. However, the outer planet is assigned an inclination that is drawn randomly from a uniform distribution $0{\ensuremath{^\circ}}< i_{\rm c} < 90{\ensuremath{^\circ}}$, and its mass is scaled by a factor $1 / \sin{i_{\rm c}}$. As such, the outer planet can have a mass that is significantly above the minimum values used in the coplanar case. The longitude of ascending node for the outer planet is drawn randomly from a uniform distribution between $0$ and $2\pi$. We then proceed as in the coplanar case, integrating the systems forward in time to understand whether the initial conditions chosen can give rise to long-term stable systems.
@rauch:1999 demonstrated that $\sim 20$ timesteps per inner-most orbit is sufficient to ensure numerical stability in symplectic integrations. As the inner planet has a period ${\mathord{\sim}}50$ days, we use a timestep of $1$ day in all of our simulations, ensuring that our integrations will comfortably maintain the desired energy conservation and hence numerical accuracy.
Coplanar Stability {#sec:nbody_coplanar}
------------------
We begin by taking a random selection of $10^5$ of the $10^6$ solutions from and integrating them for a period of $10^4$ yr ($3 \times 10^6$ timesteps). While this is [*not*]{} a particularly long integration period compared with the period of the planets (${\mathord{\sim}}50$ and ${\mathord{\sim}}400$ days), we demonstrate that even during this relatively short integration, approximately half of the systems become unstable. Tellingly, the unstable systems all tend to be the systems in which the outer planet has particularly high eccentricity. We illustrate this in [Figure \[fig:stability\]]{}, where we plot the mass-eccentricity plane for the outer planet ($m_{\rm
c},e_{\rm c}$) and plot the separation between the two planets in the system at $t=10^4$ yr. As described above, separations greater than ${\mathord{\sim}}10$AU indicate that the system has suffered an instability. This initial simulation clearly demonstrates that at $t=10^4$ yr essentially all systems with $e_{\rm c} > 0.8$ are unstable, all those with $e_{\rm c} < 0.45$ are stable, and those with $0.45 < e_{\rm c} < 0.8$ are “mixed,” with some being stable and some being unstable.
Given this promising demonstration that the dynamical integrations can restrict the set of solutions, we go on to integrate the systems for increasingly longer periods of time. To save on integration time/cost, we only select the stable solutions from the previous step, and then extend the integration time by an order of magnitude, perform a stability analysis, and repeat. By this method, the $10^5$ systems at $t=0$ are reduced to ${\mathord{\sim}}4.4 \times 10^4$ stable systems at $10^4$yr, ${\mathord{\sim}}2.5 \times 10^4$ stable systems at $10^5$yr, and ${\mathord{\sim}}9.4 \times 10^3$ stable systems at $10^6$yr. We illustrate in [Figure \[fig:stability\]]{} the successive restriction of the parameter space in the mass-eccentricity plane for the outer planet ($m_{\rm
c},e_{\rm c}$) as the integration timescales increase. We find that the long-term stable systems occupy a significantly smaller region of parameter space in the $m_{\rm c},e_{\rm c}$ plane. In particular, we see the eccentricity of the outer planet is restricted to $e_{\rm c}
\lsim 0.55$.
Using the ${\mathord{\sim}}9.4 \times 10^3$ systems which remained stable at $10^6$yr, we use the same method as in to determine a new set of best-fit orbital paramters. That is, we use a Gaussian kernel density estimator to smooth the posterior distribution and select the mode (i.e., the value with the largest probability density). The errors correspond to values with equal probability density that enclose the mode and $68.3\%$ of the PDF. The most significant changes in the best-fit parameters occurred for the outer planet, most notably for eccentricity and velocity semi-amplitude, resulting in smaller and more symmetric error bars and a revision in the best-fit minimum mass for planet c.
Inclined-system Stability {#sec:nbody_inclined}
-------------------------
In a manner similar to the coplanar analysis of , we begin by taking all $10^6$ solutions from and integrating them for a period of $10^4$yr ($3
\times 10^6$ timesteps). However, in these inclined system integrations, the outer planet has an inclination that is not edge-on, and hence a mass for the outer planet that is inflated compared to its minimum edge-on value, as described in .
A much larger fraction of the inclined systems become unstable in the first $10^4$yr, and we find that the $N=10^6$ systems at $t=0$ are reduced to $N {\mathord{\sim}}1.5 \times 10^5$ stable systems at $10^4$yr. Integrating these stable systems while keeping the pre-assigned inclinations for the outer planet (i.e., we do [*not*]{} re-randomize the inclinations) we integrate this subset on to $10^5$yr, and find that the number of stable systems subsequently reduces to $N {\mathord{\sim}}10^4$. We plot these stability results as a function of mass and inclination of the outer planet in [Figure \[fig:inclined\]]{}.
We find that the vast majority of the stable solutions are (somewhat unsuprisingly) restricted to the range of parameter space with relative inclinations $\lsim 70{\ensuremath{^\circ}}$ (i.e., $i_{\rm c} \gsim 20{\ensuremath{^\circ}}$ in [Figure \[fig:inclined\]]{}). There is a small population with significantly higher eccentricities and relative inclinations that remains stable at $t=10^5$yr, and it is possible that these systems exhibit strong Kozai oscillations, but their long-term ($>10^5$yr) behavior has not been investigated. We emphasize that the inclinations and longitudes of the ascending node were assigned [*randomly*]{}, and hence it is possible that specifically chosen orbital alignments could allow for enhanced stability in certain cases. Unfortunately, because some systems remain stable even for highly inclined outer orbits ($i_c{\mathord{\sim}}0{\ensuremath{^\circ}}$), we are unable to strongly constrain the mutual inclination of the planetary orbits.
We also note that the presence of a distant stellar companion (as detected in our AO images) or additional, as yet undetected, planets (as may be suggested by the excess RV scatter) could influence the long-term stability of the systems simulated herein. The evolution of the star on the red giant branch is also likely to affect the orbital evolution (especially for the inner planet; see ), but it is not important over the $10^6$-year timescales of these simulations. Simulating the effect of poorly characterized or hypothetical orbits, or the interactions between expanding stars and their planets, is beyond the scope of the current paper, and instead we simply remind the reader of these complications. We present planetary properties derived with and without constraints from our simulations ([Table \[tab:planetary\]]{}) so that the cautious reader may choose to adopt the more conservative (and poorly determined) parameters of the outer orbit in any subsequent analysis.
Discussion {#sec:disc}
==========
The properties of the star and planets of [[*Kepler*]{}-432]{} are unusual in several ways among the known exoplanets, which makes it a valuable system to study in detail. For example, it is a planetary system around an intermediate mass star (and an evolved star), it hosts a planet of intermediate period, and it hosts at least one very massive transiting planet. Close examination of the system may provide insight into the processes of planet formation and orbital evolution in such regimes. [[*Kepler*]{}-432]{} is also the first planet orbiting a giant star to have its eccentricity independently determined by RVs and photometry, which helps address a concern that granulation noise in giants can inhibit such photometric measurements.
Comparing the Eccentricity from AP versus RVs
---------------------------------------------
AP provides an independent technique for measuring orbital eccentricities with photometry alone, via the so-called photoeccentric effect [@dawson:2012], and can be used as a tool to evaluate the quality of planet candidates [@tingley:2011]. It was originally envisioned as a technique for measuring eccentricities [@kipping:2012], but other effects, such as a background blend, can also produce AP effects [@kipping:2014a]. Given that we here have an independent radial velocity orbital solution, there is an opportunity to compare the two independent solutions, allowing us to comment on the utility of AP.
![ The marginalized and joint posterior distributions of $e$ and $\omega$, as determined by AP and RVs. The three panels in the lower left are from AP; the three in the upper right (the narrow histograms and well-constrained joint distribution) are from RVs, which also show the AP distributions overplotted as thin gray dotted lines for comparison. \[fig:ew\]](fig13.pdf){width="48.00000%"}
[[*Kepler*]{}-432]{}[b]{} was previously analyzed as part of an ensemble AP analysis by @sliski:2014. In that work, the authors concluded that [[*Kepler*]{}-432]{}[b]{} displayed a strong AP effect, either because the candidate was a false-positive or because it exhibited a strong photoeccentric effect with $e\geq0.488_{-0.051}^{+0.025}$. Our analysis, using slightly more data and a full $\omega$ marginalization is excellent agreement with that result, finding $e=0.507^{+0.039}_{-0.114}$ and $\omega=76^{+59}_{-24}$deg. The AP results may be compared to that from our radial velocity solution, where again we find excellent agreement, since RVs yield $e=0.5134^{+0.0098}_{-0.0089}$ and $\omega=64.1^{+1.6}_{-1.5}$deg. The results may be visually compared in [Figure \[fig:ew\]]{}. We note that this is not the first time AP and RVs have been shown to yield self-consistent results, with @dawson:2012 demonstrating the same for the Sun-like star HD 17156b. However, this is the first time that this agreement has been established for a giant host star. This is particularly salient in light of the work of @sliski:2014, who find that the AP deviations of giant stars are consistently excessively large. The authors proposed that many of the KOIs around giant stars were actually orbiting a different star, with the remaining being eccentric planets around giant stars. [[*Kepler*]{}-432]{}[b]{} falls into the latter category, consistent with its original ambiguous categorization as being either a false positive or photoeccentric.
The analysis presented here demonstrates that AP can produce accurate results for giant stars. This indicates that the unusually high AP deviations of giant stars observed by @sliski:2014 cannot be solely due to time-correlated noise caused by stellar granulation, which has recently been proposed by @barclay:2014 to explain the discrepancy between AP and radial velocity data for [*Kepler*]{}-91b. However, time-correlated noise may still be an important factor in giant host stars which are more evolved than [[*Kepler*]{}-432]{} (such as [*Kepler*]{}-91), for which granulation becomes more pronounced [@mathur:2011]. To further investigate these hypotheses, we advocate for further observations of giant planet-candidate host stars to resolve the source of the AP anomalies.
A Benchmark for Compositions of Super-Jupiters
----------------------------------------------
![ The masses and radii of known giant planets (red triangles) and [[*Kepler*]{}-432]{}[b]{} (blue circle). Overplotted as solid black lines are a series of planetary models [@fortney:2007] with age $3.5$ Gyr, an appropriate semi-major axis, and core masses $0$, $10$, $25$, $50$, and $100~{\ifmmode{M_\oplus}\else $M_\oplus$\fi}$ (from top to bottom). Also plotted are models for young ($100$Myr), hot ($0.02$AU), coreless planets (dotted gray line); and old ($10$Gyr), cool ($1$AU) planets with massive ($100~{\ifmmode{M_\oplus}\else $M_\oplus$\fi}$) cores (dashed gray line). These roughly illustrate the range of sizes possible for a given mass. \[fig:massradius\]](fig14.pdf){width="48.00000%"}
[[*Kepler*]{}-432]{}[b]{} has a measured mass, radius, and age, which allows us to investigate its bulk composition. Among transiting planets—i.e., those for which interior modeling is possible—there are only $11$ more massive than $4.5~{\ifmmode{M_{\rm Jup}}\else $M_{\rm Jup}$\fi}$. We also point out the gap in the center of the plot in [Figure \[fig:massradius\]]{}; no planets with masses between $4$ and $7.25~{\ifmmode{M_{\rm Jup}}\else $M_{\rm Jup}$\fi}$ also have measured radii, so [[*Kepler*]{}-432]{}[b]{} immediately becomes a valuable data point to modelers. Moreover, most of the super-Jupiters are highly irradiated, which further complicates modeling and interpretation of planetary structure. [[*Kepler*]{}-432]{}[b]{} receives only about $20\%$ of the insolation of a $3$-day hot Jupiter orbiting a Sun-like star. Because of this, it may prove to be an important benchmark for planetary interior models—for example, as a means of checking the accuracy of models of super-Jupiters without the complication of high levels of incident flux. In [Figure \[fig:massradius\]]{}, we compare the mass and radius of [[*Kepler*]{}-432]{}[b]{} to age- and insolation-appropriate planetary models [@fortney:2007] of varying core mass. We interpolate the models to a planet with the age of [[*Kepler*]{}-432]{} ($3.5$ Gyr) in a circular orbit of $0.092$ AU around a Sun-like star (the insolation of which is identical to the time-averaged insolation of [[*Kepler*]{}-432]{}[b]{} on its more distant eccentric orbit around a more luminous star). The radius is apparently slightly inflated, but is somewhat consistent ($1$-$\sigma$) with that of a planet lacking a core of heavy elements. This is not an iron-clad result, though; the radius even agrees with the prediction for a planet with a $50~{\ifmmode{M_\oplus}\else $M_\oplus$\fi}$ core to within $1.5$-$\sigma$. Nevertheless, we interpret this as evidence that [[*Kepler*]{}-432]{}[b]{} most likely has only a small core of heavy elements.
Jupiters Do (Briefly) Orbit Giants Within $0.5$AU
-------------------------------------------------
As discussed previously, we do not know of many transiting giant planets orbiting giant stars. This is not because giant planets orbiting the progenitors to giant stars are rare; due in part to survey strategies, giant stars are, on average, more massive than the known main-sequence planet hosts, and massive stars seem to be [*more*]{} likely to harbor massive planets. However, because very few giant stars host planets inside $1$AU, the a priori probability of a transit is low for these systems. In fact, there are only two other giant planets with a measured mass ($M_p > 0.5~{\ifmmode{M_{\rm Jup}}\else $M_{\rm Jup}$\fi}$) transiting a giant star ($\log{g_\star} < 3.9$)—[*Kepler*]{}-56c [@huber:2013b], and the soon-to-be-swallowed hot Jupiter [*Kepler*]{}-91b [@barclay:2014; @lillo:2014a; @lillo:2014b]. This makes [[*Kepler*]{}-432]{}[b]{} important in at least two respects: it is a rare fully characterized giant planet transiting a giant star, and it is a short-period outlier among giant planets orbiting giant stars (see [Figure \[fig:mass\_semi\]]{}). [[*Kepler*]{}-432]{}[b]{} thus gives us a new lens through which to examine the dearth of short and intermediate period Jupiters ($<0.5$AU) around giant stars. [[*Kepler*]{}-432]{}[c]{}, on the other hand, appears to be very typical of Jupiters around evolved stars, in both mass and orbital separation.
After noticing that [[*Kepler*]{}-432]{}[b]{} sits all alone in the $M_{\rm p}$ versus $a$ parameter space, it is natural to wonder why, and we suggest two potential explanations, each of which may contribute to this apparent planetary desert. In the first, we consider that [[*Kepler*]{}-432]{}[b]{} may simply be a member of the tail of the period distribution of planets around massive and intermediate-mass main-sequence stars. That is, perhaps massive main-sequence stars simply harbor very few planets with separations less than $1$AU. If this occurrence rate is a smooth function of stellar mass, then because [[*Kepler*]{}-432]{} would most accurately be called intermediate mass, it might not be so surprising that it harbors a planet with a separation of $0.3$AU while its more massive counterparts do not. If this is the case, then as we detect more giant planets orbiting giant stars (and main-sequence stars above the Kraft break), we can expect to find a sparsely populated tail of planets interior to $1$AU. This may ultimately prove to be responsible for the observed distribution, and would have important implications for giant planet formation and migration around intermediate mass stars in comparison to Sun-like stars, but it cannot be confirmed now. It will require additional planet searches around evolved stars and improvements in detecting long period planets orbiting rapidly rotating main-sequence stars.
![ All planets with minimum masses $M_{\rm p}\sin{i} >
0.5~{\ifmmode{M_{\rm Jup}}\else $M_{\rm Jup}$\fi}$ orbiting giant stars ($\log{g_\star} < 3.9$), plotted as red triangles. [[*Kepler*]{}-432]{}[b]{} and c are labeled and plotted as large blue circles. \[fig:mass\_semi\]](fig15.pdf){width="48.00000%"}
In the second scenario, we consider that [[*Kepler*]{}-432]{}[b]{} may be a member of a more numerous group of planets interior to ${\mathord{\sim}}0.5$AU that exist around main-sequence F- and A-type stars, but do not survive through the red giant phase when the star expands to a significant fraction of an AU. If this is the case, then [[*Kepler*]{}-432]{}[b]{} is fated to be swallowed by its star through some combination of expansion of the stellar atmosphere and orbital decay, and we only observe it now because [[*Kepler*]{}-432]{} has only recently begun its ascent up the red giant branch. Suggestively, the resulting system that would include only [[*Kepler*]{}-432]{}[c]{} would look much more typical of giant stars. In this scenario, planets must begin their orbital decay very soon after the star evolves off of the main sequence, or we would expect to observe many more of them. Assuming, then, that [[*Kepler*]{}-432]{}[b]{} is close enough to its host to have started its orbital decay, we may be able to detect evidence of tidal or magnetic SPI [see, e.g., @shkolnik:2009]. As the star expands those interactions would strengthen, which should lead to more rapid orbital decay. While typically tidal interaction leading to circularization and orbital decay has been modeled to depend only on tides in the planet, @jackson:2008 show that tides in the star (which are strongly dependent on the stellar radius) can also influence orbital evolution. This provides a tidal mechanism for more distant planets to experience enhanced orbital evolution as the star expands. Recent simulations by @strugarek:2014 further demonstrate that in some scenarios (especially with strong stellar magnetic fields and slow rotation), magnetic SPI can be as important to orbital migration as tidal SPI, and can lead to rapid orbital decay. [[*Kepler*]{}-432]{} does rotate slowly, and the results of a two-decade survey by @konstantinova-antova:2013 reveal that giants can exhibit field strengths up to ${\mathord{\sim}}100$ G. Encouraged by these findings, we search for evidence of SPI in the [[*Kepler*]{}-432]{} system in .
The hypothetical scenario in which planets inside ${\mathord{\sim}}0.5$AU around massive and intermediate-mass stars ultimately get destroyed may also help explain the properties of the observed distribution of planets orbiting evolved stars. If planets form on circular orbits, then in order to excite large eccentricities, a planet must experience dynamical interaction with another planet [e.g., @rasio:1996; @juric:2008] or another star [e.g., @fabrycky:2007; @naoz:2012]. More massive stars (like many of those that have become giants) are more likely than lower mass, Sun-like stars to form giant planets, and they are also more likely to have binary companions [e.g., @derosa:2014; @raghavan:2010], so one might expect the typical planet orbiting a giant star to be [*more*]{} eccentric, not less. However, the most eccentric planets around evolved stars would have pericenter distances near, or inside the critical separation at which planets get destroyed. This could lead either to destruction by the same mechanisms as the close-in planets, or to partial orbital decay and circularization. The consequence of either process would be more circular orbits on average and an overabundance of planets at separations near the critical separation ($0.5\,{\rm AU} \lsim a \lsim
1\,{\rm AU}$), which is indeed observed. For similar arguments and additional discussion, see @jones:2014. Counterarguments articulated by those authors include that massive subgiants do not seem to host many planets inside $1$AU either, but subgiants are not large enough to have swallowed them yet. This might point toward a primordial difference in the planetary period distribution between Sun-like and more massive stars, rather than a difference that evolves over time.
Implications for Giant Planet Formation and Migration
-----------------------------------------------------
Assuming that giant planets form beyond the ice line [located at ${\mathord{\sim}}4.9$AU for a star with $M{\mathord{\sim}}1.4~{\ifmmode{M_{\odot}}\else $M_{\odot}$\fi}$, using the approximation of @ida:2005], both [[*Kepler*]{}-432]{}[b]{} and c have experienced significant inward migration. At first glance, their large orbital eccentricities would suggest that they have experienced gravitational interactions, perhaps during one or more planet–planet scattering events that brought them to their current orbital distances, or through the influence of an outer companion (like that detected in our AO images). However, the apparent alignment between the stellar spin axis and the orbit of the inner planet presents a puzzle; if the (initially circular, coplanar) planets migrated via multi-body interactions, we would expect both the eccentricities [*and*]{} the inclinations to grow. It is possible, of course, that by chance the inner planet remained relatively well-aligned after scattering, or that the planets experienced coplanar, high-eccentricity migration, which has recently been suggested as a mechanism for producing hot Jupiters [@petrovich:2014]. Measuring the mutual inclination between the planets could lend credence to one of these scenarios. Unfortunately, because we are unable to constrain the mutual inclination of the planets, we cannot say whether they are mutually well-aligned (which would argue against planet-planet scattering) or not.
![ An underlying distribution of true obliquities, $\psi$, corresponding to a hypothetical measured stellar inclination, $i_\star$. A system with edge-on rotation has $i_\star = 90{\ensuremath{^\circ}}$, and $\psi = 0{\ensuremath{^\circ}}$ indicates perfect alignment between stellar spin and planetary orbital angular momentum. The long tail of high obliquities illustrates that even with a nearly edge-on stellar spin, a transiting planet may still be misaligned. \[fig:obliquity\]](fig16.pdf){width="48.00000%"}
At this point, it is prudent to reiterate that the stellar spin and orbital angular momenta are measured to be nearly aligned only as projected along the line of sight. The true obliquity is unknown, and may be significantly non-zero. In [Figure \[fig:obliquity\]]{}, we illustrate this by simulating the underlying obliquity distribution that corresponds to an example $i_\star$ distribution with a mean of $83{\ensuremath{^\circ}}$, similar to that measured from asteroseismology. To generate this obliquity distribution, we draw from $i_\star$ and a random uniform azimuthal angle, $\theta$, and calculate the true obliquity, $\psi$. As is apparent from the $\psi$ distribution, while it is likely that the true obliquity is low, there is a tail of highly misaligned systems (representing those with spin axes lying nearly in the plane of the sky but not coincident with the orbital angular momentum) that could produce the stellar inclination we observe. Ideally, we would also measure the sky-projected spin–orbit angle, $\lambda$, in order to detect any such conspiring misalignment and calculate the true obliquity. However, there is not an obvious way to do this: the slow rotation, long period, and lack of star spots render current techniques to measure the sky-projected obliquity ineffective. We are left with an inner planet that is [*likely*]{}, but not certainly, aligned with the stellar spin axis, and an unknown inclination of the outer orbital plane.
The likely alignment of the inner planet seems at odds with the large orbital eccentricities and the assumption that multi-body interactions are responsible for their migration, but we have thus far neglected interaction between the planets and the host star. In their investigation of hot Jupiter obliquities, @winn:2010 suggested that well-aligned hot Jupiters may have been misaligned previously, and that subsequent tidal interaction may have realigned the stellar rotation with the planetary orbit, perhaps influencing the convective envelope independently of the interior. This idea was furthered by @albrecht:2012, who found that systems with short tidal-dissipation timescales are likely to be aligned, while those with long timescales are found with a wide range of obliquities. In the following section, we explore the possibility that [[*Kepler*]{}-432]{}[b]{} has undergone a similar evolution, obtaining an inclined orbit through its inward migration, but subsequently realigning the stellar spin.
Evidence for Spin–Orbit Evolution {#sec:tidal_ev}
---------------------------------
During the main-sequence lifetime of [[*Kepler*]{}-432]{}, the current position of the planets would be too far from the star to raise any significant tides, but the star is now several times its original size, such that at periastron, the inner planet passes within $7.7$ stellar radii of the star. The unusually large planetary mass also helps strengthen tidal interactions. Following @albrecht:2012 [who in turn used the formulae of @zahn:1977], for [[*Kepler*]{}-432]{} we calculate the tidal timescale for a star with a convective envelope:
$$\tau_{\rm CE} = (10^{10}~{\rm yr}) q^{-2} \left(\frac{a/R_\star}{40}\right)^{6}.$$
For a radiative atmosphere, as [[*Kepler*]{}-432]{} would have had on the main sequence, the appropriate timescale would be
$$\tau_{\rm RA} = (1.25 \times 10^9~{\rm yr}) q^{-2} (1 + q)^{-5/6} \left(\frac{a/R_\star}{6}\right)^{17/2}.$$
We repeat the words of caution from @albrecht:2012: these equations are calibrated to star–star interactions rather than star–planet interactions, and are timescales for spin–orbit synchronization, not realignment. As such, the timescales are assumed to be valid only as a relative metric. The authors arbitrarily divided the resulting values by $5 \times 10^9$ to express this point, and to set the values on a convenient scale. In [Figure \[fig:tidal\]]{}, we have reproduced a version of Figure 24 of @albrecht:2012, showing the relative tidal dissipation timescales for all hot Jupiters with measured spin–orbit angles. We have excluded the same systems they did, and added $9$ recent projected obliquity measurements, including that of [[*Kepler*]{}-432]{}[b]{}. Their result still holds true; systems with short tidal dissipation timescales are well aligned, while those with long tidal timescales display a wide range of obliquities. We also note that recent work by @valsecchi:2014, which includes a more detailed treatment of the convection and stellar evolution of each star, similarly concludes that the observed obliquity distribution can be explained by tidal evolution. While the angle measured for [[*Kepler*]{}-432]{}[b]{} is the line of sight projection rather than the sky-plane projection, and the efficiency of realignment may be different for evolved stars given their different internal structure, [[*Kepler*]{}-432]{}[b]{} does have a short tidal timescale, and it sits in a region with hot Jupiter systems that are mostly well-aligned. We interpret this result as evidence that the spin axis of [[*Kepler*]{}-432]{} has been realigned to the orbit of the inner planet by the same mechanism responsible for hot Jupiter realignment.
Alternative mechanisms have been proposed to explain the alignment for hot Jupiters orbiting cool stars. One such alternative, suggested by @dawson:2014, is that the orbital alignment trend with stellar temperature is due to strong versus weak magnetic breaking (for stars below and above the Kraft break), rather than tidal dissipation efficiency as in @albrecht:2012. This has the attractive quality of being part of a theory that is also able to predict the observed mass cut-off for retrograde planets and the trend between planet mass and host star rotation periods. In the @dawson:2014 framework, one may still conclude that [[*Kepler*]{}-432]{} would realign during its red giant phase, but this depends in part on the unknown magnetic properties of the star: the @konstantinova-antova:2013 finding of strong magnetic fields in some giants may indeed indicate that [[*Kepler*]{}-432]{} possesses a short enough magnetic braking timescale to allow for realignment in this framework, too.
![ An adapted version of Figure 24 from @albrecht:2012. We plot the projected obliquity of giant planets versus the relative tidal timescale. We include planets from @albrecht:2012 as well as $8$ more recent spin–orbit measurements from the literature (red triangles) and [[*Kepler*]{}-432]{}[b]{} (large blue circle). Because it sits in a region with well-aligned hot Jupiters, we posit that [[*Kepler*]{}-432]{}[b]{} has also realigned the spin of its host star. \[fig:tidal\]](fig17.pdf){width="48.00000%"}
In the event that the envelope and the core have different spin axis inclinations, this might manifest itself in the asteroseismic modes. Because $g$-mode-dominated $\ell=1$ modes are most sensitive to core rotation while $p$-mode-dominated mixed modes are most sensitive to the surface, one might obtain different inclination measurements from the different modes. However, our analysis of individual $\ell=1$ modes in showed they were consistent with ${\mathord{\sim}}90{\ensuremath{^\circ}}$ inclination for both core and envelope. This would argue [*against*]{} a realigned envelope for [[*Kepler*]{}-432]{}, but a modest misalignment might be undetectable given the degraded precision when fitting modes independently. Furthermore, as shown for the obliquity and spin axis in [Figure \[fig:obliquity\]]{}, even if the two spin axes both lie in the sky plane, there is a chance they could be significantly misaligned. Regardless of its efficacy in the case of [[*Kepler*]{}-432]{}, asteroseismology may be able to detect a significant misalignment between core and envelope in other systems hosting giant planets with short tidal timescales. If detected, this would be important evidence in the interpretation of giant planet migration, and we suggest this be explored for other systems.
![ The [*Kepler*]{} light curve, binned and folded to the period of the transiting planet, with the transits removed. Given that periastron occurs just before transit and the stellar rotation period is longer than the orbital period, the brightening just after transit (orbital phase 0) is what would be expected for a bright spot that is tidally excited during the periastron passage of the planet. \[fig:fold\_spot\]](fig18.pdf){width="48.00000%"}
While the evidence presented thus far for realignment of the [[*Kepler*]{}-432]{} system is circumstantial, there may be additional evidence to support the scenario. If tidal forces are strong enough to realign the spin axis of the host star, they may also excite activity on the stellar surface. SPI need not be limited to tidal interaction, though. At closest approach, the planet is only $7.7$ stellar radii from the star, and thus magnetic interaction may also be possible, since the planet passes inside the Alfvén radius (${\mathord{\sim}}10\,R_\star$). We return to the light curve to look for signs of these interactions. When folded on the orbital period, an interesting feature emerges (see [Figure \[fig:fold\_spot\]]{}). Just after transit, we detect a brightening of ${\mathord{\sim}}0.05$mmag that maintains a coherent shape and phase throughout the mission. The simplest explanation for a light curve feature with the period of the planet is a phenomenon associated with the planet. We suggest that SPI excites a bright spot on the stellar photosphere each orbit during periastron passage. Periastron occurs $1.24$ days before transit, and because the rotation period is longer than the orbital period, the spot should rotate onto the meridian and reach maximum apparent brightness slightly after transit, as is observed.
One might argue that if the planet has realigned the stellar spin, it may have also synchronized it to the orbital period—this would cause [*any*]{} long-lived bright spot to create a peak in the light curve when folded to the orbital period. However, to obtain the observed phasing, the spot would have to conveniently occur at the longitude that coincides with the close approach of the planet. There is also no reason to expect a long-lived bright spot on the star, irrespective of its phasing. Furthermore, there exists evidence that the spin is [*not*]{} synchronized to the orbital period anyway. We have measured $R_\star$, $i_\star$, and $v\sin{i_\star}$, so we can estimate the rotation period to be $P_{\rm rot}{\mathord{\sim}}77 \pm 14$ days, which is inconsistent with the orbital period ($P_{\rm orb} =
52.5$ days). Thus, a long-lived rotating star spot should not produce a coherent photometric signal when folded to the orbital period. (Similarly, we must conclude that if $P_{\rm rot} \neq P_{\rm
orb}$, any planet-induced spot must decay in brightness relatively quickly, or the brightening feature would appear at different phases in each orbit, washing out the coherent signal that we observe.) We do caution that our $v\sin{i_\star}$ measurement for this slowly rotating giant could be biased, for example, due to the unknown macroturbulent velocity of [[*Kepler*]{}-432]{}. However, even if our measurement of the rotational velocity is incorrect and the rotation period [*is*]{} synchronized to the orbit, we believe a bright star spot phased with the periastron passage of the planet would more easily be attributed to SPI than coincidence.
One additional worry would be that granulation could produce correlated noise that happens to fold coherently on the orbital period. We find this to be an unlikely scenario given the periodogram of the [*Kepler*]{} data (see [Figure \[fig:periodogram\]]{}). If granulation produced light curve amplitudes similar to that observed, we would expect several periods to show similar power simply by chance. Instead, we observe the peak power at the second harmonic of the orbital period (${\mathord{\sim}}26.2$ days), and very little power at other periods. It is not particularly surprising that the second harmonic, rather than the first, is the strongest peak, especially given that there is some additional structure in the light curve at phases other than around transit.
The observant reader might notice a possible brightening of lower amplitude at phase ${\mathord{\sim}}0.4$ of the folded light curve. We are not confident that this is real, but if it is, it holds a wealth of information. As mentioned above, any excited spot must decay in brightness quickly, or much of the star would soon be covered in such spots, obfuscating the coherent signal that we observe. If, however, a spot excited during periastron lasted only slightly more than one rotation period, we would expect to see a second peak in the folded light curve, with an amplitude dependent on the spot lifetime and phase dependent on the rotation period. For [[*Kepler*]{}-432]{}, the second peak ${\mathord{\sim}}1.4$ orbits after the first would imply a rotation period of ${\mathord{\sim}}1.4 \times 52.5~{\rm days} \approx 73.5$ days, perfectly consistent with our independently estimated rotation period. The amplitude of the second peak is roughly half that of the first, so we could conclude that the characteristic lifetime (half-life) of such a spot is roughly the same as the rotation period, $\tau_{\rm spot}
{\mathord{\sim}}73.5$ days. We reiterate that we are not confident that the data here is strong enough to make such conclusions; we merely note the consistent rotation period it would imply.
![ A periodogram of the [*Kepler*]{} light curve, after transits were removed. The maximum power occurs at a period of ${\mathord{\sim}}26.2$ days, or half of the orbital period of the inner planet. \[fig:periodogram\]](fig19.pdf){width="48.00000%"}
Summary {#sec:summary}
=======
We have presented herein the discovery of the [[*Kepler*]{}-432]{} planetary system, consisting of two giant planets orbiting a red giant star, and a faint visual companion that is probably a physically bound M dwarf with an orbital separation of at least $750$AU. The inner planet ($P=52.5$ days) transits the star, allowing a more detailed study of its properties.
An asteroseismic analysis of the host star allows precise measurements of the stellar mass, radius, age, and spin axis inclination. This in turn leads to precise planetary properties, especially for the transiting inner planet. N-body simulations have helped constrain the properties of the outer planet by invoking stability arguments to rule out a large fraction of orbital solutions.
[[*Kepler*]{}-432]{}[b]{} is among the most massive transiting planets orbiting any type of star. Furthermore, it is not highly irradiated like many of the more common short-period transiting planets, experiencing only ${\mathord{\sim}}20\%$ of the insolation that a $3$-day hot Jupiter orbiting a Sun-like star would receive. This makes it an interesting benchmark for interior models, and a good test of planetary inflation at moderate insolation. It appears to be slightly inflated compared to the models, but is marginally consistent with having no core (or a small core) of heavy elements.
The eccentricities of both planets are high, which is suggestive of migration through multi-body interactions. Puzzlingly, the transiting planet is likely well aligned with the stellar spin, which is not an expected property of systems that have experienced such interactions. However, subsequent tidal or magnetic interaction with the host star may reconcile the two results. We find that because of the large stellar radius and planetary mass, the tidal dissipation timescale of the system is similar to that of hot Jupiters that are well-aligned. Those planets are thought to be well-aligned due to reorientation of the angular momenta after the initial inward migration; we conclude that the same process that realigns hot Jupiter systems may have also realigned this system, despite its long period.
Under the assumption that the star and planet are interacting strongly enough to realign the stellar spin, we searched for evidence of this interaction in the [*Kepler*]{} photometry. The star is photometrically quiet on long timescales, which allowed us to detect a low amplitude brightening soon after periastron. We conclude that the most likely explanation is a bright spot excited (either tidally or magnetically) by the planet during its close approach to the stellar photosphere. This evidence for ongoing SPI further supports the obliquity realignment scenario.
Finally, we note that [[*Kepler*]{}-432]{}[b]{} is an outlier among giant planets orbiting giant stars. It is one of only three such planets orbiting within $0.5$AU of its host star. Either it is intrinsically rare, or it is one of a more common class of planets orbiting interior to $1$AU that get destroyed relatively quickly once the star reaches the red giant branch. Given that [[*Kepler*]{}-432]{} has only recently started its ascent up the RGB—most red giants are in a more advanced evolutionary state—and there is already evidence for SPI, it is plausible that many more red giants initially hosted planets similar to [[*Kepler*]{}-432]{}[b]{} that have subsequently been engulfed. Further investigation of this group of planets will provide more clarity and ultimately have strong implications for giant planet formation around intermediate and high mass stars, which are more difficult to study on the main sequence because of their high temperatures and rapid rotation.
We thank Russel White and an anonymous referee for valuable discussion and feedback. S.N.Q. is supported by the NSF Graduate Research Fellowship, Grant DGE-1051030. D.W.L. acknowledges partial support from NASA’s Kepler mission under Cooperative Agreement NNX11AB99A with the Smithsonian Astrophysical Observatory. D.H. acknowledges support by the Australian Research Council’s Discovery Projects funding scheme (project number DEI40101364) and support by NASA under Grant NNX14AB92G issued through the Kepler Participating Scientist Program. M.J.P. gratefully acknowledges the NASA Origins of Solar Systems Program grant NNX13A124G. Funding for the Stellar Astrophysics Centre is provided by The Danish National Research Foundation (Grant DNRF106). The research is supported by the ASTERISK project (ASTERoseismic Investigations with SONG and Kepler) funded by the European Research Council (Grant agreement no.: 267864). The research leading to the presented results has also received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 338251 (StellarAges).
This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This research has also made use of the APASS database, located at the AAVSO web site. Funding for APASS has been provided by the Robert Martin Ayers Sciences Fund.
[*Facilities:*]{} , , , ,
Aerts, C., Christensen-Dalsgaard, J., & Kurtz, D. W. 2010, Asteroseismology (Springer)
Aizenman, M., Smeyers, P., & Weigert, A. 1977, , 58, 41
Albrecht, S., Winn, J. N., Johnson, J. A., et al. 2012, , 757, 18
Appourchaux, T., Chaplin, W. J., García, R. A., et al. 2012, , 543, 54
Barclay, T., Endl, M., Huber, D., et al. 2015, , 800, 46
Barnes, J. W., Linscott, E., & Shporer, A. 2011, , 197, 10
Basu, S., Chaplin, W. J., & Elsworth, Y. 2010, , 710, 1596
Batalha, N. M., Rowe, J. F. , Bryson, S. T., et al. 2013, , 204, 24
Beck, P. G., Montalban, J., Kallinger, T., et al. 2012, , 481, 55
Beck, P. G., Hambleton, K., Vos, J., et al. 2014, , 564, 36
Bedding, T. R., Mosser, B., Huber, D. et al. 2011, , 471, 608
Belkacem, K., Goupil, M. J., Dupret, M. A., et al. 2011, , 530, 142
Benomar, O., Masuda, K., Shibahashi, H., & Suto, Y. 2014, , 66, 94
Berta, Z. K., Charbonneau, D., Désert, J.-M., et al. 2012, , 747, 35
Borucki, W. J., Koch, D., Basri, G., et al. 2010, Sci, 327, 977
Bowler, B. P., Johnson, J. A., Marcy, G. W., Henry, G. W., et al. 2010, , 709, 396
Brand[ã]{}o, I. M., Do[ğ]{}an, G., Christensen-Dalsgaard, J., et al. 2011, , 527, 37
Bressan, A., Marigo, P., Girardi, L., et al. 2012, , 427, 127
Brown, T. M., Latham, D. W., Everett, M. E., & Esquerdo, G. A. 2011, , 142, 112
Buchhave, L. A., Bakos, G. Á., Hartman, J. D., et al. 2010, , 720, 1118
Buchhave, L. A., Latham, D. W., Johansen, A., et al. 2012, Natur, 486, 375
Buchhave, L. A., Bizzarro, M., Latham, D. W., et al. 2014, Natur, 509, 593
Buchner, J., Georgakakis, A., Nandra, K., et al. 2014, , 564, 125
Casella, G., & George, E. I. 1992, American Statistician, 46, 3
Chaplin, W. J., Sanchis-Ojeda, R., Campante, T. L., et al. 2013, , 766, 101
Chaplin, W. J., Basu, S., Huber, D., et al. 2014, , 210, 1
Charbonneau, D., Brown, T. M., Noyes, R. W., & Gilliland, R. L. 2002, , 568, 377
Chen, Y., Girardi, L., Bressan, A., et al. 2014, , 444, 2525
Christensen-Dalsgaard, J. 2008a, , 316, 13
Christensen-Dalsgaard, J. 2008b, , 316, 113
Ciceri, S., Lillo-Box, J., Southworth, J., et al. 2015, , 573, L5
Collier Cameron, A., Bruce, V. A., Miller, G. R. M., et al. 2010, , 403, 151
Dawson, R. I., & Johnson, J. A. 2012, , 756, 122
Dawson, R. I. 2014, ApJL, 790, 31
de Rosa, R. J., Patience, J., Wilson, P. A., et al. 2014, , 437, 1216
Désert, J.-M., Charbonneau, D., Demory, B.-O., Ballard, S., et al. 2011, , 197, 14
Djupvik, A. A., & Andersen, J. 2010, in Highlights of Spanish Astrophysics V, ed. J. M. Diego, L. J. Goicoechea, J. I. González-Serrano, & J. Gorgas (Berlin: Springer), p. 211
Döllinger, M. P., Hatzes, A. P., Pasquini, L., Guenther, E. W., & Hartmann, M. 2009, , 505, 1311
Fabrycky, D., & Tremaine, S. 2007, , 669, 1298
Feroz, F., Hobson, M. P., & Bridges, M.. 2009, , 398, 1601
Feroz, F., Hobson, M. P., Cameron, E., & Pettitt, A. N. 2013, arXiv:1306.2144
Fortney, J. J., Marley, M. S., Barnes, J. W. 2007, , 659, 1661
Frandsen, S., & Lindberg, B. 1999, Astrophysics with the NOT, ed. H. Karttunen & V. Piirola (Piikkio: Univ. Turku), 71
Fűrész, G. 2008, Ph.D. thesis, University of Szeged, Hungary
Gai, N., Basu, S., Chaplin, W. J., & Elsworth, Y. 2011, , 730, 63
Girardi, L., Groenewegen, M. A. T., Hatziminaoglou, E., & da Costa, L. 2005, , 436, 895
Gizon, L., & Solanki, S. K. 2003, , 589, 1009
Goldreich, P., & Tremaine, S. 1980, , 241, 425
Handberg, R., & Campante, T. L. 2011, , 527, 56
Handberg, R., & Lund, M. N. 2014, , 445, 2698
Hastings, W. H. 1970, Biometrika, 57, 97
Horch, E. P., Falta, D., Anderson, L. M., et al. 2010, , 139, 205
Houdek, G., Balmforth, N. J., Christensen-Dalsgaard, J., & Gough, D. O. 1999, , 351, 582
Howell, S. B., Everett, M. E., Sherry, W., et al. 2011, , 142, 19
Huber, D., Chaplin, W. J., Christensen-Dalsgaard, J., et al. 2013a, , 767, 127
Huber, D., Carter, J. A., Barbieri, M., et al. 2013b, Sci, 342, 331
Ida, S., & Lin, D. N. C 2005, , 626, 1045
Jackson, B., Greenberg, R., & Barnes, R. 2008, , 678, 1396
Jiang, C., Jiang, B. W., Christensen-Dalsgaard, J., et al. 2011, , 742, 120
Johnson, J. A., Fischer, D. A., Marcy, G. W., Wright, J. T., et al. 2007, , 665, 785
Johnson, J. A., Aller, K. M., Howard, A. W., & Crepp, J. R. 2010, , 122, 905
Johnson, J. A., Clanton, C., Howard, A. W., Bowler, B. P., et al. 2011, , 197, 26
Johnson, M. C., Cochran, W. D., Albrecht, S., et al. 2014, , 790, 30
Jones, M. I., Jenkins, J. S., Bluhm, P., Rojo, P., & Melo, C. H. F. 2014, , 566, 113
Juric, M., & Tremaine, S. 2008, , 686, 603
Kallinger, T., Weiss, W. W., Barban, C., et al. 2010, , 509, 77
Kipping, D. M. 2008, , 389, 1383
Kipping, D. M. 2010, , 408, 1758
Kipping, D. M., Dunn, W. R., Jasinski, J. M., & Manthri, V. P. 2012, , 421, 1166
Kipping, D. M., Hartman, J., Buchhave, L. A., et al. 2013, , 770, 101
Kipping, D. M. 2013a, , 435, 2152
Kipping, D. M. 2013b, , 434, 51
Kipping, D. M. 2014a, , 440, 2164
Kipping, D. M. 2014b, , 444, 2263
Kjeldsen, H., & Bedding, T. R. 1995, , 293, 87
Kjeldsen, H., Bedding, T. R., & Christensen-Dalsgaard, J. 2008, ApJL, 683, 175
Knutson, H. A., Charbonneau, D., Allen, L. E., et al. 2008, , 673, 526
Koch, D. G., Borucki, W. J., Basri, G., et al. 2010, ApJL, 713, 79
Konstantinova-Antova, R., Aurére, M., Charbonnel, C., Wade, G., et al. 2013, BlgAJ, 19, 14
Kraft, R. P. 1967, , 150, 551
Kurucz, R. L. 1992, IAUS, 149, 225
Lillo-Box, J., Barrado, D., Moya, A., et al. 2014a, , 562, 109
Lillo-Box, J., Barrado, D., Henning, T., et al. 2014b, , 568, L1
Lin, D. N. C., & Papaloizou, J. 1986, , 309, 846
Lomb, N. R. 1976, , 39, 447
Lovis, C., & Mayor, M. 2007, , 472, 657
Lund, M. N., Lundkvist, M., Silva Aguirre, V., et al. 2014, , 570, 54
Mandel, K., & Agol, E. 2002, ApJL, 580, 171
Mandushev, G., Torres, G., Latham, D. W., et al. 2005, , 621, 1061
Mathur, S., Hekker, S., Trampedach, R., et al. 2011, , 741, 119
McLaughlin, D. B. 1924, , 60, 22
Metcalfe, T. S., Monteiro, M. J. P. F. G., Thompson, M. J., et al. 2010, , 723, 1583
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., et al. 1953, JhPh, 21, 1087
Mosser, B., Goupil, M. J., Belkacem, K., et al. 2012a, , 548, 10
Mosser, B., Goupil, M. J., Belkacem, K., et al. 2012b, , 540, 143
Naoz, S., Farr, W. M., & Rasio, F. A. 2012, ApJL, 754, 36
Nidever, D. L., Marcy, G. W., Butler, P. R., et al. 2002, , 141, 503
Nutzman, P. A., Fabrycky, D. C., & Fortney, J. J. 2011, ApJL, 740, 10
Ortiz, M., Gandolfi, D., Reffert, S., et al. 2015, , 573, L6
Paxton, B., Cantiello, M., Arras, P., et al. 2013, , 208, 4
Payne, M. J., Deck, K. M., Holman, M. J., & Perets, H. B. 2013, , 775, 44
Petrovich, C. 2014, , submitted, arXiv:1409.8296
Poppenhaeger, K., Schmitt, J. H. M. M., & Wolk, S. J. 2013, , 773, 62
Queloz, D., Henry, G. W., Sivan, J. P., et al. 2001, , 379, 279
Quinn, S. N., White, R. J., Latham, D. W., et al. 2014, , 787, 27
Raghavan, D., McAlister, H. A., Henry, T. J., et al. 2010, , 190, 1
Rasio, F. A., & Ford, E. B. 1996, Sci, 274, 954
Rauch, K. P., & Holman, M. 1999, , 117, 1087
Rossiter, R. A. 1924, , 60, 15
Saar, S. H., & Donahue, R. A. 1997, , 485, 319
Sanchis-Ojeda, R., Winn, J. N., Holman, M. J., Carter, J. A., et al. 2011, , 733, 127
Sato, B., Izumiura, H., Toyota, E., Kambe, E., et al. 2008, , 60, 539
Scargle, J. D., , 263, 835
Schlafly, E. F., & Finkbeiner, D. P. 2011, , 737, 103
Schlegel, D. J., Finkbeiner, D.P., & Davis, M. 1998, , 500, 525
Shkolnik, E., Aigrain, S., Cranmer, S., et al. 2009, AIP Conf. Series 1094, Cool Stars, Stellar Systems, and the Sun, ed. E. Stempels (Melville, NY: AIP), 275
Silva Aguirre, V., Casagrande, L., Basu, S., et al. 2012, , 757, 99
Silva Aguirre, V., Ruchti, G. R., Hekker, S., et al. 2014, , 784, 16
Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, , 131, 1163
Sliski, D. H., & Kipping, D. M. 2014, , 788, 148
Steffen, J. H., Ragozzine, D., Fabrycky, D. C., et al. 2012, PNAS, 109, 7982
Stello, D., Bruntt, H., Preston, H., & Buzasi, D. 2008, , 674, 53
Stello, D., Chaplin, W. J., Bruntt, H., et al. 2009, , 700, 1589
Stello, D., Huber, D., Bedding, T. R., et al. 2013, , 765, 41
Strugarek, A., Brun, A. S., Matt, S. P., & Réville, V. 2014, , 795, 86
Tingley, B., Bonomo, A. S., & Deeg, H. J. 2011, , 726, 112
Torres, G., Konacki, M., Sasselov, D. D., & Jha, S. 2005, , 619, 558
Torres, G., Fressin, F., Batalha, N. M., et al. 2011, , 727, 24
Torres, G., Fischer, D. A., Sozzetti, A., et al. 2012, , 757, 161
Townsend, R. H. D., & Teitler, S. A. 2013, , 435, 3406
Valsecchi, F., & Rasio, F. A. 2014, , 786, 102
van Cleve, J. 2008, [ *Kepler*]{} Instrument Handbook (available at http://archive.stsci.edu/kepler/)
Van Eylen, V., Lund, M. N., Silva Aguirre, V., et al. 2014, , 782, 14
Weiss, A., & Schlattl, H. 2008, , 316, 99
Winn, J. N., Fabrycky, D., Albrecht, S., & Johnson, J. A. 2010, ApJL, 718, 145
Wu, Y., & Murray, N. 2003, , 589, 605
Wu, Y., & Lithwick, Y. 2011, , 735, 109
Zacharias, N., Finch, C. T., Girard, T. M., et al. 2013, , 145, 44
Zahn, J.-P. 1977, , 57, 383
[^1]: Among the other $95$ such planets listed in The Exoplanet Orbit Database (exoplanets.org), none transit.
[^2]: Observations labeled as SAP\_FLUX in FITS files retrieved from the Barbara A. Mikulski Archive for Space Telescopes (MAST).
[^3]: The [*Kepler*]{} Community Follow-up Observing Program, CFOP, http://cfop.ipac.caltech.edu, publicly hosts spectra, images, data analysis products, and observing notes for [[*Kepler*]{}-432]{} and many other Kepler Objects of Interest (KOIs).
|
---
abstract: 'It is shown that the translational degrees of freedom of a large variety of molecules, from light diatomic to heavy organic ones, can be cooled sympathetically and brought to rest (crystallized) in a linear Paul trap. The method relies on endowing the molecules with an appropriate positive charge, storage in a linear radiofrequency trap, and sympathetic cooling. Two well–known atomic coolant species, ${}^9{\hbox{Be}}^+$ and ${}^{137}{\hbox{Ba}}^+$, are sufficient for cooling the molecular mass range from 2 to 20,000 amu. The large molecular charge required for simultaneous trapping of heavy molecules and of the coolant ions can easily be produced using electrospray ionization. Crystallized molecular ions offer vast opportunities for novel studies.'
author:
- Stephan Schiller
- Claus Lämmerzahl
title: Molecular Dynamics Simulation of Sympathetic Crystallization of Molecular Ions
---
After the enormous success achieved in the field of cold atom manipulation, significant efforts are under way to develop similar methods for molecules. Samples of trapped ultracold molecules could be used for performing studies of molecular structure, chemical reactions, quantum optics and molecular Bose–Einstein condensates.
While methods for trapping of molecules, such as magnetic traps [@Weinstein98], electrostatic traps [@Bethlem], radiofrequency (Paul) or electromagnetic (Penning) traps [@Gosh95] or dipole traps [@Takeoshi] have been demonstrated and are in part already well developed, translational cooling of molecules is a field still under intense development. Direct laser cooling is not applicable due to lack of closed transitions. A technique demonstrated early on for cooling of neutral and charged molecules is by means of cryogenic buffer gas such as ${}^4$He, see e.g. [@buffergas1; @buffergas2; @BarlowDunnSchauer84]. Its extension to ${}^3\hbox{He}$ [@Weinstein98] allowed reaching temperatures below 1K. Two recently developed methods are the production of ultracold diatomic neutral molecules by photoassociation of ultra-cold atoms [@Pillet], and the deceleration of polar molecules by time-dependent electric fields [@Meijer_dec].
Another powerful method is sympathetic cooling of ”sample” particles of one species by an ensemble of directly cooled (often by laser cooling) particles of another species via their mutual interaction. This technique, first demonstrated for ions in Penning traps [@NIST1; @NIST2], is being applied to an increasingly wide variety of neutral and charged particles (atoms, molecules, elementary particles) in various trap types for applications ranging from mass spectrometry to quantum computing.
In both Penning and radiofrequency traps, first studies showed that molecular ions could be sympathetically cooled (sc) by laser-cooled (lc) atomic ions to temperatures in the range of several K [@earlySC_mol1; @earlySC_mol2], and mass ratios down to $m_{\rm sc}/m_{\rm lc} = 2/3$ were achieved [@BabaWaki01]. An important aspect of ion traps is that for sufficiently strong cooling the formation of an ordered structure (Coulomb crystal) results [@Wakietal92; @Raizenetal92]. Recently it has been shown that sympathethic crystallization of molecular ions is possible in a linear rf trap where the molecules are stably incorporated into the atomic Coulomb crystal [@MoelhaveDrewsen00; @Hornekaer00]. The molecular ions included MgH$^+$, MgD$^+$ (cooled by ${\hbox{Mg}}^+$) and ${}^{16}{\hbox{O}}_2^+$ (cooled by ${}^{40}{\hbox{Ca}}^+$ or ${}^{24}{\hbox{Mg}}^+$), i.e. mass ratios down to 0.6. In sympathetic crystallization of atomic ions, a mass ratio range of 0.8 - 1.8 has been achieved [@Kai; @Drewsen03].
The range of molecular mass that can be sympathetically cooled to temperatures of the order of 10K, where the ion ensemble is still in a gas state, has very recently been debated in theoretical work [@BabaWaki02; @BabaWaki02a; @Harmonetal02]. A molecular mass range from 8 to 192 amu (mass ratios 1/3 - 8) was found using molecular dynamics simulations to be accessible using ${}^{24}{\hbox{Mg}}^+$ as coolant ion.
In the present simulations we study a much larger mass range and the regime of much lower temperatures ($\ll 1\,$K) and study whether sympathetic crystallization can be reached. We find that essentially all molecular masses can be sympathetically crystallized by one of two commonly used species of laser–coolable ions, ${}^9{\hbox{Be}}^+$ and ${}^{137}{\hbox{Ba}}^+$. The only requirement for this general method is an appropriate charge state for the molecules. Single positive charges are sufficient for small molecules (mass 2 – 2000 amu). For heavier molecules higher positive charge states are required in order to allow for reliable simultaneous trapping. These are easily produced using electrospray ionization [@Fenn]. The ability to store molecules in an almost motionless state in a collision-free ultra-high vacuum environment for essentially unlimited time is expected to open up vast opportunities for high precision spectroscopy and the study of slow molecular processes.
In order to model sympathetic cooling in rf traps, it is crucial to take collisions into account precisely. Noninteracting ions perform an oscillation at the frequency of the applied rf field, but their cycle-averaged energy is constant if ion–ion interactions can be neglected. When interactions, i.e. collisions, between the ions are taken into account, energy gain from the rf field (rf–heating) and transfer of energy from one species to another, i.e. sympathetic heating and cooling, occurs. Approximate models have been proposed to describe these mechanisms [@BabaWaki02; @BabaWaki02a]. However, it is highly desirable to perform calculations that are essentially free of approximations. We therefore choose a molecular dynamics (MD) approach [@Prestageetal91; @Schifferetal00].
The simulations are based on solving Newton’s equations of motion for the laser–cooled (lc) and sympathetically cooled ions (sc) $$m_i \ddot{\bf x}_i = Q_i {\bf E}_{\rm trap}({\bf x}_i,t)+{\bf F}_{{\rm C},i}(\{{\bf x}_j\})+{\bf
F}_{\rm L}(\dot{\bf x}_i,t) \, ,$$ where $i=1,\ldots, N_{\rm lc}+N_{\rm sc}$ ($N_{\rm lc}$ and $N_{\rm sc}$ are the numbers of laser–cooled and sympathetically cooled particles, respectively). Positions, charges and masses are ${\bf x}_i$, $Q_i$, and $m_i$, the Coulomb force ${\bf F}_{{\rm C},i}=(Q_i/4\pi\epsilon_0)\nabla_i\sum_{j}Q_j/r_{ij}$, where $r_{ij}$ is the distance between particles $i$ and $j$. Here ${\bf E}_{\rm trap} = \mbox{\boldmath$\nabla$}((x^2-y^2)
\cos(\Omega t)V_{\rm rf}/2r_0^2+(z^2-x^2/2-y^2/2)U_{\rm dc}/d^2)$ is the electric field in a linear ion trap [@Gosh95], where $z$ is along the trap axis. The radial and axial dimensions of the trap are $r_0$ and $d$. A necessary condition for stable trapping of noninteracting ions is a $q$–parameter, $q_i = 2 (Q_i/m_i) V_{\rm
rf}/(\Omega^2 r_0^2)$ less than 0.9. However, it is well-known experimentally and theoretically that operation at significantly smaller $q$ is favourable since rf–heating is less pronounced [@Harmonetal02; @Prestageetal91]. On the other hand, a lower limit is given by experimental considerations, $q_{\rm min} \simeq 0.05$. We therefore choose $q$–parameters in the range 0.05 – 0.4 for both atomic and molecular ions. As a consequence, the simultaneous storage of heavy molecules ($m_{\rm sc}
\gg m_{\rm lc}$) requires a molecular charge exceeding unity.
Laser cooling of the lc particles is described by the force ${\bf F}_{\rm L}$. In actual experiments, its strength is such that cooling may require minutes. Such durations correspond to $10^8$ or more rf periods, and cannot be simulated in high–precision MD when particle numbers are large. In order to compromise between a reasonable computing time and realistic laser cooling strength we have used stronger forces ${\bf F}_{\rm L}$ to speed up cooling. Most simulations were performed with a simple linear viscous damping ${\bf F}_{\rm L}=-\beta \dot{\bf
x}$ with friction coefficients in the range $\beta = (1.2 - 8)
\cdot10^{-22}$kg/s. These are well below the maximum value (at optimum laser detuning from the cooling transition), $\beta_{\rm
max}\simeq\pi^2\hbar/\lambda^2\simeq$ $4\cdot10^{-21}$kg/s for transitions at optical wavelengths $\lambda$. For some simulations, we have used more realistic forces, see below.
The diffusion of the lc ion momenta in momentum space due to recoils upon spontaneous emission gives rise to the Doppler cooling limit. This diffusion is included in the simulations.
The equations of motion are solved using a high–order Runge–Kutta method with adaptive step size. Initial conditions were in part chosen so as to give initial temperatures below room temperature, again in order to reduce the computational effort.
In order to characterize the state of the plasma, from the trajectories of the particles we calculate, for each species, the average kinetic energy per particle, $E_{\rm kin}^{\rm tot} = (2 N_\kappa)^{-1}\sum_\kappa m_i
\langle {\bf v}_i^2(t) \rangle$, where $\langle\cdot\rangle$ is the time–average over one period of the rf field, the time–averaged secular energy per particle $E_{\rm kin}^{\rm sec} = (2 N_\kappa)^{-1}\sum_\kappa {m_i|\langle {\bf
v}_i(t)\rangle|^2}$, and the average interaction energy per particle (at the end of a rf period). Here $\kappa$ means lc or sc, respectively, and $\sum_\kappa$ denotes summation over the corresponding species. The secular energy, where the micromotion oscillation is averaged out, can be taken as an indication of the temperature of the sample, since it arises from the “disordered” motion of the interacting ions in the time–averaged trapping potential. In contrast, the micromotion contribution to the total energy arises from a regular motion. We note that in simulations where the rf potential is replaced by the pseudopotential, i.e. the time–averaged trap potential $V_{\rm pseudo}(\rho)= \rho^2 Q^2
V_{\rm rf}^2/(4 m \Omega^2 r_0^2)$ experienced by the particles in radial direction, the cooling is moderately faster. This implies that rf heating is nonnegligible even at the relatively low $q$–values chosen here.
In the following we describe three mass regimes of sympathetic crystallization.
#### Cooling of molecular hydrogen isotopomers {#cooling-of-molecular-hydrogen-isotopomers .unnumbered}
One challenging goal in ultracold molecule studies will be the precision spectroscopy of the simplest (i.e. one-electron) molecules, the hydrogen ions ${\hbox{H}}_2^+$, ${\hbox{HD}}^+$, ${\hbox{D}}_2^+$. The experimental accuracy can potentially be improved by several orders, surpassing by far the current theoretical precision [@27neu]. It will then become possible to test and challenge calculation methods, especially of the relativistic and QED contributions. Moreover, since the vibrational energies depend explicitly on the electron–to–proton mass ratio $m_e/m_p$ [@27b], their measurement might allow to determine the value of this fundamental constant by spectroscopic means, providing an alternative and potentially more accurate approach than mass measurements in Penning traps [@Beieretal02].
A precise measurement will require cold trapped molecules in order to minimize Doppler broadening. The heteronuclear ${\hbox{HD}}^+$ is of particular interest since it has dipole–allowed vibrational transitions [@Wingetal76] that could be excited by infrared lasers such as optical parametric oscillators [@Kovalchuk01] or diode lasers.
A simulation of sympathetic cooling of 5 ${\hbox{HD}}^+$–ions by 20 laser-cooled ${}^9{\hbox{Be}}^+$–ions is shown in Fig.1. The laser quickly cools the ${\hbox{Be}}^+$–ions to a liquid state, characterized by comparable Coulomb interaction energy and secular energy (plasma parameter $\Gamma \sim 2$). The atomic ion temperature remains constant while the molecular ions are sympathetically cooled. Only once the molecular secular energy becomes comparable to the atomic secular energy does the latter decrease further. The secular energy of the ${\hbox{Be}}^+$–ions finally reaches the Doppler limit. The secular energy of the molecules reaches that level significantly more slowly, since the cooling power of the atomic ions becomes smaller as they settle into the crystalline state.
The spatial structure resulting from the cooling (Fig. \[fig1\]b) can be understood by considering the different pseudopotentials felt by the two particle species [@Hornekaeretal01]. It is three times larger for the lighter ${\hbox{HD}}^+$–ions as compared to the ${}^9{\hbox{Be}}^+$–ions. The total energy of the ensemble is minimized if the ${\hbox{HD}}^+$–molecules lie on-axis. The ${\hbox{Be}}^+$–ions form a shell structure around them. It is moderately prolate, since the radial pseudopotential and axial potential are similar in strength (${\hbox{Be}}^+$–oscillation frequencies $\omega_\rho/2\pi=340\,$kHz, $\omega_z/2\pi=285\,$kHz). The axial arrangement of the ${\hbox{HD}}^+$–ions is favourable for spectroscopic investigations, since on–axis the micromotion is zero, with a corresponding simplification of their transition spectrum. Since the ${\hbox{Be}}^+$–ions form a three–dimensional structure, their micromotion energy remains relatively high compared to their secular energy (Fig.\[fig1\]a). This is because the off–axis locations of the ions imply corresponding micromotion velocities, proportional to the radial distances.
Simulations with a more realistic (i.e. weaker) cooling force [^1] were also performed. Qualitatively, the same behaviour resulted, however, as expected, sympathetic crystallization was reached after a substantially longer time, about 10 times slower than in Fig.\[fig1\]a.
(0,0)(12,12)(0,4.6)[![ Simulation with $N_{\rm lc} = 20$ laser-cooled ${}^9{\hbox{Be}}^+$–ions and $N_{\rm sc}=5$ ${\hbox{HD}}^+$–molecules. [**a:**]{} Energies per particle. [**b:**]{} Spatial structure in the crystalline state (left: projection onto $x$–$z$–plane, right: $x$–$y$–projection); solid circles: ${}^9{\hbox{Be}}^+$, circles: ${\hbox{HD}}^+$. The trap parameters are $q_{\rm lc} = 0.13$ $q_{\rm sc}=0.39$, rf frequency $\Omega/2\pi=8.5$MHz, $V_{\rm rf}/r_0^2=17.6\,\hbox{V}/{\hbox{mm}}^2$, $U_{\rm dc}/d^2 =30\,\hbox{V}/{\hbox{cm}}^2$. Viscous laser cooling with $\beta=2.4\times10^{-22}\hbox{kg/s}$ was used. []{data-label="fig1"}](1a.eps "fig:"){width="12.5cm"}]{} (10,10.1)[$E_{\rm kin}^{\rm tot}({}^{9}{\hbox{Be}}^+)$]{} (6.8,7)[$E_{\rm kin}^{\rm sec}({}^{9}{\hbox{Be}}^+)$]{} (9,11)[$E_{\rm kin}^{\rm tot}({\hbox{HD}}^+)$]{} (10.9,8.5)[$E_{\rm kin}^{\rm sec}({\hbox{HD}}^+)$]{} (7.8,11)(6.5,10.7) (9.7,8.5)(9,7.3) (1.4,0) (0,12)[a)]{} (0,3.9)[b)]{}
A sample of molecular ions is not always sympathetically cooled in its entirety. For example, in a simulation of 40 ${\hbox{Be}}^+$–ions and 10 ${\hbox{HD}}^+$–ions, the final state of the system contained 6 on-axis crystallized ${\hbox{HD}}^+$–ions embedded in a prolate ${\hbox{Be}}^+$–crystal, while 4 ${\hbox{HD}}^+$–ions remained hot. It is clear that for small particle numbers the number of cooled particles depends on details of the initial conditions.
The above simulations for ${\hbox{HD}}^+$ are of course also applicable to the astrophysically important ${\hbox{H}}_3^+$. We have also performed simulations for other hydrogen ions. We have found (partial) sympathetic crystallization for all masses, from 2 amu (${\hbox{H}}_2^+$) to 5 amu (${\hbox{DT}}^+$).
#### Cooling of dye molecules {#cooling-of-dye-molecules .unnumbered}
Dye molecules are interesting model systems for studies of complex (i.e. polyatomic) ultracold molecule manipulation because they are well–characterized and can easily be excited optically. Various dye molecules have masses exceeding a few times the mass of heavy atoms. We consider here the case of cooling Rhodamine 6G, a molecule of mass 493 that can be transferred to the gas phase singly charged by means of electrospray ionization. The coolant ion is chosen to be ${}^{137}{\hbox{Ba}}^+$. In this case, the larger mass–to–charge ratio of the molecules leads to a shell structure with the atomic ions in the center and on–axis. The small fraction of the molecules which is well embedded in the atomic ensemble cools and crystallizes on the same timescale as the atomic ions. The remainder experiences a much weaker cooling due to the absence of a ”cageing” effect. The timescale for cooling and crystallization of all molecules was found to be 10 times larger than for the atomic ions.
#### Cooling of large molecules {#cooling-of-large-molecules .unnumbered}
(0,0)(12,13) (-0.2,5.5)[![ Sympathetic cooling of 10 molecular ions of mass 20000 and charge $20\, e$ by 30 ${}^{137}{\hbox{Ba}}^+$–ions. (a) Energies. (b) Spatial structure. (c) Spatial structure for a simulation where the axial potential is weaker. Parameters: $q_{\rm lc} = 0.33$ $q_{\rm sc}=0.045$, $\Omega/2\pi=1.6\;\hbox{MHz}$, $V_{\rm rf}/2r_0^2=23.8\,\hbox{V}/{\hbox{mm}}^2$, $U_{\rm dc}/d^2=12\,\hbox{V}/{\hbox{cm}}^2$ for (b), $3\,\hbox{V}/{\hbox{cm}}^2$ for (c). Viscous laser cooling with $\beta=4.8\times10^{-22}\hbox{kg}/\hbox{s}$ was used. []{data-label="fig_large_molecules"}](Bild2aneu.eps "fig:"){width="12.5cm"}]{} (10.2,8.4)[$E_{\rm kin}^{\rm tot}({}^{137}{\hbox{Ba}}^+)$]{} (7,7.5)[$E_{\rm kin}^{\rm sec}({}^{137}{\hbox{Ba}}^+)$]{} (10,11.5)[$E_{\rm kin}^{\rm tot}(\hbox{mol})$]{} (7.8,10.2)[$E_{\rm kin}^{\rm sec}(\hbox{mol})$]{} (8.8,8.4)(8.2,9) (6,7.9)(6.8,8.7) (0.02,3)[![ Sympathetic cooling of 10 molecular ions of mass 20000 and charge $20\, e$ by 30 ${}^{137}{\hbox{Ba}}^+$–ions. (a) Energies. (b) Spatial structure. (c) Spatial structure for a simulation where the axial potential is weaker. Parameters: $q_{\rm lc} = 0.33$ $q_{\rm sc}=0.045$, $\Omega/2\pi=1.6\;\hbox{MHz}$, $V_{\rm rf}/2r_0^2=23.8\,\hbox{V}/{\hbox{mm}}^2$, $U_{\rm dc}/d^2=12\,\hbox{V}/{\hbox{cm}}^2$ for (b), $3\,\hbox{V}/{\hbox{cm}}^2$ for (c). Viscous laser cooling with $\beta=4.8\times10^{-22}\hbox{kg}/\hbox{s}$ was used. []{data-label="fig_large_molecules"}](Bild2Alllo.eps "fig:"){width="9.9cm"}]{} (10,3.03)[![ Sympathetic cooling of 10 molecular ions of mass 20000 and charge $20\, e$ by 30 ${}^{137}{\hbox{Ba}}^+$–ions. (a) Energies. (b) Spatial structure. (c) Spatial structure for a simulation where the axial potential is weaker. Parameters: $q_{\rm lc} = 0.33$ $q_{\rm sc}=0.045$, $\Omega/2\pi=1.6\;\hbox{MHz}$, $V_{\rm rf}/2r_0^2=23.8\,\hbox{V}/{\hbox{mm}}^2$, $U_{\rm dc}/d^2=12\,\hbox{V}/{\hbox{cm}}^2$ for (b), $3\,\hbox{V}/{\hbox{cm}}^2$ for (c). Viscous laser cooling with $\beta=4.8\times10^{-22}\hbox{kg}/\hbox{s}$ was used. []{data-label="fig_large_molecules"}](Bild2Allro.eps "fig:"){width="2.545cm"}]{} (0,0)[![ Sympathetic cooling of 10 molecular ions of mass 20000 and charge $20\, e$ by 30 ${}^{137}{\hbox{Ba}}^+$–ions. (a) Energies. (b) Spatial structure. (c) Spatial structure for a simulation where the axial potential is weaker. Parameters: $q_{\rm lc} = 0.33$ $q_{\rm sc}=0.045$, $\Omega/2\pi=1.6\;\hbox{MHz}$, $V_{\rm rf}/2r_0^2=23.8\,\hbox{V}/{\hbox{mm}}^2$, $U_{\rm dc}/d^2=12\,\hbox{V}/{\hbox{cm}}^2$ for (b), $3\,\hbox{V}/{\hbox{cm}}^2$ for (c). Viscous laser cooling with $\beta=4.8\times10^{-22}\hbox{kg}/\hbox{s}$ was used. []{data-label="fig_large_molecules"}](Bild2Alllu.eps "fig:"){height="3cm"}]{} (10,0.05)[![ Sympathetic cooling of 10 molecular ions of mass 20000 and charge $20\, e$ by 30 ${}^{137}{\hbox{Ba}}^+$–ions. (a) Energies. (b) Spatial structure. (c) Spatial structure for a simulation where the axial potential is weaker. Parameters: $q_{\rm lc} = 0.33$ $q_{\rm sc}=0.045$, $\Omega/2\pi=1.6\;\hbox{MHz}$, $V_{\rm rf}/2r_0^2=23.8\,\hbox{V}/{\hbox{mm}}^2$, $U_{\rm dc}/d^2=12\,\hbox{V}/{\hbox{cm}}^2$ for (b), $3\,\hbox{V}/{\hbox{cm}}^2$ for (c). Viscous laser cooling with $\beta=4.8\times10^{-22}\hbox{kg}/\hbox{s}$ was used. []{data-label="fig_large_molecules"}](Bild2Allru.eps "fig:"){height="2.91cm"}]{} (0,0)(12.5,0)(12.5,0.7)(1,0.7)(1,5.3)(0,5.3) (01.2,0.45)[-400]{} (9.5,0.45)[400]{} (5.4,0.2)[$z \;[\mu\hbox{m}]$]{} (10.3,0.45)[-100]{} (12.3,0.45)[100]{} (11.25,0.2)[$y \;[\mu\hbox{m}]$]{} (0.55,0.9)[-100]{} (0.55,2.8)[100]{} (0.55,3.2)[-100]{} (0.55,5.1)[100]{} (0.4,1.8)[$x \;[\mu\hbox{m}]$]{} (0.4,4.1)[$x \;[\mu\hbox{m}]$]{} (-0.4,13)[a)]{} (-0.4,5.5)[b)]{} (-0.4,2.8)[c)]{}
Large molecules such as amino acids or proteins can be transferred into vacuum in high charge states by electrospray ionization [@Fenn]. In this method, protons are attached to molecules that emerge from a solution spray. Large molecules are produced with a distribution of charge states, with typical specific charges in the range $m/Q=700 - 1500$. For example, the charge states for a protein of mass 17,000 can range from 11 to 25 [@Cole97]. Such specific charges are sufficiently near those of laser–coolable heavy atomic ions (${}^{137}{\hbox{Ba}}^+$, ${}^{172}{\hbox{Yb}}^+$, ${}^{199}{\hbox{Hg}}^+$). Since the $q$–parameters $\sim Q_i/m_i$ are within an order of magnitude, simultaneous trapping is possible. Moreover, the large charge ensures a good coupling between molecular ensemble and atomic ensemble. Finally, the pseudopotential of the molecular ions ($\sim Q_{\rm sc}^2/m_{\rm sc}$) can be comparable or even larger than that of the atoms, thanks to the large molecular charge. A good spatial overlap of the subensembles is therefore ensured. Fig.\[fig\_large\_molecules\] shows two simulations for molecular ions of mass 20,000 amu and charge 20$e$. As can be seen, the molecular ions are strongly cooled as far as their secular energy is concerned. However, a large micromotion energy remains in the crystallized state because of the nonaxial arrangement of the molecular ions and their large mass. This nonaxial arrangement occurs in spite of the fact that the pseudopotential of the molecules is larger than that of the atoms. Indeed, the Coulomb interaction energy is an important contribution that influences the overall arrangement of the particles. This can be seen by noting that the Coulomb energy of a given spatial structure is not invariant under particle exchange lc$\leftrightarrow$sc, due to the differing molecular and atomic charges. For a sufficiently large axial potential (Fig.\[fig\_large\_molecules\]a,b) the total energy is minimized by forcing the molecular ions off axis resulting in a shell structure. For smaller axial potential (Fig.\[fig\_large\_molecules\]c) the structure is string–like, but molecular ions that are adjacent bulge out of the axis because of their strong repulsion. Where several molecular ions are crystallized in adjacent positions, they form sections of a helix [@HasseSchiffer90]. Here again we find that molecules that are well embedded in the atomic ion ensemble, i.e. those that are individually located between atomic ions (e.g. the two sc ions in Fig.\[fig\_large\_molecules\]c furthest from the center in z-direction), cool as fast as the lc ions while the off-axis molecular ions cool much more slowly.
We have also performed simulations for molecular masses 2000, 5000, and 10,000 amu, with the same mass to charge ratio of 1000 amu/$e$. We find a similar behaviour of crystallization as in the examples above. As the charge becomes smaller only molecular Coulomb structures of the shell–type occur, and due to the weaker lc–sc interaction, a fraction of the molecules remain uncooled.
The above examples show that sympathetic crystallization is possible for ion masses significantly larger or smaller than the coolant ion mass. Based on the present and previous [@Harmonetal02] results, it can be stated that simply charged molecular ions of mass between that of ${}^9{\hbox{Be}}^+$ and ${}^{137}{\hbox{Ba}}^+$ can also be sympathetically crystallized.
In conclusion, we have shown that a wide variety of charged molecules can be trapped and cooled to mK temperatures with masses from 2 to 20,000 amu, i.e. from diatomic molecules to polymers and proteins. The molecules are incorporated into an ordered structure. The spatial arrangement within the Coulomb crystal depends on the masses and charges of the coolant and the molecular ions. We have pointed out that sympathetic crystallization should be very advantageous for precision spectroscopy of the various ions of molecular hydrogen.
Since emphasis was placed on using a laser cooling force of realistic magnitude and on employing a high–precision numerical code, the simulations were performed for small particle numbers (up to 50). Experimentally, this regime is both accessible and suited for detailed studies of molecular properties. In future, the simulations can be extended to larger numbers, by developing faster but approximate algorithms or using more powerful computers, and to other types of traps (multipole-rf traps [@buffergas2] and Penning traps). This should allow detailed comparisons of dynamics and structure of sympathetic crystallization between theory and experiments currently under way.
#### Acknowledgment {#acknowledgment .unnumbered}
We thank M. Drewsen, E. Göklü, M. Hochbruck and V. Grimm for stimulating discussions and D. Schumacher for providing computer support.
[35]{}
J. D. Weinstein [*et al*]{}, [*Nature*]{} **395**, 148 (1998).
H. Bethlem [*et al.*]{}, [*Nature*]{} **406**, 491 (2000).
P. Gosh, *Ion Traps* (Clarendon Press, Oxford 1995).
T. Takekoshi [*et al.*]{}, [*Phys. Rev. Lett.*]{} **81**, 5105 (1998).
J. Pearson [*et al.*]{}, [*Phys. Rev. Lett.*]{} **75**, 2940 (1995).
W. Paul [*et al.*]{}, [*Chem. Phys.*]{} **209**, 265 (1996).
S. Barlow [*et al.*]{}, [*Phys. Rev. Lett.*]{} **52**, 902 (1984).
A. Fioretti [*et al.*]{}, [*Phys. Rev. Lett.*]{} **80**, 4402 (1998).
H. Bethlem [*et al.*]{}, [*Phys. Rev. Lett.*]{} **83**, 1558 (1999).
R. Drullinger [*et al.*]{}, [*Appl. Phys.*]{} **22**,365 (1980).
D. Larson [*et al.*]{}, [*Phys. Rev. Lett.*]{} **57**, 70 (1986).
T. Baba and I. Waki, [*Jpn. J. Appl. Phys., Part 2*]{} **35**, L1134 (1996).
M. van Eijkelenborg [*et al.*]{}, [*Phys. Rev.*]{} **A 60**, 3903 (1999).
T. Baba and I. Waki, [*J. Appl. Phys.*]{} **89**, 4592 (2001).
M. Raizen [*et al.*]{}, [*Phys. Rev.*]{} **A 45**, 6493 (1992).
I. Waki [*et al.*]{}, [*Phys. Rev. Lett.*]{} **68**, 2007 (1992).
K. M[ø]{}lhave and M. Drewsen, [*Phys. Rev.*]{} **A 62**, 011401(R) (2000).
L. Hornek[æ]{}r, [*unpublished*]{}, Aarhus Univ. (2000).
Y. Kai [*et al.*]{}, [*Jap. J. Appl. Phys.*]{} **40**, 5136 (2001).
M. Drewsen, [*private communication*]{}.
T. Baba and I. Waki, [*Appl. Phys.*]{} **B 74**, 375 (2002).
T. Baba and I. Waki, [*J. Appl. Phys.*]{} **92**, 4109 (2002).
T. Harmon [*et al.*]{}, [*Instability heating of symathetically–cooled ions in a linear [P]{}aul trap*]{}, physics/0208069.
J. Fenn [*et al.*]{}, [*Science*]{} **246**, 64 (1989).
J. Prestage [*et al.*]{}, [*Phys. Rev. Lett.*]{} **66**, 2964 (1991).
J. Schiffer [*et al.*]{}, [*Proc. Nat. Acad. Sci. USA*]{} **97**, 10697 (2000).
C.A. Leach and R. E. Moss, [*Annu. Rev. Phys. Chem*]{} [**46**]{}, 55 (1995).
Hilico, N. Billy, B. Gremaud, D. Delande, [*Eur. Phys. J.*]{} [**D 12**]{}, 449 (2000).
T. Beier [*et al*]{}, [*Phys. Rev. Lett.*]{} **88**, 011603 (2002).
W. Wing [*et al.*]{}, [*Phys. Rev. Lett.*]{} **36**, 1488 (1976).
E. Kovalchuk [*et al.*]{}, [*Opt. Lett.*]{} **26**, 1430 (2001).
L. Hornek[æ]{}r [*et al.*]{}, [*Phys. Rev. Lett.*]{} **86**, 1994 (2001).
H. Metcalf and P. van der Straten, [*Laser Cooling and Trapping*]{} (Springer, New York 1999).
R. Cole (ed.): [*Electrospray Ionization Mass Spectrometry*]{}, (Wiley 1997).
R. Hasse and J. Schiffer, [*Ann. Phys.*]{} (N.Y.) **203**, 419 (1990).
[^1]: The semiclassical cooling force with full dependence on particle velocity and detuning of the laser frequency from resonance [@Metcalf] was used, but enhanced by a factor $\sim 8$. The detuning was repetitively scanned in a sawtooth manner from far red–detuned to $-\gamma/2$ where $\gamma$ is the cooling transition linewidth. The laser beam direction was at 54.7$\deg$ with respect to the trap symmetry axes.
|
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author:
- '*ANTONIO GIORGILLI Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, 20133 — Milano, Italy and Istituto Lombardo Accademia di Scienze e Lettere*'
- '*MARCO SANSOTTERA naXys, Namur Center for Complex Systems, FUNDP, Rempart de la Vierge 8, B5000 — Namur, Belgium.*'
title: Methods of algebraic manipulation in perturbation theory
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form chrform
1
=
[Overview]{} Algebraic manipulation on computer is a tool that has been developed quite soon, about one decade after the birth of computers, the first examples dating back to the end of the fifties of the last century. General purpose packages began to be developed during the sixties, and include, e.g., Reduce (1968), Macsyma (1978), muMath (1980), Maple (1984), Scratchpad (1984), Derive (1988), Mathematica (1988), Pari/GP (1990) and Singular (1997) (the dates refer to the first release). However, most of the facilities of these general purpose manipulators are simply ignored when dealing with perturbation methods in Celestial Mechanics. For this reason, the job of developing specially devised manipulation tools has been undertaken by many people, resulting in packages that have limited capabilities, but are definitely more effective in practical applications. Producing a list of these packages is a hard task, mainly because most of them are not publicly available. A list of “old time” packages may be found in Henrard and Laskar . In recent times a manipulator developed by J. Laskar and M. Gastineau has become quite known.
Finding references to the methods implemented in specially devised packages is as difficult as giving a list. We know only a few papers by Broucke and Garthwaite , Broucke , Rom , Henrard and , Laskar , Jorba and Biscani . A complete account of the existing literature on the subject goes beyond the limits of the present note. The present work introduces some ideas that have been used by the authors in order to implement a package named .
As a matter of fact, most of the algebraic manipulation used in Celestial Mechanics makes use of the so called “Poisson series”, namely series with a general term of the form $$x_1^{j_1}\cdot\ldots\cdot x_n^{j_n}
{{\cos}\atop{\sin}} (k_1\phi_1+\ldots+k_m\phi_m)\ ,$$ (with obvious meaning of the symbols). Thus, a very minimal set of operations is required, namely sums, products and derivatives of polynomials and/or trigonometric polynomials. Traditionally, also the operation of inversion of functions, usually made again via series expansion, was required. However, the expansion methods based on Lie series and Lie transforms typically get rid of the latter operation (see, e.g., ).
Writing a program doing algebraic manipulation on series of the type above leads one to be confronted with a main question, namely how to represent a polynomial, trigonometric polynomial or Poisson series on a computer. The papers quoted above actually deal with this problem, suggesting some methods. In these lectures we provide an approach to this problem, followed by a few examples of applications.
In sect. we include a brief discussion about the construction of normal form for a Hamiltonian system in the neighborhood of an elliptic equilibrium. We do not attempt to give a complete discussion, since it is available in many papers. We rather try to orient the reader’s attention on the problem of representing perturbation series.
In sect. – we introduce a method which turns out to be quite useful for the representation of a function as an array of coefficients. The basic idea has been suggested to one of the authors by the paper of Gustavson (who, however, just mentions that he used an indexing method, without giving any detail about its implementation). One introduces an [indexing function]{} which transforms an array of exponents in a polynomial (or trigonometric polynomial) in a single index within an array. The general scheme is described in sect. . The basics behind the construction of an indexing function are described in sect. . The details concerning the representation of polynomials and trigonometric polynomials are reported in sects. and , respectively. In sect. we include some hints about the case of sparse series, that may be handled by combining the indexing functions above with a tree representation. Finally, sect. is devoted to three applications, by giving a short account of the contents of published papers.
1.a
===
[A common problem in perturbation theory]{} A typical application of computer algebra is concerned with the construction of first integrals or of a normal form for a Hamiltonian system. A nontrivial example, which however may be considered as a good starting point, is the calculation of a normal form for the celebrated model of Hénon and Heiles , which has been done by Gustavson . Some results on this model are reported in sect. .
We assume that the reader is not completely unfamiliar with the concept of normal form for a (possibly Hamiltonian) system of differential equations. Thus, let us briefly illustrate the problem by concentrating our attention on the algorithmic aspect and by explaining how algebraic manipulation may be introduced.
1.a.1
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[Computation of a normal form]{} Let us consider a canonical system of differential equations in the neighborhood of an elliptic equilibrium. The Hamiltonian may typically be given the form of a power series expansion $$H(x,y) = H_0(x,y) + H_1(x,y) +\ldots\ ,\quad
H_0(x,y) = \sum_{j=1}^{n} \frac{\omega_j}{2}(x_j^2+y_j^2)\ ,
\formula{nrm.1}$$ where $H_{s}(x,y)$ for $s\ge 1$ is a homogeneous polynomial of degree $s+2$ in the canonical variables $(x,y)\in\reali^{2n}$. Here $\omega\in\reali^n$ is the vector of the frequencies, that are assumed to be all different from zero.
In such a case the system is said to be in Birkhoff normal form in case the Hamiltonian takes the form $$H(x,y) = H_0(x,y) + Z_1(x,y) + Z_2(x,y)+\ldots\quad
{\rm with}\quad L_{H_0} Z_s =0\ ,
\formula{nrm.2}$$ where $L_{H_0}\cdot = \{H_0,\cdot\}$ is the Lie derivative with respect to the flow of $H_0$, actually the Poisson bracket with $H_0$.
The concept of Birkhoff normal form is better understood if one assumes also that the frequencies are non resonant, i.e., if $$\langle k,\omega\rangle \ne 0\quad {\rm for\ all\ } k\in\interi^n
\>,\ k\ne 0\ ,$$ where $\langle k,\omega\rangle = \sum_{j}k_j\omega_j$. For, in this case the functions $Z_s(x,y)$ turn out to be actually function only of the $n$ actions of the system, namely of the quantities $$I_j = \frac{x_j^2+y_j^2}{2}\ ,\quad j=1,\ldots,n\ .$$ It is immediate to remark that $I_1,\ldots,I_n$ are independent first integrals for the Hamiltonian, an that they are also in involution, so that, by Liouville’s theorem, the system turns out to be integrable. The definition of normal form given in is more general, since it includes also the case of resonant frequencies.
The calculation of the normal form may be performed using the Lie transform method, which turns out to be quite effective. We give here the algorithm without proof. A complete description may be found, e.g., in , and the description of a program implementing the method via computer algebra is given in . The corresponding [FORTRAN]{} program is available from the CPC library.
The Lie transform is defined as follows. Let a [generating sequence]{} $\chi_1(x,y)$, $\chi_2(x,y),\ldots$ be given, and define the operator $$T_{\chi} = \sum_{s\ge 0} E_s
\formula{nrm.3}$$ where the sequence $E_0,E_1,\ldots$ of operators is recursively defined as $$E_0=1\ ,\quad
E_s = \sum_{j=1}^{s}\frac{j}{s} L_{\chi_j}E_{s-j}
\formula{nrm.3a}$$ This is a linear operator that is invertible and satisfies the interesting properties $$T_{\chi} \{ f,g\} = \{T_{\chi}f,T_{\chi}g\}\ ,\quad
T_{\chi} (f\cdot g) = T_{\chi}f\cdot T_{\chi}g\ .
\formula{nrm.4}$$ Let now $Z(x,y) = H_0(x,y) + Z_1(x,y) + Z_2(x,y)+\ldots\ $ be a function such that $$T_{\chi} Z = H\ ,
\formula{nrm.5}$$ where $H$ is our original Hamiltonian, and let $Z$ possess a first integral $\Phi$, i.e., a function satisfying $\{Z,\Phi\}=0$. Then one has also $$T_{\chi}\{Z,\Phi\} = \{T_{\chi}Z,T_{\chi}\Phi\} = \{H,T_{\chi}\Phi\} =0\ ,$$ which means that [if $\Phi$ is a first integral for $Z$ then $T_{\chi}\Phi$ is a first integral for $H$.]{}
The question now is: [can we find a generating sequence $\chi_1,\,\chi_2,\ldots$ such that the function $Z$ satisfying is in Birkhoff normal form?]{}
The answer to this question is in the positive, and the generating sequence may be calculated via an explicit algorithm that can be effectively implemented via computer algebra. We include here the algorithm, referring to, e.g., for a complete deduction. Here we want only to stress that all operations that are required may be actually implemented on a computer.
The generating sequence is determined by solving for $\chi$ and $Z$ the equations $$Z_s - L_{H_0}\chi_s = H_s + Q_s\ ,\quad
s \ge 1\ ,
\formula{nrm.6}$$ where $Q_s$ is a known homogeneous polynomial of degree $s+2$ given by $Q_1=0$ and $$Q_s = -\sum_{j=1}^{s-1}
\bigl( E_j Z_{s-j} + \frac{j}{s}\{\chi_j, E_{s-j}H_0\}\bigr)
\ ,\quad s\gt 1\ .$$ In order to solve it is convenient to introduce complex variables $\xi,\eta$ via the canonical transformation $$x_j = \frac{1}{\sqrt{2}} (\xi_j+i\eta_j)\ ,\quad
y_j = \frac{i}{\sqrt{2}} (\xi_j-i\eta_j)$$ which transforms $H_0 = i\sum_{j}\omega_j\xi_j\eta_j$. In these variables the operator $L_{H_0}$ takes a diagonal form, since $$L_{H_0} \xi^j\eta^k = i\langle k-j,\omega\rangle \xi^j\eta^k\ ,$$ where we have used the multi-index notation $\xi^j=\xi_1^{j_1}\cdot\ldots\cdot\xi_n^{j_n}$, and similarly for $\eta$. Thus, writing the r.h.s. of as a sum of monomials $c_{j,k}\xi^j\eta^k$ the most direct form of the solution is found by including in $Z$ all monomials with $\langle
k-j,\omega\rangle=0$, and adding $\frac{c_{j,k}}{i\langle
k-j,\omega\rangle}\xi^j\eta^k$ to $\chi_s$ for all monomials with $\langle k-j,\omega\rangle\ne 0$. This is the usual way of constructing a normal form for the system .
Let us now examine in some more detail the algebraic aspect. With some patience one can verify that involves only homogeneous polynomials of degree $s+2$. Thus, one should be able to manipulate this kind of functions. Moreover, a careful examination of the algorithm shows that there are just elementary algebraic operations that are required, namely:
[(i)]{}sums and multiplication by scalar quantities;
[(ii)]{}Poisson brackets, which actually require derivatives of monomials, sums and products;
[(iii)]{}linear substitution of variables, which may still be reduced to calculation of sums and products without affecting the degree of the polynomial;
[(iv)]{}solving equation , which just requires a division of coefficients.
These remarks should convince the reader that implementing the calculation of the normal form via algebraic manipulation on a computer is just matter of being able of representing homogeneous polynomials in many variables and performing on them a few elementary operations, such as sum, product and derivative.
1a.2
----
[A few elementary considerations]{} In order to have an even better understanding the reader may want to consider the elementary problem of representing polynomials in one single variable. We usually write such a polynomial of degree $s$ (non homogeneous, in this case) as $$f(x) = a_0 + a_1 x + \ldots + a_s x^s\ .$$ A machine representation is easily implemented by storing the coefficients $a_0$, $a_1,\ldots$,$\,a_n$ as a one-dimensional array of floating point quantities, either real or complex. E.g., in [FORTRAN]{} language one can represent a polynomial of degree 100 by just saying, e.g., [DIMENSION F(101)]{} and storing the coefficient $a_j$ as [F(j+1)]{} (here we do not use the extension of [FORTRAN]{} that allows using zero or even negative indices for an array). Similarly in a language like [C]{} one just says, e.g., [double f\[101\]]{} and stores $a_j$ as [f\[j\]]{}.
The operation of sum is a very elementary one: if $f,\,g$ are two polynomials and the coefficients are stored in the arrays [f,g]{} (in [C]{} language) then the sum $h$ is the array [h]{} with elements [h\[j\] = f\[j\] + g\[j\]]{}. The derivative of $f$ is the array [fp]{} with elements [fp\[j\] = (j+1)\*f\[j+1\]]{}. In a similar way one can calculate the product, by just translating in a programming language the operations that are usually performed by hand.
The case of polynomials in two variables is just a bit more difficult. A homogeneous polynomial of degree $s$ is usually written as $$f(x,y) = a_{s,0}x^s + a_{s-1,1}x^sy +\ldots+a_{0,s}y^s\ .$$ The naive (not recommended) representation would use an array with two indices (a matrix), by saying, e.g., [DIMENSION F(101,101)]{} and storing the coefficient $a_{j,k}$ as [F(j+1,k+1)]{}. Then the algebra is just a straightforward modification with respect to the one-dimensional case.
Such a representation is not recommended for at least two reasons. The first one is that arrays with arbitrary dimension are difficult to use, or even not allowed, in programming languages. The second and more conclusive reason is that such a method turns out to be very effective in wasting memory space. E.g., in the two dimensional case a polynomial of degree up to $s$ requires a matrix with $(s+1)^2$ elements, while only $(s+1)(s+2)/2$ are actually used. Things go much worse in higher dimension, as one easily realizes.
The arguments above should have convinced the reader that an effective method of representing polynomials is a basic tool in order to perform computer algebra for problems like the calculation of normal form. Once such a method is available, the rest is essentially known algebra, that needs to be translated in a computer language.
The problem for Poisson series is a similar one, as the reader can easily imagine. The following sections contains a detailed discussion of indexing methods particularly devised for polynomials and for Poisson series. The underlying idea is to represent the coefficients as a one-dimensional array by suitably packing them in an effective manner, so as to avoid wasting of space.
2
=
[General scheme]{} The aim of this section is to illustrate how an appropriate algebraic structure may help in representing the particular classes of functions that usually appear in perturbation theory. We shall concentrate our attention only on polynomials and trigonometric polynomials, which are the simplest and most common cases. However, the reader will see that most of the arguments used here apply also to more general cases.
funrep.1.1
----------
[Polynomials and power series]{} Let $\Pscr$ denote the vector space of polynomials in the independent variables $x=(x_1,\ldots,x_n)\in\reali^{n}$. A basis for this vector space is the set $\{u_k(x)\}_{k\in\interi^n_+}$, where $$u_k(x)=x^k\equiv x_1^{k_1}\cdot\ldots\cdot x_n^{k_n}\ .
\formula{funrep.1}$$ In particular, we shall consider the subspaces $\Pscr_s$ of $\Pscr$ that contain all homogeneous polynomials of a given degree $s\ge 0$; the subspace $\Pscr_0$ is the one-dimensional space of constants, and its basis is $\{1\}$. The relevant algebraic properties are the following:
[(i)]{}every subspace $\Pscr_s$ is closed with respect to sum and multiplication by a number, i.e., if $f\in\Pscr_s\ \wedge\
g\in\Pscr_s$ then $f+g\in\Pscr_s$ and $\alpha f\in\Pscr_s$;
[(ii)]{}the product of homogeneous polynomials is a homogeneous polynomial, i.e., if $f\in\Pscr_r\ \wedge\ g\in\Pscr_s$ then $fg\in\Pscr_{r+s}$;
[(iii)]{}the derivative with respect to one variable maps homogeneous polynomials into homogeneous polynomials, i.e., if $f\in\Pscr_s$ then $\partial_{x_j}f\in\Pscr_{s-1}$; if $s=0$ then $\partial_{x_j}f=0$, of course.
These three properties are the basis for most of the algebraic manipulations that are commonly used in perturbation theory.
A power series is represented as a sum of homogeneous polynomials. Of course, in practical calculations the series will be truncated at some order. Since every homogeneous polynomial $f\in\Pscr_s$ can be represented as $$f(x) = \sum_{|k|=s} f_k u_k(x)\ ,$$ it is enough to store in a suitable manner the coefficients $f_k$. A convenient way, particularly effective when most of the coefficients are different from zero, is based on the usual lexicographic ordering of polynomials (to be pedantic, inverse lexicographic). E.g., a homogeneous polynomial of degree $s$ in two variables is ordered as $$a_{s,0} x_1^s + a_{s-1,1}x_1^{s-1}x_2 + \ldots + a_{0,s}x_2^s\ .$$ The idea is to use the position of a monomial $x^{k}$ in the lexicographic order as an index $I(k_1,\ldots,k_n)$ in an array of coefficients. We call $I$ and [indexing function]{}. Here we illustrate how to use it, deferring to sect. the actual construction of the function.
The method is illustrated in table . Let $f$ be a power series, truncated at some finite order $s$. A memory block is assigned to $f$. The size of the block is easily determined as $I\bigl((0,\ldots,0,s)\bigr)$. For, $(0,\ldots,0,s)$ is the last vector of length $s$. The starting address of the block is assigned to the coefficient of $u_{(0,0,\ldots,0)}$; the next address is assigned to the coefficient of $u_{(1,0,\ldots,0)}$, because $(1,0,\ldots,0)$ is the first vector of length $1$, and so on. Therefore, the address assigned to the coefficient of $u_{(k_1,\ldots,k_n)}$ is the starting address of the block incremented by $I\bigl((k_1,\ldots,k_n)\bigr)$. If $f$ is a homogeneous polynomial of degree $s$ the same scheme works fine with a few minor differences: the length of the block is $I\bigl((0,\ldots,0,s)\bigr)-I\bigl((0,\ldots,0,s-1)\bigr)$, the starting address of the block is associated to the coefficient of $u_{(s,0,\ldots,0)}$, and the coefficient of $u_{(k_1,\ldots,k_n)}$ is stored at the relative address $I\bigl((k_1,\ldots,k_n)\bigr)-I\bigl((0,\ldots,0,s-1)\bigr)$. This avoids leaving an empty space at the top of the memory block.
In view of the form above of the representation a function is identified with a set of pairs $(k,f_k)$, where $k\in\interi_+^n$ is the vector of the exponents, acting as the label of the elements of the basis, and $f_k$ is the numerical coefficient. Actually the vector $k$ is not stored, since it is found via the index. The algebraic operations of sum, product and differentiation can be considered as operations on the latter set.
[(i)]{}If $f,g\in\Pscr_s$ then the operation of calculating the sum $f+g$ is represented as
$$\left.
\vcenter{\openup1\jot\halign{
\hfil$\displaystyle({#},$
&$\displaystyle{#})$\hfil
\cr
k & f_k\cr
k & g_k\cr
}}
\right\}\mapsto
(k,f_k+g_k)\ ,$$
to be executed over all $k$ such that $|k|=s$.
[(ii)]{}If $f\in\Pscr_r$ and $g\in\Pscr_s$ then the operation of calculating the product $fg$ is represented as
$$\left.
\vcenter{\openup1\jot\halign{
\hfil$\displaystyle({#},$
&$\displaystyle{#})$\hfil
\cr
k & f_k\cr
k' & g_{k'}\cr
}}
\right\}\mapsto
(k+k',f_kg_{k'})\ ,$$
to be executed over all $k,k'$ such that $|k|=r$ and $|k'|=s$.
[(iii)]{}If $f\in\Pscr_s$ then the operation of differentiating $f$ with respect to, e.g., $x_1$ is represented as
$$(k,f_k) \mapsto
\left\{
\vcenter{\openup1\jot\halign{
$\displaystyle{#}$\hfil
&\quad{\rm for\ }$\displaystyle{#}$\hfil
\cr
\emptyset & k_1=0\ ,\cr
(k',k_1 f_k) & k_1\ne 0\ ,
\cr
}}
\right.$$
where $k'=(k_1-1,k_2,\ldots,k_n)$.
It is perhaps worthwhile to spend a few words about how to make the vector $k$ to run over all its allowed values. In the case of sum, we do not really need it: since the indexes of both addends and of the result are the same, the operation can actually be performed no matter which $k$ is involved: just check that the indexes are in the correct range.[^1] In order to perform product and differentiation it is essential to know the values of $k$ and $k'$. To this end, we can either use the inverse of the indexing function, or generate the whole sequence by using a function that gives the vector next to a given $k$.
funrep.1.2
----------
[Fourier series]{} Let us denote by $\phi=(\phi_1\ldots,\phi_n)\in\toro^n$ the independent variables. The Fourier expansion of a real function on $\toro^n$ takes the form $$f(\phi)=\sum_{k\in\interi^n}\left(a_k\cos\langle k,\phi\rangle
+b_k\sin\langle k,\phi\rangle\right)\ ,
\formula{funrep.5}$$ where $a_k$ and $b_k$ are numerical coefficients. In this representation there is actually a lot of redundancy: in view of $\cos(-\alpha)=\cos\alpha$ and $\sin(-\alpha)=-\sin\alpha$ the modes $-k$ and $k$ can be arbitrarily interchanged. On the other hand, it seems that we actually need two different arrays for the sin and cos components, respectively. A straightforward way out is to use the exponential representation $\sum_{k} a_k e^{i\langle k,\phi\rangle}$, but a moment’s thought leads us to the conclusion that the redundancy is not removed at all. However, we can at the same time remove the redundancy and reduce the representation to a single array by introducing a suitable basis $\{u_k(\phi)\}_{k\in\interi^n}\,$. Let $k\in\interi^n$; we shall say that $k$ is [even]{} if the first non zero component of $k$ is positive, and that $k$ is [odd]{} if the first non zero component of $k$ is negative. The null vector $k=0$ is said to be even. Then we set $$u_k(\phi)=\left\{
\vcenter{\openup1\jot\halign{
$\displaystyle{#}$\hfil
&\quad{\rm for\ }$\displaystyle{#}$\hfil
\cr
\cos\langle k,\phi\rangle & k{\ \rm even}\ ,\cr
\sin\langle k,\phi\rangle & k{\ \rm odd}\ .\cr
}}
\right.
\formula{funrep.6}$$ This makes the representation $f(\phi)=\sum_{k\in\interi^n}\phi_k
u_{k}(\phi)$ unique and redundancy free. It may be convenient to remark that the notation for the sin function may create some confusion. Usually, working with one variable, we write $\sin\phi$. The convention above means that we should rather write $-\sin(-\phi)$, which is correct, but a bit funny. This should be taken into account when, after having accurately programmed all the operations, we discover that our manipulator says, e.g., that $\der{}{\phi}\cos\phi =
-\sin(-\phi)$.
In view of the discussion in the previous section it should now be evident that a truncated Fourier expansion of a function $f(\phi)$ can easily be represented by storing the coefficient of $u_{k}(\phi)$ at an appropriate memory address, as calculated by the indexing function $I(k)$ of sect. .
The considerations of the previous section can be easily extended to the problem of calculating the sum and/or product of two functions, and of differentiating a function. Let us identify any term of the Fourier expansion of the function $f$ with the pair $(k,f_k)$. Let us also introduce the functions $\odd(k)$ and $\even(k)$ as follows: if $k$ is odd, then $\odd(k)=k$ and $\even(k)=-k$; else $\odd(k)=-k$ and $\even(k)=k$. That, is, force $k$ to be odd or even, as needed, by possibly changing its sign.
[(i)]{}Denoting by $(k,f_k)$ and $(k,g_k)$ the same Fourier components of two functions $f$ and $g$, respectively, the sum is computed as
$$\left.
\vcenter{\openup1\jot\halign{
\hfil$\displaystyle({#},$
&$\displaystyle{#})$\hfil
\cr
k & f_k\cr
k & g_k\cr
}}
\right\}\mapsto
(k,f_k+g_k)\ .
\formula{funrep.7}$$
[(ii)]{}Denoting by $(k,f_k)$ and $(k',g_{k'})$ any two terms in the Fourier expansion of the functions $f$ and $g$, respectively, the product is computed as
$$\left.
\vcenter{\openup1\jot\halign{
\hfil$\displaystyle({#},$
&$\displaystyle{#})$\hfil
\cr
k & f_k\cr
k' & g_{k'}\cr
}}
\right\}\mapsto
\left\{
\vcenter{\openup1\jot\halign{
\hfil$\displaystyle{#}$
&$\displaystyle{#}$\hfil
&\ {\rm for}$\>\displaystyle{#}$\hfil
\cr
% 1 %
\left(\even(k+k'),\frac{f_kg_{k'}}{2}\right)
& \cup \left(\even(k-k'),\frac{f_kg_{k'}}{2}\right)
& k\>{\rm even}\,,\>k'\>{\rm even}\>,
\cr
% 2 %
\left(\odd(k+k'),\frac{f_kg_{k'}}{2}\right)
& \cup \left(\odd(k-k'),-\frac{f_kg_{k'}}{2}\right)
& k\>{\rm even}\,,\>k'\>{\rm odd}\>,
\cr
% 3 %
\left(\odd(k+k'),\frac{f_kg_{k'}}{2}\right)
& \cup \left(\odd(k-k'),\frac{f_kg_{k'}}{2}\right)
& k\>{\rm odd}\,,k'\>{\rm even}\>,
\cr
% 4 %
\left(\even(k+k'),-\frac{f_kg_{k'}}{2}\right)
& \cup \left(\even(k-k'),\frac{f_kg_{k'}}{2}\right)
& k\>{\rm odd}\,,\>k'\>{\rm odd}\>.
\cr
}}
\right.
\formula{funrep.8}$$ Remark that the product always produces two distinct terms, unless $k=0$ or $k'=0$.
[(iii)]{}Denoting by $(k,f_k)$ any term in the Fourier expansion of a function $f$, differentiation with respect to, e.g., $\phi_1$ is performed as
$$(k,f_k) \mapsto
\left\{
\vcenter{\openup1\jot\halign{
$\displaystyle{#}$\hfil
&\quad{\rm for\ }$\displaystyle{#}$\hfil
\cr
(-k,-k_1 f_k) & k{\ \rm even}\ ,\cr
(-k, k_1 f_k) & k{\ \rm odd}\ .\cr
}}
\right.
\formula{funrep.9}$$ All these formul[æ]{} follow from well known trigonometric identities.
3
=
[Indexing functions]{} The basic remark for constructing an index function is the following. Suppose that we are given a countable set $\Ascr$. Suppose also that $\Ascr$ is equipped with a relation of complete ordering, that we shall denote by the symbols $\prec$, $\preceq$, $\succ$ and $\succeq$. So, for any two elements $a,b\in\Ascr$ exactly one of the relations $a\prec b$, $a=b$ and $b\succ a$ is true. Suppose also that there is a minimal element in $\Ascr$, i.e., there is $a_0\in\Ascr$ such that $a\succ a_0$ for all $a\in\Ascr$ such that $a\ne a_0$. Then an index function $I$ is naturally defined as $$I(a)=\#\{b\in\Ascr\>:\>b\prec a\}\ .
\formula{indexf.0}$$ If $\Ascr$ is a finite set containing $N$ elements, then $I(\Ascr)=\{0,1,\ldots,N-1\}$. If $\Ascr$ is an infinite (but countable) set, then $I(\Ascr)=\interi_+$, the set of non negative integers. For instance, the trivial case is $\Ascr=\interi_+$ equipped with the usual ordering relation. In such a case the indexing function is just the identity.
Having defined the function $I(a)$, we are interested in performing the following basic operations:
[(i)]{}for a given $a\in\Ascr$, find the index $I(a)$;
[(ii)]{}for a given $a\in\Ascr$, find the element next (or prior) to $a$, if it exists;
[(iii)]{}for a given $l\in I(\Ascr)$, find $I^{-1}(l)$, i.e., the element $a\in\Ascr$ such that $I(a)=l$.
The problem here is to implement an effective construction of the index for some particular subsets of $\interi^{n}$ that we are interested in. In order to avoid confusions, we shall use the symbols $\prec$, $\preceq$, $\succ$ and $\succeq$ when dealing with an ordering relation in the subset of $\interi^{n}$ under consideration. The symbols $\lt$, $\le$, $\ge$ and $\gt$ will always denote the usual ordering relation between integers.
As a first elementary example, let us consider the case $\Ascr=\interi$. The usual ordering relation $\lt$ does not fulfill our requests, because there is no minimal element. However, we can construct a different ordering satisfying our requests as follows.
The resulting order is $0,1,-1,2,-2,\ldots\,$, so that $0$ is the minimal element.
Constructing the indexing function in this case is easy. Indeed, we have $$I(0)=0\ ,\quad
I(a)=\left\{
\vcenter{\openup1\jot\halign{
$\displaystyle{#}$\hfil
&\quad{\rm for\ }$\displaystyle{#}$\hfil
\cr
2a-1&a\gt 0\ ,
\cr
-2a&a\lt 0\ .
\cr
}}
\right.
\formula{indexf.0.1}$$ The inverse function is also easily constructed: $$I^{-1}(0)=0\ ,\quad
I^{-1}(l)=
\left\{
\vcenter{\openup1\jot\halign{
$\displaystyle{#}$\hfil
&\quad{\rm for\ }$\displaystyle{#}$\hfil
\cr
(l+1)/2& l\ {\rm odd}\ ,
\cr
-l/2&l\ {\rm even}\ .
\cr
}}
\right.
\formula{indexf.0.2}$$ In the rest of this section we show how an indexing function can be constructed for two particularly interesting cases, namely polynomials and trigonometric polynomials. However, we stress that the procedure we are using is a quite general one, so it can be extended to other interesting situations.
indexf.1
========
[The polynomial case]{} Let us first take $\Ascr_n=\interi^n_+$, i.e., integer vectors with non negative components; formally $$\Ascr_n=\{k=(k_1,\ldots,k_n)\in\interi^n\>:\>k_1\ge 0,\ldots,k_n\ge
0\}\ .$$ The index $n$ in $\Ascr_n$ denotes the dimension of the space. This case is named “polynomial” because it occurs precisely in the representation of homogeneous polynomials, and so also in the Taylor expansion of a function of $n$ variables: the integer vectors $k\in\Ascr_n$ represent all possible exponents.
We shall denote by $|k|=k_1+\ldots+k_n$ the length (or norm) of the vector $k\in\interi^n_+$. Furthermore, to a given vector $k=(k_1,\ldots,k_n)\in\Ascr_n$ we shall associate the vector $t(k)\in\Ascr_{n-1}$ (the [tail]{} of $k$) defined as $t(k)=(k_2,\ldots,k_n)$. This definition is meaningful only if $n\gt
1$, of course.
indexf.1.1
----------
[Ordering relation]{} Pick a fixed $n$, and consider the finite family of sets $\Ascr_{1}=\interi_+,\ldots,\Ascr_{n}=\interi^n_+$.
In table the ordering resulting from this definition is illustrated for the cases $m=2,3,4,5$.
If $m=1$ then only (i) applies, and this ordering coincides with the natural one in $\interi_+$. For $m\gt 1$, if (i) and (ii) are both false, then (iii) means that one must decrease the dimension $n$ by replacing $k$ with its tail $t(k)$, and retry the comparison. For this reason the ordering has been established for $1\le m\le
n$. Eventually, one ends up with $m=1$, to which only (i) applies.
It is convenient to define $\Pscr_n(k)$ as the set of the elements which precede $k$; formally: $$\Pscr_n(k)=\{k'\in\Ascr_n\>:\>k'\prec k\}\ .$$ With the latter notation the indexing function is simply defined as $I(k)=\#\Pscr_n(k)$. The following definitions are also useful. Pick a vector $k\in\Ascr_n$, and define the sets $\Bscr_n^{(i)}(k)$, $\Bscr_n^{(ii)}(k)$ and $\Bscr_n^{(iii)}(k)$ as the subsets of $\Ascr_n$ satisfying (i), (ii) and (iii), respectively, in the ordering algorithm above. Formally: $$\eqalign{ \Bscr_n^{(i)}(0) &=\Bscr_n^{(ii)}(0)=\Bscr_n^{(iii)}(0)
=\Bscr_1^{(ii)}(k)=\Bscr_1^{(iii)}(k)=\emptyset\ ,
\cr
\Bscr_n^{(i)}(k)
&=\{k'\in\Ascr_n\>:\>|k'|\lt |k|\}\ ,
\cr
\Bscr_n^{(ii)}(k)
&=\{k'\in\Ascr_n\>:\>|k'|=|k|\ \wedge\ k'_1\gt k_1\}\ ,
\cr
\Bscr_n^{(iii)}(k)
&=\{k'\in\Ascr_n\>:\>|k'|=|k|\ \wedge\ k'_1=k_1
\ \wedge\ t(k')\lt t(k)\}\ .
\cr
}
\formula{indexf.7}$$ The sets $\Bscr_n^{(i)}(k)$, $\Bscr_n^{(ii)}(k)$ and $\Bscr_n^{(iii)}(k)$ are pairwise disjoint, and moreover $$\Bscr_n^{(i)}(k)\cup \Bscr_n^{(ii)}(k) \cup \Bscr_n^{(iii)}(k)
=\Pscr_n(k)\ .$$ This easily follows from the definition.
indexf.1.2
----------
[Indexing function for polynomials]{} Let $k\in\Ascr_n$. In view of the definitions and of the properties above the index function, defined as in , turns out to be $$I(0)=0\ ,\quad
I(k)= \#\Bscr_n^{(i)}(k) +\#\Bscr_n^{(ii)}(k)
+\#\Bscr_n^{(iii)}(k)\ .
\formula{indexf.8}$$
Let us introduce the functions $$\eqalign{
J(n,s)&=\#\{k\in\Ascr_n\>:\>|k|=s\}\ ,\cr
N(n,s)&=\sum_{j=0}^{s}J(n,j)\qquad\qquad\qquad{\rm for\ }n\ge
1\,,\>s\ge 0\ .\cr
}
\formula{indexf.10}$$ These functions will be referred to in the following as [$J$-table and $N$-table]{}.
We claim that the indexing function can be recursively computed as $$\eqalign{
I(0)&=0\ ,
\cr
I(k)&=\left\{
\vcenter{\openup1\jot\halign{
$\displaystyle{#}$\hfil
&\quad{\rm for\ }$\displaystyle{#}$\hfil
\cr
N(n,|k|-1) & k_1=|k|\ ,
\cr
N(n,|k|-1)+I(t(k)) & k_1\lt|k|\ .
\cr
}}\right.
\cr
}
\formula{indexf.100}$$ The claim follows from $$\formdef{indexf.20}
\formdef{indexf.21}
\formdef{indexf.22}
\leqalignno{
\#\Bscr_n^{(i)}(k) &= N(n,|k|-1)\ ;
&\frmref{indexf.20}
\cr
\#\Bscr_n^{(ii)}(k)
&=\left\{\vcenter{\openup1\jot\halign{
\hbox to 13pc{$\displaystyle{#}$\hfil}\hfil
&\quad{\rm for\ }$\displaystyle{#}$\hfil
\cr
0 & k_1=|k|\ ,
\cr
N(n-1,|k|-k_1-1) & k_1\lt |k|\ ;
\cr
}}\right.
&\frmref{indexf.21}
\cr
\#\Bscr_n^{(iii)}(k)
&=\left\{\vcenter{\openup1\jot\halign{
\hbox to 13pc{$\displaystyle{#}$\hfil}\hfil
&\quad{\rm for\ }$\displaystyle{#}$\hfil
\cr
I(t(k)) & k_1=|k|\ ,
\cr
I(t(k))-N(n-1,|k|-k_1-1) & k_1\lt |k|\ .
\cr
}}\right.
&\frmref{indexf.22}
\cr
}$$ The equality is a straightforward consequence of the definition of the $N$-table. The equality follows from . Indeed, for $|k|=k_1$ we have $\Bscr_n^{(ii)}(k)=\emptyset$, and for $|k|\gt k_1$ we have $$\eqalign{
\Bscr_n^{(ii)}(k)
&= \bigcup_{k_1\lt j\le |k|}
\{k'\in\Ascr_n\>:\>k'_1=j\ \wedge\ |t(k')|=|k|-j\}
\cr
&= \bigcup_{0\le l\lt |k|-k_1}
\{k'\in\Ascr_n\>:\>k'_1=|k|-l\ \wedge\ |t(k')|=l\}\ .
\cr
}$$ Coming to , first remark that $$\Bscr_n^{(iii)}(k)
=\{k'\in\Ascr_n\>:\>k'_1=k_1\ \wedge\ |t(k')|=|k|-k_1
\ \wedge\ t(k')\prec t(k)\}\ ,$$ so that $$\#\Bscr_n^{(iii)}(k)=\#\{\lambda\in\Ascr_{n-1}\>:\>|\lambda|=|k|-k_1
\ \wedge\ \lambda\prec t(k)\}\ .$$ Then, the equality follows by remarking that $$\Pscr_{n-1}(t(k))=
\{\lambda\in\Ascr_{n-1}\>:\>|\lambda|=|k|-k_1
\ \wedge\ \lambda\prec t(k)\}
\cup
\{\lambda\in\Ascr_{n-1}\>:\>|\lambda|\lt |k|-k_1\}\ .$$ Adding up all contributions follows.
indexf.1.3
----------
[Construction of the tables]{} In view of and the indexing function is completely determined in explicit form by the $J$-table. We show now how to compute the $J$-table recursively. For $n=1$ we have, trivially, $J(1,s)=1$ for $s\ge 0$. For $n\gt 1$ use the elementary property $$\{k\in\Ascr_n\>:\>|k|=s\}=
\bigcup_{0\le j\le s}\{k\in\Ascr_n\>:\>k_1=s-j
\ \wedge\ |t(k)|=j\}\ .$$ Therefore $$\eqalign{
J(1,s)&=1\ ,
\cr
J(n,s)&=\sum_{j=0}^{s}J(n-1,j)\quad {\rm for\ }n\gt 1\ .
\cr
}
\formula{indexf.11}$$ This also means that, according to , we have $N(n,s)=J(n+1,s)\,$.
By the way, one will recognize that the $J$-table is actually the table of binomial coefficients, being $J(n,s)=\left({n+s-1}\atop{n-1}\right)\,$.
indexf.1.4
----------
[Inversion of the index function]{} The problem is to find the vector $k\in\interi_+^n$ corresponding to a given index $l$.
For $n=1$ we have $I^{-1}(l)=l$, of course. Therefore, let us assume $n>1$. We shall construct a recursive algorithm which calculates the inverse function by just showing how to determine $k_1$ and $I\bigl(t(k)\bigr)$.
[(i)]{}If $l=0$, then $k=0$, and there is nothing else to do.
[(ii)]{}If $l>0$, find an integer $s$ satisfying $N(n,s-1)\le l
\lt N(n,s)$. In view of we have $|k|=s$ and $I(t(k))=l-N(n,s-1)$. Hence, by the same method, we can determine $\bigl|t(k)\bigr|$, and so also $k_1=s-\bigl|t(k)\bigr|$.
indexf.5
--------
[An example of implementation]{} We include here an example of actual implementation of the indexing scheme for polynomials. This is part of a program for the calculation of first integrals that is fully described in . The complete computer code is also available from the CPC program library.
We should mention that the [FORTRAN]{} code included here has been written in 1976. Hence it may appear a little strange to programmers who are familiar with the nowadays compilers, since it does not use many features that are available in [FORTRAN 90 ]{} or in the current versions of the compiler. It rather uses the standard of [FORTRAN II]{}, with the only exception of the statement [PARAMETER]{} that has been introduced later.
The [PARAMETER]{}s included in the code allow the user to control the allocation of memory, and may be changed in order to adapt the program to different needs.
[NPMAX]{} is the maximum number of degrees of freedom
[NORDMX]{} is the maximal polynomial degree that will be used
[NBN1]{} and [NBN2]{} are calculated from the previous parameters, and are used in order to allocate the correct amount of memory for the table of binomial coefficients. Here are the statements:
PARAMETER (NPMAX=3) PARAMETER (NORDMX=40) PARAMETER (NBN2=2\*NPMAX) PARAMETER (NBN1=NORDMX+NBN2)
As explained in the previous sections, the indexing function for polynomials uses the table of binomial coefficients. The table is stored in a common block named [BINTAB]{} so that it is available to all program modules. In the same block there are also some constants that are used by the indexing functions and are defined in the subroutine [BINOM]{} below. Here is the statement that must be included in every source module that uses these data:
COMMON /BINTAB/ IBIN(NBN1,NBN2),NPIU1,NMEN1,NFAT,NBIN
Subroutine [BINOM]{} fills the data in the common block [BINTAB]{}. It must be called at the beginning of the execution, so that the constants become available. Forgetting this call will produce unpredictable results. The calling arguments are the following.
[NLIB]{} : the number of polynomial variables. In the Hamiltonian case considered in the present notes it must be set as $2n$, where $n$ is the number of degrees of freedom. It must not exceed the value of the parameter [NPMAX]{}.
[NORD]{} : the wanted order of calculation of the polynomials, which in our case is the maximal order of the normal form. It must not exceed the value of the parameter [NFAT]{}.
The subroutine checks the limits on the calling arguments; if the limits are violated then the execution is terminated with an error message. The calculation of the part of the table of binomial coefficients that will be used is based on well known formul[æ]{}.
SUBROUTINE BINOM(NLIB,NORD) C C Compute the table of the binomial coefficients. C COMMON /BINTAB/ IBIN(NBN1,NBN2),NPIU1,NMEN1,NFAT,NBIN C NFAT=NORD+NLIB NBIN=NLIB IF(NFAT.GT.NBN1.OR.NBIN.GT.NBN2) GO TO 10 NPIU1 = NLIB+1 NMEN1 = NLIB-1 DO 1 I=1,NFAT IBIN(I,1) = I DO 1 K=2,NBIN IF(I-K) 2,3,4 2 IBIN(I,K) = 0 GO TO 1 3 IBIN(I,K) = 1 GO TO 1 4 IBIN(I,K) = IBIN(I-1,K-1)+IBIN(I-1,K) 1 CONTINUE RETURN 10 WRITE(6,1000) NFAT,NBIN STOP 1000 FORMAT(//,5X,15HERROR SUB BINOM,2I10,//) END
Function [INDICE]{} implements the calculation of the indexing function for polynomials. The argument [J]{} is an integer array of dimension [NLIB]{} which contains the exponents of the monomial. It must contain non negative values with sum not exceeding the value [NORD]{} initially passed to the subroutine [BINOM]{}. These limits are not checked in order to avoid wasting time: note that this function may be called several millions of times in a program. The code actually implements the recursive formula using iteration. Recall that recursion was not implemented in [FORTRAN II]{}.
FUNCTION INDICE(J,NLIB) C C Compute the relative address I corresponding to the C exponents J. C COMMON /BINTAB/ IBIN(NBN1,NBN2),NPIU1,NMEN1,NFAT,NBIN DIMENSION J(NLIB) C NP=NLIB+1 INDICE = J(NLIB) M = J(NLIB)-1 DO 1 I=2,NLIB IB=NP-I M = M + J(IB) IB=M+I INDICE = INDICE + IBIN(IB,I) 1 CONTINUE RETURN END
Subroutine [ESPON]{} is the inverse of the indexing function. Given the index [N]{} it calculates the array [J]{} of dimension [NLIB]{} which contains the exponents. The value of [N]{} must be positive (not checked) and must not exceed the maximal index implicitly introduced by the initial choice of [NLIB]{} and [NORD]{} passed to [BINOM]{}. The latter error is actually checked (this does not increase the computation time). The code implements the recursive algorithm described in sect. , again using iteration.
SUBROUTINE ESPON(N,J,NLIB) C C Compute the exponents J correponding to the C index N. C COMMON /BINTAB/ IBIN(NBN1,NBN2),NPIU1,NMEN1,NFAT,NBIN DIMENSION J(NLIB) C NM=NLIB-1 NP=NLIB+1 DO 1 K=NLIB,NFAT IF (N.LT.IBIN(K,NLIB)) GO TO 2 1 CONTINUE WRITE(6,1000) STOP 2 NN = K-1 M = N-IBIN(NN,NLIB) IF(NLIB-2) 8,6,7 7 DO 3 I = 2,NM L = NP-I DO 4 K=L,NFAT IF(M.LT.IBIN(K,L)) GO TO 5 4 CONTINUE 5 IB=NLIB-L J(IB) = NN-K NN = K-1 M = M - IBIN(NN,L) 3 CONTINUE 6 J(NM) = NN-M-1 J(NLIB) = M RETURN 8 J(1)=N RETURN 1000 FORMAT(//,5X,15HERROR SUB ESPON,//) END
The code described here is the skeleton of a program performing algebraic manipulation on polynomial. Such a program must include a call to [BINOM]{} in order to initialize the table of binomial coefficients.
In order to store the coefficient of a monomial with exponents [J]{} (an integer array with dimension [NLIB]{} the user must include a statement like
K = INDICE(J,NLIB)
and then say, e.g., [F(K)=$\ldots$]{} which stores the coefficient at the address [K]{} of the array [F]{}.
Suppose instead that we must perform an operation on all coefficients of degree [IORD]{} of a given function [F]{}. We need to perform a loop on all the corresponding indices and retrieve the corresponding exponents. Here is a sketch of the code.
C Compute the minimum and maximum index NMIN and NMAX C of the coefficients of order IORD. C IB=IORD+NMEN1 NMIN = IBIN(IB,NLIB) IB=IORD+NLIB NMAX = IBIN(IB,NLIB) - 1 C C Loop on all coefficients C DO 1 N = NMIN,NMAX CALL ESPON(N,J,NLIB) ... [more code to operate on the coefficient]{} F(N) ... 1 CONTINUE
Let us add a few words of explanation. According to , the index of the first coefficient of degree $s$ in $n$ variables is $I(s,0,\ldots,0)=N(n,s-1)$, and we also have $N(n,s-1)=\left({n+s-1}\atop{n}\right)$ as explained at the end of sect. . This explains how the limits [NMIN]{} and [NMAX]{} are calculated as $N(n,s-1)$ and $N(n,s+1)-1$, respectively. The rest of the code is the loop that retrieves the exponents corresponding to the coefficient of index [N]{}.
indexf.2
========
[Trigonometric polynomials]{} Let us now consider the more general case $\Ascr_n=\interi^n$. The index $n$ in $\Ascr_n$ denotes again the dimension of the space. The name used in the title of the section is justified because this case occurs precisely in the representation of trigonometric polynomials, as explained in sect. .
We shall now denote by $|k|=|k_1|+\ldots+|k_n|$ the length (or norm) of the vector $k\in\interi^n$. The tail $t(k)$ of a vector $k$ will be defined again as $t(k)=(k_2,\ldots,k_n)$.
indexf.2.1
----------
[Ordering relation]{} Pick a fixed $n$, and consider the finite family of sets $\Ascr_{1}=\interi,\ldots,\Ascr_{n}=\interi^n$.
In table the order resulting from this definition is illustrated for the cases $m=2,3,4$.
If $m=1$ this ordering coincides with the ordering in $\interi$ introduced in sect . For $m\gt 1$, if (i), (ii) and (iii) do not apply, then (iv) means that one must decrease the dimension $n$ by replacing $k$ with its tail $t(k)$, and retry the comparison. Eventually, one ends up with $m=1$, falling back to the one dimensional case to which only (i) and (iii) apply.
The ordering in this section has been defined for the case $\Ascr_n=\interi^n$. However, it will be useful to consider particular subsets of $\interi^n$. The natural choice will be to use again the ordering relation defined here. For example, the case of integer vectors with non negative components discussed in sect. can be considered as a particular case: the restriction of the ordering relation to that case gives exactly the order introduced in sect. . Just remark that the condition (iii) above becomes meaningless in that case, so that it can be removed.
The set $\Pscr_n(k)$ of the elements preceding $k\in\Ascr^n$ in the order above is defined as in sect. . Following the line of the discussion in that section it is also convenient to give some more definitions. Pick a vector $k\in\Ascr_n$, and define the sets $\Bscr_n^{(i)}(k)$, $\Bscr_n^{(ii)}(k)$, $\Bscr_n^{(iii)}(k)$ and $\Bscr_n^{(iv)}(k)$ as the subsets of $\Ascr_n$ satisfying (i), (ii), (iii) and (iv), respectively, in the ordering algorithm above. Formally, $$\eqalign{
\Bscr_n^{(i)}(0)
&=\Bscr_n^{(ii)}(0)=\Bscr_n^{(iii)}(0)=\Bscr_n^{(iv)}(0)
=\Bscr_1^{(ii)}(k)=\Bscr_1^{(iv)}(k)=\emptyset\ ,
\cr
\Bscr_n^{(i)}(k)
&=\{k'\in\Ascr_n\>:\>|k'|\lt |k|\}\ ,
\cr
\Bscr_n^{(ii)}(k)
&=\{k'\in\Ascr_n\>:\>|k'|=|k|\ \wedge\ |k'_1|\gt |k_1|\}\ ,
\cr
\Bscr_n^{(iii)}(k)
&=\{k'\in\Ascr_n\>:\>|k'|=|k|\ \wedge\ |k'_1|=|k_1|
\ \wedge\ k'_1\gt k_1\}\ ,
\cr
\Bscr_n^{(iv)}(k)
&=\{k'\in\Ascr_n\>:\>|k'|=|k|\ \wedge\ k'_1=k_1
\ \wedge\ t(k')\lt t(k)\}\ .
\cr
}
\formula{indexf.17}$$ The sets $\Bscr_n^{(i)}(k)$, $\Bscr_n^{(ii)}(k)$, $\Bscr_n^{(iii)}(k)$ and $\Bscr_n^{(iv)}(k)$ are pairwise disjoint, and moreover $$\Bscr_n^{(i)}(k)\cup \Bscr_n^{(ii)}(k) \cup \Bscr_n^{(iii)}(k)
\cup \Bscr_n^{(iv)}(k)
=\Pscr(k)\ .$$ This easily follows from the definition.
indexf.2.2
----------
[Indexing function for trigonometric polynomials]{} Let $k\in\Ascr_n$. In view of the definitions and of the properties above the index function, defined as in , turns out to be $$I(0)=0\ ,\quad
I(k)= \#\Bscr_n^{(i)}(k) +\#\Bscr_n^{(ii)}(k)
+\#\Bscr_n^{(iii)}(k) +\#\Bscr_n^{(iv)}(k)\ .
\formula{indexf.108}$$
Let us introduce the $J$-table and the $N$-table as $$\eqalign{
J(n,s)&=\#\{k\in\Ascr_n\>:\>|k|=s\}\ ,\cr
N(n,s)&=\sum_{j=0}^{s}J(n,j)\qquad\qquad\qquad{\rm for\ }n\ge
1\,,\>s\ge 0\ .\cr
}
\formula{indexf.110}$$ We claim that the index function can be recursively computed as $$\eqalign{
I(0)&=0\ ,
\cr
I(k)&=\left\{
\vcenter{\openup1\jot\halign{
$\displaystyle{#}$\hfil
&\quad{\rm for\ }$\displaystyle{#}$\hfil
\cr
N(n,|k|-1) & |k_1|=|k|\ \wedge\ k_1\ge 0\ ,
\cr
N(n,|k|-1)+1 & |k_1|=|k|\ \wedge\ k_1\lt 0\ ,
\cr
N(n,|k|-1)+N(n-1,|k|-|k_1|-1)+I(t(k))
& |k_1|\lt |k|\ \wedge\ k_1\ge 0\ ,
\cr
N(n,|k|-1)+N(n-1,|k|-|k_1|)+I(t(k))
& |k_1|\lt |k|\ \wedge\ k_1\lt 0\ .
\cr
}}\right.
\cr
}
\formula{indexf.200}$$ This formula follows from $$\formdef{indexf.120}
\formdef{indexf.121}
\formdef{indexf.122}
\formdef{indexf.123}
\leqalignno{
%1%
\qquad\#\Bscr_n^{(i)}(k)
&=N(n,|k|-1)\ ;
&\frmref{indexf.120}
\cr
%2%
\qquad\#\Bscr_n^{(ii)}(k)
&=\left\{\vcenter{\openup1\jot\halign{
\hbox to 12pc{$\displaystyle{#}$\hfil}\hfil
&\quad{\rm for\ }$\displaystyle{#}$\hfil
\cr
0 & |k_1|=|k|\ ,
\cr
2 N(n-1,|k|-|k_1|-1) & |k_1|\lt |k|\ ;
\cr
}}\right.
&\frmref{indexf.121}
\cr
%3%
\qquad\#\Bscr_n^{(iii)}(k)
&=\left\{\vcenter{\openup1\jot\halign{
\hbox to 12pc{$\displaystyle{#}$\hfil}\hfil
&\quad{\rm for\ }$\displaystyle{#}$\hfil
\cr
0 & |k_1|\le |k|\ \wedge\ k_1\ge 0\ ,
\cr
J(n-1,|k|-|k_1|) & |k_1|\le |k|\ \wedge\ k_1\lt 0\ ;
\cr
}}\right.
&\frmref{indexf.122}
\cr
%4%
\qquad\#\Bscr_n^{(iv)}(k)
&=\left\{\vcenter{\openup1\jot\halign{
\hbox to 12pc{$\displaystyle{#}$\hfil}\hfil
&\quad{\rm for\ }$\displaystyle{#}$\hfil
\cr
I(t(k)) & |k_1|=|k|\ ,
\cr
I(t(k))-N(n-1,|k|-|k_1|-1) & |k_1|\lt|k|\ .
\cr
}}\right.
&\frmref{indexf.123}
\cr
}$$ The equality is a straightforward consequence of the definition . The equality follows by remarking that for $|k_1|=|k|$ we have $\Bscr_n^{(ii)}(k)=\emptyset$, and for $|k_1|\lt |k|$ we have $$\Bscr_n^{(ii)}(k)=B_n^{+}(k)\cup B_n^{-}(k)\ ,\quad
B_n^{+}(k)\cap B_n^{-}(k)=\emptyset\ ,$$ with $$\eqalign{
B_n^{+}(k)
&=\bigcup_{0\le l\lt |k|-|k_1|}
\{k'\in\Ascr_n\>:\>k'_1=|k|-l\ \wedge\ |t(k)|=l\}\ ,
\cr
B_n^{-}(k)
&=\bigcup_{0\le l\lt |k|-|k_1|}
\{k'\in\Ascr_n\>:\>k'_1=l-|k|\ \wedge\ |t(k)|=l\}\ ;
\cr
}$$ use also $\#B_n^{+}(k)=\#B_n^{-}(k)$. The equality follows from $$\Bscr_n^{(iii)}(k)=\left\{
\vcenter{\openup1\jot\halign{
$\displaystyle{#}$\hfil
&\quad{\rm for\ }$\displaystyle{#}$\hfil
\cr
\emptyset & |k_1|=|k|\ \wedge\ k_1\ge 0\ ,
\cr
\{k'\in\Ascr_n\>:\>k'_1=|k_1|\ \wedge\ |t(k')|=|k|-|k_1|\}
&|k_1|=|k|\ \wedge\ k_1\lt 0\ .
\cr
}}\right.$$ Coming to , remark that $$\Bscr_n^{(iv)}(k)=
\{k'\in\Ascr_n\>:\> |t(k')|=|k|-|k_1|\ \wedge\ t(k')\prec t(k)\}\ .$$ Proceeding as in the polynomial case we find again $$\#\Bscr_n^{(iv)}(k)=\#\{\lambda\in\Ascr_{n-1}\>:\>|\lambda|=|k|-|k_1|
\ \wedge\ \lambda\prec t(k)\}\ ,$$ and follows by remarking that $$\Pscr_{n-1}(t(k))=
\{\lambda\in\Ascr_{n-1}\>:\>|\lambda|=|k|-|k_1|
\ \wedge\ \lambda\prec t(k)\}
\cup
\{\lambda\in\Ascr_{n-1}\>:\>|\lambda|\lt |k|-|k_1|\}\ .$$ Adding up all contributions follows.
indexf.2.3
----------
[Construction of the tables]{} We show now how to construct recursively the $J$-table, so that the $N$-table can be constructed, too. For $n=1$ we have, trivially, $J(1,0)=1$ and $J(1,s)=2$ for $s\gt 0$. For $n\gt 1$ use the elementary property $$\{k\in\Ascr_n\>:\>|k|=s\}=
\bigcup_{-s\le j\le s}\{k\in\Ascr_n\>:\>k_1=j
\ \wedge\ |t(k)|=s-|j|\}\ .$$ Therefore $$\eqalign{
J(1,0)&=1\ ,
\cr
J(1,s)&=2\ ,
\cr
J(n,s)&=\sum_{j=-s}^{s}J(n-1,s-|j|)\quad {\rm for\ }n\gt 1\ .
\cr
}
\formula{indexf.211}$$ This completely determines the $J$-table.
indexf.2.4
----------
[Inversion of the index function]{} The problem is to find the vector $k$ of given dimension $n$ corresponding to the given index $l$. For $n=1$ the function $I(k)$ and its inverse $I^{-1}(l)$ are given by and . Therefore in the rest of this section we shall assume $n\gt 1$. We shall give a recursive algorithm, showing how to determine $k_1$ and $I\bigl(t(k)\bigr)$.
[(i)]{}If $l=0$ then $k=0$, and there is nothing else to do.
[(ii)]{}Assuming that $l\gt 0$, determine $s$ such that $$N(n,s-1)\le l \lt N(n,s)\ .$$
From this we know that $|k|=s$.
[(iii)]{}Define $l_1=l-N(n,s-1)$, so that $I\bigl(t(k)\bigr)\le
l_1$ by . If $l_1=0$ set $s_1=0$; else, determine $s'$ such that $$N(n-1,s'-1)\le l_1 \lt N(n-1,s')\ ,$$ and let $s_1=\min(s',s)$. In view of $I\bigl(t(k)\bigr)\le l_1$ we know that $|t(k)|\le s_1$. Remark also that $s_1=0$ if and only if $l_1=0$. For, if $s_1\ge 1$ then we have $l_1\ge N(n-1,0)=1$.
[(iv)]{}If $l_1=0$, then by the first of we conclude $$k_1=|k|=s\ ,\quad t(k)=0\ ,$$ and there is nothing else to do.
[(v)]{}If $l_1=1$, then by the second of we conclude $$k_1= -|k|= -s\ ,\quad t(k)=0\ ,$$
and there is nothing else to do.
[(vi)]{}If $l_1\gt 1$ and $s_1\gt 0$, we first look if we can set $0\le k_1\lt |k|$. In view of the third of we should have $$|k|-k_1=s_1\ ,\quad
\bigl|t(k)\bigr|=s_1\ ,\quad
I\bigl(t(k)\bigr)=l_1-N(n-1,s_1-1)\ .$$
This can be consistently made provided the conditions $$s_1\gt 0\quad {\rm and}\quad I\bigl(t(k)\bigr)\ge N(n-1,s_1-1)$$
are fulfilled. The condition $s\gt 0$ is already satisfied. By , the second condition is fulfilled provided $l_1\ge 2 N(n-1,s_1-1)$. This has to be checked. If the second condition is true, then set $k_1=|k|-s_1$, and recall that $\bigl|t(k)\bigr|=s_1$. Hence, we can replace $n$, $l$, and $s$ by $n-1$, $l_1-N(n-1,s_1-1)$ and $s_1$, respectively, and proceed by recursion restarting again from the point (iii). If the second condition is false, then we proceed with the next point.
[(vii)]{}Recall that $l_1\gt 1$, and remark that we have also $s_1\gt 1$. Indeed, we already know $s_1\gt 0$, so we have to exclude the case $s_1=1$. Let, by contradiction, $s_1=1$. Then we have $l_1\ge 2= 2N(n-1,s_1-1)$, which is the case already excluded by (vi). We conclude $s_1\gt 1$. We look now for the possibility of setting $|k_1|\lt |k|$ and $k_1\lt 0$. In view of the fourth of we should have $$|k|+k_1=s_1-1\ ,\quad
\bigl|t(k)\bigr|=s_1-1\ ,\quad
I\bigl(t(k)\bigr)=l_1-N(n-1,s_1-1)\ .$$
This can be consistently made provided the conditions $$s_1\gt 1\quad {\rm and}\quad I\bigl(t(k)\bigr)\ge N(n-1,s_1-2)$$
are fulfilled. The condition $s_1\gt 1$ is already satisfied. As to the second condition, by it is fulfilled provided $l_1\gt N(n-1,s_1-1)+N(n-1,s_1-2)$. This has to be checked. If the second condition is true, then set $k_1=-|k|+s_1-1$, and recall that $\bigl|t(k)\bigr|=s_1-1$. Hence, we can replace $n$, $l$, and $s$ by $n-1$, $l_1-N(n-1,s_1-2)$ and $s_1-1$, respectively, and proceed by recursion restarting again from the point (iii). If the second condition is false we must decrease $s_1$ by one and start again with the point (vi); remark that $s_1\gt
1$ implies $s_1-1\gt 0$, which is the first of the two conditions to be satisfied at the point (vi), hence the recursion is correct.
Since $l_1\gt 1$ we have $l_1\gt 2N(n-1,0)$, so that the conditions of point (vi) are satisfied for $s=1$. Hence the algorithm above does not fall into an infinite loop between points (vi) and (vii). On the other hand, for $n=1$ either (iii) or (iv) applies, so that the algorithm stops at that point.
4
=
[Storing the coefficients for sparse functions]{} The method of storing the coefficient using the index, as illustrated in sect. , is the most direct one, but reveals to be ineffective when most of the coefficients of a function are zero (sparse function). For, allocating memory space for all coefficients results in a wasting of memory.
A method that we often use is to store the coefficients using a tree structure based on the index. However we should warn the reader that the method described here has the advantage of being easily programmed, but does not pretend to be the most effective one. Efficient programming of tree structure is described, e.g., in the monumental books [The art of computing programming]{}, by D.E. Knuth .
4.1
---
[The tree structure]{} The first information we need is how many bits are needed in order to represent the maximum index for a function. We shall refer to this number as the [length of the index]{}. In the scheme that we are presenting here this is actually the length of the path from the root of the tree to its leave, where the coefficient is found.
In fig. we illustrate the scheme assuming that 4 bits are enough, i.e., there are at most 16 coefficients indexed from $0$ to $15$. The case is elementary, of course, but the method is the general one, and is extended to, e.g., several millions of coefficients (with a length a little more than 20) in a straightforward manner. The bits are labeled by their position, starting from the less significant one (choosing the most significant one as the first bit is not forbidden, of course, and sometimes may be convenient). The label of the bit corresponds to a level in the tree structure, level $0$ being the root and level $3$ being the last one, in our case. At level zero we find a cell containing two pointers, corresponding to the digit $0$ and $1$, respectively. To each digit we associate a cell of level $1$, which contains a pair of pointers, and so on until we reach the last level ($3$ in our case). Every number that may be represented with $4$ bits generates a unique path along the tree, and the last cell contains pointers to the coefficient. The example in the figure represents the path associated with the binary index $1010$, namely $10$ in decimal notation.
Let us also illustrate how this structure may be represented in memory, trying to avoid wasting of space. We use two separate arrays, the first one for pointers and the second one for the coefficients, as illustrated in fig. . The cells containing pairs of pointers are allocated in the first array, the root of the tree having label zero. The label of a cell is always even: the first element corresponds to the zero bit, the next one (with odd label) to the bit one.
The arrays are initially allocated with appropriate size, and are cleared. A good method is to fill the array of pointers with $-1$ (denoting an unused pointer) and the coefficients table with zeros. We also keep track of the first unused cell in the array, which initially is set to $2$ because the root cell is considered to be in use, and of the first free coefficient, which initially is $0$.
We shall use the following notations: ${\tt cell}(2j)$ is the cell with even label $2j$ in the array; ${\tt cell}(2j,0)$ and ${\tt cell}(2j,1)$ are the pointers corresponding to a bit 0 or 1 which are stored at locations $2j$ and $2j+1$, respectively, in the array of pointers; ${\tt
coef}(j)$ is the $j$-th element of the array of coefficients; ${\tt
cc}$ is the current cell and ${\tt cb}$ is the current bit (see below for the meaning); ${\tt fp}$ is the label of the first free (unused) cell of pointers; ${\tt fc}$ is the label of the first free coefficient; $\ell$ is the length of the index.
4.2
---
[Storing the first coefficient]{} Let us describe how the first coefficient is stored. Suppose we want to store the value $x$ as the coefficient corresponding to a given index. Here is the scheme.
[(i)]{}[Initialization:]{} set ${\tt cc}=0$ and ${\tt
cb}=0$. The values of ${\tt fp}=2$ and ${\tt fc}=0$ have already been set when during the array allocation.
[(ii)]{}[Creating a path:]{} repeat the following steps until ${\tt cb}$ equals $\ell-1$:
if the bit at position ${\tt cb}$ in the index is 0, then redefine ${\tt cell}({\tt
cc},0)={\tt fp}$; else redefine ${\tt cell}({\tt cc},1)={\tt fp}$;
set ${\tt cc}={\tt fp}$ and increment ${\tt fp}$ by $2$ (point to the next free cell);
increment ${\tt cb}$ by $1$ (next bit).
[(iii)]{}[Store the coefficient:]{}
if the bit at position ${\tt cb}$ in the index is 0, then redefine ${\tt cell}({\tt
cc},0)={\tt fc}$; else redefine ${\tt cell}({\tt cc},1)={\tt fc}$;
set ${\tt coef}({\tt fc})=x$;
increment ${\tt fc}$ by $1$ (point to the next free coefficient).
Programming this algorithm in a language such as [C]{} or [FORTRAN]{} requires some $10$ to $20$ statements.
Let us see in detail what happens if we want to store the coefficient $0.6180339$ with index $1010$ and $\ell=4$, as illustrated in fig. . Here is the sequence of operations actually made $$\vcenter{\openup1\jot\halign{
{#}\hfil
&\quad$\displaystyle{#}$\ ,\hfil
&\quad$\displaystyle{#}$\ ,\hfil
&\quad$\displaystyle{#}$\ ,\hfil
&\quad$\displaystyle{#}$\hfil
&\quad {#}\hfil
\cr
step\ (i): & {\tt cc}=0 & {\tt cb}=0 & {\tt fp}=2 & {\tt fc}=0\ ;
\cr
step\ (ii): & {\tt cell(0,0)}=2 & {\tt cc}=2 & {\tt fp}=4 & {\tt cb}=1\ ,& then\ ,
\cr
& {\tt cell(2,1)}=4 & {\tt cc}=4 & {\tt fp}=6 & {\tt cb}=2\ ,& then\ ,
\cr
& {\tt cell(4,0)}=6 & {\tt cc}=6 & {\tt fp}=8 & {\tt cb}=3\ ,& end of loop\ ;
\cr
step\ (iii): & {\tt cell(6,1)}=0 & \multispan 2\quad{\tt coef}(0)=0.6180339\ , & {\tt fc}=1\ , & end of game\ .
\cr
}}$$ After this, the contents of the arrays are as represented in fig. .
4.2
---
[Retrieving a coefficient]{} The second main operation is to retrieve a coefficient, which possibly has never been stored. In the latter case, we assume that the wanted coefficient is zero. Here is a scheme.
[(i)]{}[Initialization:]{} set ${\tt cc}=0$ and ${\tt cb}=0$.
[(ii)]{}[Follow a path:]{} repeat the following steps until ${\tt cb}$ equals $\ell$:
save the current value of ${\tt cc}$;
if the bit at position ${\tt cb}$ in the index is 0, then redefine ${\tt cc}$ as ${\tt cell}({\tt
cc},0)$; else redefine ${\tt cc}$ as ${\tt cell}({\tt cc},1)$;
if ${\tt cc}=-1$ then the coefficient is undefined. Return $0$ as the value of the coefficient;
increment ${\tt cb}$ by $1$ (next bit).
[(iii)]{}[Coefficient found:]{} return the coefficient ${\tt coef}({\tt cc})$.
Let us give a couple of examples in order to better illustrate the algorithm. Suppose that we are looking for the coefficient corresponding to the binary index $1010$. By following the algorithm step by step, and recalling that in our example the length of the index is $4$, the reader should be able to check that the sequence of operations is the following: $$\vcenter{\openup1\jot\halign{
{#}\quad\hfil
&$\displaystyle{#}$\hfil
&\quad$\displaystyle{#}$\hfil
&\quad {#}\hfil
\cr
step\ (i): & {\tt cc}=0\ , & {\tt cb}=0\ ;
\cr
step\ (ii): & {\tt cc}=2\ , & {\tt cb}=1\ ,& then\ ,
\cr
& {\tt cc}=4\ , & {\tt cb}=2\ ,& then\ ,
\cr
& {\tt cc}=6\ , & {\tt cb}=3\ ,& then\ ,
\cr
& {\tt cc}=0 & {\tt cb}=4\ ,& end of path\ ;
\cr
step\ (iii): & & & return\ $0.6180339$\ .
\cr
}}$$ The returned value is that of ${\tt coef}(0)$, stored in the location $0$ of the coefficients array.
Suppose now that we are looking for the coefficient corresponding to the binary index $1110$. Here is the actual sequence of operations: $$\vcenter{\openup1\jot\halign{
{#}\quad\hfil
&$\displaystyle{#}$\hfil
&\quad$\displaystyle{#}$\hfil
&\quad {#}\hfil
\cr
step\ (i): & {\tt cc}=0\ , & {\tt cb}=0\ ;
\cr
step\ (ii): & {\tt cc}=2\ , & {\tt cb}=1\ ,& then\ ,
\cr
& {\tt cc}=4\ , & {\tt cb}=2\ ,& then\ ,
\cr
& {\tt cc}=-1\ , & {\tt cb}=2\ ,& return zero\ .
\cr
}}$$ Here the algorithm stops because a coefficient has not been found.
4.4
---
[Other operations]{} Having implemented the two operations above, the reader should be able to implement also the following operations:
[(i)]{}storing a new coefficient corresponding to a given index;
[(ii)]{}adding something to a given coefficient;
[(iii)]{}multiplying a given coefficient by a number.
These are the basic operations that we need in order to perform an elementary computer algebra. Let us add a few hints.
Storing a new coefficient requires perhaps some moment of thinking. Using the index, one should follow the corresponding path in the tree (as in the operation of retrieving) until either happens: the coefficient is found, or the search fails at some point. If the coefficient is found, then it can be overwritten if the new value has to replace the old one. On failure, the path must be completed by appropriately defining the pointers (as in the case of the first coefficient), and then the coefficient can be stored in the appropriate location. As an exercise, suppose that we want to store the coefficient $1.4142136$ corresponding to the binary index $1110$. After completing the operation the memory should look as in fig. .
Adding something to a given coefficient is not very different from the previous operation. Just follow the path. If the coefficient is found, then add the wanted value to it. On failure, just change the “add” operation to a “store” one, and proceed as in the case (i).
Multiplying a coefficient by a constant is even easier. If the coefficient is found, then do the multiplication. On failure, just do nothing.
Further operations can be imagined, but we think that we have described the basic ones. There are just a couple of remarks.
The method illustrated here uses an amount of memory that clearly depends on the number of non zero coefficients of a function. However, this amount is typically not known in advance. Thus, enough memory should be allocated at the beginning in order to assure that there is enough room. When a function is filled, and we know that it will not be changed, the excess of memory can be freed and reused for other purposes. Every operating system and language provides functions that allow the programmer to allocate memory blocks and resize them on need.
A second remark is that other storing methods can be imagined. E.g., once a function is entirely defined it may be more convenient to represent it as a sequential list of pairs (index, coefficient). This is definitely a very compact representation for a sparse function (although not the best for a crowded one).
5
=
[Applications]{} We report here some examples of application of algebraic manipulation that have been obtained by implementing the formal algorithm of sect. . We consider three cases, namely the model of Hénon and Heiles, the Lagrangian triangular equilibria for the Sun-Jupiter system and the planetary problem including Sun, Jupiter, Saturn and Uranus (SJSU).
5.1
---
[The model of Hénon and Heiles]{} A wide class of canonical system with Hamiltonian of the form $$H(x,y) = \frac{\omega_1}{2}(y_1^2+x_1^2) +
\frac{\omega_2}{2}(y_2^2+x_2^2)+x_1^2 x_2
\formula{espnum.5}$$ has been studied by Contopoulos, starting at the end of the fifties, for different values of the frequencies. This approximates the motion of a star in a galaxy, at different distances from the center. A wide discussion on the use of these models in galactic dynamics and on the construction of the so called “third integral” can be found in the book . The third integral is constructed as a power series $\Phi=\Phi_2+\Phi_3+\ldots$ where $\Phi_s$ is a homogeneous polynomial of degree $s$ which is the solution of the equation $\left\{H,\Phi\right\}=0$, where $\{\cdot,\cdot\}$ is the Poisson bracket (see, e.g., or ). A different method is based on the construction of the Birkhoff normal form .
A particular case with two equal frequencies and Hamiltonian $$H(x,y) = \frac{1}{2}(y_1^2+x_1^2) +
\frac{1}{2}(y_2^2+x_2^2)+x_1^2 x_2 -\fraz{1}{3} x_2^3
\formula{hh.ham}$$ has been studied by Hénon and Heiles in 1964 . This work has become famous since for the first time the existence of a chaotic behavior in a very simple system has been stressed, showing some figures. It should be remarked that the existence of chaos had been discovered by Poincaré in his memory on the problem of three bodies , but it had been essentially forgotten.
A program for the construction of the third integral has been implemented by Contopoulos since 1960. He made several comparisons between the level lines of the integral so found on the surface of constant energy and the figures given by the Poincaré sections of the orbits. A similar calculation for the case of Hénon and Heiles has been made by Gustavson , who used the normal form method. The third integral was expanded up to order 8, which may seem quite low today, but it was really difficult to do better with the computers available at that time. Here we reproduce the figures of Gustavson extending the calculation up to order 58, which is now easily reached even on a PC.
In fig. we show the Poincaré sections for the values of energy used by Hénon and Heiles in their paper. As stressed by the authors, an essentially ordered motion is found for $E\lt {\scriptstyle\frac{1}{12}}$, while the chaotic orbits become predominant at higher energies.
The comparison with the level lines of the third integral at energy $E={\scriptstyle\frac{1}{100}}$ is reported in fig. . The correspondence with the Poincaré sections is evident even at order 8, as calculated also by Gustavson. We do not produce the figures for higher orders because they are actually identical with the one for order 8. This may raise the hope that the series for the first integral is a convergent one.
Actually, a theorem of Siegel states that for the Birkhoff normal form divergence is a typical case . A detailed numerical study has been made in and , showing the mechanism of divergence. Moreover, it was understood by Poincaré that perturbations series typically have an asymptotic character (see , Vol. II). Estimates of this type have been given, e.g., in and .
For energy $E={\scriptstyle\frac{1}{12}}$ (fig. ) the asymptotic character of the series starts to appear. Indeed already at order 8 we have a good correspondence between the level lines and the Poincaré section, as was shown also Gustavson’s paper. If we increase the approximation we see that the correspondence remains good up to order 32, but then the divergence of the series shows up, since at order 43 an unwanted “island” appears on the right side of the figure which has no correspondent in the actual orbits, and at order 58 a bizarre behavior shows up.
The phenomenon is much more evident for energy $E={\scriptstyle\frac{1}{8}}$ (fig. ). Here some rough correspondence is found around order 9, but then the bizarre behavior of the previous case definitely appears already at order 27.
The non convergence of the normal form is illustrated in fig. . Writing the homogeneous terms of degree $s$ of the third integral as $\Phi_s=\sum_{j,k}\phi_{j,k}x^jy^k$, we may introduce the norm $$\bigl\|\Phi_{s}\bigr\| = \sum_{j,k} |\phi_{j,k}|\ .$$ Then an indication of the convergence radius may be found by calculating one of the quantities $$\bigl\|\Phi_{s}\bigr\|^{1/s}\ ,\quad
\frac{\bigl\|\Phi_{s}\bigr\|}{\bigl\|\Phi_{s-1}\bigr\|}\ ,\quad
\left(\frac{\bigl\|\Phi_{s}\bigr\|}{\bigl\|\Phi_{s-2}\bigr\|}\right)^{1/2}\ .$$ The first quantity corresponds to the root criterion for power series. The second one corresponds to the ratio criterion. The third one is similar to the ratio criterion, but in the present case turns out to be more effective because it takes into account the peculiar behavior of the series for odd and even degrees. The values given by the root criterion are plotted in the left panel of fig. . The data for the ratio criterion are plotted in the right panel, where open dots and solid dots refer to the second and third quantities in the formula above, respectively. In all cases it is evident that the values steadily increase, with no tendency to a definite limit. The almost linear increase is consistent with the behavior $\bigl\|\Phi_{s}\bigr\|\sim s!$ predicted by the theory.
5.2
---
[The Trojan asteroids]{} The asymptotic behavior of the series lies at the basis of Nekhoroshev theory on exponential stability. The general result, referring for simplicity to the case above, is that in a ball of radius $\rho$ and center at the origin one has $$\bigl| \Phi(t)-\Phi(0)\bigr| \lt O(\rho^3) \quad
{\rm for}\ |t|\lt O(\exp(1/\rho^a))\ ,$$ for some positive $a\le 1$. This is indeed the result given by the theory (see, e.g., ). In rough terms the idea is the following. Due to the estimate $\bigl\|\Phi_{s}\bigr\|\sim s!$ and remarking that $\dot\Phi=\bigl\{H,\Phi\bigr\}$ starts with terms of degree $s+1$, one gets $\bigl|\dot\Phi\bigr| = O(s!\rho^{s+1})$. Then one looks for an optimal degree $s$ which minimizes the time derivative, i.e., $s\sim 1/\rho$. By truncating the integrals at the optimal order one finds the exponential estimate.
However, the theoretical estimates usually give a value of $\rho$ which is useless in practical applications, being definitely too small. Realistic results may be obtained instead if the construction of first integrals for a given system if performed by computer algebra. That is, one constructs the expansion of the first integral up to an high order, compatibly with the computer resources available, and then looks for the optimal truncation order by numerical evaluation of the norms.
The numerical optimization has been performed for the expansion of the Hamiltonian in a neighborhood of the Lagrangian point $L_4$, in the framework of the planar circular restricted problem of three bodies in the Sun-Jupiter case. This has a direct application to the dynamics of the Trojan asteroids (see ).
The two first integrals which are perturbations of the harmonic actions have been constructed up to order $34$ (close to the best possible with the computers at that time). The estimate of the time of stability is reported in fig. . The lower panel gives the optimal truncation order vs. $\log_{10}\rho$. In the upper panel we calculate the stability time as follows: for an initial datum inside a ball of radius $\rho_0$ we determine the minimal time required for the distance to increase up to $2\rho_0$. Remark that the vertical scale is logarithmic. The units are chosen so that $\rho=1$ is the distance of Jupiter from the Sun, and $t=2\pi$ is the period of Jupiter. With this time unit the estimated age of the universe is about $10^9$. The figure shows that the obtained data are already realistic, although, due to the unavoidable approximations, only four of the asteroids close to $L_4$ known at the time of that work did fall inside the region of stability for a time as long as the age of the Universe.
5.3
---
[The SJSU system]{} As a third application we consider the problem of stability for the planar secular planetary model including the Sun and three planets, namely Jupiter, Saturn and Uranus. The aim is evaluate how long the semi-major axes and the eccentricities of the orbits remain close to the current value (see ).
The problem here is much more difficult than in the previous cases. The Hamiltonian must be expanded in Poincaré variables, and is expressed in action-angle variables for the fast motions and in Cartesian variables for the slow motions, for a total of 9 polynomial and 3 trigonometric variables. The expansion of the Hamiltonian in these variables clearly is a major task, that has been handled via computer algebra.
The reduction to the secular problem actually removes the fast motions, so that we get an equilibrium corresponding to an orbit of eccentricity zero close to a circular Keplerian one, and a Hamiltonian expanded in the neighborhood of the equilibrium, which is still represented as a system of perturbed harmonic oscillators, as in the cases above. Thus, after a long preparatory work, we find a problem similar to the previous one, that can be handled with the same methods.
The results are represented in fig. , where we report again the optimal truncation order and the estimated stability time, in the same sense as above. The time unit here is the year, and the distance is chosen so that $\rho_0=1$ corresponds to the actual eccentricity of the three planets. The result is still realistic, although a stability for a time of the order of the age of Universe holds only inside a radius corresponding roughly to $70\%$ of the real one.
The work of M. S. is supported by an FSR Incoming Post-doctoral Fellowship of the Académie universitaire Louvain, co-funded by the Marie Curie Actions of the European Commission.
[^1]: For a homogeneous polynomial of degree $s$ the first vector is $(s,0,\ldots,0)$, and the last one is $(0,\ldots,0,s)$. The indexes of these two vectors are the limits of the indexes in the sum.
|
---
abstract: 'A new integrable class of Davey–Stewartson type systems of nonlinear partial differential equations (NPDEs) in $2+1$ dimensions is derived from the matrix Kadomtsev–Petviashvili equation by means of an asymptotically exact nonlinear reduction method based on Fourier expansion and spatio-temporal rescaling. The integrability by the inverse scattering method is explicitly demonstrated, by applying the reduction technique also to the Lax pair of the starting matrix equation and thereby obtaining the Lax pair for the new class of systems of equations. The characteristics of the reduction method suggest that the new systems are likely to be of applicative relevance. A reduction to a system of two interacting complex fields is briefly described.'
---
\[maccari-firstpage\]
Introduction
============
New classes of evolution nonlinear partial differential equations (NPDEs) integrable by the inverse scattering method (S-integrable) have been found in the last years. These equations are known to be applicable to various branches of physics such as fluid dynamics, nonlinear optics, condensed matter physics and so on. The most famous examples are the Korteweg-de Vries and the nonlinear Schrodinger equations in $1+1$ dimensions and the Kadomtsev–Petviashvili and the Davey–Stewartson equations in $2+1$ dimensions \[1\].
A simple explanation of this coincidence (integrability and applicative relevance) is based on the observation that very large classes of evolution NPDEs in $1+1$ and $2+1$ dimensions, with a dispersive linear part, can be reduced, by a limiting procedure involving the wave modulation induced by weak nonlinear effects, to a very limited number of “universal” evolution NPDEs. Moreover, the same model equations obtained in this way appear in many applicative situations (for instance in plasma physics, nonlinear optics, hydrodynamics, etc.), where weakly nonlinear effects are important \[2–5\].
The reduction method preserves integrability and therefore the model equations are likely to be integrable. For example, it is sufficient that the very large class of equations from which they are obtainable contains just one S-integrable equation, provided the limiting procedure preserves integrability, so that the property of S-integrability is inherited through this limiting technique. Obviously, the last statement about the integrability is based on heuristic considerations and could not be characterized as a rigorous theorem. No precise definition of integrability is available for evolution NPDEs, there being much difference between finding the general solution of a NPDE or solving an initial-value problem with given input data and boundary conditions. It would be possible to derive the spectral transform of the Davey–Stewartson equation from the spectral transform of the Kadomtsev–Petviashvili equation.
Thus this approach, besides explaining why certain model equations are integrable and applicable, provides a powerful tool to investigate the relation among different integrable equations, to test the integrability of nonlinear evolution PDEs and, most importantly, to identify integrable evolution equations that are likely to be of applicative relevance.
In previous papers, we applied this method to certain integrable equations in $2+1$ dimensions. The most interesting results are that the Davey-Stewarston equation \[6–7\] is the typical model equation in $2+1$ dimensions, while new integrable NPDEs can be obtained together with their Lax pair \[8–11\]. Moreover, we used the reduction method to derive two equations of applicative relevance in plasma physics \[12–13\].
The basic idea of the reduction method is to consider a nonlinear evolution PDE whose linear part is dispersive; as it is well known the linear evolution is most appropriately described in terms of Fourier modes and each Fourier mode evolves with constant amplitude and an associated group velocity, that represents the speed with which a wave packet peaked at that Fourier mode would move in configuration space. To evaluate the weak nonlinear effects it is convenient to consider a specific Fourier mode and follow it by going over to a frame of reference that moves with its group velocity. The weak nonlinear effects give rise to a modulation of the amplitude of that Fourier mode (that would remain constant in the absence of nonlinear effects). The modulation is best described in terms of rescaled “coarse-grained” and “slow” variables, that display the weak nonlinear effects on larger space and time scales; indeed, the first step of the reduction method is to use a moving frame of reference with the introduction of the slow variables: $$\begin{gathered}
\xi = \varepsilon ^{p}(x - V_{1} t),\qquad \eta = \varepsilon ^{p}(y - V_{2} t),
\qquad \tau = \varepsilon ^{q}t, \nonumber\\
p > 0,\qquad q > 0, \label{maccari:eq1.1}\end{gathered}$$ where $V_{1} = V_{1} (K_{1} ,K_{2} )$, $V_{2} = V_{2} (K_{1} ,K_{2} )$ are the components of the group velocity $\underline {V} \left( {\underline {K}}
\right) \equiv \left( V_{1} (K_{1} ,K_{2} ),V_{2} (K_{1} ,K_{2} ) \right)$ of the linearized equation, i.e. of the equation obtained by neglecting all the nonlinear terms, and $\varepsilon $ is a “small” expansion parameter.
It is thereby seen that the function that represents the amplitude modulation satisfies, in terms of the rescaled, slow, variables, evolution equations having a universal character; since the coarse-grained nature of the new variables implies that only certain general features of the nonlinear interaction are important.
In this paper we expose an interesting extension of this approach and consider the matrix Kadomtsev–Petviashvili equation \[14–15\] $$\begin{gathered}
U_{t} + U_{xxx} - W_{y} + i\sqrt {3} \left[ {W,U} \right] - 3\left\{
{U,U_{x}} \right\} = 0,\nonumber\\
W_{x} = U_{y},\label{maccari:eq1.2}\end{gathered}$$ where $[A,B]=AB-BA$, $U = U(x,y,t)$, $W = W(x,y,t)$ are $N \otimes N$ complex matrices and the subscripts denote partial differentiation.
By applying the reduction method, a new class of integrable matrix systems of evolution NPDEs depending on a real parameter $\lambda$ is obtained $$\begin{gathered}
i\Psi _{\tau} + L\Psi - \lambda \left[ \Psi ,\Lambda \right] - \left[
\Omega ,\Psi \right] + \sqrt {3} \left\{ \Lambda ,\Psi \right\} -
\left\{ \Phi ,\Psi ^{2} \right\} = 0,\nonumber\\
i\Phi _{\tau} - L\Phi - \lambda \left[ \Phi ,\Lambda \right] - \left[
\Omega ,\Phi \right] - \sqrt {3} \left\{ \Lambda ,\Phi \right\} +
\left\{ \Psi ,\Phi ^{2} \right\} = 0,\nonumber\\
\left(3 - \lambda ^{2}\right)\Lambda _{\xi} + 2\lambda \Lambda _{\eta} - \Omega
_{\eta} - \sqrt {3} \left\{ \Psi ,\Phi \right\}_{\xi} + \left[ \Psi
,\Phi \right]_{\eta} + \lambda \left[ \Phi ,\Psi \right]_{\xi} = 0,\nonumber\\
\Lambda _{\eta} = \Omega _{\xi},\label{maccari:eq1.3}\end{gathered}$$ where $\{ A,B\} = AB + BA$, the linear differential operator $L$ is given by $$\label{maccari:eq1.4}
L = - \left(3 + \lambda ^{2}\right){\frac{{\partial ^{2}}}{{\partial \xi ^{2}}}} +
2\lambda {\frac{{\partial ^{2}}}{{\partial \xi \partial \eta} }} -
{\frac{{\partial ^{2}}}{{\partial \eta ^{2}}}},$$ and $\Psi = \Psi (\xi ,\eta ,\tau )$, $\Phi = \Phi (\xi ,\eta ,\tau )$, $\Lambda = \Lambda (\xi ,\eta ,\tau )$ and $\Omega = \Omega (\xi ,\eta ,\tau
)$are $N \otimes N$ complex matrices.
The paper is organized as follows. In the next section we apply the reduction method to the starting equation (\[maccari:eq1.2\]) and obtain the new system of matrix equations (\[maccari:eq1.3\])–(\[maccari:eq1.4\]). Moreover, we reduce the matrix system of equations to a new integrable two-component complex fields system of nonlinear equations, which, in the one-component case, reduces to the standard Davey–Stewartson equation. In Section 3 we discuss in some detail how the reduction method can be applied to the Lax pair of the equation (\[maccari:eq1.2\]) and we derive the Lax pair of the system of matrix equations (\[maccari:eq1.3\])–(\[maccari:eq1.4\]). Finally in the last section we recapitulate the most important results and indicate some possible extensions.
A new integrable matrix system in dimensions
============================================
The linear dispersive part of the starting equation (\[maccari:eq1.2\]) admits as a solution a Fourier mode, with a group velocity $\underline {V} (\underline {K} ) = \left( V_{1} (K_{1} ,K_{2} ), V_{2}
(K_{1} ,K_{2} ) \right)$, $$\begin{gathered}
V_{1} (K_{1} ,K_{2} ) = - 3K_{1}^{2} + \frac{K_{2}^{2}}{K_{1}^{2}},\qquad V_{2} (K_{1} ,K_{2} ) = - 2{\frac{{K_{2}} }{{K_{1}} }},\label{maccari:eq2.2}\end{gathered}$$ where $$*\label{maccari:eq2.3}
\underline {V} (\underline {K} ) = {\frac{{\partial \omega} }{{\partial
\underline {K}} }}$$ and $\omega = \omega (K_{1} ,K_{2} ) = - K_{1}^{3} - {\frac{{K_{2}^{2}} }{{K_{1}
}}}$ is the dispersion relation.
It is sufficient to substitute the plane wave into the linear part of the matrix KP equation.
We use the transformation (\[maccari:eq1.1\]) and introduce the following formal asymptotic Fourier expansion $$\label{maccari:eq2.4}
U(x,y,t) = {\sum\limits_{n = - \infty} ^{ + \infty} {\varepsilon ^{\gamma
_{n}} \psi _{n} (\xi ,\eta ,\tau ;\varepsilon )\exp {\left\{ {i(nz)}
\right\}}}},$$ where $z = K_{1} x + K_{2} y - \omega t$, $\gamma _{n} = {\left| {n}
\right|}$ for $n \ne 0$, and $\gamma _{0} = r$ is a non negative rational number which will be fixed later. The unknown functions $\psi _{n} $ depend on $\varepsilon $ and it is supposed that their limit for $\varepsilon
\to 0$ exists and is finite; in the following this limit will be denoted with $\psi _{n} (\xi ,\eta ,\tau )$. Moreover we suppose that they can be expanded in power series of , i.e. $$\psi _{n} (\xi ,\eta ,\tau ;\varepsilon ) = {\sum\limits_{i = 0}^{\infty}
{\varepsilon ^{i}}} \psi _{n}^{(i)} (\xi ,\eta ,\tau ),
\qquad
\psi _{n} (\xi ,\eta ,\tau ) = \psi _{n}^{(0)} (\xi ,\eta ,\tau ).$$ We now introduce an analogous Fourier expansion $$\label{maccari:eq2.6}
W(x,y,t) = {\sum\limits_{n = - \infty} ^{ + \infty} {\varepsilon ^{\tilde
{\gamma} _{n}} \varphi _{n} (\xi ,\eta ,\tau ;\varepsilon )\exp {\left\{
{i(nz)} \right\}}}}$$ and obtain $$\varphi _{n} = (K_{2} )(K_{1} )^{ - 1}\psi _{n} + O\left(\varepsilon ^{p}\right).$$ In the following for simplicity we use the abbreviations $\psi _{1}^{(0)} =
\Psi $, $\psi _{ - 1}^{(0)} = \Phi $, $\psi _{0}^{(0)} = \Lambda $ (and $\phi _{n}^{(0)} = \phi _{n} $, $\phi _{0}^{(0)} = \Omega $).
The final goal is to obtain the evolution equation satisfied by the modulation amplitudes $\Psi = \Psi (\xi ,\eta ,\tau )$ and $\Phi (\xi ,\eta ,\tau )$ and to understand how it is modified by choosing different wave numbers.
The matrix system (\[maccari:eq1.3\])–(\[maccari:eq1.4\]) can be obtained applying the reduction technique to the matrix Kadomtsev-Petviashvili system (\[maccari:eq1.2\]).
We insert the expansions (\[maccari:eq2.4\]) and (\[maccari:eq2.6\]) into the equation (\[maccari:eq1.2\]) and consider the different equations obtained by considering the coefficients of the Fourier modes.
[It is convenient to separate the contributions of the linear and nonlinear parts by writing $$\varepsilon ^{\gamma _{n}} D_{n} \psi _{n} = \varepsilon ^{2}F_{n},$$ where $D_{n} $ is a linear differential operator acting on $\psi _{n} (\xi ,\eta ,\tau )$ and $F_{n} $ is the contribution of the nonlinear part. The operator $D_{n}$ is $$\begin{gathered}
D_{n} = \left( - in\omega + \varepsilon ^{q}\partial _{\tau} - V_{1}
\varepsilon ^{p}\partial _{\xi} - V_{2} \varepsilon ^{p}\partial _{\eta} \right)
+ \left(inK_{1} + \varepsilon ^{p}\partial _{\xi} \right)^{3}
- \left(inK_{2} + \varepsilon ^{p}\partial _{\eta} \right)\nonumber\\
\qquad {} - (i / K_{1} )\left(\varepsilon
^{p}\partial _{\eta} - (K_{2} / K_{1} )\partial _{\xi} \right)
+ \left(1 / K_{1}^{2} \right)\varepsilon ^{2p}
\left(\partial _{\xi \eta} - (K_{2} / K_{1} )\partial _{\xi \xi} \right). \end{gathered}$$ $F_{n} $ can be derived, by assessing the importance of the different terms, which originate from the nonlinear interaction of the Fourier amplitudes $\psi _{n} (\xi ,\eta ,\tau )$: $$\begin{gathered}
F_{2} = 6iK_{1} \Psi ^{2} + O\left(\varepsilon ^{p}\right),\\
F_{0} = \varepsilon ^{p}\left(3{\left\{ {\Psi ,\Phi} \right\}}_{\xi} - i\sqrt
{3} {\left[ {\varphi _{1}^{(p)} ,\Phi} \right]} - i\sqrt {3} {\left[
{\varphi _{ - 1}^{(p)} ,\Psi} \right]}\right) + O\left(\varepsilon ^{2p},\varepsilon
^{2}\right),\\
F_{1} = \varepsilon ^{r - 1}\left( - i\sqrt {3} {\left[ {\Omega ,\Psi} \right]}
- i\sqrt {3} {\frac{{K_{2}} }{{K_{1}} }}{\left[ {\Psi ,\Lambda} \right]} +
3iK_{1} {\left\{ {\Lambda ,\Psi} \right\}}\right) \nonumber\\
\qquad {}+ 3i\varepsilon K_{1} {\left\{ {\psi _{2} ,\Phi} \right\}}
+ O\left(\varepsilon ^{r + p - 1},\varepsilon ^{3}\right),\end{gathered}$$ and so on.]{}
By setting $q = 2$, $p = 1$, $r = 2$ for the proper balance of terms, we obtain the equations for the Fourier modes at the lowest order for $n = 0$, $n = 1$ and $n = 2$: $$\begin{gathered}
\psi _{2} = - {\frac{{1}}{{K_{1}^{2}} }}\Psi ^{2},\\
\left(3K_{1}^{2} - {\frac{{K_{2}^{2}} }{{K_{1}^{2}} }}\right)\Lambda _{\xi} +
2{\frac{{K_{2}} }{{K_{1}} }}\Lambda _{\eta} - \Omega _{\eta} - 3{\left\{
{\Psi ,\Phi} \right\}}_{\xi} \nonumber\\
\qquad {}+ i\sqrt {3} {\left[ {\varphi _{1}^{(p)}
,\Phi} \right]} + i\sqrt {3} {\left[ {\varphi _{ - 1}^{(p)} ,\Psi} \right]} = 0,\\
\Psi _{\tau} + i\left(3K_{1} + {\frac{{K_{2}^{2}} }{{K_{1}^{3}} }}\right)\Psi _{\xi
\xi} + {\frac{{i}}{{K_{1}} }}\Psi _{\eta \eta} - 2i{\frac{{K_{2}
}}{{K_{1}^{2}} }}\Psi _{\xi \eta} \nonumber\\
\qquad {}+ i\sqrt {3} {\left[ {\Omega ,\Psi} \right]} + i\sqrt {3} {\frac{{K_{2}
}}{{K_{1}} }}{\left[ {\Psi ,\Lambda} \right]} - 3iK_{1} {\left\{ {\Lambda
,\Psi} \right\}} - 3iK_{1} {\left\{ {\psi _{2} ,\Phi} \right\}} = 0.
\end{gathered}$$
Finally, after the cosmetic rescaling $$\begin{gathered}
\sqrt {3} K_{1} \Lambda \to \Lambda, \qquad
\sqrt {3} \Omega \to \Omega, \qquad
\sqrt {{\frac{{3}}{{K_{1}} }}} \Psi \to \Psi ,\nonumber\\
\sqrt {{\frac{{3}}{{K_{1}} }}} \Phi \to \Phi ,
\qquad
\lambda = {\frac{{K_{2}} }{{K_{1}^{2}} }},
\qquad
{\xi} ' = \xi / \sqrt {K_{1}} , \qquad
{\eta} ' = K_{1} \eta,\label{maccari:eq2.12}\end{gathered}$$ we arrive at the matrix system of nonlinear evolution equations (\[maccari:eq1.3\])–(\[maccari:eq1.4\]).
This matrix system must be integrable by the spectral transform, because it has been derived from an S-integrable equation. This is explicitly demonstrated in the next section.
Let us now look in more detail at the integrable NPDEs implied these results. If we take $\Phi = \Psi ^{ *} $, $N=1$, we obtain the equation $$\label{maccari:eq2.13}
i\Psi _{\tau} + L_{1} \Psi + \chi \Psi = 0,
\qquad
L_{2} \chi = 2L_{1} {\left| {\Psi} \right|}^{2},$$ with $$\begin{gathered}
\chi _{\eta} = 2{\left| {\Psi} \right|}_{\eta} ^{2} + 2\sqrt {3} \Omega _{\xi},\\
L_{1} = - \left(3 + \lambda ^{2}\right){\frac{{\partial ^{2}}}{{\partial \xi ^{2}}}} +
2\lambda {\frac{{\partial ^{2}}}{{\partial \xi \partial \eta} }} -
{\frac{{\partial ^{2}}}{{\partial \eta ^{2}}}},\\
L_{2} = \left(\lambda ^{2} - 3\right){\frac{{\partial ^{2}}}{{\partial \xi ^{2}}}} -
2\lambda {\frac{{\partial ^{2}}}{{\partial \xi \partial \eta} }} -
{\frac{{\partial ^{2}}}{{\partial \eta ^{2}}}}.\end{gathered}$$ The NPDE (\[maccari:eq2.13\]), up to trivial rescalings, coincides with the Davey–Stewartson equation \[6\], whose integrability is well known \[7\]. Note that the S-integrable equations found in \[10–11\] are different from the standard Davey–Stewartson equation and then not connected with the S-integrable system (\[maccari:eq1.3\])–(\[maccari:eq1.4\]).
In the case $N=2$, we get a nonlinear system for eight interacting fields. However, an interesting reduction is possible, if we set $$\Psi = \left(\!\begin{array}{cc}
{\psi _{1}} & {\psi _{2}} \\
{\psi _{2}} & {\psi _{1}}
\end{array} \!\right)\!,
\quad
\Phi = \left(\!\begin{array}{cc}
{\varphi _{1}} & {\varphi _{2}} \\
{\varphi _{2}} & {\varphi _{1}}
\end{array} \!\right)\!,
\quad
\Lambda = \left(\! \begin{array}{cc}
{\Lambda _{1}} & {\Lambda _{2}}\\
{\Lambda _{2}} & {\Lambda _{1}}
\end{array} \!\right)\!,
\quad
\Omega = \left(\!\begin{array}{cc}
{\Omega _{1}} & {\Omega _{2}} \\
{\Omega _{2}} & {\Omega _{1}}
\end{array} \!\right)\!.$$
From the matrix system (\[maccari:eq1.3\])–(\[maccari:eq1.4\]), we obtain $\Psi = \Phi ^{ *} $ and $$\begin{gathered}
i\psi _{1,\tau} + L_{1} \psi _{1} + 2\sqrt {3} (\Lambda _{1} \psi _{1} +
\Lambda _{2} \psi _{2} ) - 2\left({\left| {\psi _{1}} \right|}^{2}\psi _{1} +
\psi _{1}^{ *} \psi _{2}^{2} + 2\psi _{1} {\left| {\psi _{2}}
\right|}^{2}\right) = 0,\nonumber\\
i\psi _{2,\tau} + L_{1} \psi _{2} + 2\sqrt {3} (\Lambda _{1} \psi _{2} +
\Lambda _{2} \psi _{1} ) - 2\left({\left| {\psi _{2}} \right|}^{2}\psi _{2} +
\psi _{1}^{2} \psi _{2}^{ *} + 2\psi _{2} {\left| {\psi _{1}}
\right|}^{2}\right) = 0,\nonumber\\
L_{2} \Lambda _{1} = L_{3} \left({\left| {\psi _{1}} \right|}^{2} + {\left|
{\psi _{2}} \right|}^{2}\right),
\qquad
L_{2} \Lambda _{2} = L_{3} \left(\psi _{1} \psi _{2}^{ *} + \psi _{2} \psi _{1}^{ *} \right),\label{maccari:eq2.15}\end{gathered}$$ where $$\begin{gathered}
L_{1} = - \left(3 + \lambda ^{2}\right)\partial _{\xi} ^{2} + 2\lambda \partial _{\xi
\eta} ^{2} - \partial _{\eta} ^{2} ,\\
L_{2} = \left(3 - \lambda ^{2}\right)\partial _{\xi} ^{2} + 2\lambda \partial _{\xi
\eta} ^{2} - \partial _{\eta} ^{2},
\qquad
L_{3} = 2\sqrt {3} \partial _{\xi} ^{2}.\end{gathered}$$
Integrable Davey–Stewartson type equations and system of equations have been extensively investigated by many authors \[16–20\]. A detailed list of Davey-Stewartson systems and equations integrable by the inverse scattering method has been recently given \[21\]. The system of equations (\[maccari:eq2.15\]) does not appear in these papers. We expect that this new system be integrable by the inverse scattering method, because it has been obtained from an integrable equation and the property of integrability is expected to be maintained through the application of the reduction method. The integrability of the system of equations (\[maccari:eq1.3\])–(\[maccari:eq1.4\]), and of the system (\[maccari:eq2.15\]) which is a particular case, is demonstrated in the next section.
The Lax pair for the integrable system of equations
===================================================
In this section we apply the reduction method also to the Lax pair of the starting matrix equation (\[maccari:eq1.2\]), to demonstrate explicitly the integrability by the spectral transform of the matrix system (\[maccari:eq1.3\])–(\[maccari:eq1.4\]), and we thereby identify the Lax pair for the system of equations (\[maccari:eq1.3\])–(\[maccari:eq1.4\]).
The Lax operators $(L, A)$ of the matrix KP equation are $$\begin{gathered}
\label{maccari:eq3.1}
L = {\frac{{i}}{{\sqrt {3}} }}{\frac{{\partial} }{{\partial y}}} +
{\frac{{\partial ^{2}}}{{\partial x^{2}}}} - U(x,y,t),
\qquad
L\phi (x,y,t) = 0,\\
\label{maccari:eq3.2}
A = 4{\frac{{\partial ^{3}}}{{\partial x^{3}}}} - 6U(x,y,t){\frac{{\partial
}}{{\partial x}}} - 3U_{x} (x,y,t) + i\sqrt {3} W,\end{gathered}$$ with $$\label{maccari:eq3.3}
\phi _{t} (x,y,t) + A\phi (x,y,t) = 0.$$
It can be verified by direct substitution that the operator relation $$L_{t} = i[L,A] = i(LA - AL)$$ reproduces equation (\[maccari:eq1.2\]).
The matrix system (\[maccari:eq1.3\])–(\[maccari:eq1.4\]) is S-integrable and its Lax pair $(L, A)$ is $$\label{maccari:eq3.5}
L\hat {\phi} = 0,$$ where $$\begin{gathered}
L = \left(\begin{array}{cc}
{L_{11}} & {L_{12}} \\
{L_{21}} & {L_{22}}
\end{array}\right),\qquad
L_{11} = I\left(i\partial _{\eta} + i(\sqrt {3} - \lambda )\partial _{\xi} \right),
\qquad
L_{12} = - \Psi ,\nonumber\\
L_{21} = - \Phi ,
\qquad
L_{22} = I\left(i\partial _{\eta} - i(\sqrt {3} + \lambda )\partial _{\xi} \right),\qquad
\hat {\phi} = \left(\begin{array}{c}
{\phi _{ +} } \\
{\phi _{ -} }
\end{array} \right),\label{maccari:eq3.6}\end{gathered}$$ $I$ is the $N \otimes N$ unit matrix and $$\begin{gathered}
\label{maccari:eq3.7}
A = \left(\begin{array}{cc}
{A_{11}} & {A_{12}} \\
{A_{21}} & {A_{22}}
\end{array} \right),\end{gathered}$$ where $$\begin{gathered}
A_{11} = 6i\partial _{\xi} ^{2} I + i\Omega - i\left(\sqrt {3} + \lambda \right)\Lambda
+ i\left(1 + {\frac{{\lambda \sqrt {3}} }{{12}}}\right)\Phi \Psi ,\\
A_{12} = - 2\sqrt {3} \Psi \partial _{\xi} - \left(\sqrt {3} + \lambda \right)\Psi
_{\xi} + \Psi _{\eta} ,\\
A_{12} = - 2\sqrt {3} \Phi \partial _{\xi} - \left(\sqrt {3} - \lambda\right )\Phi
_{\xi} - \Phi _{\eta} ,\\
A_{22} = - 6iI\partial _{\xi} ^{2} + i\Omega + i\left(\sqrt {3} - \lambda
\right)\Lambda + i\left({\frac{{\lambda \sqrt {3}} }{{12}}} - 1\right)\Psi \Phi .\end{gathered}$$
Let us apply the reduction method to the Lax pair (\[maccari:eq3.1\])–(\[maccari:eq3.3\]) of equation (\[maccari:eq1.2\]).
The components $\phi _{j} (x,y,t)$, $j=1,\ldots, N$, of the column vector $\phi (x,y,t)$ can be expanded in Fourier modes as follows $$\label{maccari:eq3.8}
\phi _{j} (x,y,t) = {\sum\limits_{n = - \infty} ^{ + \infty} {\varepsilon
^{\gamma _{n}} \phi _{j,n} (\xi ,\eta ,\tau ;\varepsilon )\exp {\left[
{i\left( {(\lambda _{1} x + \lambda _{2} y + \lambda _{3} t) +
{\frac{{n}}{{2}}}z} \right)} \right]}}},$$ where $z = K_{1} x + K_{2} y - \omega t$, the $\phi _{j,n} (\xi ,\eta ,\tau
;\varepsilon )$ depend parametrically on $\varepsilon $ and remain finite when $\varepsilon \to 0$, the $\gamma _{n} $ are non negative rational numbers and $\lambda _{m}$, $m = 1,\ldots,3$, are real constants to be determined.
Inserting now the expression for $\phi _{j} (x,y,t)$ in (\[maccari:eq3.1\]), we derive a series of relations which are generated by the coefficients of the Fourier modes. Each relation must be valid for a given order of approximation in $\varepsilon $.
In particular, for the fundamental harmonics $n = \pm 1,$considering terms $O\left(\varepsilon ^{0}\right)$ in (\[maccari:eq3.1\]) and (\[maccari:eq3.3\]), we obtain $$\begin{gathered}
{\frac{{i}}{{\sqrt {3}} }}\left(\pm {\frac{{iK_{2}} }{{2}}} + i\lambda _{2} \right) +
\left(\pm {\frac{{iK_{1}} }{{2}}} + i\lambda _{1}\right)^{3} = 0,\\
\left( \mp {\frac{{i\omega} }{{2}}} + i\lambda _{3}\right) + 4\left(\pm {\frac{{iK_{1}
}}{{2}}} + i\lambda _{1} \right)^{3} = 0,\end{gathered}$$ and then $$\lambda _{1} = - {\frac{{K_{2}} }{{2K_{1} \sqrt {3}} }},
\qquad
\lambda _{2} = - {\frac{{\sqrt {3}} }{{4}}}\left({\frac{{K_{2}^{2}} }{{3K_{1}^{2}
}}} + K_{1}^{2} \right),
\qquad
\lambda _{3} = - {\frac{{K_{2}^{3}} }{{6K_{1}^{3} \sqrt {3}} }} -
{\frac{{\sqrt {3}} }{{2}}}K_{1} K_{2}.$$
We thereby understand that the harmonics $$\label{maccari:eq3.11}
\phi _{j,1} ,\qquad \phi _{j, - 1}, \qquad j=1,\ldots,N,$$ are fundamental, i.e. for them $\gamma _{n} $ assumes the smallest value, $\gamma _{n} = 0$.
The successive order $\varepsilon $ for the equation (\[maccari:eq3.1\]) allow us to obtain the new spectral problem, because all the $\phi _{j,n} $ may be expressed by means of the fundamental harmonics (\[maccari:eq3.11\]), which are connected through the relations: $$\begin{gathered}
\label{maccari:eq3.12}
i\phi _{ + ,\eta} + i\left(\sqrt {3} - \lambda \right)\phi _{ + ,\xi} - \Psi \phi _{ -} = 0,\\
i\phi _{ - ,\eta} - i\left(\sqrt {3} + \lambda\right)\phi _{ - ,\xi} - \Phi \phi _{ +} = 0,\label{maccari:eq3.13}\end{gathered}$$ where we set $(\phi _{j,1};\ j = 1,\ldots,N) = \phi _{ +} $, $(\phi _{j, - 1} ;\ j = 1,\ldots,N) = \phi _{ -}$.
By means of the variable rescaling (\[maccari:eq2.12\]), and by introducing the $2N \otimes
2N$ matrix operator $L$, we arrive at the final form (\[maccari:eq3.5\])–(\[maccari:eq3.6\]).
To calculate the temporal evolution, we must insert the expression (\[maccari:eq3.8\]) in (\[maccari:eq3.3\]) and consider the relation obtained for the different harmonics $n$ and for a given order of approximation in $\varepsilon $. If we consider the first order in $\varepsilon $, we obtain again the spectral problem (\[maccari:eq3.5\])–(\[maccari:eq3.6\]). Only if we take into account the next orders of approximation of equation (\[maccari:eq3.3\]), i.e. the order $\varepsilon ^{2}$, the temporal evolution can be determined. However, new quantities, the corrections $\tilde {\phi} _{\pm} (\xi ,\eta ,\tau )$ of order $\varepsilon $ to the fundamental harmonics $\phi _{\pm} (\xi ,\eta ,\tau )$, appear. These unknown quantities can be eliminated in the equation (\[maccari:eq3.3\]) by taking advantage of the relation obtained from equation (\[maccari:eq3.1\]), considering terms of order $\varepsilon ^{2}$. This elimination is possible only because equations (\[maccari:eq3.1\]) and (\[maccari:eq3.3\]) are identical at the order $\varepsilon $. In particular, if we consider (\[maccari:eq3.3\]) calculated to the order $\varepsilon ^{2}$ for $n = \pm 1$, we get $$\begin{gathered}
\label{maccari:eq3.14}
\phi _{ + ,\tau} + 12i\left({\frac{{K_{1}} }{{2}}} + \lambda _{1} \right)\phi _{ +
,\xi \xi} - 6\Psi \phi _{ - ,\xi} - 3\Psi _{\xi} \phi _{ -}
+ {\frac{{\sqrt {3}} }{{K_{1}} }}\left(\Psi _{\eta} - {\frac{{K_{2}} }{{K_{1}
}}}\Psi _{\xi} \right)\phi _{ -} \nonumber\\
\qquad {} + i\sqrt {3} \Omega- 6i\Lambda \left({\frac{{K_{1} }}{{2}}} + \lambda _{1} \right)\phi _{ +}
+ i\left( - 6\lambda _{1} - 6K_{1} + \sqrt {3} {\frac{{K_{2}} }{{K_{1}} }}\right)\Phi
\phi _{ + 3} \nonumber\\
\qquad {} - {\frac{{2iK_{2} \sqrt {3}} }{{K_{1}} }}\left({\frac{{i}}{{\sqrt {3}} }}\tilde
{\phi} _{ + ,\eta} + i\left(K_{1} - {\frac{{K_{2}} }{{K_{1} \sqrt {3}} }}\right)\tilde
{\phi} _{ + ,\xi} - \Psi \tilde {\phi} _{ -} \right) = 0, \\
\phi _{ - ,\tau} + 12i\left( - {\frac{{K_{1}} }{{2}}} + \lambda _{1} \right)\phi _{ -
,\xi \xi} - 6\Phi \phi _{ + ,\xi} - 3\Phi _{\xi} \phi _{ +}
- {\frac{{\sqrt {3}} }{{K_{1}} }}\left(\Phi _{\eta} - {\frac{{K_{2}} }{{K_{1}
}}}\Phi _{\xi} \right)\phi _{ +} \nonumber\\
\qquad {} + i\sqrt {3} \Omega- 6i\Lambda \left( -
{\frac{{K_{1}} }{{2}}} + \lambda _{1} \right)\phi _{ -}
+ i\left( - 6\lambda _{1} + 6K_{1} + \sqrt {3} {\frac{{K_{2}} }{{K_{1}} }}\right)\Psi
\phi _{ - 3} \nonumber\\
\qquad {}- {\frac{{2iK_{2} \sqrt {3}} }{{K_{1}} }}\left({\frac{{i}}{{\sqrt {3}} }}\tilde
{\phi} _{ - ,\eta} - i\left(K_{1} + {\frac{{K_{2}} }{{K_{1} \sqrt {3}} }}\right)\tilde
{\phi} _{ - ,\xi} - \Phi \tilde {\phi} _{ +} \right) = 0.
\end{gathered}$$
To evaluate this expression we took advantage of the fact that $\phi _{\pm
3} $ are connected with the fundamental harmonics (these relations are obtained from (\[maccari:eq3.1\]) for $n = \pm 3$ at the lower order in $\varepsilon $): $$\label{maccari:eq3.15}
\phi _{ + 3} = \left( {{\frac{{ - 1}}{{2K_{1}^{2}} }}} \right)\Psi \phi _{ + } ,
\qquad
\phi _{ - 3} = \left( {{\frac{{ - 1}}{{2K_{1}^{2}} }}} \right)\Phi \phi _{ - } .$$
We now consider the equation (\[maccari:eq3.1\]) at the order $\varepsilon ^{2}$ for $n =
\pm 1$, which provides the corrections $\tilde {\phi} _{ +} (\xi ,\eta
,\tau )$, $\tilde {\phi} _{ -} (\xi ,\eta ,\tau )$. Via the transformation (\[maccari:eq2.12\]) and after a lengthy calculation we arrive at the final form (\[maccari:eq3.7\]) for the $2N \otimes 2N$ matrix operator $A$, which satisfies the equation $$\label{maccari:eq3.16}
\hat {\phi} _{\tau} + A\hat {\phi} = 0.$$
The determination of the Lax pair (\[maccari:eq3.6\]) and (\[maccari:eq3.7\]), which satisfies the equations (\[maccari:eq3.5\]) and (\[maccari:eq3.16\]), demonstrates the S-integrability of the system (\[maccari:eq1.3\])–(\[maccari:eq1.4\]).
Conclusion
==========
We have derived a new, integrable, and presumably of applicative interest, nonlinear matrix system of evolution equations of Davey–Stewartson type from the integrable matrix equation (\[maccari:eq1.2\]), by means of an extension of the reduction method based on Fourier expansion and space-time rescalings. It reduces to the standard Davey–Stewartson equation in the single mode case and to a new integrable system of two interacting fields in the $N=2$ case. Moreover, we have applied the reduction method to the Lax pair (\[maccari:eq3.1\])–(\[maccari:eq3.3\]) of the original equation and have demonstrated the integrability property of the new matrix system of equations, by exhibiting the corresponding Lax pair (\[maccari:eq3.11\])–(\[maccari:eq3.12\]) and (\[maccari:eq3.15\])–(\[maccari:eq3.16\]).
We have outlined the approach that permits to obtain such system of equations and the next steps will be the explicit resolution of the spectral problem and the possible identification of localized or asymptotically finite solutions.
It is also convenient to push the approach beyond its “leading order” application by considering different rescalings in the two spatial variables or looking at special cases when some key parameters vanish, in analogy to the case of the model equations treated in \[8–9\].
[99]{}
Ablowitz M J and Clarkson P A, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, 1991.
Calogero F and Eckhaus W, Nonlinear Evolution Equations, Rescalings, Model PDEs and Their Integrability. I, *Inverse Problems* **3** (1987), 229–262.
Calogero F and Eckhaus W, Nonlinear Evolution Equations, Rescalings, Model PDEs and Their Integrability. II, *Inverse Problems* **4** (1988), 11–33.
Calogero F and Maccari A, Equations of Nonlinear Schrödinger Type in $1+1$ and $2+1$ Dimensions Obtained from Integrable PDEs’, in Inverse Problems: an Interdisciplinary Study, Proceedings of the Meeting on Inverse Problems, Montpellier, 1986, Editor: Sabatier P C, *Advances in Electronics and Electron Physics*, Academic Press - New York, **19** (1988), 463–480.
Calogero F, “Why are Certain Nonlinear PDEs Both Widely Applicable and Integrable, in What is Integrability (for Nonlinear PDEs), Editor: Zakharov V E, Springer-Verlag, 1991.
Davey A and Stewartson K, On Three-Dimensional Packets of Surface Waves”, *Proc. R. Soc. London* **A338** (1974), 101–110.
Anker D and Freeman N C, On the Soliton Solutions of the Davey–Stewartson Equation for Long Waves”, *Proc. R. Soc. London* **A360** (1978), 529–540.
Maccari A, The Kadomtsev–Petviashvili Equation as a Source of Integrable Model Equations, *J. Math. Phys.* **37** (1996), 6207–6212.
Maccari A, A Generalized Hirota Equation in $2+1$ Dimensions, *J. Math. Phys.* **39** (1998), 6547–6551.
Maccari A, A New Integrable Davey–Stewartson Type Equation, *J. Math. Phys.* **40** (1999), 3971–3977.
Maccari A, On the Integrability of a New Davey–Stewartson Type Equation in $2+1$ Dimensions”, *Physics Letters* **A265** (2000), 187–193.
Maccari A, A Model Equation for Non-Resonant Interacting Ion Acoustic Plasma Waves, *Journal of Plasma Physics* **60** (1998), 275–288.
Maccari A, Non-Resonant Interacting Ion Acoustic Waves in a Magnetized Plasma, *J. Phys.* **A32** (1999), 693–709.
Kadomtsev B B and Petviashvili V I, On the Stability of Solitary Waves in Weakly Dispersive Media, [*Sov. Phys. Dokl.*]{} **15** (1970), 539–541.
Zakharov V E and Kusnetsov E A, Multiscale Expansions in the Theory of Systems Integrable by the Inverse Scattering Transform, *Physica* **D18** (1986), 455–463.
Shulman E I, On the Integrability of Equations of Davey–Stewartson Type, *Theor. Math. Phys.* **56** (1983), 720–724 (*Teor. Mat. Fiz.* **56** (1983), 131–136).
Fokas A S and Santini P M, Recursion Operators and Bi-Hamiltonian Structures in Multidimensions. I, *Comm. Math. Phys.* **115** (1988), 375–419.
Pempinelli F, Boiti M and Sabatier P C, First and Second Order Nonlinear Evolution Equations from an Inverse Spectral Problem, *Inverse Problems* **9** (1993), 1–37.
Fokas A S, On the Simplest Integrable Equation in $2+1$, *Inverse Problems* **10** (1994), L19–L22.
Radha R and Lakshmanan M, Localized Coherent Structures and Integrability in a Generalized $(2+1)$-Dimensional Nonlinear Schrödinger Equation, *Chaos, Solitons & Fractals* **8** (1997), 17–25.
Folguera A, Davey–Stewartson Type Equations of the Second and Third Order: Derivation and Classification, *J. Math. Phys.* **39** (1998), 2947–2968.
\[maccari-lastpage\]
|
---
abstract: 'Multilayer and multiplex networks are becoming common network data sets in recent times. We consider the problem of identifying the common community structure for a special type of multilayer networks called multi-relational networks. We consider extensions of the spectral clustering methods for multi-relational networks and give theoretical guarantees that the spectral clustering methods recover community structure consistently for multi-relational networks generated from multilayer versions of both stochastic and degree-corrected block models even with dependence between network layers. The methods are shown to work under optimal conditions on the degree parameter of the networks to detect both assortative and disassortative community structures with vanishing error proportions even if individual layers of the multi-relational network has the network structures below community detectability threshold. We reinforce the validity of the theoretical results via simulations too.'
address:
- |
Department of Statistics\
239 Weniger Hall\
Corvallis, OR, 97331\
- |
Department of Mathematics\
North Academic Center 8/133\
160 Convent Ave\
New York, NY, 10031\
author:
-
-
bibliography:
- 'Biom1.bib'
title: 'General Community Detection with Optimal Recovery Conditions for Multi-relational Sparse Networks with Dependent Layers'
---
Introduction {#sec_intro}
============
Statistical analysis of network data has now become a well-studied field within statistics (see [@goldenberg2010survey; @kolaczyk2014statistical] for reviews). Methods for network data analysis are being developed not only in the discipline of statistics but also in computer science, physics, and mathematics. Network datasets show up in several disciplines. Examples include networks originating from biosciences such as gene regulation networks [@emmert2014gene], protein-protein interaction networks [@de2010protein], structural [@rubinov2010complex] and functional networks [@friston2011functional] of brain and epidemiological networks [@reis2007epidemiological]; networks originating from social media such as Facebook, Twitter and LinkedIn [@faloutsos2010online]; citation and collaboration networks [@lehmann2003citation]; information and technological networks such as internet-based networks [@adamic2005political], power networks [@pagani2013power] and cell-tower networks [@isaacman2011identifying]. There are several active areas of research in developing statistical inference methods for network data analysis and also deriving the theoretical properties of the statistical methods. Examples of inferential questions that have received a lot of attention in current research include fitting of random graph models to the network data sets [@goldenberg2010survey], finding stochastic properties of summary statistics of networks like subgraph counts [@bickel2011method], community detection [@fortunato2010community] and link prediction [@liben2007link].
In this paper, we focus on the problem of recovering a common community structure present in a finite sequence of (possibly sparse) networks. The community detection problem can be thought of as a *vertex clustering problem*, in which the goal is to divide the set of vertices of a given network (or a finite sequence of networks) into groups based on some common properties of the vertices. The main goal in community detection is to partition the vertices of a graph (or a finite sequence of graphs) into groups such that the average numbers of connections within the groups are *significantly different* than that between groups. Communities in networks are usually called *assortative* (see \[sec\_model\] for more details) if the average number of connections within communities is *significantly higher* than that between communities. Communities in networks are usually called *disassortative*, if the average number of connections within communities is *significantly lesser* than the average number of connections between communities. A network may consist of both assortative and disassortative communities (see [@newman2003mixing; @newman2004finding]). Since many works on community detection only deal with assortative community detection, to avoid ambiguity we have referred our goal as *general* community detection. *In this paper, we do not restrict ourselves to any specific type of community structure*. Several random graph models have been proposed in the literature with a mathematically rigorous definition of community labels for vertices. Examples of random graph models for a single network with community structure include stochastic block models [@holland1983stochastic], degree-corrected block models [@karrer2011stochastic] and random dot product models [@young2007random]. Many methods have been proposed in the statistics and machine learning literature to recover community labels (see [@fortunato2010community] for a review) for a given single network. The methods can be broadly classified into two types, namely (i) *model-based approaches* (e.g., different likelihood-based methods [@bickel2009nonparametric]), where the methods are developed assuming a specific generative model for the given network, and (ii) model agnostic approaches (e.g. modularity based methods [@newman2004finding], spectral clustering methods [@MR2893856], label propagation [@gregory2010finding]), where the methods are developed without a specific generative model in mind.
Most of the research on network data in statistics literature has focused on questions based on a single observed network as data. However, multiple network datasets (a finite sequence of networks) are currently becoming common in many applications. Examples of applications include, neuroscience [@bassett2017network; @thompson2017static], economics [@bargigli2015multiplex], sociology and social networks [@heaney2014multiplex; @lewis2012social], ecology [@pilosof2017multilayer], epidemiology [@zhao2014immunization], and technological networks [@sen2014identification; @zignani2014exploiting]. Depending on the structure and interconnectivity among a finite sequence of networks, various kinds of multiple networks have been considered in the literature, e.g., multilayer networks, multiplex networks, multi-relational networks, multidimensional networks, time-evolving networks, dynamic networks, and hypergraphs [@boccaletti2014structure; @kivela2014multilayer]. A *multi-relational network* consists of a finite set of networks (each such network is called a *network layer*) having the same vertex set but possibly different edge sets in different layers. Temporal networks having the same vertex set and time-evolving edge sets can also be considered as *multi-relational networks*. *We consider the problem of community detection based on *multi-relational* network datasets, which is a generalization of its analog for a single-layer network*.
Community detection using the spectral decomposition of matrices associated with graphs is a common statistical method. Spectral clustering has several advantages - firstly, the method is model agnostic. Secondly, spectral clustering is highly scalable, as scalable implementations of matrix factorization algorithms is an active research topic in the numerical analysis literature [@blackford1997scalapack]. Thirdly, spectral clustering methods have also been shown to work in recovering community labels for single-layer networks under various probabilistic models and analyzed in many subsequent papers (see [@shi2000normalized], [@ng2002spectral], [@MR2396807], [@MR2893856], [@sussman2012consistent], [@lei2015consistency], [@bhattacharyya2014community], [@gao2017achieving]). Also for a single-layer network, many of the proposed community detection methods [@chin2015stochastic; @joseph2016impact; @abbe2017community; @gao2017achieving; @le2017concentration; @gao2018community] in the literature has been shown to recover community labels for sparse networks, but still the scalability of the methods have rarely been addressed.
Most of the statistical and probabilistic models for multiple networks that appear in the literature are extensions of random graph models for a single network into the multiple networks setup. Examples of such models include extension of latent space models [@sarkar2005dynamic], [@sewell2014latent], mixed membership block models [@ho2011evolving], random dot-product models [@tang2013attribute], stochastic block models [@xu2014dynamic], [@xu2015stochastic], [@matias2017statistical], [@ghasemian2016detectability], [@corneli2016exact], [@zhang2017random], [@pensky2019dynamic], and Erdós-Rényi graph models [@crane2015time]. Also, some Bayesian models and associated inference procedures have been proposed in the context of multiple networks [@yang2011detecting], [@durante2014nonparametric]. *In this paper, for theoretical analysis we have considered a multilayer version of stochastic and degree-corrected block models which has been used in some of the previous works [@han2015consistent].*
Several recent works have focused on developing statistical inference procedures based on different versions of multilayer networks [@xu2014dynamic; @han2015consistent; @matias2017statistical; @zhang2017random; @paul2016consistent]. Some model-agnostic methods have also been proposed to detect communities in multilayer networks [@tang2009clustering; @kumar2010co; @dong2012clustering; @chen2017multilayer; @paul2017spectral; @pensky2019spectral]. However, only a few of the recently proposed algorithms [@han2015consistent; @paul2016consistent; @taylor2016enhanced; @paul2017spectral; @chen2017multilayer; @pensky2019spectral] attempts to evaluate the performance of the proposed community recovery procedures theoretically when the multilayer network is sampled from some random network generating model. None of the proposed methods have been proven to work for multilayer networks in which an aggregation of individual networks is sparse, namely when the total degree of a typical vertex in the aggregated network goes to infinity arbitrarily slowly. Also, some recent works like [@mercado2018power], considers power of Laplacian matrices for community detection in multilayer networks but they only consider networks generated from special cases of multilayer stochastic block model. *So, to the best of our knowledge, no known polynomial-time community detection algorithm with proven theoretical guarantee to consistently recover community labels within a general class of sparse multilayer networks has been proposed.* *Also, to the best of our knowledge, none of the recently proposed community detection algorithms in the literature have been shown to recover community labels under general dependence structures between the network layers.*
Contributions of our work {#sec_contribution}
-------------------------
We address some of the limitations of current works in this paper, so, we propose and analyze two spectral clustering algorithms for finding the common community structure within a given finite sequence of networks with possible dependence structures. The proposed algorithms are naturally scalable and model agnostic, and they work for a single network as well as for multilayer networks, irrespective of edge density of individual networks as well as their aggregated versions. To evaluate the performance of the proposed community recovery algorithms theoretically and see when they perform consistently, we consider a particular case of multilayer networks, *multi-relational networks* [@cai2005community; @kivela2014multilayer] generated from a multilayer generalization of stochastic and degree-corrected block models [@han2015consistent].
The main contributions of our work are the following.
- We propose two novel methods based on spectral clustering of sum of squared adjacency matrices for recovering community labels in multi-relational networks with a common community structure. The methods can be used for community detection in a single-layer network too.
- We also prove analytically that, under the mildest (necessary) parametric conditions, the proposed spectral clustering methods identify communities in the networks generated from single-layer or multilayer stochastic block models and degree-corrected block models consistently. We show analytically that in the multi-relational networks generated from multilayer versions of stochastic and degree-corrected block models, our spectral clustering methods can recover the common community structure consistently even if each of the individual network layers has fixed size and is highly sparse (e.g., has a constant average degree) and has connectivity below the community detectability threshold as long as the aggregated version of the network satisfies certain conditions.
- It has been theoretically shown that the proposed community detection methods are flexible enough to work for both sparse and dense networks. It has been theoretically shown that the methods are flexible enough to identify both assortative and disassortative community structures even when the community structures vary between layers.
- It has been theoretically shown that the proposed community detection methods recover community labels even in the presence of dependence between network layers.
- We also propose a method for detecting the number of communities in the multi-relational networks. The proposed method has been shown to recover the correct number of communities asymptotically.
Structure of the paper {#sec_organization}
----------------------
The remainder of the paper is organized as follows. In \[sec\_model\], we introduce the multiple network models. In \[sec\_method\], we describe the spectral clustering methods. In \[sec\_theory\], we state the theoretical results regarding the performance of the proposed spectral clustering methods. In \[sec\_simulation\], we demonstrate the effectiveness of the methods for simulated datasets.
Multi-relational Network Data and Model {#sec_model}
=======================================
Multi-relational networks data {#sec_multiple}
------------------------------
In this paper, we suppose that the observed data consists of a single network or a *multi-relational network*. The formal definition of a *multi-relational network* is given below.
A *multi-relational network* consists of a finite sequence of unlabeled graphs $\{G_n^{(t)}; t=1, \ldots, T\}$ on the same vertex set $V_n=\{v_1, v_2, \ldots, v_n\}$ having $n$ vertices but the edge sets of the graphs may be different. $G_n^{(t)}$ is referred as the the $t$-th *network layer*.
A *multi-relational network* can also be considered as an *edge-colored multi-graph*, where different colors correspond to edge sets of different network layers. The $t$-th layer $G_n^{(t)}$ is represented by the corresponding adjacency matrix ${\mathbf{A}}^{(t)}_{n\times n}$ whose elements are ${\mathbf{A}}^{(t)}_{ij}\in\{0,1\}$. ${\mathbf{A}}^{(t)}_{ij} = 1$ if node $v_i$ is linked to node $v_j$ at time $t$, and ${\mathbf{A}}^{(t)}_{ij} = 0$ otherwise. Thus, the numerical data for the community detection problem consists of $T{\geqslant}1$ adjacency matrices $\left\{{\mathbf{A}}^{(1)}_{n\times n}, \ldots, {\mathbf{A}}^{(T)}_{n\times n}\right\}$. We shall only consider undirected and unweighted graphs in this paper. However, the conclusions of the paper can be extended to positively weighted graphs with non-random weights in a quite straightforward way by considering weighted adjacency matrices. The theoretical analysis in this paper can easily be extended to positively weighted adjacency matrices. Also, in this paper we consider that the multi-relational network has a *common community structure*. So, the multi-relational network $\{G_n^{(t)}\}_{t=1}^T$ has the *same* community structure in every layer with $K$ as the number of communities. Let us denote ${\mathbf{Z}}_{n\times K}$ to be the actual common community membership matrix of the nodes in each of the graphs $G_n^{(t)}$, where, ${\mathbf{Z}}_{ik} = 1$ if the $i$-th node belongs to the $k$-th community for all $G_n^{(t)}$ and zero otherwise.
Notations {#sec_notation}
---------
Let $[n] := \{1, 2, \ldots, n\}$ for any positive integer $n$, ${\mathscr{M}}_{m,n}$ be the set of all $m\times n$ matrices which have exactly one 1 and $n-1$ 0’s in each row. ${ \mathds R}^{m\times n}$ denotes the set of all $m\times n$ real matrices. $||\cdot||_2$ is used to denote Euclidean $\ell_2$-norm for vectors in ${ \mathds R}^{m\times 1}$. $||\cdot||$ is the spectral norm on ${ \mathds R}^{m\times n}$. $||\cdot||_F$ is the Frobenius norm on ${ \mathds R}^{m\times n}$, namely $||M||_F := \sqrt{trace(M^T M)}$. ${\mathbf{1}}_{n} \in { \mathds R}^{n\times 1}$ consists of all 1’s, $\mathbf 1_A$ denotes the indicator function of the event $A$. ${\mathbf{I}}_n$ is the $n\times n$ identity matrix and ${\mathbf{J}}_n := {\mathbf{1}}_n{\mathbf{1}}_n^T$ is the $n\times n$ matrix of all 1’s. For ${\mathbf{A}}\in{\mathds{R}}^{n\times n}$, ${\mathcal{C}}({\mathbf{A}})$ and ${\mathcal{N}}({\mathbf{A}})$ denote its column space and null space of ${\mathbf{A}}$ respectively, and $\gl_1({\mathbf{A}}), \gl_1^+({\mathbf{A}})$ denote the smallest and smallest positive eigenvalues of ${\mathbf{A}}$. If ${\mathbf{A}}\in{ \mathds R}^{m\times n}$, $I\subset [m]$ and $j\in [n]$, then ${\mathbf{A}}_{I,j}$ (resp. ${\mathbf{A}}_{I,*}$) denotes the submatrix of ${\mathbf{A}}$ corresponding to row index set $I$ and column index $j$ (resp. index set $[n]$). For ${\mathbf{A}}\in { \mathds R}^{n\times n}$, ${\left\langle {\mathbf{A}}\right\rangle}$ denotes the matrix ${\mathbf{A}}$ with its diagonal zeroed out: ${\left\langle A\right\rangle}_{i,j}=A_{i,j}$ if $i \ne j, i,j\in [n]$ and ${\left\langle A\right\rangle}_{i,i} = 0$ for $i\in [n]$. For a random variable (real valued or matrix valued) $X$, we write ${\left\llbracket X\right\rrbracket}:=X-\operatorname{\mathds{E}}(X)$. For two random variables $X$ and $Y$, we write $X{\preccurlyeq }Y$ to denote that $X$ is stochastically dominated by $Y$. $\gl_i({\mathbf{W}}), i\in[n],$ will denote the $i$-th largest eigenvalue of ${\mathbf{W}}\in{ \mathds R}^{n\times n}$.
Multilayer Stochastic Block Model
---------------------------------
The first model that we consider is an extension of stochastic block model (SBM) for generating multi-relational networks. We will refer to this model as [*multilayer stochastic block model*]{} (MSBM) in the paper. MSBM for $K$ communities (${\mathcal{C}}_1, \ldots, {\mathcal{C}}_K$) can be described in terms of three parameters: (i) the membership vector ${\ensuremath{\boldsymbol{z}}\xspace}=(z_1, \ldots, z_n)$, where each $z_i \in \{1, \ldots, K\}$; (ii) the $K\times K$ connectivity probability matrices ${\mathbf{B}}:=\left({\mathbf{B}}^{(t)}: 1{\leqslant}t{\leqslant}T\right)$ and (iii) the $K\times 1$ probability vector of allocation in each community, ${\ensuremath{\boldsymbol{\pi}}\xspace} = (\pi_1, \ldots, \pi_K)$. The MSBM having parameters $({\ensuremath{\boldsymbol{z}}\xspace}, {\ensuremath{\boldsymbol{\pi}}\xspace}, {\ensuremath{\boldsymbol{B}}\xspace})$ is given by $$\begin{aligned}
\label{eq_sbm0}
{\mathbf{z}}_1, \ldots, {\mathbf{z}}_n & \stackrel{iid}{\sim} & \mbox{Mult}(1;(\pi_1,\ldots,\pi_K)),\\
\label{eq_sbm1}
{ \mathds P}\left(A^{(t)}_{ij} = 1 | {\mathbf{z}}_i, {\mathbf{z}}_j\right) & = & B^{(t)}_{{\mathbf{z}}_i {\mathbf{z}}_j}\ \ \ \text{for } i >j,\ i,j\in [n] .
\end{aligned}$$
Suppose ${\mathbf{Z}}\in {\mathscr{M}}_{n,K}$ denotes the actual membership matrix. ${\mathbf{Z}}$ is unknown and we wish to estimate it. If for $i \in [n]$ the corresponding community index is ${\mathbf{z}}_i \in [K]$, then clearly $${\mathbf{Z}}_{ij} = \mathbf 1_{\{{\mathbf{z}}_i=j\}},$$ In a MSBM$({\mathbf{z}}, {\ensuremath{\boldsymbol{\pi}}\xspace}, {\mathbf{B}})$, independent edge formation is assumed given the edge probability matrices ${\mathbf{P}}^{(t)}:=(P^{(t)}_{ij})_{i,j\in[n]}$. So, for $i, j \in [n]$ with $i \ne j$ and for $t \in [T]$ \[A\^t bmdef\] A\^[(t)]{}\_[i,j]{} \~Bernoulli(P\^[(t)]{}\_[i,j]{}), \^[(t)]{} := \^[(t)]{}\^T.
Multilayer Degree Corrected Block Model
---------------------------------------
Multilayer degree-corrected block model is an extension of the degree corrected block model (DCBM) for generating multi-relational networks. The multilayer degree-corrected block model (MDCBM) for $K$ communities (${\mathcal{C}}_1, \ldots, {\mathcal{C}}_K$) can be described in terms of four sets of parameters: (i) the membership vector ${\ensuremath{\boldsymbol{z}}\xspace}=(z_1, \ldots, z_n)$, where each $z_i \in \{1, \ldots, K\}$, (ii) the $K\times K$ connectivity probability matrices ${\mathbf{B}}:=\left({\mathbf{B}}^{(t)}: 1{\leqslant}t{\leqslant}T\right)$, (iii) a given set of *degree parameters* ${\ensuremath{\boldsymbol{\psi}}\xspace} = (\psi_1, \ldots, \psi_n)$ and (iv) the $K\times 1$ probability vector of allocation in each community, ${\ensuremath{\boldsymbol{\pi}}\xspace}=(\pi_1, \ldots, \pi_K)$. The MDCBM having parameters $({\ensuremath{\boldsymbol{z}}\xspace}, {\ensuremath{\boldsymbol{\pi}}\xspace}, {\ensuremath{\boldsymbol{B}}\xspace}, {\ensuremath{\boldsymbol{\psi}}\xspace})$ is given by $$\begin{aligned}
\label{eq_dcbm0}
{\mathbf{z}}_1, \ldots, {\mathbf{z}}_n & \stackrel{iid}{\sim} & \mbox{Mult}(1;(\pi_1,\ldots,\pi_K)),\\
\label{eq_dcbm1}
{ \mathds P}\left(A^{(t)}_{ij} = 1\right| {\mathbf{z}}_i, {\mathbf{z}}_j) & = & \psi_i\psi_jB^{(t)}_{{\mathbf{z}}_i {\mathbf{z}}_j} \ \ \ \text{for } i >j,\ i,j\in [n].
\end{aligned}$$
The inclusion of ${\ensuremath{\boldsymbol{\psi}}\xspace}$ involves the obvious issue of identifiability. In order to avoid this issue we assume that [@lei2015consistency] $$\begin{aligned}
\label{eq_dcbm_id}
\max_{i\in{\mathcal{C}}_k} \psi_i=1 \text{ for all } k\in\{1, 2, \ldots, K\}.
\end{aligned}$$
In an MDCBM$({\mathbf{z}}, {\ensuremath{\boldsymbol{\psi}}\xspace}, {\ensuremath{\boldsymbol{\pi}}\xspace}, {\mathbf{B}})$ also independent edge formation is assumed given the edge probability matrices $\tilde{\mathbf{P}}^{(t)}$. Here also, for $i, j \in [n]$ with $i \ne j$ and for $t \in [T]$ \[A\^t def\] A\^[(t)]{}\_[i,j]{} \~Bernoulli(P\^[(t)]{}\_[i,j]{}), \^[(t)]{} := ([$\boldsymbol{\psi}$]{})\^[(t)]{}\^T([$\boldsymbol{\psi}$]{}) where, ${\mathfrak{D}}({\ensuremath{\boldsymbol{\psi}}\xspace}) = \text{diag}({\ensuremath{\boldsymbol{\psi}}\xspace})$.
Community Structure {#sec_comm}
-------------------
The *assortative* and *disassortative* community structures can be defined formally using the parameter structures of multilayer stochastic block models and degree-corrected block models, specially, the connectivity probability matrices $\{{\mathbf{B}}^{(t)}\}_{t=1}^T$.
\[AssoDisso\] For a multi-relational network generated from MSBM or MDCBM with connectivity probability matrices $\{{\mathbf{B}}^{(t)}\}_{t=1}^T$, the $t$-th layer is said to have - (i) *assortative* structure if all the eigenvalues of ${\mathbf{B}}^{(t)}$ are positive; (ii) *disassortative* structure if at least one of the eigenvalues of ${\mathbf{B}}^{(t)}$ is negative.
In this paper, we consider the case where, the community membership does not change between the layers of multi-relational network but the connectivity structure can change arbitrarily between layers and the layers can have either assortative or disassortative community structures.
Community Detection Algorithms {#sec_method}
==============================
Spectral clustering using sum of squared adjacency matrices {#sec_algo_1}
-----------------------------------------------------------
Let ${\mathbf{Z}}\in {\mathscr{M}}_{n,K}$ denote the actual community membership matrix of the nodes, where, if ${\mathbf{Z}}_{ik} = 1$ ($i\in[n]$ and $k\in[K]$), then, node $i$ belongs to $k$-th community. The goal of the statistical methods is estimation of ${\mathbf{Z}}$ based on the adjacency matrix data ${\mathbf{A}}^{(1)}, \ldots, {\mathbf{A}}^{(T)}$. We apply the spectral clustering method to a matrix which is derived from the **sum of the squared adjacency matrices** ${\mathbf{A}}_0^{[2]}:=\sum_{t\in[T]} \left({\mathbf{A}}^{(t)}\right)^2$. We zero out the diagonal of ${\mathbf{A}}_0^{[2]}$ to obtain ${\left\langle {\mathbf{A}}_0^{[2]}\right\rangle}$.
The squared adjacency matrices capture both assortative and disassortative community structures in different network layers. The squared adjacency matrices maintain the community structure in form of an assortative structure, since the non-zero elements of squared adjacency matrices represent paths of length two between the corresponding nodes. So, summing up squared adjacency matrices maintain both assortative and disassortative community structures in different network layers in an assortative form.
Now, we prune $\langle{\mathbf{A}}^{[2]}_0\rangle$ so that the empirical spectrum of the pruned matrix captures the community structure even if the networks are sparse. For each node $i$, get the number of max-one-neighbors (resp. total-two-neighbors) $D^{[1]}_i$ (resp. $D^{[2]}_i$). Also, get the average number of two-neighbors $\bar{d^2}$ of the nodes. $$D^{[1]}_i :=\max_{t\in[T]}\sum_{j\in[n]} A^{(t)}_{i,j}, D^{[2]}_i:=\sum_{ j\in[n]}{\left\langle A^{[2]}_0\right\rangle}_{i,j} \text{ for } i\in[n], \bar{d^2} = \frac{1}{nT}\sum_{i\in[n]} D^{[2]}_i$$ Then we sort the numbers $(D^{[l]}_i, i\in[n])$ to get the order statistics $D^{[l]}_{(1)}{\leqslant}\cdots{\leqslant}D^{[l]}_{(n)}$ for both $l=1, 2$. Let $n'$ be the number of nodes and $1{\leqslant}k_1<k_2<\cdots<k_{n'}{\leqslant}n$ be the node indices having at most $D^{[1]}_{(n+1-\gC_1)}$ many max-one-neighbor and at most $D^{[2]}_{(n+1-\gC_2)}$ many total-two-neighbors, where $$\begin{aligned}
\gC_1:=\left\lceil n\exp\left(-\frac 12T^{1/2}\left[\bar{d^2}\right]^{3/4}\right)\right\rceil, \gC_2:=\left\lceil n\exp\left(-\frac 13T\left[\bar{d^2}\right]^{1/2}\right)\right\rceil. \label{gC def}
\end{aligned}$$
\
Let ${\mathbf{A}}^{[2]}\in{ \mathds R}^{n'\times n'}$ be the submatrix of $\langle{\mathbf{A}}_0^{[2]}\rangle$ such that $A^{[2]}_{i,j} := \langle A_0^{[2]}\rangle_{k_i,k_j}$ for $i, j\in [n']$. Next, we obtain the leading $K$ eigenvectors of ${\mathbf{A}}^{[2]}$ corresponding to its largest eigenvalues. Suppose $\hat{\mathbf{U}}\in { \mathds R}^{n'\times K}$ contains those eigenvectors as columns. Then, we use an $(1+{\epsilon})$-approximate $K$-means clustering algorithm on the row vectors of $\hat{\mathbf{U}}$ to obtain $\hat{\mathbf{Z}}\in {\mathscr{M}}_{n',K}$ and $\hat {\mathbf{X}}\in { \mathds R}^{K \times K}$ such that $$\begin{aligned}
\label{eq:kmeans}
{\left\Vert\hat{\mathbf{Z}}\hat{\mathbf{X}}- \hat{\mathbf{U}}\right\Vert}_F^2 {\leqslant}(1+{\epsilon}) \min_{{\ensuremath{\boldsymbol{\gC}}\xspace} \in {\mathscr{M}}_{n'\times K}, {\mathbf{X}}\in { \mathds R}^{K\times K}} {\left\Vert{\ensuremath{\boldsymbol{\gC}}\xspace} {\mathbf{X}}- \hat{\mathbf{U}}\right\Vert}_F^2.
\end{aligned}$$ Finally, $\hat{\mathbf{Z}}$ is extended to $\hat{\mathbf{Z}}_0\in{\mathscr{M}}_{n,K}$ by taking $(\hat{\mathbf{Z}}_0)_{k_j,*}:=\hat{\mathbf{Z}}_{j,*}$ for all $j\in[n']$, and filling in the remaining rows arbitrarily. One simple choice would be assigning all the pruned nodes to the first community. $$(\hat{\mathbf{Z}}_0)_{i,*} := \begin{cases} \hat{\mathbf{Z}}_{j,*} & \text{ if $i=k_j$ for some $j\in[n']$} \\ (e^K_1)^T & \text{otherwise}\end{cases}$$ $\hat{\mathbf{Z}}_0$ is the estimate of ${\mathbf{Z}}$ from this method.
The reason for using an $(1+{\epsilon})$-approximate $K$-means clustering algorithm is completely theoretical. $K$-means clustering is originally an NP-hard problem with any $K$-means clustering algorithm generating an approximate solution. However, we need a guarantee on the error of $K$-means clustering algorithm. So, we choose to use the $K$-means algorithms that can give us a guarantee on the error of the optimized objective function like algorithms proposed in [@kumar2004simple; @feldman2007ptas].
Spherical Spectral Clustering Algorithm for Sum of Squared Adjacency Matrices {#sec_algo_2}
-----------------------------------------------------------------------------
The goal is to estimate the community membership matrix ${\mathbf{Z}}$ based on the adjacency matrices ${\mathbf{A}}^{(1)}, \ldots, {\mathbf{A}}^{(T)}$. We apply the spherical spectral clustering method, which is a modification of Algorithm 1. The modification is motivated from the works [@jin2015fast] and [@lei2015consistency]. Let ${\mathbf{A}}_0^{[2]}$, ${\mathbf{A}}^{[2]}$ and $\hat{\mathbf{U}}$ be as in \[sec\_algo\_1\]. For $\hat{\mathbf{U}}$, let $n''$ be the number of nonzero rows (with indices $1{\leqslant}l_1 < l_2 <\cdots <l_{n''}{\leqslant}n'$). Let $\hat{\mathbf{U}}^+ \in { \mathds R}^{n''\times K}$ consist of the normalized nonzero rows of $\hat{\mathbf{U}}$, i.e. $\hat{\mathbf{U}}^+_{i,*}=(||\hat{\mathbf{U}}_{l_i,*}||^{-1}_2)\hat{\mathbf{U}}_{l_i,*}$ for $i\in [n'']$. Apply an $(1+\gee)$ approximate $K$-means clustering algorithm on the rows of $\hat{\mathbf{U}}^+$ to get $\check{\mathbf{Z}}^+ \in {\mathscr{M}}_{n'',K}$ and $\check {\mathbf{X}}\in { \mathds R}^{K \times K}$ so that \[k mean\_Frob\] [\^+- \^+]{}\_F (1+) \_[[$\boldsymbol{\gC}$]{} \_[n”K]{}, [ R]{}\^[KK]{}]{} [- \^+]{}\_F. Finally, $\check{\mathbf{Z}}^+$ is extended to $\check{\mathbf{Z}}\in{\mathscr{M}}_{n',K}$, and then $\check{\mathbf{Z}}$ is extended to $\check{\mathbf{Z}}_0\in{\mathscr{M}}_{n,K}$ by taking $\check{\mathbf{Z}}_{l_j,*}:=\check{\mathbf{Z}}^+_{j,*}, j\in [n''],$ and $(\check{\mathbf{Z}}_0)_{k_j,*}:=\check{\mathbf{Z}}_{j,*}, j\in [n'],$ and filling in the remaining rows arbitrarily. $\check{\mathbf{Z}}_0$ is the estimate of ${\mathbf{Z}}$ from this method. Unlike in Algorithm 1, we use the normalized nonzero rows of $\hat{\mathbf{U}}$ in Algorithm 2 (see ) instead of all rows of $\hat{\mathbf{U}}$ in Algorithm 1 (see ). However, like in Algorithm 1, the reason for using an $(1+{\epsilon})$-approximate $K$-means clustering algorithm in Algorithm 2 is also purely theoretical as we need theoretical guarantee on the solutions of the heuristic algorithms used to solve the $K$-means problem as given in works like [@kumar2004simple; @feldman2007ptas].
\
Selection of $K$ {#sec_k_sel}
----------------
In both Algorithm 1 in \[sec\_algo\_1\] and Algorithm 2 in \[sec\_algo\_2\], the number of communities $K$ were considered to be known. However, number of communities can also be estimated using the absolute eigenvalues of the matrix ${\mathbf{A}}^{[2]}$ by using the thresholding methods as in [@chatterjee2015matrix; @bickel2016hypothesis; @le2015estimating]. More work needs to be done to get a better estimate of number of communities $K$ in the multiple network context. Extensions of methods in [@bickel2016hypothesis], [@wang2017likelihood], [@chen2018network], and [@le2015estimating] seem to be the first step for further research on this topic.
Here, we give an intuitive method for detection of number of communities based on the eigenvalues of ${\mathbf{A}}^{[2]}$. Using the concentration result of ${\mathbf{A}}^{[2]}$ to $\operatorname{\mathds{E}}{\mathbf{A}}^{[2]}$ used in proof of Theorem \[ConsSum1\], we can get a threshold on the eigenvalues of ${\mathbf{A}}^{[2]}$ corresponding to the zero eigenvalues of ${\mathbf{A}}_0^{[2]}$.
\
Theoretical Justification {#sec_theory}
=========================
Consistency of spectral clustering label $\hat{\mathbf{Z}}_0$ under multilayer stochastic block model {#sec_proof_hatZ_supp}
-----------------------------------------------------------------------------------------------------
In order to state the theoretical results on the estimated community membership matrix, $\hat{\mathbf{Z}}_0$, we first need to define certain quantities and conditions on the parameters of multilayer stochastic block model. The following parameters are functions of $({\mathbf{z}}, {\ensuremath{\boldsymbol{\pi}}\xspace}, \{{\mathbf{B}}^{(t)}\}_{t=1}^T)$: (i) [*$d=n(\max_{a,b\in[K], t\in[T]}B^{(t)}_{ab})$ is the maximum expected degree*]{} of a node at any layer; (ii) [*$\gl=T^{-1} \sum_{t\in[T]}\gl_K\left((\frac nd {\mathbf{B}}^{(t)})^2\right)>0$ is the average of the smallest eigenvalues*]{} of squared normalized probability matrices $\{{\mathbf{B}}^{(t)}\}_{t=1}^T$; and (iii) $n_{\text{min}}$ is the size of the smallest community.
\[ConsSum1\] Let $({\mathbf{A}}^{(t)}, t\in[T])$ be the adjacency matrices of the networks generated from the multilayer stochastic block model with parameters $({\mathbf{z}}, {\ensuremath{\boldsymbol{\pi}}\xspace}, \{{\mathbf{B}}^{(t)}\}_{t=1}^T)$. For $a\in[K]$, let $f_a$ denote the proportion of nodes having community label $a$, which are misclassified in Algorithm 1. For any $\gee>0$ and $\gD>8$, there are constants $C=C(\gee), C'>0$ such that if $$\begin{aligned}
\label{eq_sum_ass}
\gl \left(\frac{n_{\text{min}}}{n}\right)^2 >\max\left\{\frac 7n, \frac{C\gD\sqrt K}{ (Td)^{1/4}}\right\}, \text{ then }
\end{aligned}$$ $$\begin{aligned}
\operatorname{\mathds{P}}\left(\sum_{a\in[K]} f_a {\leqslant}\left[\frac{C\gD\sqrt K}{(Td)^{1/4} \gl \left(\frac{n_{\text{min}}}{n}\right)^2}\right]^2\right) {\geqslant}1- \frac{C'+2nK}{n(Td)^{3/4}} - 2n^{5-\gD^2/12}. \label{eq_mis_error}
\end{aligned}$$ Therefore, in the special case, when (i) $K$ is a constant and (ii) the community sizes are balanced, i.e. $n_{\text{max}}/n_{\text{min}} = O(1)$, then the proportion of misclassified nodes in $\hat{\mathbf{Z}}_0$ is arbitrarily small (resp. goes to zero) with probability $1-o(1)$ if $(Td)^{1/4}\gl$ is large enough (resp. $(Td)^{1/4}\gl\to\infty$).
\[rem\_condition\_asymp\] Note that the result in equation involves the interplay of the parameters $n$, $T$, $K$ and $\{{\mathbf{B}}^{(t)}\}_{t=1}^T$ and does not assume any apriori condition on any of the parameters except equation . Also, the result in equation is a non-asymptotic result, but, it can be made into an asymptotic result. We need the condition $ \sqrt{K}(Td)^{-1/4}\gl^{-1}\rightarrow 0$ and $\gl > \frac{cn}{n_{min}^2}$ for $c > 7$ for having an asymptotically vanishing mis-classification error with probability $1-o(1)$. The asymptotics can be with respect to $T\to \infty$ and/or $d\to \infty$ (as $n \to \infty$). Also, for the asymptotic case of $n\to\infty$, $\gD$ is a constant. But for $n$ fixed and $T\to\infty$, $\gD$ has to be chosen such that it satisfies both $n^{5-\gD^2/12}\to 0$ and the condition in equation . For example $\gD$ can be taken as, $\gD = \left(\frac{(Td)^{1/4} n_{min}^2}{\sqrt{K}n^2}\right)^{\gd}$ for any constant $\gd$ with $0 < \gd < 1$.
\[rem\_condition\_optimal\] The condition “$(Td)^{1/4}\gl\to\infty$" for the special case, is necessary and sufficient in order to have a consistent estimator of ${\mathbf{Z}}$. Theorem \[ConsSum1\] proves the sufficiency. The necessity of the condition follows from the work of [@ZZ16]. Consider a stochastic block model (so $T=1$), where (i) there are two communities having equal size $n$ and (ii) the within (resp. between) community connection probability is $a/n$ (resp. $b/n$) for some constants $a>b>0$. In this case $(Td)^{1/4}\gl=\frac{a^{1/4}(a-b)}{a}$ is a constant. [@ZZ16] states that in the above setup, there is a constant $c>0$ such that if $$\frac{(a-b)^2}{a+b} <c\log\frac 1\gc$$ for some constant $\gc$ (e.g. $\gc=e^{-(a-b)/c}$), then the expected proportion of misclassification for every algorithm will be at least $\gc$. In other words, no algorithm can give consistent estimator of ${\mathbf{Z}}$. So, the condition “$(Td)^{1/4}\gl\to\infty$" becomes an *optimal* condition for consistent recovery of community labels.
\[rem\_condition\_eig\] The assumption in equation makes sure that there is a community structure in the aggregated network. The condition in is quite relaxed. In the balanced case with constant $K$, it does not require $O(T)$ many matrices among $\{{\mathbf{B}}^{(t)}\}_{t=1}^T$ to have full-rank but only requires $\frac{T}{(Td)^{1/4}}$, which is $o(T)$, many matrices among $\{{\mathbf{B}}^{(t)}\}_{t=1}^T$ to have all nonzero eigenvalues or full-rank. Note that according to the condition in equation , the number of necessary informative (full-rank) $\{{\mathbf{B}}^{(t)}\}_{t=1}^T$ matrices (a) should increase as $T$ increases for fixed but large $d$ (or $n$); (b) should decrease as $d$ (or $n$) increases for fixed $T$. This behavior is illustrated in Scenario 2 of simulation in \[sec\_simulation\].
Extensions to the case of dependent adjacency matrices {#sec:DepAdj}
------------------------------------------------------
In this section, we will consider a general situation where ${\mathcal{B}}:= ({\mathbf{B}}^{(t)}, t\in[T])$ is a stochastic process and the distribution of $({\mathbf{A}}^{(t)}, t\in[T])$ is conditionally independent as described in . Now, let us define some important functions of the stochastic process $({\mathbf{B}}^{(t)}, t\in[T])$ which will be useful in quantifying the mis-classification error of Algorithm 1.
(a) The smallest eigenvalue $\gl_{K,t}:=\gl_K([{\mathbf{B}}^{(t)}]^2)$ is also a random variable and cumulative distribution function of $\gl_{K,t}$ is given by $F_t(x):=\operatorname{\mathds{P}}(\gl_{K,t}{\leqslant}x)$ for $x{\geqslant}0$. Let $b_t:=\mathbf 1_{\{\gl_{K,t}=0\}}$ be the indicator random variable for the event of rank-deficient $[{\mathbf{B}}^{(t)}]^2$. Let $F^+_t(x):=\frac{F_t(x)-F_t(0)}{1-F_t(0)}$ be the distribution function corresponding to the truncated positive part of $\gl_{K,t}$ and $\tilde\gl_{K,t}\sim F^+_t(x)$ for all $t$ is an independent copy generated from the truncated distribution. So, $\tilde\gl_{K,t}$ is independent of ${\mathcal{B}}$. Then, we can define the random variable $$\gl^+_{K,t}:=\begin{cases}\gl_{K,t} \text{ if } b_t=0, \\\tilde\gl_{K,t}\text{ if } b_t=1.\end{cases}$$ So $\gl_{K,t}= b_t{\boldsymbol\delta}_0+(1-b_t)\gl^+_{K,t}$, $b_t\sim Ber(F_t(0)), \gl^+_{K,t}\sim F^+_t$. Lastly, it follows from elementary probability calculations that $\mathfrak b=(b_t, t\in[T])$ and $\boldsymbol\gl=(\gl^+_{K,t}, t\in[T])$ are independent.
(b) The *maximal degree variable*, ${\underline{d}}_n({\varepsilon})$ for any ${\varepsilon}> 0$, is defined in the following way - $$\begin{aligned}
{\underline{d}}_n({\varepsilon}):=\sup\left\{x\in[0, n]: \operatorname{\mathds{P}}\left(\max_{t\in[T], \; a,b\in[K]} n B^{(t)}_{ab}{\leqslant}x\right){\leqslant}{\varepsilon}\right\}.
\end{aligned}$$
(c) **(Mixing condition)** We consider a decreasing function $\ga_\da:{ \mathds Z}_+\mapsto[0,1]$ to reflect the decay of correlation (at any rate) between two events of non-informative (smallest eigenvalue of ${\mathbf{B}}^{(t)}$ being zero) ${\mathbf{B}}^{(t)}$ matrices, like ${\mathbf{B}}^{(t_1)}$ and ${\mathbf{B}}^{(t_2)}$, where, $ t_1,t_2 \in [T], t_1\neq t_2$. $$\begin{aligned}
\left|\operatorname{\mathds{P}}\left(\cap_{i\in[2]}\{\gl_{K,t_i}=0\}\right) - \prod_{i\in[2]}\operatorname{\mathds{P}}(\gl_{K,t_i}=0)\right| {\leqslant}\ga_\da(|t_1-t_2|) \label{StMix}
\end{aligned}$$ with $\ga_\da$ having the property $$\ga_\da(s) \da 0 \text{ as $ s\ua \infty$, and } \ga_\da(0) = 1.$$ This decay of correlation is necessary to have consistent recovery of communities.
(d) We consider a function $\psi_{\ua\da}:{ \mathds N}\times{ \mathds R}_+\mapsto[0, 1]$ in terms of $T$ and ${\underline{d}}_n({\varepsilon})$, which captures the probability that network layers are non-informative, that is, $$\begin{aligned}
\max_{t \in [T]}\operatorname{\mathds{P}}\left(\{\gl_{K,t} = 0\}\right) \leq \psi_{\ua\da}(T, {\underline{d}}_n({\varepsilon})) . \label{ind_net}
\end{aligned}$$ $\psi_{\ua\da}(T, {\underline{d}})$ is a function which captures the behavior that on one hand $\psi_{\ua\da}$ increases to 1 as $T$ increases and ${\underline{d}}$ remains constant. But, on the other hand $\psi_{\ua\da}$ decreases to 0 as ${\underline{d}}$ increases and the number of networks $T$ stays the same, that is, $$\lim_{T\ua\infty}\psi_{\ua\da}(T,{\underline{d}}) = 1\ \ \ \text{and }\ \ \psi_{\ua\da}(T,{\underline{d}}) \da 0 \text{ as $ {\underline{d}}\ua \infty$. }$$
(e) We consider a decreasing and convex function $\phi_\da:(0,\infty)\mapsto(0,\infty)$, which controls the lower tail behavior of the smallest eigenvalues of the probability matrices $({\mathbf{B}}^{(t)}, t\in[T])$ near 0 with the property $$\phi_\da(x)\ua\infty \text{ as $ x\da 0$. }$$
Based on the random variables $\gl_{K,t}^+$ and ${\underline{d}}_n({\varepsilon})$, and the functions $\ga_\da$, $\psi_{\ua\da}$ and $\phi_\da$ defined above, we place the following conditions on the stochastic process ${\mathcal{B}}$.\
**Assumption A:** Let ${\mathcal{B}}= ({\mathbf{B}}^{(t)}, t\in[T])$ be a stochastic process with the following properties - $$\begin{aligned}
& & (a)\; \psi_{\ua\da}(T, {\underline{d}}_n({\varepsilon})){\leqslant}1-\left[\frac{\sqrt T}{T}+\ga_\da(\sqrt T)\right]^{1/2-\gd}\vee\frac{1}{(T{\underline{d}}_n({\varepsilon}))^{\frac{1}{60}}} \notag \\
& & (b)\; \max_{t\in[T]} F_t(0){\leqslant}\psi_{\ua\da}(T,{\underline{d}}_n(1/2)), \text{and } \label{lamAss} \\
& & (c)\; \max_{t\in[T]}\operatorname{\mathds{E}}\phi_\da(\gl^+_{K,t}) {\leqslant}C_1 \notag
\end{aligned}$$ for any ${\varepsilon}>0$ and for some constants $C_1<\infty \text{ and } \gd<1/2$.
\[Gen B\^t\] Let $({\mathbf{B}}^{(t)},t\in[T])$ be any stochastic process satisfying Assumption A of , $({\mathbf{A}}^{(t)}, t\in[T])$ be the adjacency matrices satisfying , $(f_a, a\in[K])$ and $C'$ be as in Theorem \[ConsSum1\]. For any ${\varepsilon}>0, \gd\in(0,1/2)$, $$\begin{aligned}
\operatorname{\mathds{P}}\left(\sum_{a\in[K]} f_a > (T{\underline{d}}({\varepsilon}))^{-1/6}\right) {\leqslant}{\varepsilon}+ \frac{C_1}{\phi_\da\left(\frac{2n^2}{n^2_{\text{min}}}\left[T{\underline{d}}_n({\varepsilon})\right]^{-1/15}\right)} \label{eq_mis_error_mart}\\
+\min\left\{4\left[\frac{\sqrt T}{T}+\ga_\da(\sqrt T)\right]^{2\gd}, T\psi_{\ua\da}(T,{\underline{d}}_n({\varepsilon}))\right\} +\frac{2C'+2nK}{n}\left([T{\underline{d}}_n({\varepsilon})]^{-3/4}+{\varepsilon}\right). \notag
\end{aligned}$$ Therefore, in the special case, when (i) $K$ is a constant and (ii) the community sizes are balanced, i.e. $n_{\text{max}}/n_{\text{min}} = O(1)$, then the proportion of misclassified nodes in $\hat{\mathbf{Z}}_0$ is arbitrarily small (resp. goes to zero) with probability $1-o(1)$ if $T{\underline{d}}_n({\varepsilon})$ is large enough (resp. $T{\underline{d}}_n({\varepsilon})\to\infty$) and ${\varepsilon}$ is small enough (resp. ${\varepsilon}\to 0$).
The statement of Theorem \[Gen B\^t\] is pretty general, flexible and involves many components which can be fine-tuned to capture a wide-variety of aspects of the community detection problem under dependence between layers. 1. The assumption on $\ga_\da$ reflects the decay of correlation (at any rate) between the two events of non-informative (smallest eigenvalue of ${\mathbf{B}}^{(t)}$ being zero) ${\mathbf{B}}^{(t)}$ matrices, like ${\mathbf{B}}^{(t_1)}$ and ${\mathbf{B}}^{(t_2)}$, where, $ t_1,t_2 \in [T], t_1\neq t_2$. This decay of correlation is necessary to have consistent recovery of communities. Faster decay rate implies smaller error rate in terms of $T$ for community recovery.
2\. The assumption about $\psi_{\ua\da}$ shows that consistent recovery of communities is possible by bounding the probability that an individual network is non-informative (i.e. corresponding ${\mathbf{B}}^{(t)}$ is singular).
3\. The assumption on $\phi_\da$ describes the lower tail behavior of the smallest eigenvalues of the probability matrices $({\mathbf{B}}^{(t)}, t\in[T])$ near 0. The larger value of $\phi_\da$ implies smallest eigenvalue being further away from zero and thus smaller error rate.
4\. The reason for taking ${\underline{d}}_n({\varepsilon})$ as a measure of sparsity is the following. When $T$ is small, in order to have a consistent algorithm it is necessary for not only the mean, but also all quantiles of the distribution of the maximum degree to increase to infinity. If $T$ is small and the distribution of $\max_{t,a,b} B^{(t)}_{ab}$ has non-vanishing probability for any subinterval of ${ \mathds R}_+$, then consistency cannot hold. Note that, if $d^{(T)} := \max_{t\in[T], \; a,b\in[K]} n B^{(t)}_{ab}$ concentrates, then, $\operatorname{\mathds{E}}(d^{(T)})$ or $\text{Median}(d^{(T)})$ can replace ${\underline{d}}_n({\varepsilon})$ in the Theorem.
5\. The $\sqrt T$ appearing in can be replaced by any $o(T)$ term.
6\. The exponents $1/6$ and $1/15$ appearing in vary between 0 and $1/2$. If $1/6$ is replaced by $\eta$, then $1/15$ can be replaced by any number $<1/4-\eta/2$.
7\. The asymptotics in Theorem \[Gen B\^t\] can be with respect to $T\to \infty$ and/or $d\to \infty$ (as $n \to \infty$). The rates of decay of functions $\ga_\da$, $\psi_{\ua\da}$ and $\phi_\da$ also become crucial for mis-classification error to vanish with probability $1-o(1)$ as $T\to \infty$ and/or $d\to \infty$ (as $n \to \infty$).
\[cor\_Gen\_Bt\] If $n$ is constant and $({\mathbf{B}}^{(t)}, t\in[T])$ is jointly stationary and ergodic process with $\gl:=\operatorname{\mathds{E}}\gl_K([{\mathbf{B}}^{(1)}]^2)>0$, then there is a sequence of fractions $\{{\varepsilon}_T\}_{T{\geqslant}1}$ satisfying ${\varepsilon}_T\da 0$ as $T\ua \infty$ such that $\operatorname{\mathds{P}}(\sum_{a\in[K]} f_a<cKT^{-1/4}\gl^{-2}){\geqslant}1-(c_1+c_2K)T^{-3/4}-{\varepsilon}_T$, where $c, c_1, c_2$ are constants.
In the setup of Theorem \[ConsSum1\] and \[Gen B\^t\], if $\{{\mathbf{B}}^{(t)}\}_{t=1}^T$ is a piecewise constant stochastic process with $k(T)$ many change-points, and the adjacency matrices remain unchanged between successive change-points and we apply Algorithm 1 on the distinct adjacency matrices, then all the communities can be recovered consistently if $(k(T)d)^{1/4}\gl\to \infty$ (resp. $(k(T){\underline{d}}_n({\varepsilon}))^{1/4}(\min_{t\in[k(T)]}\gl_{K,t})\to \infty$) in the case of Theorem \[ConsSum1\] (resp. \[Gen B\^t\]).
Consistency of Spherical Spectral Clustering Labels $\check{\mathbf{Z}}_0$ under multilayer degree-corrected block model {#sec_mdsbm}
------------------------------------------------------------------------------------------------------------------------
In order to state the theoretical results on the estimated community membership matrix, $\check{\mathbf{Z}}_0$, we first need to define certain quantities and conditions on the parameters of multilayer degree-corrected block model. The following parameters are functions of $({\mathbf{z}}, {\ensuremath{\boldsymbol{\pi}}\xspace}, \{{\mathbf{B}}^{(t)}\}_{t=1}^T, {\ensuremath{\boldsymbol{\psi}}\xspace})$: (i) $d=n(\max_{a,b\in[K], t\in[T]}B^{(t)}_{ab}))$ is the maximum expected degree of a node at any snapshot; (ii) for $a\in[K]$, $\tilde n_a:=\sum_{i\in{\mathcal{C}}_a\}}\psi_i^2$ and $\tau_a := \sum_{i\in{\mathcal{C}}_a} \psi_i^2 \sum_{i\in{\mathcal{C}}_a} \psi_i^{-2}$ is a measure of heterogeneity of ${\ensuremath{\boldsymbol{\psi}}\xspace}$; (iii) $\psi_{\text{min}}:=\min_{i\in[n]} \psi_i$; (iv) $\tilde n_{\text{max}}=\max_{a\in[K]}\tilde n_a, \tilde n_{\text{min}}=\min_{a\in[K]}\tilde n_a$; and (v) $\gl=T^{-1} \sum_{t\in[T]}\gl_K\left((\frac nd {\mathbf{B}}^{(t)})^2\right)>0$ the average of the smallest eigenvalues of the squared normalized probability matrices.
\[ConsSum2\] Let $({\mathbf{A}}^{(t)}, t\in[T])$ be the adjacency matrices of the networks (having $n$ nodes and $K$ communities) generated from the multilayer degree-corrected block model with parameters $({\mathbf{z}}, {\ensuremath{\boldsymbol{\pi}}\xspace}, \{{\mathbf{B}}^{(t)}\}_{t=1}^T, {\ensuremath{\boldsymbol{\psi}}\xspace})$ satisfying . For any $\gee>0$ and $\gD>8$, there are constants $C(\gee), C'>0$ such that if $$\begin{aligned}
\label{lambda_assump_alt}
\gl \left(\frac{\tilde n_{\text{min}}}{n}\right)^2 >\frac 7n,\text{ and }
n_{\text{min}}> \frac{C(K\tilde n_{\text{max}})^3\gl^{-2}}{\psi^2_{\text{min}}\tilde n^4_{\text{min}}} + \frac{C\gD\sqrt{K\sum_{a\in[K]}\tau_a}}{ (Td)^{1/4}\gl \left(\frac{\tilde n_{\text{min}}}{n}\right)^2},\end{aligned}$$ then the total number of misclassified nodes in $\check{\mathbf{Z}}_0$ is at most \[overall misclassify\] + with probability at least $1-(C'/n+2K)(Td)^{-3/4}-2n^{5-\gD^2/12}$.
Therefore, in the special case, when (i) $K$ is a constant, (ii) the community sizes are balanced, i.e. $n_{\text{max}}/n_{\text{min}} = O(1)$ and (iii) $\psi_i=\alpha_i/\max\{\alpha_j: z_i=z_j\}$, where $(\alpha_i)_{i=1}^n$ are i.i.d. positive weights, then consistency holds for $\check{\mathbf{Z}}_0$ with probability $1-o(1)$ if $\operatorname{\mathds{E}}[\max\{\alpha_1^2,\alpha_1^{-2}\}]<\infty$ and $(Td)^{1/4}\gl\to\infty$.
The condition “$(Td)^{1/4}\gl\to\infty$" for the special case, is necessary and sufficient in order to have a consistent estimator of ${\mathbf{Z}}$. Theorem \[ConsSum2\] proves the sufficiency. The necessity of the condition also follows from the work of [@ZZ16] by considering the special case of stochastic block model. So, the condition “$(Td)^{1/4}\gl\to\infty$" becomes an *optimal* condition for consistent recovery of community labels.
Like in Remark \[rem\_condition\_asymp\], the statement in Theorem \[ConsSum2\] is also non-asymptotic, but it can be viewed as an asymptotic statement in terms of $T \to \infty$ and/or $d\to\infty$ (as $n \to \infty$) under conditions and .
In the special case for $(\alpha_i)_{i=1}^n$ as i.i.d. positive weights, condition of $\operatorname{\mathds{E}}[\max\{\alpha_1^2,\alpha_1^{-2}\}]<\infty$ is satisfied for a large class of distributions, such as $\text{Uniform}(c, d)$ with $c > 0$, $\text{Pareto}(\ga)$ with $\ga > 2$ and $\text{Gamma}(\ga, \gb)$ with $\ga > 2$.
Consistency of Estimated Number of Communities $\hat{K}$ in Algorithm 3 {#sec_Khat_th}
-----------------------------------------------------------------------
In \[sec\_k\_sel\], we give a method for estimating number of communities in Algorithm 3. In order to prove consistency of the estimated number of communities, $\hat{K}$, obtained from Algorithm 3, we consider that the multi-relational network has been generated from the multiple stochastic block model with parameters $({\mathbf{z}}, {\ensuremath{\boldsymbol{\pi}}\xspace}, \{{\mathbf{B}}^{(t)}\}_{t=1}^T)$.
\[Khat\_thm\] Let $({\mathbf{A}}^{(t)}, t\in[T])$ be the adjacency matrices of the networks (having $n$ nodes and $K$ communities) generated from the multilayer stochastic block model with parameters $({\mathbf{z}}, {\ensuremath{\boldsymbol{\pi}}\xspace}, \{{\mathbf{B}}^{(t)}\}_{t=1}^T)$ and $\hat{K}$ be the estimate of $K$ from Algorithm 3. Let us consider the special case when $K$ is a constant. There are constants $C, C'>0$ such that if $\gD>8, \gl(n_{\text{min}}/n)^2>\max\{7/n, 3(Td)^{-1/8}\}$ and $Td > C$, then $\operatorname{\mathds{P}}\left(\hat{K} \neq K\right) \leq \frac{C'/n+2K}{(Td)^{3/4}} + 2n^{5-\gD^2/12}$.
Simulation Results {#sec_simulation}
==================
We simulate multilayer networks in several different scenarios for empirically testing the community detection performance of the methods proposed in the paper.
We compare six different algorithms -
(i) *Sum:* spectral clustering with sum of adjacency matrices with truncation for high-degree nodes [@bhattacharyya2018spectral; @bhattacharyya2020consistent].
(ii) *Spectral sum:* clustering the rows of sum of eigen-spaces $\sum_{t=1}^T U^{(t)}$ of each network snapshot (where, ${\mathbf{U}}^{(t)}_{n\times K}$ is the matrix formed by the eigenvectors of top $K$ eigenvectors of ${\mathbf{A}}^{(t)}$). It was shown empirically in [@paul2017spectral] to have a good community detection performance.
(iii) *Sum (Spherical):* spherical spectral clustering with sum of adjacency matrices with truncation for high-degree nodes [@bhattacharyya2018spectral; @bhattacharyya2020consistent].
(iv) *Co-regularized spectral clustering:* the method was proposed in [@kumar2010co] and shown empirically in [@paul2017spectral] to have a good community detection performance.
(v) *Algorithm 1* of the paper.
(vi) *Algorithm 2* of the paper.
Note that all the algorithms are not compared in every experiment of the scenarios. We consider three different scenarios with different combinations of $n, T, {\ensuremath{\boldsymbol{\psi}}\xspace}$, and ${\mathbf{B}}$ to generate multilayer stochastic block models and multilayer degree-corrected block models. The performance on community detection is reported in terms of normalized mutual information (NMI) metric between true and estimated community labels. The value of NMI is between 0 and 1 and higher value of NMI implies better community detection performance.
*Scenario 1:* In this scenario, we consider a situation where the interaction between some communities change their nature from disassortative to assortative between layers where as interaction between some communities remain assortative throughout all the layers. We simulate such multilayer networks from multilayer stochastic block model (SBM) and multilayer degree-corrected block model (DCBM) under the framework of and of \[sec\_model\]. We consider four experiments under this scenario. Each experiment is repeated 25 times and the results are averages over the repetitions.
1\. *Changing number of nodes ($n$) for multilayer SBM:* We vary node size $n$ from $1000$ to $15000$ with other parameters being $K=4$, ${\mathbf{B}}^{(t)}_{4\times 4} = 3\frac{(\log n)^{3/4}}{n}\left({\ensuremath{\boldsymbol{I}}\xspace}_2\otimes{\ensuremath{\boldsymbol{J}}\xspace}_2 + {\underline{b}}_t{\mathbf{I}}_4\right)$, ${\ensuremath{\boldsymbol{\pi}}\xspace} = \frac{1}{4}{\ensuremath{\boldsymbol{1}}\xspace}_{4\times 1}$, and $T=11$, where, ${\underline{b}}_t = -1 + 0.2(t-1)$ for $t \in [T]$. We compare the algorithms (i), (ii), (iv), and (v). We also apply Algorithm 3 to estimate the number of communities with changing $n$. The results on average NMI and average $\hat{K}$ are given in Figure \[fig\_msbm\_disass\].
2\. *Changing number of layers ($T$) for multilayer SBM:* We vary number of layers $T$ from $5$ to $55$ with other parameters being $K=4$, ${\mathbf{B}}^{(t)}_{4\times 4} = \frac{5}{n}\left({\ensuremath{\boldsymbol{I}}\xspace}_2\otimes{\ensuremath{\boldsymbol{J}}\xspace}_2 + {\underline{b}}_t{\mathbf{I}}_4\right)$, ${\ensuremath{\boldsymbol{\pi}}\xspace} = \frac{1}{4}{\ensuremath{\boldsymbol{1}}\xspace}_{4\times 1}$, and $n=2000$, where, ${\underline{b}}_t = -1 + 0.2(t-1)$ for $t \in [T]$. We compare the algorithms (i), (ii), (iv), and (v). The results on average NMI are given in Figure \[fig\_msbm\_mdcbm\_disassT\](a).
3\. *Changing number of nodes ($n$) for multilayer DCBM:* We vary node size $n$ from $1000$ to $15000$ with other parameters remaining the same as experiment 1 with the only addition of degree parameters $\psi_i \stackrel{iid}{\sim} U(0.5, 1)$ for $i \in [n]$. We compare the algorithms (ii), (iii), (iv), and (vi). The results on average NMI are given in Figure \[fig\_mdcbm\_disass\].
4\. *Changing number of layers ($T$) for multilayer DCBM:* We vary number of layers $T$ from $5$ to $55$ with other parameters remaining the same as experiment 2 except ${\mathbf{B}}^{(t)}_{4\times 4} = \frac{10}{n}\left({\ensuremath{\boldsymbol{I}}\xspace}_2\otimes{\ensuremath{\boldsymbol{J}}\xspace}_2 + {\underline{b}}_t{\mathbf{I}}_4\right)$ and the addition of degree parameters $\psi_i \stackrel{iid}{\sim} U(0.5, 1)$ for $i \in [n]$. We compare the algorithms (ii), (iii), (iv), and (vi). The results on average NMI are given in Figure \[fig\_msbm\_mdcbm\_disassT\](b).
*Scenario 2:* In this scenario, we consider a situation where only one layer has a disassortative community structure, where as all other network layers are uninformative in terms of the community structure. We simulate such multilayer networks under the framework of and of \[sec\_model\]. We consider four experiments under this scenario. Each experiment is repeated 25 times and the results are averages over the repetitions.
1\. *Changing number of nodes ($n$) for multilayer SBM:* We vary node size $n$ from $2000$ to $10000$ with other parameters being $K=4$, ${\mathbf{B}}^{(1)}_{4\times 4} = \frac{(\log n)^{4/3}}{n}\left({\ensuremath{\boldsymbol{J}}\xspace}_4 - {\ensuremath{\boldsymbol{I}}\xspace}_4 \right)$, ${\mathbf{B}}^{(2)}_{4\times 4} = \frac{(\log n)^{4/3}}{n}{\ensuremath{\boldsymbol{J}}\xspace}_4$, ${\mathbf{B}}^{(t)}_{4\times 4} = \frac{(\log n)^{4/3}}{nT}{\ensuremath{\boldsymbol{J}}\xspace}_4 $ for $t=3, \ldots, T$, ${\ensuremath{\boldsymbol{\pi}}\xspace} = \frac{1}{4}{\ensuremath{\boldsymbol{1}}\xspace}_{4\times 1}$, and $T=11$. We compare the algorithms (i), (ii), (iv), and (v). We also apply Algorithm 3 to estimate the number of communities with changing $n$. The results on average NMI and average $\hat{K}$ are given in Figure \[fig\_msbm\_dis\].
2\. *Changing number of layers ($T$) for multilayer SBM:* We vary number of layers $T$ from $5$ to $55$ with other parameters being $K=4$, ${\mathbf{B}}^{(1)}_{4\times 4} = \frac{(\log n)^{4/3}}{n}\left({\ensuremath{\boldsymbol{J}}\xspace}_4 - {\ensuremath{\boldsymbol{I}}\xspace}_4 \right)$, ${\mathbf{B}}^{(2)}_{4\times 4} = \frac{(\log n)^{4/3}}{n}{\ensuremath{\boldsymbol{J}}\xspace}_4$, ${\mathbf{B}}^{(t)}_{4\times 4} = \frac{(\log n)^{4/3}}{nT}{\ensuremath{\boldsymbol{J}}\xspace}_4$ for $t=3, \ldots, T$, ${\ensuremath{\boldsymbol{\pi}}\xspace} = \frac{1}{4}{\ensuremath{\boldsymbol{1}}\xspace}_{4\times 1}$, and $n=5000$. We compare the algorithms (i), (ii), (iv), and (v). The results on average NMI are given in Figure \[fig\_msbm\_mdcbm\_disT\](a).
3\. *Changing number of nodes ($n$) for multilayer DCBM:* We vary node size $n$ from $2000$ to $10000$ with other parameters being $K=4$, ${\mathbf{B}}^{(1)}_{4\times 4} = \frac{(\log n)^{3/2}}{n}\left({\ensuremath{\boldsymbol{J}}\xspace}_4 - {\ensuremath{\boldsymbol{I}}\xspace}_4 \right)$, ${\mathbf{B}}^{(2)}_{4\times 4} = \frac{(\log n)^{3/2}}{n}{\ensuremath{\boldsymbol{J}}\xspace}_4$, ${\mathbf{B}}^{(t)}_{4\times 4} = \frac{(\log n)^{3/2}}{nT}{\ensuremath{\boldsymbol{J}}\xspace}_4$ for $t=3, \ldots, T$, $\psi_i \stackrel{iid}{\sim} U(0.5, 1)$ for $i \in [n]$, and $T=11$. We compare the algorithms (ii), (iii), (iv), and (vi). The results on average NMI are given in Figure \[fig\_mdcbm\_dis\].
4\. *Changing number of layers ($T$) for multilayer DCBM:* We vary number of layers $T$ from $5$ to $55$ with other parameters being $K=4$, ${\mathbf{B}}^{(1)}_{4\times 4} = \frac{(\log n)^{3/2}}{n}\left({\ensuremath{\boldsymbol{J}}\xspace}_4 - {\ensuremath{\boldsymbol{I}}\xspace}_4 \right)$, ${\mathbf{B}}^{(2)}_{4\times 4} = \frac{(\log n)^{3/2}}{n}{\ensuremath{\boldsymbol{J}}\xspace}_4$, ${\mathbf{B}}^{(t)}_{4\times 4} = \frac{(\log n)^{3/2}}{nT}{\ensuremath{\boldsymbol{J}}\xspace}_4$ for $t=3, \ldots, T$, ${\ensuremath{\boldsymbol{\pi}}\xspace} = \frac{1}{4}{\ensuremath{\boldsymbol{1}}\xspace}_{4\times 1}$, $\psi_i \stackrel{iid}{\sim} U(0.5, 1)$ for $i \in [n]$, and $n=5000$. We compare the algorithms (ii), (iii), (iv), and (vi). The results on average NMI are given in Figure \[fig\_msbm\_mdcbm\_disT\](b).
*Scenario 3:* In this scenario, we consider a situation where ${\mathcal{B}}=\{B^{(t)}: t\in [T]\}$ is a stochastic process. For each $t$ ($t\in [T]$), given, $B^{(t)}$, we simulate multilayer networks under the framework of and of \[sec\_model\]. We consider two experiments under this scenario. Each experiment is repeated 25 times and the results are averages over the repetitions.
1\. *Changing number of layers ($T$) for multilayer SBM:* We vary number of layers $T$ from $5$ to $55$ with other parameters being $K=4$, ${\ensuremath{\boldsymbol{\pi}}\xspace} = \frac{1}{4}{\ensuremath{\boldsymbol{1}}\xspace}_{4\times 1}$, $n=5000$, ${\mathbf{B}}^{(t)}_{4\times 4} = \frac{1}{n} \left(2{\ensuremath{\boldsymbol{I}}\xspace}_4 + {\underline{b}}_t{\ensuremath{\boldsymbol{J}}\xspace}_4 \right)$ for $t \in [5]$ where ${\underline{b}}_t = -7 + 7(t-1)/T$ for $t \in [T]$, and $${\mathbf{B}}^{(t)}_{i,j} = \frac{20}{n(1+\exp(n{\mathbf{B}}^{(t-5)}_{i, j} + {\underline{{\varepsilon}}}_t))}\ \ \ \text{where, } {\underline{{\varepsilon}}}_t \stackrel{iid}{\sim} N(0, 0.05).$$ We compare the algorithms (i), (ii), and (v). The results on average NMI are given in Figure \[fig\_msbm\_mdcbm\_depT\](a).
2\. *Changing number of layers ($T$) for multilayer DCBM:* We vary number of layers $T$ from $5$ to $55$ with other parameters being same as previous experiment except ${\underline{b}}_t = -12 + 12(t-1)/T$ for $t \in [T]$. We compare the algorithms (ii), (iii), and (vi). The results on average NMI are given in Figure \[fig\_msbm\_mdcbm\_depT\](b).
We see that Algorithm 1 works better in recovering community labels in all the scenarios compared to other algorithms for networks generated from multilayer stochastic block models as either $n \to \infty$ or $T \to \infty$. We also see that Algorithm 2 works better in recovering community labels in all the scenarios compared to other algorithms for networks generated from multilayer degree-corrected block models as either $n \to \infty$ or $T \to \infty$. Algorithm 1 and Algorithm 2 is also shown to recover community labels under dependent network layers. Algorithm 3 also recovers correct number of communities as $n \to \infty$. The simulation results are in concert with the theoretical results in Theorem \[ConsSum1\], Theorem \[ConsSum2\], Theorem \[Gen B\^t\], and Theorem \[Khat\_thm\].
Conclusion and Future Works
===========================
In this paper, we consider the problem of community detection for multi-relational networks with constant community memberships and changing connectivity matrices. We consider spectral clustering on aggregate versions of squared adjacency matrices. It is shown in the paper that under multilayer stochastic block model and multilayer degree-corrected block model, spectral clustering based on the sum of squared adjacency matrices has guarantee of consistent community recovery under weakest conditions on the degree parameters of the block models. We establish our claims both theoretically and empirically in the paper.
Future Works
------------
Several extensions are possible from the current work. Some possible extensions of our work will include considering the cases where community memberships change with layers and the dependence of the network layers are more general, such as, dependence of probability of edge formation of a specific network layer on edge structure and community memberships of other network layers. Methods for community recovery with theoretical guarantee are quite rare for general multilayer networks and it would be good to investigate such problems in later works.
Acknowledgements
================
We thank Peter Bickel, Paul Bourgade, Ofer Zeitouni and Harrison Zhou for helpful discussions and comments.
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---
abstract: 'We study three wave function optimization methods based on energy minimization in a variational Monte Carlo framework: the Newton, linear and perturbative methods. In the Newton method, the parameter variations are calculated from the energy gradient and Hessian, using a reduced variance statistical estimator for the latter. In the linear method, the parameter variations are found by diagonalizing a nonsymmetric estimator of the Hamiltonian matrix in the space spanned by the wave function and its derivatives with respect to the parameters, making use of a strong zero-variance principle. In the less computationally expensive perturbative method, the parameter variations are calculated by approximately solving the generalized eigenvalue equation of the linear method by a nonorthogonal perturbation theory. These general methods are illustrated here by the optimization of wave functions consisting of a Jastrow factor multiplied by an expansion in configuration state functions (CSFs) for the C$_2$ molecule, including both valence and core electrons in the calculation. The Newton and linear methods are very efficient for the optimization of the Jastrow, CSF and orbital parameters. The perturbative method is a good alternative for the optimization of just the CSF and orbital parameters. Although the optimization is performed at the variational Monte Carlo level, we observe for the C$_2$ molecule studied here, and for other systems we have studied, that as more parameters in the trial wave functions are optimized, the diffusion Monte Carlo total energy improves monotonically, implying that the nodal hypersurface also improves monotonically.'
author:
- Julien Toulouse
- 'C. J. Umrigar'
bibliography:
- 'biblio.bib'
title: Optimization of quantum Monte Carlo wave functions by energy minimization
---
Introduction {#sec:intro}
============
Quantum Monte Carlo (QMC) methods (see e.g. Refs. ) constitute an alternative to standard [*ab initio*]{} methods of quantum chemistry for accurate calculations of the electronic structure of atoms, molecules and solids. The two most commonly used variants, variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC), rely on an explicitly correlated trial wave function, generally consisting for atoms and molecules of a Jastrow factor multiplied by a short expansion in configuration state functions (CSFs), each consisting of a linear combination of Slater determinants, a form capable of encompassing most of the electron correlation effects. To fully benefit from the considerable flexibility in the form of the wave function, it is crucial to be able to efficiently optimize the parameters in these wave functions.
Variance minimization in correlated sampling [@UmrWilWil-PRL-88; @UmrWilWil-INC-88; @Umr-IJQC-89] has become the most frequently used method in QMC for optimizing wave functions because it is far more efficient than [*straightforward*]{} energy minimization on a finite Monte Carlo sample. However, while the method works relatively well for the optimization of the Jastrow factor, it is much less effective for the optimization of the determinantal part of the wave function (though still possible [@UmrWilWil-PRL-88; @FilUmr-JCP-96; @HuaUmrNig-JCP-97]). Further, there is some evidence that energy-optimized wave functions give on average better expectation values for other observables than variance-optimized ones (see, e.g., Refs. ). As a result, a lot of effort has recently been devoted to developing efficient methods for the optimization of QMC wave functions by energy minimization. On the other hand, it should be mentioned that variance-minimized wave functions often have a smaller time-step error in DMC.
We now summarize some of the major approaches that have been proposed for energy minimization in VMC. The most efficient method to minimize the energy with respect to linear parameters, such as the CSF coefficients, is to solve the associated generalized eigenvalue equation using a non-symmetric estimator of the Hamiltonian matrix [@NigMel-PRL-01]. The energy fluctuation potential (EFP) method [@Fah-INC-99; @FilFah-JCP-00; @PreBevFah-PRB-02; @SchFah-JCP-02; @SchFil-JCP-04] is very efficient for optimizing some nonlinear parameters and has been applied very successfully to the optimization of the orbitals [@FilFah-JCP-00; @SchFil-JCP-04] and CSF coefficients [@SchFah-JCP-02; @SchFil-JCP-04]. It has also been applied to the optimization of Jastrow factors in periodic solids [@PreBevFah-PRB-02]. The perturbative EFP method, a simplification of the EFP method, retains the same convergence rate for the optimization of the orbitals and CSF coefficients while decreasing the computational cost [@SceFil-PRB-06]. The stochastic reconfiguration (SR) method, originally developed for lattice systems [@Sor-PRB-01], has been applied to the full optimization of atomic and molecular wave functions consisting of an antisymmetrized geminal power part multiplied by a Jastrow factor [@CasSor-JCP-03; @CasAttSor-JCP-04]. It is related to the perturbative EFP method and is simpler but less efficient [@SchFil-JCP-04; @SceFil-PRB-06]. The Newton method is a conceptually simple and general optimization method but a straightforward implementation of it in QMC is rather inefficient [@LinZhaRap-JCP-00; @LeeMelRap-JCP-05]. However, an improved version of it, making use of a reduced variance estimator of the Hessian matrix [@UmrFil-PRL-05], is very efficient for the optimization of Jastrow factors. Another modified version of the Newton method with an approximate Hessian, named stochastic reconfiguration with Hessian acceleration (SRH), has been applied to lattice models [@Sor-PRB-05].
In this work, we investigate the three best energy minimization methods for the optimization of the Jastrow, CSF and orbital parameters of QMC wave functions: the Newton, linear and perturbative methods. The Newton method has already been applied very successfully to the optimization of Jastrow factors by Umrigar and Filippi [@UmrFil-PRL-05], and in this paper it is also applied to the optimization of the determinantal part of the wave function. The linear method is an extension of the zero-variance generalized eigenvalue equation approach of Nightingale and Melik-Alaverdian [@NigMel-PRL-01] to arbitrary nonlinear parameters: at each step of the iterative procedure, the wave function is linearized with respect to the parameters and the optimal values of the parameters are found by diagonalizing the Hamiltonian in the space spanned by the current wave function and its derivatives with respect to the parameters. This method is briefly presented in Ref. . The perturbative method coincides with the perturbative EFP method of Scemama and Filippi [@SceFil-PRB-06] for the optimization of the CSF and orbital parameters. Here, we put this approach on more general grounds by recasting it as a simplification of the linear method where the generalized eigenvalue equation is solved approximately by a nonorthogonal perturbation theory. The Newton and linear methods are very efficient for the optimization of the Jastrow, CSF and orbital parameters. The perturbative method is a good alternative for the optimization of just the CSF and orbital parameters.
The paper is organized as follows. In Sec. \[sec:wfparam\], the parametrization of the trial wave function is presented. The energy minimization procedures are discussed in Sec. \[sec:emin\], and their realizations in VMC are discussed in Sec \[sec:vmc\]. Sec. \[sec:compdetails\] contains computational details of the calculations performed on the C$_2$ molecule to test the optimization methods, and in Sec. \[sec:results\] we present the results. Sec. \[sec:conclusion\] contains our conclusions.
Hartree atomic units (Ha) are used throughout this work.
Wave function parametrization and derivatives {#sec:wfparam}
=============================================
We begin by describing the form of the wave function used, the actual parametrization chosen for the optimization, and the corresponding derivatives of the wave function with respect to the parameters.
Form of the wave function
-------------------------
We use an $N$-electron wave function of the usual Jastrow-Slater form that is denoted at each iteration of the optimization procedure by $$\begin{aligned}
{\ensuremath{\vert \Psi_0 \rangle}} = \hat{J}(\bm{\alpha}^0) {\ensuremath{\vert \Phi_0 \rangle}},
\label{Psi0}\end{aligned}$$ where $\hat{J}(\bm{\alpha}^0)$ is a Jastrow operator depending on the current parameters $\alpha_i^0$ and ${\ensuremath{\vert \Phi_0 \rangle}}$ is a multi-determinantal wave function. For notational convenience, we assume that the wave function ${\ensuremath{\vert \Psi_0 \rangle}}$ is always normalized to unity, i.e. ${\ensuremath{\langle \Psi_0 \vert \Psi_0 \rangle}}=1$. In practice, ${\ensuremath{\vert \Psi_0 \rangle}}$ can have arbitrary normalization.
The wave function ${\ensuremath{\vert \Phi_0 \rangle}}$ is a linear combination of $N_{{\ensuremath{\text{CSF}}}}$ orthonormal configuration state functions (CSFs), ${\ensuremath{\vert C_{I} \rangle}}$, with current coefficients $c_{I}^{0}$, $$\begin{aligned}
{\ensuremath{\vert \Phi_0 \rangle}} = \sum_{I=1}^{N_{{\ensuremath{\text{CSF}}}}} c_{I}^{0} {\ensuremath{\vert C_{I} \rangle}}.\end{aligned}$$ Each CSF is a short linear combination of products of spin-up and spin-down Slater determinants, ${\ensuremath{\vert D_{{\ensuremath{\mathbf{k}}}}^\uparrow \rangle}}$ and ${\ensuremath{\vert D_{{\ensuremath{\mathbf{k}}}}^\downarrow \rangle}}$, ${\ensuremath{\vert C_{I} \rangle}} = \sum_{{\ensuremath{\mathbf{k}}}} d_{I,{\ensuremath{\mathbf{k}}}}
{\ensuremath{\vert D_{{\ensuremath{\mathbf{k}}}}^{\uparrow} \rangle}} {\ensuremath{\vert D_{{\ensuremath{\mathbf{k}}}}^{\downarrow} \rangle}}$, where the coefficients $d_{I,{\ensuremath{\mathbf{k}}}}$ are fully determined by the spatial and spin symmetries of the state considered (see, e.g., Ref. ). The use of CSFs is important to decrease the number of coefficients to be optimized and to ensure the correct symmetry of the wave function after optimization in the presence of statistical noise. The $N_{\uparrow}$-electron and $N_{\downarrow}$-electron spin-assigned Slater determinants are generated from a set of current orthonormal orbitals, ${\ensuremath{\vert D_{{\ensuremath{\mathbf{k}}}}^\uparrow \rangle}} = \hat{a}^{\dag}_{k_1\uparrow}
\hat{a}^{\dag}_{k_2\uparrow} \cdots \, \hat{a}^{\dag}_{k_{N_\uparrow}
\uparrow} {\ensuremath{\vert {\ensuremath{\text{vac}}}\rangle}}$ and ${\ensuremath{\vert D_{{\ensuremath{\mathbf{k}}}}^\downarrow \rangle}} =
\hat{a}^{\dag}_{k_{N_\uparrow+1} \downarrow}
\hat{a}^{\dag}_{k_{N_\uparrow+2} \downarrow} \cdots \,
\hat{a}^{\dag}_{k_N \downarrow} {\ensuremath{\vert {\ensuremath{\text{vac}}}\rangle}}$, where $\hat{a}^{\dag}_{k
\, \sigma}$ (with $\sigma=\uparrow, \downarrow$) is the fermionic creation operator for the spatial orbital ${\ensuremath{\vert \phi_{k}^0 \rangle}}$ in the spin-$\sigma$ determinant, and ${\ensuremath{\vert {\ensuremath{\text{vac}}}\rangle}}$ is the vacuum state of second quantization. The (occupied and virtual) spatial orbitals are written as linear combinations of $N_{{\ensuremath{\text{bas}}}}$ basis functions ${\ensuremath{\vert \chi_{\mu} \rangle}}$ (e.g., Slater or Gaussian functions) with current coefficients $\lambda_{k,\mu}^{0}$, ${\ensuremath{\vert \phi_{k}^0 \rangle}} =
\sum_{\mu=1}^{N_{{\ensuremath{\text{bas}}}}} \lambda_{k,\mu}^{0} {\ensuremath{\vert \chi_{\mu} \rangle}}$.
The $N$-electron Jastrow operator, $\hat{J}(\bm{\alpha}^0)$, is defined by its matrix elements in the $N$-electron position basis ${\ensuremath{\vert {\ensuremath{\mathbf{R}}} \rangle}}={\ensuremath{\vert {\ensuremath{\mathbf{r}}}_1,{\ensuremath{\mathbf{r}}}_2,...,{\ensuremath{\mathbf{r}}}_N \rangle}}$ $$\begin{aligned}
{\ensuremath{\langle {\ensuremath{\mathbf{R}}} \vert}} \hat{J}(\bm{\alpha}^0) {\ensuremath{\vert {\ensuremath{\mathbf{R}}}' \rangle}} = J(\bm{\alpha}^0;{\ensuremath{\mathbf{R}}}) \delta({\ensuremath{\mathbf{R}}}-{\ensuremath{\mathbf{R}}}'),\end{aligned}$$ where $J(\bm{\alpha}^0;{\ensuremath{\mathbf{R}}})$ is the spin-assigned Jastrow factor, a real positive function of ${\ensuremath{\mathbf{R}}}$ which is symmetric under the exchange of two same-spin electrons. Its action on an arbitrary $N$-electron state ${\ensuremath{\vert \Phi \rangle}}$ is given by ${\ensuremath{\langle {\ensuremath{\mathbf{R}}} \vert}}
\hat{J}(\bm{\alpha}^0) {\ensuremath{\vert \Phi \rangle}} = J(\bm{\alpha}^0;{\ensuremath{\mathbf{R}}}) \Phi({\ensuremath{\mathbf{R}}}) $ where $\Phi({\ensuremath{\mathbf{R}}}) = {\ensuremath{\langle {\ensuremath{\mathbf{R}}} \vert \Phi \rangle}}$. The Jastrow operator is Hermitian, $\hat{J}(\bm{\alpha}^0)^{\dag} =
\hat{J}(\bm{\alpha}^0)$. We use flexible Jastrow factors consisting of the exponential of the sum of electron-nucleus, electron-electron and electron-electron-nucleus terms, written as systematic polynomial or Padé expansions [@Umr-UNP-XX] (see also Refs. ).
Wave function parametrization
-----------------------------
We want to optimize the Jastrow parameters $\alpha_i$, the CSF coefficients $c_I$ and the orbital coefficients $\lambda_{k,\mu}$. Some parameters in the Jastrow factor are fixed by imposing the electron-nucleus and electron-electron cusp conditions [@Kat-CPAM-57] on the wave function; the other Jastrow parameters are varied freely. Due to the arbitrariness of the overall normalization of the wave function, only $N_{{\ensuremath{\text{CSF}}}}-1$ CSF coefficients need be varied, e.g., the coefficient of the first configuration can be kept fixed. The situation is more involved for the orbital coefficients which are not independent due to the invariance properties of determinants under elementary row operations. To easily retain only unconstrained, nonredundant orbital parameters in the optimization, it is convenient to vary the orbital coefficients by performing rotations among the (occupied and virtual) orbitals with a unitary operator parametrized as an exponential of an anti-Hermitian operator. This parametrization is used in multi-configuration self-consistent-field (MCSCF) calculations (for a recent and general review of MCSCF theory, see Ref. ). More specifically, we use the following parametrization of the wave function depending on $N^{\ensuremath{\text{opt}}}_{\ensuremath{\text{Jas}}}$ Jastrow parameters $\bm{\alpha}$, $N^{\ensuremath{\text{opt}}}_{\ensuremath{\text{CSF}}}= N_{{\ensuremath{\text{CSF}}}}-1$ free CSF coefficients ${\ensuremath{\mathbf{c}}}$ ($c_1$ is fixed), and $N^{\ensuremath{\text{opt}}}_{\ensuremath{\text{orb}}}$ orbital rotation parameters $\bm{\kappa}$ $$\begin{aligned}
{\ensuremath{\vert \Psi(\bm{\alpha},{\ensuremath{\mathbf{c}}},\bm{\kappa}) \rangle}} = \hat{J}(\bm{\alpha}) e^{\hat{\kappa}(\bm{\kappa})} \sum_{I=1}^{N_{\ensuremath{\text{CSF}}}} c_I {\ensuremath{\vert C_I \rangle}},
\label{Psi}\end{aligned}$$ where $e^{\hat{\kappa}(\bm{\kappa})}$ is the unitary operator that performs rotations in orbital space (see, e.g., Refs. ). More elaborate parametrizations of the CSF coefficients, such as a unitary parametrization [@Dal-CPL-79], are often used in MCSCF theory (see, e.g., Ref. ), but we have not found any decisive advantage to using them for our purpose.
The rotations in orbital space are generated by the anti-Hermitian real singlet orbital excitation operator [@DalJor-JCP-78] $$\begin{aligned}
\hat{\kappa}(\bm{\kappa}) = \sum_{k<l} \kappa_{kl} \, \hat{E}_{kl}^{-},
\label{}\end{aligned}$$ where the sum is over all nonredundant orbital pairs, $\hat{E}_{kl}^{-} = \hat{E}_{kl} - \hat{E}_{lk}$, and $\hat{E}_{kl} =
\hat{a}_{k \uparrow}^{\dag} \hat{a}_{l \uparrow} + \hat{a}_{k
\downarrow}^{\dag} \hat{a}_{l \downarrow} $ is the singlet excitation operator from orbital $l$ to orbital $k$. In Eq. (\[Psi\]), the action of the operator $e^{\hat{\kappa}(\bm{\kappa})}$ is to rotate each occupied orbital in the Slater determinants as $$\begin{aligned}
{\ensuremath{\vert \phi_k \rangle}} = e^{\hat{\kappa}(\bm{\kappa})} {\ensuremath{\vert \phi_k^{0} \rangle}} = \sum_{l} (e^{\bm{\kappa}})_{kl} {\ensuremath{\vert \phi_l^{0} \rangle}},
\label{phik}\end{aligned}$$ where the sum is over all (occupied and virtual) orbitals, and $(e^{\bm{\kappa}})_{kl}$ are the elements of the orthogonal matrix $e^{\bm{\kappa}}$ constructed from the real anti-symmetric matrix $\bm{\kappa}$ with elements $\kappa_{kl}$. More generally, any unitary matrix can be written as an exponential of an anti-Hermitian matrix, the off-diagonal upper triangular part of the anti-Hermitian matrix realizing a nonredundant parametrization of the unitary matrix. To maintain the orthonormality of the entire set of orbitals, the operator $e^{\hat{\kappa}(\bm{\kappa})}$ is applied to the virtual orbitals as well. For a single Slater determinant wave function, the orbitals can be partitioned into three sets referred to as *closed* (i.e., doubly occupied), *open* (i.e., singly occupied) and *virtual* (i.e., unoccupied). The nonredundant excitations to consider are then: closed $\to$ open, closed $\to$ virtual and open $\to$ virtual. For a multiconfiguration wave function, the orbitals can be partitioned into three sets referred to as *inactive* (i.e., occupied in all determinants), *active* (i.e., occupied in some determinants and unoccupied in the others) and *secondary* (i.e., unoccupied in all determinants). For a multiconfiguration complete active space (CAS) wave function [@RooTaySie-CP-80], the nonredundant excitations are then: inactive $\to$ active, inactive $\to$ secondary and active $\to$ secondary. For a single-determinant and multi-determinant CAS wave function, the action of the reverse excitation from orbital $k$ to $l$ ($\hat{E}_{lk}$) in $\hat{E}_{kl}^{-}=\hat{E}_{kl} - \hat{E}_{lk}$ is always zero. For a general multiconfiguration wave function (not CAS), some active $\to$ active excitations must also be included. Consequently, the action of the reverse excitation $\hat{E}_{lk}$ in $\hat{E}_{kl}^{-}=\hat{E}_{kl} - \hat{E}_{lk}$ does not generally vanish. Only excitations between orbitals of the same spatial symmetry have to be considered. In the super configuration interaction approach [@GreCha-CPL-71] where the orbitals are optimized by adding the single excitations of the (multiconfiguration) reference wave function to the variational space, pioneered in QMC by Filippi and coworkers [@SchFil-JCP-04; @SceFil-PRB-06], an alternative linear parametrization of the orbital space is chosen, ${\ensuremath{\vert \phi_k \rangle}} = \left( \hat{1} + \hat{\kappa}(\bm{\kappa}) \right) {\ensuremath{\vert \phi_k^{0} \rangle}}$, instead of the unitary parametrization of Eq. (\[phik\]). In that case, the optimized orbitals are not orthonormal.
In the following, we will collectively refer to the Jastrow, CSF and orbital parameters as ${\ensuremath{\mathbf{p}}}=(\bm{\alpha},{\ensuremath{\mathbf{c}}},\bm{\kappa})$. The wave function of Eq. (\[Psi0\]) is thus simply ${\ensuremath{\vert \Psi_0 \rangle}} = {\ensuremath{\vert \Psi({\ensuremath{\mathbf{p}}}^0) \rangle}}$ where ${\ensuremath{\mathbf{p}}}^0=(\bm{\alpha}^0,{\ensuremath{\mathbf{c}}}^0,\bm{\kappa}^0=\bm{0})$ are the current parameters. We will designate by $N^{\ensuremath{\text{opt}}}= N^{\ensuremath{\text{opt}}}_{\ensuremath{\text{Jas}}}+ N^{\ensuremath{\text{opt}}}_{\ensuremath{\text{CSF}}}+ N^{\ensuremath{\text{opt}}}_{\ensuremath{\text{orb}}}$ the total number of parameters to be optimized.
First-order wave function derivatives {#sec:deriv1}
-------------------------------------
We now give the expressions for the first-order derivatives of the wave function ${\ensuremath{\vert \Psi({\ensuremath{\mathbf{p}}}) \rangle}}$ of Eq. (\[Psi\]) with respect to the parameters $p_i$ at ${\ensuremath{\mathbf{p}}}={\ensuremath{\mathbf{p}}}^0$ $$\begin{aligned}
{\ensuremath{\vert \Psi_i \rangle}} = \left( \frac{\partial {\ensuremath{\vert \Psi({\ensuremath{\mathbf{p}}}) \rangle}}}{\partial p_i} \right)_{{\ensuremath{\mathbf{p}}}={\ensuremath{\mathbf{p}}}^0},
\label{Psii}\end{aligned}$$ which collectively designate the derivatives with respect to the Jastrow parameters $$\begin{aligned}
{\ensuremath{\vert \Psi_{\alpha_i} \rangle}}= \frac{\partial \hat{J}(\bm{\alpha}^0)}{\partial \alpha_i} {\ensuremath{\vert \Phi_0 \rangle}},
\label{}\end{aligned}$$ with respect to the CSF parameters $$\begin{aligned}
{\ensuremath{\vert \Psi_{c_I} \rangle}} &=& \hat{J}(\bm{\alpha}^0) {\ensuremath{\vert C_I \rangle}},
\label{}\end{aligned}$$ and with respect to the orbital parameters $$\begin{aligned}
{\ensuremath{\vert \Psi_{\kappa_{kl}} \rangle}} &=& \hat{J}(\bm{\alpha}^0) \hat{E}_{kl}^{-} {\ensuremath{\vert \Phi_0 \rangle}}.
\label{}\end{aligned}$$ The first-order orbital derivatives are thus generated by the single excitations of orbitals out of the state ${\ensuremath{\vert \Phi_0 \rangle}}$.
Second-order wave function derivatives
--------------------------------------
The second-order derivatives with respect to the parameters $p_i$ at ${\ensuremath{\mathbf{p}}}={\ensuremath{\mathbf{p}}}^0$, which are needed only for the Newton method, are $$\begin{aligned}
{\ensuremath{\vert \Psi_{ij} \rangle}}= \left( \frac{\partial^2 {\ensuremath{\vert \Psi({\ensuremath{\mathbf{p}}}) \rangle}}}{\partial p_i \partial p_j} \right)_{{\ensuremath{\mathbf{p}}}={\ensuremath{\mathbf{p}}}^0},
\label{Psiij}\end{aligned}$$ which collectively designate the Jastrow-Jastrow derivatives $$\begin{aligned}
{\ensuremath{\vert \Psi_{\alpha_i\alpha_j} \rangle}}= \frac{\partial^2 \hat{J}(\bm{\alpha}^0)}{\partial \alpha_i \partial \alpha_j} {\ensuremath{\vert \Phi_0 \rangle}},
\label{}\end{aligned}$$ the Jastrow-CSF derivatives $$\begin{aligned}
{\ensuremath{\vert \Psi_{\alpha_i c_I} \rangle}}= \frac{\partial \hat{J}(\bm{\alpha}^0)}{\partial \alpha_i} {\ensuremath{\vert C_I \rangle}},
\label{}\end{aligned}$$ the Jastrow-orbital derivatives $$\begin{aligned}
{\ensuremath{\vert \Psi_{\alpha_i \kappa_{kl}} \rangle}}= \frac{\partial \hat{J}(\bm{\alpha}^0)}{\partial \alpha_i} \hat{E}_{kl}^{-} {\ensuremath{\vert \Phi_0 \rangle}},
\label{}\end{aligned}$$ the CSF-orbital derivatives $$\begin{aligned}
{\ensuremath{\vert \Psi_{c_I \kappa_{kl}} \rangle}} &=& \hat{J}(\bm{\alpha}^0) \hat{E}_{kl}^{-} {\ensuremath{\vert C_I \rangle}},
\label{}\end{aligned}$$ and the orbital-orbital derivatives $$\begin{aligned}
{\ensuremath{\vert \Psi_{\kappa_{kl} \kappa_{mn}} \rangle}} &=& \hat{J}(\bm{\alpha}^0) \hat{E}_{kl}^{-} \hat{E}_{mn}^{-} {\ensuremath{\vert \Phi_0 \rangle}}.
\label{}\end{aligned}$$ Notice that the wave function form of Eq. (\[Psi\]) is linear in the CSF parameters and therefore the CSF-CSF derivatives are zero, ${\ensuremath{\vert \Psi_{c_I c_J} \rangle}} = 0$. The orbital-orbital derivatives correspond to double excitations of orbitals out of the state ${\ensuremath{\vert \Phi_0 \rangle}}$. Since we usually start the optimization with reasonably good initial orbitals coming from a standard MCSCF calculation we set these second derivatives to zero, ${\ensuremath{\vert \Psi_{\kappa_{kl} \kappa_{mn}} \rangle}} = 0$, in order to reduce the computational cost per iteration during Newton minimization. Nevertheless, it takes only a few steps to optimize the orbitals as discussed in Sec. \[sec:results\].
Energy minimization procedures {#sec:emin}
==============================
In this section, we present the three methods investigated in this work to minimize the variational energy with respect to the wave function parameters ${\ensuremath{\mathbf{p}}}$ $$\begin{aligned}
E = \min_{{\ensuremath{\mathbf{p}}}} E({\ensuremath{\mathbf{p}}}),
\label{Evar}\end{aligned}$$ where $E({\ensuremath{\mathbf{p}}}) = {\ensuremath{\langle \Psi({\ensuremath{\mathbf{p}}}) \vert}} \hat{H} {\ensuremath{\vert \Psi({\ensuremath{\mathbf{p}}}) \rangle}} /{\ensuremath{\langle \Psi({\ensuremath{\mathbf{p}}}) \vert \Psi({\ensuremath{\mathbf{p}}}) \rangle}}$ and $\hat{H} =\hat{T} + \hat{W}_{ee} + \hat{V}_{ne}$ is the electronic Hamiltonian, including the kinetic, electron-electron interaction and nuclei-electron interaction terms. The Hamiltonian can also include a nonlocal pseudopotential, enabling one to avoid the explicit treatment of core electrons. The energy corresponding to the current parameters ${\ensuremath{\mathbf{p}}}^0$ will be denoted by $E_0 = E({\ensuremath{\mathbf{p}}}^0)$.
Newton optimization method
--------------------------
The Newton method was first applied to the optimization of QMC wave functions by Rappe and coworkers [@LinZhaRap-JCP-00; @LeeMelRap-JCP-05]. It has been considerably improved by Umrigar and Filippi [@UmrFil-PRL-05], and by Sorella [@Sor-PRB-05], by making use of a lower variance statistical estimator of the Hessian matrix and by employing stabilization techniques. In Ref. the correct Hessian was used, whereas in Ref. an approximate Hessian, which reduces to the exact Hessian for parameters that are linear in the exponent, was used. We now recall the basic working equations.
The energy $E({\ensuremath{\mathbf{p}}})$ is expanded to second-order in the parameters ${\ensuremath{\mathbf{p}}}$ around ${\ensuremath{\mathbf{p}}}^0$ $$\begin{aligned}
E^{[2]} ({\ensuremath{\mathbf{p}}}) = E_0 + \sum_{i=1}^{N^{\ensuremath{\text{opt}}}} g_i \Delta p_i + \frac{1}{2} \sum_{i=1}^{N^{\ensuremath{\text{opt}}}} \sum_{j=1}^{N^{\ensuremath{\text{opt}}}} h_{ij} \Delta p_i \Delta p_j,
\nonumber\\
\label{E2}\end{aligned}$$ where the sums are over all the parameters to be optimized, $\Delta p_i = p_i - p_i^0$ are the components of the vector of parameter variations $\Delta {\ensuremath{\mathbf{p}}}$, $$\begin{aligned}
g_i = \left( \frac{\partial E ({\ensuremath{\mathbf{p}}})}{\partial p_i} \right)_{{\ensuremath{\mathbf{p}}} = {\ensuremath{\mathbf{p}}}^0},\end{aligned}$$ are the components of the energy gradient vector ${\ensuremath{\mathbf{g}}}$, and $$\begin{aligned}
h_{ij} = \left( \frac{\partial^2 E ({\ensuremath{\mathbf{p}}})}{\partial p_i \partial p_j} \right)_{{\ensuremath{\mathbf{p}}} = {\ensuremath{\mathbf{p}}}^0},\end{aligned}$$ are the elements of the energy Hessian matrix ${\ensuremath{\mathbf{h}}}$. Imposition of the stationarity condition on the expanded energy expression, $\partial
E^{[2]} ({\ensuremath{\mathbf{p}}}) /\partial p_i = 0$, gives the following standard solution for the parameter variations $$\begin{aligned}
\Delta {\ensuremath{\mathbf{p}}}= - {\ensuremath{\mathbf{h}}}^{-1} \cdot {\ensuremath{\mathbf{g}}},
\label{deltaalphanewton}\end{aligned}$$ where ${\ensuremath{\mathbf{h}}}^{-1}$ is the inverse of the Hessian matrix. In practice, the energy gradient and Hessian are calculated in VMC with the statistical estimators given in Sec. \[sec:gradhess\], yielding the parameter variations $\Delta {\ensuremath{\mathbf{p}}}$ of Eq. (\[deltaalphanewton\]) that are used to update the current wave function, ${\ensuremath{\vert \Psi_0 \rangle}} \to {\ensuremath{\vert \Psi({\ensuremath{\mathbf{p}}}^0 + \Delta
{\ensuremath{\mathbf{p}}}) \rangle}}$. It simply remains to iterate until convergence.
[*Stabilization.*]{} As explained in Ref. , the stabilization of the Newton method is achieved by adding a positive constant, $a_{{\ensuremath{\text{diag}}}} \geq 0$, to the diagonal of the Hessian matrix ${\ensuremath{\mathbf{h}}}$, i.e., $h_{ij} \to h_{ij} + a_{{\ensuremath{\text{diag}}}} \delta_{ij}$. As $a_{{\ensuremath{\text{diag}}}}$ is increased, the parameter variations $\Delta {\ensuremath{\mathbf{p}}}$ become smaller and rotate from the Newtonian direction to the steepest descent direction. A good value of $a_{{\ensuremath{\text{diag}}}}$ is automatically determined at each iteration by performing three very short Monte Carlo calculations using correlated sampling with wave function parameters obtained with three trial values of $a_{{\ensuremath{\text{diag}}}}$ and predicting by parabolic interpolation the value of $a_{{\ensuremath{\text{diag}}}}$ that minimizes the energy [@UmrTouFilSorHeg-JJJ-XX], with some bounds imposed. The use of correlated sampling makes it possible to calculate energy differences with much smaller statistical error than the energies themselves. This procedure helps convergence if one is far from the minimum or if the statistical noise is large in the Monte Carlo evaluation of the gradient and Hessian.
We have found that adding in a multiple of the unit matrix to the Hessian as described above works well, but there exist other possible choices of positive definite matrices that could be added in. For instance, Sorella [@Sor-PRB-05] adds in a multiple of the overlap matrix of the first-order derivatives of the wave function. Another possible choice is a multiple of the Levenberg-Marquardt approximation to the Hessian of the variance of the local energy.
Linear optimization method {#sec:linear}
--------------------------
The most straightforward way to energy-optimize linear parameters in wave functions, such as the CSF parameters, is to diagonalize the Hamiltonian in the variational space that they define, leading to a generalized eigenvalue equation. This has been done in QMC for example in Refs. . The linear method that we present now is an extension of the approach of Ref. to arbitrary nonlinear parameters. This method is also presented in Ref. , using slightly different but equivalent conventions.
For notational convenience, we first introduce the normalized wave function $$\begin{aligned}
{\ensuremath{\vert {\ensuremath{\overline{\Psi}}}({\ensuremath{\mathbf{p}}}) \rangle}} = \frac{{\ensuremath{\vert \Psi({\ensuremath{\mathbf{p}}}) \rangle}}}{\sqrt{{\ensuremath{\langle \Psi({\ensuremath{\mathbf{p}}}) \vert \Psi({\ensuremath{\mathbf{p}}}) \rangle}}}}.
\label{Psin}\end{aligned}$$ The idea is then to expand this normalized wave function ${\ensuremath{\vert {\ensuremath{\overline{\Psi}}}({\ensuremath{\mathbf{p}}}) \rangle}}$ to first-order in the parameters ${\ensuremath{\mathbf{p}}}$ around the current parameters ${\ensuremath{\mathbf{p}}}^0$, $$\begin{aligned}
{\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}({\ensuremath{\mathbf{p}}}) \rangle}} = {\ensuremath{\vert \Psi_0 \rangle}} + \sum_{i=1}^{N^{{\ensuremath{\text{opt}}}}} \Delta p_i \, {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_i \rangle}},
\label{Psib1}\end{aligned}$$ where the wave function at ${\ensuremath{\mathbf{p}}}={\ensuremath{\mathbf{p}}}^0$ is simply ${\ensuremath{\vert {\ensuremath{\overline{\Psi}}}({\ensuremath{\mathbf{p}}}^0) \rangle}} = {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_0 \rangle}} = {\ensuremath{\vert \Psi_0 \rangle}}$ (chosen to be normalized to $1$) and, for $i \geq 1$, ${\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_i \rangle}}$ are the derivatives of ${\ensuremath{\vert {\ensuremath{\overline{\Psi}}}({\ensuremath{\mathbf{p}}}) \rangle}}$ that are orthogonal to ${\ensuremath{\vert \Psi_0 \rangle}}$ $$\begin{aligned}
{\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_i \rangle}}= \left( \frac{\partial {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}({\ensuremath{\mathbf{p}}}) \rangle}}}{\partial p_i} \right)_{{\ensuremath{\mathbf{p}}}={\ensuremath{\mathbf{p}}}^0} = {\ensuremath{\vert \Psi_i \rangle}} - S_{0i} \, {\ensuremath{\vert \Psi_0 \rangle}},\end{aligned}$$ where $S_{0i}= {\ensuremath{\langle \Psi_0 \vert \Psi_i \rangle}}$. The minimization of the energy calculated with this linear wave function $$\begin{aligned}
E_{\ensuremath{\text{lin}}}= \min_{{\ensuremath{\mathbf{p}}}} E_{\ensuremath{\text{lin}}}({\ensuremath{\mathbf{p}}}),
\label{E1min}\end{aligned}$$ where $$\begin{aligned}
E_{\ensuremath{\text{lin}}}({\ensuremath{\mathbf{p}}}) = \frac{{\ensuremath{\langle {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}({\ensuremath{\mathbf{p}}}) \vert}} \hat{H} {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}({\ensuremath{\mathbf{p}}}) \rangle}}}{{\ensuremath{\langle {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}({\ensuremath{\mathbf{p}}}) \vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}({\ensuremath{\mathbf{p}}}) \rangle}}},
\label{Elin}\end{aligned}$$ leads to the stationary condition of the associated Lagrange function $$\begin{aligned}
\nabla_{{\ensuremath{\mathbf{p}}}} \left[ {\ensuremath{\langle {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}({\ensuremath{\mathbf{p}}}) \vert}} \hat{H} {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}({\ensuremath{\mathbf{p}}}) \rangle}} - E_{\ensuremath{\text{lin}}}{\ensuremath{\langle {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}({\ensuremath{\mathbf{p}}}) \vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}({\ensuremath{\mathbf{p}}}) \rangle}} \right] = 0,
\nonumber\\
\label{nablaL}\end{aligned}$$ where $E_{\ensuremath{\text{lin}}}$ acts as a Lagrange multiplier for the normalization condition. The Lagrange function is quadratic in ${\ensuremath{\mathbf{p}}}$ and Eq. (\[nablaL\]) leads to the following generalized eigenvalue equation $$\begin{aligned}
\overline{{\ensuremath{\mathbf{H}}}} \cdot \Delta {\ensuremath{\mathbf{p}}} = E_{\ensuremath{\text{lin}}}\, \overline{{\ensuremath{\mathbf{S}}}} \cdot \Delta {\ensuremath{\mathbf{p}}},
\label{GEQ}\end{aligned}$$ where $\overline{{\ensuremath{\mathbf{H}}}}$ is the matrix of the Hamiltonian $\hat{H}$ in the $(N^{{\ensuremath{\text{opt}}}}+1)$-dimensional basis consisting of the current normalized wave function and its derivatives $\{{\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_0 \rangle}}, {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_1 \rangle}},{\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_2 \rangle}}, \cdots, {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{N^{{\ensuremath{\text{opt}}}}} \rangle}} \}$, with elements $\overline{H}_{ij} = {\ensuremath{\langle {\ensuremath{\overline{\Psi}}}_i \vert}}\hat{H}{\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_j \rangle}}$, $\overline{{\ensuremath{\mathbf{S}}}}$ is the overlap matrix of this $(N^{{\ensuremath{\text{opt}}}}+1)$-dimensional basis, with elements $\overline{S}_{ij} = {\ensuremath{\langle {\ensuremath{\overline{\Psi}}}_i \vert {\ensuremath{\overline{\Psi}}}_j \rangle}}$ (note that $\overline{S}_{00}=1$ and $\overline{S}_{i0}=\overline{S}_{0i}=0$ for $i \geq 1$), and $\Delta {\ensuremath{\mathbf{p}}}$ is the $(N^{{\ensuremath{\text{opt}}}}+1)$-dimensional vector of parameter variations with $\Delta p_0=1$. The linear method consists of solving the generalized eigenvalue equation of Eq. (\[GEQ\]), for the lowest (physically reasonable) eigenvalue and associated eigenvector denoted by $\Delta \overline{{\ensuremath{\mathbf{p}}}}$. The overlap and (non-symmetric) Hamiltonian matrices are computed in VMC using the statistical estimators given in Sec. \[sec:ovlpham\]. Although we focus here on the optimization of the ground-state wave function, solving Eq. (\[GEQ\]) also gives upper bound estimates of excited state energies of states with the same spatial and spin symmetries.
However, there is an arbitrariness in the previously described procedure: we have found the parameter variations $\Delta {\ensuremath{\mathbf{{\ensuremath{\overline{p}}}}}}$ from the expansion of the wave function ${\ensuremath{\vert {\ensuremath{\overline{\Psi}}}({\ensuremath{\mathbf{p}}}) \rangle}}$ of Eq. (\[Psin\]), but another choice of the normalization of the wave function will lead to different parameter variations. To see that, consider a differently-normalized wave function $${\ensuremath{\vert {\ensuremath{\overline{\overline{\Psi}}}}({\ensuremath{\mathbf{p}}}) \rangle}} = N({\ensuremath{\mathbf{p}}}) {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}({\ensuremath{\mathbf{p}}}) \rangle}},
\label{Psibb}$$ where the normalization function $N({\ensuremath{\mathbf{p}}})$ is chosen to satisfy $N({\ensuremath{\mathbf{p}}}^0)=1$ so as to leave unchanged the normalization at ${\ensuremath{\mathbf{p}}}={\ensuremath{\mathbf{p}}}^0$, i.e. ${\ensuremath{\vert {\ensuremath{\overline{\overline{\Psi}}}}({\ensuremath{\mathbf{p}}}^0) \rangle}} = {\ensuremath{\vert \Psi_0 \rangle}}$. The derivatives of this new wave function are $$\begin{aligned}
{\ensuremath{\vert {\ensuremath{\overline{\overline{\Psi}}}}_i \rangle}}= \left( \frac{\partial {\ensuremath{\vert {\ensuremath{\overline{\overline{\Psi}}}}({\ensuremath{\mathbf{p}}}) \rangle}}}{\partial p_i} \right)_{{\ensuremath{\mathbf{p}}}={\ensuremath{\mathbf{p}}}^0} = {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_i \rangle}} + N_{i} \, {\ensuremath{\vert \Psi_0 \rangle}},\end{aligned}$$ where $N_{i} =\left( \partial N({\ensuremath{\mathbf{p}}}) / \partial p_i \right)_{{\ensuremath{\mathbf{p}}}={\ensuremath{\mathbf{p}}}^0}$, i.e. their projections onto the current wave function ${\ensuremath{\vert \Psi_0 \rangle}}$ depend on the normalization. Consequently, the first-order expansion of this new wave function $$\begin{aligned}
{\ensuremath{\vert {\ensuremath{\overline{\overline{\Psi}}}}_{\ensuremath{\text{lin}}}({\ensuremath{\mathbf{p}}}) \rangle}} = {\ensuremath{\vert \Psi_0 \rangle}} + \sum_{i=1}^{N^{{\ensuremath{\text{opt}}}}} \Delta p_i \, {\ensuremath{\vert {\ensuremath{\overline{\overline{\Psi}}}}_i \rangle}},
\label{Psibb}\end{aligned}$$ leads, after optimization of the energy, to different optimal parameter variations $\Delta {\ensuremath{\mathbf{{\ensuremath{\overline{\overline{p}}}}}}}$. As the two wave functions ${\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}({\ensuremath{\mathbf{p}}}) \rangle}}$ and ${\ensuremath{\vert {\ensuremath{\overline{\overline{\Psi}}}}_{\ensuremath{\text{lin}}}({\ensuremath{\mathbf{p}}}) \rangle}}$ lie in the same variational space, they must be proportional after minimization of the energy, which implies that the new optimal parameter variations $\Delta {\ensuremath{\mathbf{{\ensuremath{\overline{\overline{p}}}}}}}$ are actually related to the original optimal parameter variations $\Delta {\ensuremath{\mathbf{{\ensuremath{\overline{p}}}}}}$ by a uniform rescaling $$\begin{aligned}
\Delta {\ensuremath{\mathbf{{\ensuremath{\overline{\overline{p}}}}}}} = \frac{\Delta {\ensuremath{\mathbf{{\ensuremath{\overline{p}}}}}}}{1-\sum_{i=1}^{N^{\ensuremath{\text{opt}}}} N_i \Delta {\ensuremath{\overline{p}}}_i}.
\label{deltapbb}\end{aligned}$$ Any choice of normalization does not necessarily give good parameter variations. For the CSF parameters, it is obvious that the best choice is the normalization of the wave function of Eq. (\[Psi\]) in order to keep the linear dependence on these parameters, ensuring convergence of the linear method in a single step. This is achieved by choosing ${\ensuremath{\vert {\ensuremath{\overline{\overline{\Psi}}}}_i \rangle}}={\ensuremath{\vert \Psi_i \rangle}}$ which gives $$\begin{aligned}
N_i = S_{i0}, \,\,\,\, \text{for linear parameters}.
\label{Nil}\end{aligned}$$ For the nonlinear Jastrow and orbital parameters, several criteria are possible. We have found that a good one is to choose the normalization by imposing that, for the variation of the nonlinear parameters, each derivative ${\ensuremath{\vert {\ensuremath{\overline{\overline{\Psi}}}}_i \rangle}}$ is orthogonal to a linear combination of ${\ensuremath{\vert \Psi_0 \rangle}}$ and ${\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}\rangle}}$, i.e. ${\ensuremath{\langle {\ensuremath{\overline{\overline{\Psi}}}}_i \vert \xi \Psi_0 + (1-\xi) {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}/||{\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}|| \rangle}}=0$, where $\xi$ is a constant between $0$ and $1$, resulting in $$\begin{aligned}
N_i &=& - \frac{(1-\xi) \sum_j^\text{nonlin} \Delta {\ensuremath{\overline{p}}}_j \overline{S}_{ij}}{(1-\xi) +\xi \sqrt{1+\sum_{j,k}^\text{nonlin} \Delta {\ensuremath{\overline{p}}}_j \Delta {\ensuremath{\overline{p}}}_k \overline{S}_{jk} }} ,
\nonumber\\
&&
\text{for nonlinear parameters},
\label{Ninl}\end{aligned}$$ where the sums are only over the nonlinear Jastrow and orbital parameters. The simple choice $\xi=1$ first used by Sorella [@Sor-PRB-01] in the context of the SR method leads in many cases to good parameter variations, but in some cases can result in parameter variations that are too large. The choice $\xi=0$ making the norm of the linear wave function change $\vert\vert {\ensuremath{\overline{\overline{\Psi}}}}_{{\ensuremath{\text{lin}}}} - \Psi_0 \vert\vert$ minimum is safer but in some cases can yield parameter variations that are too small. In those cases, the choice $\xi=1/2$, imposing $\vert\vert {\ensuremath{\overline{\overline{\Psi}}}}_{\ensuremath{\text{lin}}}\vert\vert = \vert\vert \Psi_0 \vert\vert$, avoids both too large and too small parameter variations. In particular, if $\Delta {\ensuremath{\mathbf{{\ensuremath{\overline{p}}}}}} = \infty$, meaning that ${\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}$ is orthogonal to $\Psi_0$, it follows from Eqs. (\[deltapbb\]) and (\[Ninl\]) that $\Delta {\ensuremath{\mathbf{{\ensuremath{\overline{\overline{p}}}}}}}$ is zero for $\xi=0$ but $\Delta {\ensuremath{\mathbf{{\ensuremath{\overline{\overline{p}}}}}}}$ is nonzero and finite for $\xi=1/2$. In practice, all these three choices for $\xi$ usually lead to a very rapid convergence of the nonlinear parameters. In contrast, choosing the original derivatives, i.e. $N_i = S_{i0}$, leads to slowly converging or diverging Jastrow parameters.
[*Stabilization.*]{} Similarly to the procedure used for the Newton method, we stabilize the linear method by adding a positive constant, $a_{{\ensuremath{\text{diag}}}} \geq 0$, to the diagonal of $\overline{{\ensuremath{\mathbf{H}}}}$ except for the first element, i.e. $\overline{H}_{ij} \to \overline{H}_{ij} + a_{{\ensuremath{\text{diag}}}} \delta_{ij}
(1 -\delta_{i0})$. Again, as $a_{{\ensuremath{\text{diag}}}}$ becomes larger, the parameter variations $\Delta {\ensuremath{\mathbf{p}}}$ become smaller and rotate toward the steepest descent direction. The value of $a_{{\ensuremath{\text{diag}}}}$ is then automatically adjusted in the course of the optimization in the same way as in the Newton method. Note that if instead we were to add $a_{{\ensuremath{\text{diag}}}}$ to $\overline{{\ensuremath{\mathbf{S}}}}^{\, -1} \cdot \overline{{\ensuremath{\mathbf{H}}}}$ then it would be the “level-shift” parameter commonly used in diagonalization procedures. We prefer to add to $\overline{{\ensuremath{\mathbf{H}}}}$, in part, because it is not necessary to compute $\overline{{\ensuremath{\mathbf{S}}}}^{\, -1} \cdot \overline{{\ensuremath{\mathbf{H}}}}$ in order to solve Eq. (\[GEQ\]).
[*Connection to the EFP method.*]{} The generalized eigenvalue equation of Eq. (\[GEQ\]) can be re-written as an eigenvalue equation, $\overline{{\ensuremath{\mathbf{H}}}}' \cdot \Delta {\ensuremath{\mathbf{p}}} = E_{\ensuremath{\text{lin}}}\Delta {\ensuremath{\mathbf{p}}}$, where $\overline{{\ensuremath{\mathbf{H}}}}' = \overline{{\ensuremath{\mathbf{S}}}}^{\, -1} \cdot \overline{{\ensuremath{\mathbf{H}}}}$, i.e. with matrix elements $\overline{H}_{ij}' = \sum_{k=0}^{N^{\ensuremath{\text{opt}}}} ( \overline{{\ensuremath{\mathbf{S}}}}^{\, -1} )_{ik} {\ensuremath{\langle {\ensuremath{\overline{\Psi}}}_k \vert}} \hat{H} {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_j \rangle}}$. This form is useful to establish the connection with the EFP optimization method for the CSF and orbital parameters [@FilFah-JCP-00; @SchFah-JCP-02; @SchFil-JCP-04]. This latter approach consists of solving at each iteration the effective eigenvalue equation, $\overline{{\ensuremath{\mathbf{H}}}}^{{\ensuremath{\text{EFP}}}} \cdot \Delta {\ensuremath{\mathbf{p}}} = E^{{\ensuremath{\text{EFP}}}} \Delta {\ensuremath{\mathbf{p}}}$, where the EFP effective Hamiltonian has matrix elements $\overline{H}^{{\ensuremath{\text{EFP}}}}_{ij} = {\ensuremath{\langle \Phi_i \vert}} \hat{H} {\ensuremath{\vert \Phi_i \rangle}} \delta_{ij} + \sum_{k=1}^{N^{\ensuremath{\text{opt}}}} ( \overline{{\ensuremath{\mathbf{S}}}}^{\, -1} )_{ik} {\ensuremath{\langle {\ensuremath{\overline{\Psi}}}_k \vert}} \hat{H} {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_0 \rangle}} \left[ (1-\delta_{i0}) \delta_{0j} + \delta_{i0} (1-\delta_{0j}) \right]$, where ${\ensuremath{\vert \Phi_i \rangle}}$ designates the current wave function and its derivatives without the Jastrow factor, i.e. ${\ensuremath{\vert \Psi_i \rangle}} = \hat{J}(\bm{\alpha}^0) {\ensuremath{\vert \Phi_i \rangle}}$, and ${\ensuremath{\langle {\ensuremath{\overline{\Psi}}}_k \vert}} \hat{H} {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_0 \rangle}}$ are just the components of half the gradient of the energy. Hence, in the EFP method, only the off-diagonal elements in the first column and first row calculated from the components of the energy gradient are retained in $\overline{{\ensuremath{\mathbf{H}}}}^{{\ensuremath{\text{EFP}}}}$.
[*Connection to the Newton and SRH methods.*]{} In the linear method, the energy expression that is minimized at each iteration, $E_{\ensuremath{\text{lin}}}({\ensuremath{\mathbf{p}}})$, contains all orders in the parameter variations because of the presence of the denominator in Eq. (\[Elin\]), though only the zeroth- and first-order terms match those of the expansion of the exact energy $E({\ensuremath{\mathbf{p}}})$. In contrast, in the Newton method, the energy expression of Eq. (\[E2\]), $E^{[2]}({\ensuremath{\mathbf{p}}})$, is truncated at second order in $\Delta {\ensuremath{\mathbf{p}}}$ but is exact up to this order. Now, if instead of solving the generalized eigenvalue equation (\[GEQ\]), one expands the energy expression of Eq. (\[Elin\]) to second order in $\Delta {\ensuremath{\mathbf{p}}}$, one recovers the Newton method with an approximate (symmetric) Hessian $h_{ij}^{{\ensuremath{\text{lin}}}}=\overline{H}_{ij}+\overline{H}_{ji} - 2 E_0 \overline{S}_{ij}$ corresponding exactly to the SRH method with $\beta=0$ of Ref. . The SRH method is much less stable and converges more slowly than either our linear method or our Newton method for the systems studied here.
Perturbative optimization method
--------------------------------
The perturbative method discussed next is identical to the perturbative EFP approach of Scemama and Filippi [@SceFil-PRB-06] for the optimization of the CSF and orbital parameters, provided that the same choice is made for the energy denominators (see below). We give here an alternate proof without introducing the concept of energy fluctuations that in principle extends the method to other kinds of parameters as well.
Instead of calculating the optimal linearized wave function ${\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}\rangle}}$ by diagonalizing the Hamiltonian $\hat{H}$ in the subspace spanned by $\{{\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_0 \rangle}}, {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_1 \rangle}}, {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_2 \rangle}}, \cdots, {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{N^{{\ensuremath{\text{opt}}}}} \rangle}} \}$, we formulate a nonorthogonal perturbation theory for ${\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}\rangle}}$. The textbook formulation of perturbation theory starts from the Hamiltonian $\Hhat$ whose eigenstates we wish to compute, and a zeroth order Hamiltonian $\Hhat^{(0)}$ whose eigenstates are known. Instead, here we start with $\Hhat$ and the states $\{{\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_0 \rangle}}, {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_1 \rangle}}, {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_2 \rangle}}, \cdots, {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{N^{{\ensuremath{\text{opt}}}}} \rangle}} \}$, and define a zeroth order operator $\Hhat^{(0)}$ for which these states are right eigenstates. To do this, we introduce $\{{\ensuremath{\langle \tilde{\Psi}_i \vert}}\}$, the dual (biorthonormal) basis of the basis $\{{\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_i \rangle}}\}$, i.e. ${\ensuremath{\langle \tilde{\Psi}_i \vert {\ensuremath{\overline{\Psi}}}_j \rangle}}=\delta_{ij}$, given by (see, e.g., Ref. ) $${\ensuremath{\langle \tilde{\Psi}_i \vert}} = \sum_{j=0}^{N^{{\ensuremath{\text{opt}}}}} (\overline{{\ensuremath{\mathbf{S}}}}^{\, -1})_{ij} \, {\ensuremath{\langle {\ensuremath{\overline{\Psi}}}_j \vert}},$$ where $(\overline{{\ensuremath{\mathbf{S}}}}^{\, -1})_{ij}$ are the elements of the inverse of the overlap matrix $\overline{{\ensuremath{\mathbf{S}}}}$, and we introduce the non-Hermitian projector operator onto this subspace $$\hat{P} = \sum_{i=0}^{N^{{\ensuremath{\text{opt}}}}} {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_i \rangle}} {\ensuremath{\langle {\ensuremath{\tilde{\Psi}}}_i \vert}}.$$ The optimal linearized wave function, minimizing the energy \[Eq. (\[E1min\])\], satisfies the projected Schrödinger equation $$\hat{P} \hat{H} {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}\rangle}} = E_{\ensuremath{\text{lin}}}\hat{P} {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}\rangle}},
\label{PHPsiEPPsi}$$ with the normalization condition ${\ensuremath{\langle {\ensuremath{\tilde{\Psi}}}_0 \vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}\rangle}}=1$, ensuring that the coefficient of ${\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}\rangle}}$ on ${\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_0 \rangle}}={\ensuremath{\vert \Psi_0 \rangle}}$ is $1$ as in Eq. (\[Psib1\]).
To construct the perturbation theory, we now introduce a fictitious projected Schrödinger equation depending on a coupling constant ${\ensuremath{\lambda}}$ $$\hat{P} \hat{H}^{{\ensuremath{\lambda}}} {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}^{\ensuremath{\lambda}}\rangle}} = E_{\ensuremath{\text{lin}}}^{\ensuremath{\lambda}}\hat{P} {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}^{\ensuremath{\lambda}}\rangle}},
\label{PHlPsilElPPsil}$$ with the normalization condition ${\ensuremath{\langle {\ensuremath{\tilde{\Psi}}}_0 \vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}^{\ensuremath{\lambda}}\rangle}}=1$ for all ${\ensuremath{\lambda}}$, so that, for ${\ensuremath{\lambda}}=1$, Eq. (\[PHlPsilElPPsil\]) reduces to Eq. (\[PHPsiEPPsi\]): $\hat{H}^{{\ensuremath{\lambda}}=1} = \hat{H}$, ${\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}^{{\ensuremath{\lambda}}=1} \rangle}}={\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}\rangle}}$, $E_{\ensuremath{\text{lin}}}^{{\ensuremath{\lambda}}=1}=E_{\ensuremath{\text{lin}}}$, and we partition the Hamiltonian $\hat{H}^{{\ensuremath{\lambda}}}$ as follows $$\hat{H}^{{\ensuremath{\lambda}}} = \hat{H}^{(0)} + {\ensuremath{\lambda}}\hat{H}^{(1)}.$$ In this expression, $\hat{H}^{(0)}$ is a zeroth-order non-Hermitian operator $$\hat{H}^{(0)} = \sum_{i=0}^{N^{{\ensuremath{\text{opt}}}}} {\cal E}_i {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_i \rangle}} {\ensuremath{\langle \tilde{\Psi}_i \vert}},$$ where ${\cal E}_i$ are *arbitrary* energies. Clearly, $\hat{H}^{(0)}$ admits ${\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_i \rangle}}$ as right-eigenstate and ${\ensuremath{\langle \tilde{\Psi}_i \vert}}$ as left-eigenstate, with common eigenvalue ${\cal E}_i$. The non-Hermitian perturbation operator is obviously defined as $\hat{H}^{(1)}=\hat{H}-\hat{H}^{(0)}$. We expand ${\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}^{\ensuremath{\lambda}}\rangle}}$ and $E_{\ensuremath{\text{lin}}}^{\ensuremath{\lambda}}$ in powers of ${\ensuremath{\lambda}}$: ${\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}^{\ensuremath{\lambda}}\rangle}} =\sum_{k=0}^{\infty} {\ensuremath{\lambda}}^k {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}^{(k)} \rangle}}$ and $E_{\ensuremath{\text{lin}}}^{\ensuremath{\lambda}}=\sum_{k=0}^{\infty} {\ensuremath{\lambda}}^k E_{\ensuremath{\text{lin}}}^{(k)}$. The zeroth-order (right) eigenstate and energy are simply: ${\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}^{(0)} \rangle}} = {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_0 \rangle}}$ and $E_{\ensuremath{\text{lin}}}^{(0)}= {\cal E}_0$. The first-order correction to the wave function is determined by the equation $$\hat{P} \left( \hat{H}^{(0)} - {\cal E}_0 \right) {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}^{(1)} \rangle}} = - \hat{P} \left( \hat{H}^{(1)} - E_{\ensuremath{\text{lin}}}^{(1)} \right) {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_0 \rangle}}.
\label{Psi1eq}$$ To solve this equation, we define the non-Hermitian projector operator $\hat{R}=\sum_{i=1}^{N^{\ensuremath{\text{opt}}}} {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_i \rangle}}{\ensuremath{\langle {\ensuremath{\tilde{\Psi}}}_i \vert}}$ which, in comparison to the projector $\hat{P}$, also removes the component parallel to ${\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_0 \rangle}}$. Note that $\hat{R}\hat{P}=\hat{R}$, $\hat{R}$ commutes with $\hat{H}^{(0)} - {\cal E}_0$ and $\hat{R}
{\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}^{(1)} \rangle}} = {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}^{(1)} \rangle}}$ (since ${\ensuremath{\langle {\ensuremath{\tilde{\Psi}}}_0 \vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}^{{\ensuremath{\lambda}}} \rangle}}=1$ and ${\ensuremath{\langle {\ensuremath{\tilde{\Psi}}}_0 \vert {\ensuremath{\overline{\Psi}}}_0 \rangle}}=1$, implying ${\ensuremath{\langle {\ensuremath{\tilde{\Psi}}}_0 \vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}^{(1)} \rangle}}=0$), so that applying $\hat{R}$ on Eq. (\[Psi1eq\]) leads to $$\begin{aligned}
{\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}^{(1)} \rangle}} &=& - \frac{\hat{R}}{\hat{H}^{(0)} - {\cal E}_0 } \left( \hat{H}^{(1)} - E_{\ensuremath{\text{lin}}}^{(1)} \right) {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_0 \rangle}}
\nonumber\\
&=& - \sum_{i=1}^{N^{{\ensuremath{\text{opt}}}}} {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_i \rangle}} \frac{{\ensuremath{\langle {\ensuremath{\tilde{\Psi}}}_i \vert}} \hat{H} - {\cal E}_0 - E_{\ensuremath{\text{lin}}}^{(1)} {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_0 \rangle}}}{{\cal E}_i - {\cal E}_0}
\nonumber\\
&=&
- \sum_{i=1}^{N^{{\ensuremath{\text{opt}}}}} \sum_{j=1}^{N^{{\ensuremath{\text{opt}}}}} (\overline{{\ensuremath{\mathbf{S}}}}^{\, -1})_{ij} \frac{{\ensuremath{\langle {\ensuremath{\overline{\Psi}}}_j \vert}} \hat{H} {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_0 \rangle}}}{{\cal E}_i - {\cal E}_0} {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_i \rangle}},
\label{}\end{aligned}$$ where ${\cal E}_0$ and $E_{\ensuremath{\text{lin}}}^{(1)}$ in the numerator and the term $j=0$ have been dropped since, for $i \neq 0$, ${\ensuremath{\langle {\ensuremath{\tilde{\Psi}}}_i \vert {\ensuremath{\overline{\Psi}}}_0 \rangle}}=0$ and $(\overline{{\ensuremath{\mathbf{S}}}}^{\, -1})_{i0}
=0$, respectively. Therefore, the parameter variations in this first-order perturbation theory are $$\Delta {\ensuremath{\overline{p}}}_i^{(1)} = - \frac{1}{\Delta {\cal E}_i} \sum_{j=1}^{N^{{\ensuremath{\text{opt}}}}} (\overline{{\ensuremath{\mathbf{S}}}}^{\, -1})_{ij} \overline{H}_{j0},
\label{Deltaalpha1}$$ where $\overline{H}_{j0}={\ensuremath{\langle {\ensuremath{\overline{\Psi}}}_j \vert}} \hat{H} {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_0 \rangle}} = {\ensuremath{\langle \Psi_j \vert}} \hat{H} - E_0{\ensuremath{\vert \Psi_0 \rangle}} = g_j/2$ is just half the gradient of the energy and $\Delta {\cal E}_i = {\cal E}_i - {\cal E}_0$. The perturbative method consists of calculating the parameter variations $\Delta {\ensuremath{\mathbf{{\ensuremath{\overline{p}}}}}}^{(1)}$ according to Eq. (\[Deltaalpha1\]), updating the current wave function, ${\ensuremath{\vert \Psi_0 \rangle}} \to {\ensuremath{\vert \Psi({\ensuremath{\mathbf{p}}}^0 + \Delta {\ensuremath{\mathbf{{\ensuremath{\overline{p}}}}}}^{(1)}) \rangle}}$, and iterating until convergence. It is apparent from Eq. (\[Deltaalpha1\]) that the perturbative method can be viewed as the Newton method with an approximate Hessian, $h_{ij}^{{\ensuremath{\text{pert}}}} = (\overline{{\ensuremath{\mathbf{S}}}}^{\, -1})_{ij}/\Delta {\cal E}_i$, as also noted in Ref. .
The energy denominators $\Delta {\cal E}_i$ in Eq. (\[Deltaalpha1\]) remain to be chosen. Since perturbation theory works best when $\hat{H}^{(0)}$ is “close” to $\hat{H}$ we choose $\hat{H}^{(0)}$ to have the same diagonal elements as $\hat{H}$, resulting in $$\Delta {\cal E}_i = \frac{{\ensuremath{\langle {\ensuremath{\overline{\Psi}}}_i \vert}} \hat{H} {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_i \rangle}}}{{\ensuremath{\langle {\ensuremath{\overline{\Psi}}}_i \vert {\ensuremath{\overline{\Psi}}}_i \rangle}}} -E_0 = \frac{\overline{H}_{ii}}{\overline{S}_{ii}} - \overline{H}_{00}.
\label{DeltaEi}$$ In practice, only rough estimates of the $\Delta {\cal E}_i$’s are necessary for the optimization so that one can compute them for just the initial iteration and keep them fixed for the following iterations. Therefore, for these iterations, only the inverse overlap matrix, $\overline{{\ensuremath{\mathbf{S}}}}^{\, -1}$, and the gradient of the energy, $g_j = 2 \, \overline{H}_{j0}$, need to be calculated in the perturbative method, leading to an important computational speedup per iteration in comparison to the linear method.
[*Stabilization.*]{} Similarly to the linear method, the perturbative method can be stabilized by adding an adjustable positive constant, $a_{{\ensuremath{\text{diag}}}} \geq
0$, to the energy denominators, i.e. $\Delta {\cal E}_i \to \Delta
{\cal E}_i + a_{{\ensuremath{\text{diag}}}}$, which has the effect of decreasing the parameter variations $\Delta {\ensuremath{\mathbf{{\ensuremath{\overline{p}}}}}}^{(1)}$.
[*Connection to the perturbative EFP and SR methods.*]{} For the CSF and orbital parameters, if the energy denominators are chosen to be $\Delta {\cal E}_{i} = {\ensuremath{\langle \Phi_i \vert}} \hat{H}
{\ensuremath{\vert \Phi_i \rangle}} / {\ensuremath{\langle \Phi_i \vert \Phi_i \rangle}} - {\ensuremath{\langle \Phi_0 \vert}} \hat{H}
{\ensuremath{\vert \Phi_0 \rangle}} / {\ensuremath{\langle \Phi_0 \vert \Phi_0 \rangle}}$ (i.e., without the Jastrow factor), Eq. (\[Deltaalpha1\]) exactly reduces to the perturbative EFP method [@SceFil-PRB-06]. Also, Eq. (\[Deltaalpha1\]) reduces to the SR optimization method [@Sor-PRB-01; @CasSor-JCP-03; @CasAttSor-JCP-04] if the energy denominators are all chosen equal, $\Delta {\cal E}_i = \Delta {\cal E}$ for all $i$.
Variational Monte Carlo realization {#sec:vmc}
===================================
When the previously-described energy minimization procedures are implemented in VMC it is important to pay attention to the statistical fluctuations. Expressions that are equivalent in the limit of an infinite Monte Carlo sample, can in fact have very different statistical errors for a finite sample. We provide prescriptions for low variance estimators in this section.
We also note that, in order to reduce round-off noise, it can help to rescale the elements of the gradient vector, and the hessian, Hamiltonian and overlap matrices using the square root of the diagonal of overlap matrix.
At each step of the optimization, the quantum-mechanical averages are computed by sampling the probability density of the current wave function $\Psi_0({\ensuremath{\mathbf{R}}})^2$. We will denote the statistical average of a local quantity, $f({\ensuremath{\mathbf{R}}})$, by $\left\langle
f({\ensuremath{\mathbf{R}}}) \right\rangle = (1/M) \sum_{k=1}^{M} f({\ensuremath{\mathbf{R}}}_k)$ where the $M$ electron configurations ${\ensuremath{\mathbf{R}}}_k$ are sampled from $\Psi_0({\ensuremath{\mathbf{R}}})^2$.
Energy gradient and Hessian {#sec:gradhess}
---------------------------
In terms of the derivatives $\Psi_i({\ensuremath{\mathbf{R}}})$ of the wave function of Eq. (\[Psi\]), and using the Hermiticity of the Hamiltonian $\hat{H}$, an estimator of the energy gradient is [@CepCheKal-PRB-77] $$g_i = 2 \Biggl[ \left\langle \frac{\Psi_i({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}}) \right\rangle - \left\langle \frac{\Psi_i({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle \left\langle E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}}) \right\rangle \Biggl],
\label{gi}$$ where $E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}})=\left[ H({\ensuremath{\mathbf{R}}})\Psi_0({\ensuremath{\mathbf{R}}}) \right]/\Psi_0({\ensuremath{\mathbf{R}}})$ is the local energy. In the limit that $\Psi_0({\ensuremath{\mathbf{R}}})$ is an exact eigenfunction, the local energy becomes constant, $E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}})= E_{\ensuremath{\text{exact}}}$ for all ${\ensuremath{\mathbf{R}}}$, and thus the gradient of Eq. (\[gi\]) vanishes with zero variance. This leads to the following zero-variance principle for the Newton and perturbative methods: in the limit that $\Psi_0({\ensuremath{\mathbf{R}}})$ is an exact eigenfunction, the parameter variations of Eqs. (\[deltaalphanewton\]) and (\[Deltaalpha1\]) vanish with zero variance.
Taking the derivative of Eq. (\[gi\]) leads to the straightforward estimator of the energy Hessian of Lin, Zhang and Rappe (LZR) [@LinZhaRap-JCP-00] $$h_{ij}^{\text{LZR}} = A_{ij} + B_{ij} + C_{ij},
\label{hijLZR}$$ where $$\begin{aligned}
A_{ij} &=& 2 \Biggl[ \left\langle \frac{\Psi_{ij}({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}}) \right\rangle - \left\langle \frac{\Psi_{ij}({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle \left\langle E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}}) \right\rangle
\nonumber\\
&& - \left\langle \frac{\Psi_i({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \frac{\Psi_j({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}}) \right\rangle
\nonumber\\
&& + \left\langle \frac{\Psi_i({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \frac{\Psi_j({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle \left\langle E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}}) \right\rangle \Biggl],
\nonumber\\
\label{Aij}\end{aligned}$$ involving the second derivatives $\Psi_{ij}({\ensuremath{\mathbf{R}}})$ of the wave function, $$\begin{aligned}
B_{ij} &=& 4 \Biggl[\left\langle \frac{\Psi_i({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \frac{\Psi_j({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}}) \right\rangle
\nonumber\\
&& - \left\langle \frac{\Psi_i({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \frac{\Psi_j({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle \left\langle E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}}) \right\rangle \Biggl]
\nonumber\\
&&- 2 \left\langle \frac{\Psi_{i}({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle g_j
- 2 \left\langle \frac{\Psi_{j}({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle g_i
\nonumber\\
&=& 4 \, \Bigg\langle \left( \frac{\Psi_i({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} - \left\langle \frac{\Psi_i({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle \right)
\nonumber \\
&& \times \left( \frac{\Psi_j({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} - \left\langle \frac{\Psi_j({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle \right)
\left( E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}}) - \left\langle E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}}) \right\rangle \right) \Bigg\rangle,
\nonumber \\
\label{Bij}\end{aligned}$$ and $$\begin{aligned}
C_{ij} &=& 2 \left\langle \frac{\Psi_{i}({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} E_{{\ensuremath{\text{L}}},j}({\ensuremath{\mathbf{R}}}) \right\rangle,
\label{Cij}\end{aligned}$$ where $E_{{\ensuremath{\text{L}}},j}({\ensuremath{\mathbf{R}}}) = \left[ H({\ensuremath{\mathbf{R}}})\Psi_j({\ensuremath{\mathbf{R}}})
\right]/\Psi_0({\ensuremath{\mathbf{R}}}) - \left[ \Psi_j({\ensuremath{\mathbf{R}}})/\Psi_0({\ensuremath{\mathbf{R}}}) \right]
E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}})$ is the derivative of the local energy with respect to parameter $j$. In this estimator of the Hessian, the term that fluctuates the most is $C_{ij}$.
Umrigar and Filippi [@UmrFil-PRL-05] observed that the fluctuations of a covariance $\langle ab \rangle - \langle a \rangle \langle b \rangle$ are much smaller than those of $\langle ab \rangle$ if the fluctuations of $a$ are much smaller than the average of $a$, i.e., $\sqrt{\langle a^2 \rangle - \langle a \rangle ^2} \ll \vert \langle a \rangle \vert$, and, $a$ is not strongly correlated with $1/b$. In Eq. (\[Cij\]), $\Psi_{i}({\ensuremath{\mathbf{R}}})/\Psi_0({\ensuremath{\mathbf{R}}})$ is always of the same sign for parameters in the exponent and in practice its fluctuations are much smaller than its average. Further, it follows from the Hermiticity of the Hamiltonian that $\left\langle E_{{\ensuremath{\text{L}}},j}({\ensuremath{\mathbf{R}}}) \right\rangle$ vanishes in the limit of an infinite sample [@LinZhaRap-JCP-00]. Using these two observations, Umrigar and Filippi [@UmrFil-PRL-05] provided an estimator of the Hessian, $$h_{ij}^{\text{UF}} = A_{ij} + B_{ij} + D_{ij},
\label{hijUF}$$ that fluctuates much less than the straightforward LZR estimator, where the symmetrized estimator $$\begin{aligned}
D_{ij} &=& \left\langle \frac{\Psi_{i}({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} E_{{\ensuremath{\text{L}}},j}({\ensuremath{\mathbf{R}}}) \right\rangle - \left\langle \frac{\Psi_{i}({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle \left\langle E_{{\ensuremath{\text{L}}},j}({\ensuremath{\mathbf{R}}}) \right\rangle
\nonumber\\
&&+ \left\langle \frac{\Psi_{j}({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} E_{{\ensuremath{\text{L}}},i}({\ensuremath{\mathbf{R}}}) \right\rangle - \left\langle \frac{\Psi_{j}({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle \left\langle E_{{\ensuremath{\text{L}}},i}({\ensuremath{\mathbf{R}}}) \right\rangle,
\nonumber\\
\label{Dij}\end{aligned}$$ has the same average as the term $C_{ij}$ in the limit of an infinite sample, but being a covariance has much smaller fluctuations. We note that $A_{ij}$ is already a covariance and $B_{ij}$ is a tri-covariance.
Although the $A_{ij}$ and $B_{ij}$ terms vanish with zero variance in the limit that $\Psi_0({\ensuremath{\mathbf{R}}})$ is an exact eigenfunction (the $D_{ij}$ term does not), in practice for the Jastrow parameters, far from the minimum, the $B_{ij}$ fluctuates more than the $D_{ij}$ term for the Jastrow parameters in the Hessian of Eq. (\[hijUF\]). With the form of the Jastrow factors that we use, we have observed that the ratio $(B_{ij} + D_{ij})/D_{ij}$ is roughly independent of $i$ and $j$ for most $i$ and $j$ though it changes during the Monte Carlo iterations. It is typically between 1.2 and 2.5 at the initial iteration and between 0.9 and 1.1 at the final iteration. We exploit this to decrease the fluctuations by defining a new, approximate Hessian partially averaged over the Jastrow parameters $$h_{ij}^{\text{TU}} = A_{ij} + \frac{\left\langle \left\langle |B_{ij} + D_{ij}|\right\rangle \right\rangle}{\left\langle \left\langle |D_{ij}|\right\rangle \right\rangle} D_{ij},
\label{hijTU}$$ where TU are the initials of the present authors, and the average over the Jastrow parameter pairs are defined by $\left\langle \left\langle X_{ij} \right\rangle \right\rangle = (2/N^{\ensuremath{\text{opt}}}_{\ensuremath{\text{Jas}}}(N^{\ensuremath{\text{opt}}}_{\ensuremath{\text{Jas}}}+1)) \sum_{i=1}^{N^{\ensuremath{\text{opt}}}_{\ensuremath{\text{Jas}}}} \sum_{j=i}^{N^{\ensuremath{\text{opt}}}_{\ensuremath{\text{Jas}}}} X_{ij}$. The average is calculated as $\left\langle \left\langle |B_{ij} + D_{ij}| \right\rangle \right\rangle/\left\langle \left\langle |D_{ij}| \right\rangle
\right\rangle$ and not as $\left\langle \left\langle (|B_{ij} + D_{ij}|)/|D_{ij}| \right\rangle \right\rangle$ to avoid possible numerical divergences of this ratio for small $D_{ij}$. In Eq. (\[hijTU\]), $i$ and $j$ refer only to Jastrow parameters. For all the terms related to the other parameters (including all the mixed terms), the Hessian of Eq. (\[hijUF\]) is used without further modification.
Exact or approximate wave functions such as $\Psi_0({\ensuremath{\mathbf{R}}})$ go linearly to zero with the distance $d$ between ${\ensuremath{\mathbf{R}}}$ and their nodal hypersurface, i.e., $\Psi_0({\ensuremath{\mathbf{R}}}) \sim d$ for $d\to0$. The local energy $E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}})$ generally diverges as $1/d$ for $d\to0$ for approximate wave functions. In contrast to the case of the Jastrow parameters, the derivatives $\Psi_i({\ensuremath{\mathbf{R}}})$ for the CSF and orbital parameters have a different nodal hypersurface than $\Psi_0({\ensuremath{\mathbf{R}}})$ and the ratio $\Psi_i({\ensuremath{\mathbf{R}}})/\Psi_0({\ensuremath{\mathbf{R}}})$ thus also diverges as $1/d$, even if the wave function $\Psi_0({\ensuremath{\mathbf{R}}})$ is exact. Consequently, the derivative of the local energy $E_{{\ensuremath{\text{L}}},i}({\ensuremath{\mathbf{R}}})$ generally diverges as $1/d^2$ for approximate wave functions. In the expression of the Hessian, the leading divergence at the nodes of the approximate wave function $\Psi_0({\ensuremath{\mathbf{R}}})$ thus comes from the terms $\left(\Psi_i({\ensuremath{\mathbf{R}}})/\Psi_0({\ensuremath{\mathbf{R}}})\right) \left(\Psi_j({\ensuremath{\mathbf{R}}})/\Psi_0({\ensuremath{\mathbf{R}}})\right) E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}})$, $\left(\Psi_i({\ensuremath{\mathbf{R}}})/\Psi_0({\ensuremath{\mathbf{R}}})\right) E_{{\ensuremath{\text{L}}},j}({\ensuremath{\mathbf{R}}})$ and $\left(\Psi_j({\ensuremath{\mathbf{R}}})/\Psi_0({\ensuremath{\mathbf{R}}})\right) E_{{\ensuremath{\text{L}}},i}({\ensuremath{\mathbf{R}}})$ that behave as $1/d^3$. It is however easy to check that these third-order divergences cancel exactly in Eq. (\[hijUF\]).
Overlap and Hamiltonian matrices {#sec:ovlpham}
--------------------------------
The elements of the symmetric overlap matrix $\overline{{\ensuremath{\mathbf{S}}}}$ are
$$\overline{S}_{00}=1,
\label{}$$
and, for $i, j > 0$, $$\overline{S}_{i0}=\overline{S}_{0j}=0,
\label{}$$ and $$\overline{S}_{ij}=\left\langle \frac{\Psi_i({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \frac{\Psi_j({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle - \left\langle \frac{\Psi_i({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle \left\langle \frac{\Psi_j({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle.
\label{Sij}$$ \[S\]
The elements of the Hamiltonian matrix $\overline{{\ensuremath{\mathbf{H}}}}$ are
$$\overline{H}_{00}= \left\langle E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}}) \right\rangle,
\label{}$$
and, for $i, j > 0$, $$\overline{H}_{i0}= \left\langle \frac{\Psi_i({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}}) \right\rangle - \left\langle \frac{\Psi_i({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle \left\langle E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}}) \right\rangle,
\label{Hi0}$$ $$\begin{aligned}
\overline{H}_{0j} &=& \left[ \left\langle \frac{\Psi_j({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}}) \right\rangle - \left\langle \frac{\Psi_j({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle \left\langle E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}}) \right\rangle \right]
\nonumber\\
&&+ \left\langle E_{{\ensuremath{\text{L}}},j}({\ensuremath{\mathbf{R}}}) \right\rangle,
\label{H0j}\end{aligned}$$ which are two estimators of half of the energy gradient, and $$\begin{aligned}
\overline{H}_{ij} &=& \Biggl[ \left\langle \frac{\Psi_i({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \frac{\Psi_j({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}}) \right\rangle
\nonumber\\
&& - \left\langle \frac{\Psi_i({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle \left\langle \frac{\Psi_j({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}}) \right\rangle
\nonumber\\
&& - \left\langle \frac{\Psi_j({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle \left\langle \frac{\Psi_i({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}}) \right\rangle
\nonumber\\
&& + \left\langle \frac{\Psi_i({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle \left\langle \frac{\Psi_j({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle \left\langle E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}}) \right\rangle \Biggl]
\nonumber\\
&&+ \Biggl[ \left\langle \frac{\Psi_i({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} E_{{\ensuremath{\text{L}}},j}({\ensuremath{\mathbf{R}}}) \right\rangle - \left\langle \frac{\Psi_i({\ensuremath{\mathbf{R}}})}{\Psi_0({\ensuremath{\mathbf{R}}})} \right\rangle \left\langle E_{{\ensuremath{\text{L}}},j}({\ensuremath{\mathbf{R}}}) \right\rangle \Biggl].
\nonumber\\
\label{Hij}\end{aligned}$$ \[H\]
We do not use the Hermiticity of the Hamiltonian $\hat{H}$ to symmetrize the matrix $\overline{{\ensuremath{\mathbf{H}}}}$. In fact, as shown by Nightingale and Melik-Alaverdian [@NigMel-PRL-01], using the non-symmetric matrix $\overline{{\ensuremath{\mathbf{H}}}}$ of Eqs. (\[H\]) leads to a stronger zero-variance principle than the one previously-described for the Newton and perturbative methods: in the limit that the states $\{{\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_0 \rangle}}, {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_1 \rangle}}, {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_2 \rangle}}, \cdots, {\ensuremath{\vert {\ensuremath{\overline{\Psi}}}_{N^{{\ensuremath{\text{opt}}}}} \rangle}} \}$ span an invariant subspace of the Hamiltonian $\hat{H}$, i.e. in the limit that the linear wave function ${\ensuremath{\overline{\Psi}}}_{\ensuremath{\text{lin}}}({\ensuremath{\mathbf{R}}})$ of Eq. (\[Psib1\]) after optimization is an exact eigenfunction, the matrix $\overline{{\ensuremath{\mathbf{S}}}}^{\, -1} \cdot \overline{{\ensuremath{\mathbf{H}}}}$ and consequently the eigenvector solution $\Delta {\ensuremath{\mathbf{{\ensuremath{\overline{p}}}}}}$ have zero variance. In practice, even if we do not work in an invariant subspace of $\hat{H}$, using the non-symmetric matrix $\overline{{\ensuremath{\mathbf{H}}}}$ leads to smaller statistical errors on a finite sample than using its symmetrized analog. Although in principle diagonalization of a non-symmetric matrix leads to complex eigenvalues, in practice the physically reasonable (i.e., with large overlap with the current wave function) lowest eigenvectors have usually real eigenvalues. Of course, in the limit of an infinite sample $M \to \infty$ a symmetric matrix $\overline{{\ensuremath{\mathbf{H}}}}$ is recovered.
As noted in the previous subsection for the Hessian, although the terms $\left(\Psi_i({\ensuremath{\mathbf{R}}})/\Psi_0({\ensuremath{\mathbf{R}}})\right) \left(\Psi_j({\ensuremath{\mathbf{R}}})/\Psi_0({\ensuremath{\mathbf{R}}})\right) E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}})$ and $\left(\Psi_i({\ensuremath{\mathbf{R}}})/\Psi_0({\ensuremath{\mathbf{R}}})\right) E_{{\ensuremath{\text{L}}},j}({\ensuremath{\mathbf{R}}})$ in the expression of the Hamiltonian matrix of Eq. (\[Hij\]) display a third-order divergence $1/d^3$ as the distance $d$ between ${\ensuremath{\mathbf{R}}}$ and the nodal hypersurface of $\Psi_0({\ensuremath{\mathbf{R}}})$ goes to zero, again these divergences cancel exactly.
Comparison of computational cost per iteration
----------------------------------------------
At each optimization iteration, besides the calculation the current wave function $\Psi_0({\ensuremath{\mathbf{R}}})$ and the local energy $E_{{\ensuremath{\text{L}}}}({\ensuremath{\mathbf{R}}})$, the Newton method requires the computation of the first-order and second-order wave function derivatives, $\Psi_i({\ensuremath{\mathbf{R}}})$ and $\Psi_{ij}({\ensuremath{\mathbf{R}}})$, and the first-order derivatives of the local energy $E_{{\ensuremath{\text{L}}},i}({\ensuremath{\mathbf{R}}})$. The linear method requires the calculation of $\Psi_i({\ensuremath{\mathbf{R}}})$ and $E_{{\ensuremath{\text{L}}},i}({\ensuremath{\mathbf{R}}})$ but not of the second-order derivatives of the wave function with respect to the parameters. In principle, this decreases the computational cost per iteration, especially if the many orbital-orbital second-order derivatives were to be computed in the Newton method. In practice, since our implementation of the Newton method neglects these orbital-orbital derivatives, the computational cost per iteration of the Newton and linear methods is very similar.
The perturbative method requires the computation of the same quantities as the linear method. However, since the method is not very sensitive to having accurate energy denominators $\Delta {\cal E}_i$ in Eq. (\[Deltaalpha1\]), and since the energy denominators do not undergo large changes from iteration to iteration we compute these for the first iteration only. Hence it is not necessary to compute $E_{{\ensuremath{\text{L}}},i}({\ensuremath{\mathbf{R}}})$ for subsequent iterations. This leads to a computational speedup per iteration in comparison with the linear method. The precise speedup factor depends on the wave function used; typically, for the systems studied here, we have found factors ranging from about $1.5$ for a single-determinant wave function to $5.5$ for the largest multi-determinant wave function considered, for the iterations for which the $\Delta {\cal E}_i$’s are not computed.
Computational details {#sec:compdetails}
=====================
We illustrate the optimization methods by calculating the ground-state electronic energy of the all-electron C$_2$ molecule at the experimental equilibrium interatomic distance of $2.3481$ Bohr [@CadWah-ADNDT-74]. The ground-state wave function is of symmetry $^1\Sigma_g^+$ in the point group $D_{\infty h}$. The estimated exact, infinite nuclear mass, nonrelativistic electronic energy is $-75.9265(8)$ Hartree [@BytRue-JCP-05], where the number in parentheses is an estimate of the uncertainty in the last digit. This system has a strong multiconfiguration character due to the energetic near-degeneracy of the valence orbitals, making it a challenging system despite its small size.
We start by generating a standard [*ab initio*]{} wave function using the quantum chemistry program GAMESS [@SchBalBoaElbGorJenKosMatNguSuWinDupMon-JCC-93], typically a restricted Hartree-Fock (RHF) wave function or a MCSCF wave function, using the symmetry point group $D_{4h}$ which is the largest subgroup of $D_{\infty h}$ available in GAMESS.i We use the uncontracted Slater basis set form of Clementi and Roetti [@CleRoe-ADNDT-74], with exponents reoptimized at the RHF level by Koga [*et al.*]{} [@KogTatTha-PRA-93]. For carbon, the basis set contains two $1s$, three $2s$, one $3s$ and four $2p$ Slater functions, that are each approximated by a fit to six Gaussian functions [@HehStePop-JCP-69; @Ste-JCP-70] in GAMESS. Specifically, we consider the following [*ab initio*]{} wave functions: a RHF wave function, with orbital occupations $1\sigma_g^2 1\sigma_u^2 2\sigma_g^2 2\sigma_u^2 1\pi_{u,x}^2 1\pi_{u,y}^2$; a CAS(8,5) wave function, containing 6 CSFs in $D_{4h}$ symmetry made of 7 Slater determinants generated by distributing the 8 valence electrons over the 5 active valence orbitals $2\sigma_g 2\sigma_u 1\pi_{u,x} 1\pi_{u,y} 3\sigma_g$; a CAS(8,7) wave function, containing 80 CSFs made of 165 determinants with the 7 active orbitals $2\sigma_g 2\sigma_u 1\pi_{u,x} 1\pi_{u,y} 3\sigma_g 1\pi_{g,x} 1\pi_{g,y}$; a CAS(8,8) wave function, containing 264 CSFs made of 660 determinants with the 8 active orbitals $2\sigma_g 2\sigma_u 1\pi_{u,x} 1\pi_{u,y} 3\sigma_g 1\pi_{g,x} 1\pi_{g,y} 3\sigma_u$, i.e. all the valence orbitals originating from the $n=2$ shell of the C atoms. In addition, we construct a larger one-electron basis set by adding to the basis of Koga [*et al.*]{} one $d$ function with an exponent of $2.13$ optimized in RHF and we consider a wave function obtained from a restricted active space (RAS) calculation in this basis that would correspond to a CAS(8,26) calculation, using all the 26 orbitals originating from the $n=2$ and $n=3$ shells of the C atoms, but where only single (S), double (D), triple (T) and quadruple (Q) excitations are allowed in the active space. This wave function, that we denote by RAS-SDTQ(8,26), contains 110481 CSFs made of 411225 determinants.
The standard [*ab initio*]{} wave function is then multiplied by a Jastrow factor, imposing the electron-electron cusp conditions, but with essentially all other free parameters chosen to be $0$ to form our starting trial wave function. QMC calculations are performed with the program CHAMP [@Cha-PROG-XX], using this time the true Slater basis set rather than its Gaussian expansion. In comparison to GAMESS, additional symmetries outside the point group $D_{4h}$ are detected numerically which allows one to reduce the numbers of CSFs to 5, 50 and 165 for the CAS(8,5), CAS(8,7) and CAS(8,8) wave functions, respectively. For the large RAS-SDTQ(8,26) wave function, only a fraction of all the CSFs are retained in QMC by applying a variable cutoff on the CSF coefficients and an extrapolation procedure is used to estimate the QMC result if all the CSFs had been included (see Sec. \[sec:systematic\_improvement\]). For the orbital optimization, only the single excitations between orbitals of the same irreducible representation of $D_{\infty h}$ are generated. We, however, impose no restriction inside each of the two-dimensional irreducible representations $\pi_u$ and $\pi_g$. Although one can in principle identify the $\pi_x$ and $\pi_y$ components and forbid excitations between these two components to further reduce the number of free parameters, these redundancies appear to cause no problem in practice during the optimization. Also, we impose the electron-nucleus cusp condition on each orbital. The parameters of the trial wave function are optimized by the previously-described energy minimization procedures in VMC, using a very efficient accelerated Metropolis algorithm [@Umr-PRL-93; @Umr-INC-99], allowing us to simultaneously make large Monte Carlo moves in configuration space and have a high acceptance probability. Once a trial wave function has been optimized, we perform a DMC calculation within the fixed-node and the short-time approximations (see, e.g., Refs. ). We use an imaginary time step of $\tau=0.01$ H$^{-1}$ in an efficient DMC algorithm featuring very small time-step errors [@UmrNigRun-JCP-93], so that the accuracy is essentially limited by the quality of the nodal hypersurface of the trial wave function.
Results and discussion {#sec:results}
======================
Optimization of the Jastrow factor
----------------------------------
We first study the convergence behavior of the energy minimization methods for the separate optimization of the Jastrow, CSF and orbital parameters. To facilitate comparisons, we apply the VMC optimization procedures with a common fixed statistical error of the energy at each step, namely $0.5$ mHa. This is not the usual way in which we routinely perform optimizations which is described later in Sec. \[sec:optall\].
(0,0)(44,-160)
Figure \[fig:c2\_ktt\_hfj\_vmc\_emin\_j\_opt\_conv\_stab\] shows the convergence of the total VMC energy during the optimization of the 24 Jastrow parameters in a wave function composed of the RHF Slater determinant multiplied by a Jastrow factor. The linear, perturbative and Newton methods are compared. For the Newton method, we present the results obtained with the UF Hessian of Eq. (\[hijUF\]), already used in Ref. , and with the TU Hessian of Eq. (\[hijTU\]). To compare the fluctuations of these two Hessians, we have computed the quantity $\eta = 1/N^{\ensuremath{\text{opt}}}(N^{\ensuremath{\text{opt}}}+1) \sum_{i=1}^{N^{\ensuremath{\text{opt}}}} \sum_{j=i}^{N^{\ensuremath{\text{opt}}}} (\sigma(h_{ij}))^2$ where $(\sigma(h_{ij}))^2$ is the variance of the element $h_{ij}$ of the Hessian averaged over 100 Monte Carlo configurations. For the initial iteration of the optimization, far from the energy minimum, the UF Hessian fluctuates more than the TU Hessian by a factor $\eta^{\text{UF}}/\eta^{\text{TU}}=3.6$. For comparison, the LZR Hessian of Eq. (\[hijLZR\]) fluctuates more than the TU Hessian by a factor $\eta^{\text{LZR}}/\eta^{\text{TU}}=150$, more than two orders of magnitude larger even for this modest system. Near the energy minimum, the factors are $\eta^{\text{UF}}/\eta^{\text{TU}}=3.3$ and $\eta^{\text{LZR}}/\eta^{\text{TU}}=600$. These factors tend to increase with the system size. The Newton method with the UF Hessian converges reasonably fast in about 6 iterations, which is a little faster than the convergence shown in Figs. 1,2 and 4 of Ref. due to the previously-described correlated sampling adjustment of the stabilizing constant $a_{\ensuremath{\text{diag}}}$ in the course of the optimization and despite the fact that we are performing an all-electron rather than a pseudopotential calculation here[@UmrFil-PRL-05-note]. The Newton method with the TU Hessian displays an even faster convergence, the energy being essentially converged within the statistical error at iteration 3 or 4. The linear method has a similar convergence rate to the Newton method with the TU Hessian. The Newton method with the TU Hessian and the linear method are both stable even without stabilization if sufficiently large Monte Carlo samples are used. When stabilization is employed, the stabilization constant $a_{\ensuremath{\text{diag}}}$ remains small during the optimization, in this example from $10^{-3}$ for the initial iteration to $10^{-7}$ for the last iterations which is 2 or 3 orders of magnitude smaller than the values of $a_{\ensuremath{\text{diag}}}$ in the Newton method with the UF Hessian. The perturbative method, in contrast, converges very slowly. In fact, it turns out that the energy denominators for the Jastrow parameters, $\Delta {\cal E}_{\alpha_i}$, calculated according to Eq. (\[DeltaEi\]), are all of order unity and $a_{\ensuremath{\text{diag}}}$ needs to be increased to as much as $10^2$ to retain stability. In this case, the perturbative method essentially reduces to the inefficient SR optimization technique.
Optimization of the CSF coefficients
------------------------------------
Figure \[fig:c2\_ktt\_cas87j\_vmc\_emin\_csf\_opt\_conv\_stab\] shows the convergence of the total VMC energy during the optimization of the 49 CSF parameters in a wave function composed of a CAS(8,7) determinantal part multiplied by a previously-optimized Jastrow factor, using the linear, perturbative and Newton \[with the UF Hessian of Eq. (\[hijUF\])\] methods. The linear method converges in $1$ iteration, as it must, and does not require any stabilization. When stabilization is used, $a_{\ensuremath{\text{diag}}}$ remains as low as $10^{-6}$ to $10^{-8}$ during the whole optimization. The Newton and perturbative methods converge in $2$ or $3$ iterations, and are not as intrinsically stable, $a_{\ensuremath{\text{diag}}}$ being a few orders of magnitude larger for the Newton method and several orders of magnitude larger for the perturbative method. The energy denominators for the CSF parameters in the perturbative method, $\Delta {\cal
E}_{c_I}$, calculated according to Eq. (\[DeltaEi\]), span only one order of magnitude.
Optimization of the orbitals
----------------------------
Figure \[fig:c2\_ktt\_hfj\_vmc\_emin\_o\_opt\_conv\_stab\] shows the convergence of the total VMC energy during the optimization of all the 44 orbital parameters in a wave function composed of a single Slater determinant multiplied by a previously-optimized Jastrow factor, using the linear, perturbative and Newton \[with the UF Hessian of Eq. (\[hijUF\])\] methods. The three methods display very similar convergence rates, the energy being converged within the statistical error in $1$ iteration using any of the three methods. In this example, the linear and perturbative methods converged even without stabilization whereas the Newton method required stabilization. The energy denominators for the orbital parameters in the perturbative method, $\Delta {\cal E}_{kl}$, calculated according to Eq. (\[DeltaEi\]), typically span two orders of magnitude from $1$ to $100$.
(0,0)(183,-175)
(0,0)(-72,-175)
In the previous orbital optimization, we have considered a full optimization of all the orbital parameters, i.e. all the allowed excitations from the 6 closed occupied orbitals to the 30 virtual orbitals were included in the calculation. One may also consider a partial orbital optimization by restricting the excitations to the lowest several virtual orbitals, as also proposed within the EFP or perturbative EFP approaches [@Fil-PRIV-XX]. This allows one to reduce the computational effort and also to decrease the statistical noise in the calculation since it is the excitations to the highest-lying virtual orbitals that modify the most the nodal structure of the wave function, leading to large fluctuations of the ratio $\Psi_i({\ensuremath{\mathbf{R}}})/\Psi_0({\ensuremath{\mathbf{R}}})$. Fig. \[fig:c2\_ktt\_hfj\_vmc\_emin\_o\_opt3\_orb\] shows the total VMC energy with respect to the number of virtual orbitals included in the optimization for a wave function composed of a single Slater determinant multiplied by a previously-optimized Jastrow factor. Two sets of starting orbitals are compared: orbitals obtained from a RHF calculation and orbitals obtained from a restricted Kohn-Sham (RKS) calculation with the hybrid exchange-correlation functional B3LYP [@Bec-JCP-93; @SteDevChaFri-JPC-94], using the ordering given by the orbital energies. In both cases, as expected, the energy decreases monotonically within the statistical error as the number of virtual orbitals included in the optimization increases. However the slope of the energy does not change monotonically and it is necessary to include almost all the orbitals to get close to the optimal energy. From Fig. \[fig:c2\_ktt\_hfj\_vmc\_emin\_o\_opt3\_orb\] we see that for the C$_2$ molecule the B3LYP orbitals provide a better starting point than the RHF orbitals. In our experience, this is often but not always the case. It is possible that the selection of the virtual orbitals adopted here, based on the orbital energy ordering, may not be the best choice and other selections based on symmetry or chemical intuition could lead to a more rapid convergence.
Note that Fig. \[fig:c2\_ktt\_hfj\_vmc\_emin\_o\_opt3\_orb\] was obtained by just optimizing the orbital parameters for a fixed, previously-optimized Jastrow factor. If instead the Jastrow and orbital parameters are optimized simultaneously a significantly lower energy is obtained, e.g. including all 30 virtual orbitals gives an energy of -75.8069(5) Ha (see Table \[tab:energies\]) as opposed to -75.7845(5) Ha in Fig. \[fig:c2\_ktt\_hfj\_vmc\_emin\_o\_opt3\_orb\].
To summarize, the Newton and the linear methods converge very rapidly when optimizing any kind of parameter, though the linear method is more stable for the optimization of the determinantal part of the wave function. The perturbative method is a good, less expensive alternative for the optimization of the orbital parameters and, to a lesser extent, for the optimization of the CSF parameters, but is very slowly convergent for the Jastrow parameters.
It is clear from Eq. (\[Deltaalpha1\]) that the perturbative method can be viewed as a Newton method with an approximate Hessian. The poor behavior of the perturbative method for the Jastrow parameters means that this Hessian is a bad approximation to the exact Hessian, whose eigenvalues span more than 10 orders of magnitude for these parameters. In fact, any method based on an approximate Hessian that is not able to reproduce all these orders of magnitude, such as the steepest-descent method, is bound to converge very slowly. On the other hand, the eigenvalues of the Hessian for the CSF and orbital parameters span only a couple of orders of magnitude and the approximate Hessian of the perturbative method is sufficient to allow rapid convergence.
Optimization of all the parameters: simultaneous or alternated optimization? {#sec:optall}
----------------------------------------------------------------------------
After having studied the behavior of the energy minimization methods for the optimization of each kind of parameter, we now move on to the more practical problem of how to optimize all the parameters.
The most obvious possibility is to optimize *simultaneously* the Jastrow, CSF and orbital parameters using the linear method, the method having the best overall efficiency for all these parameters. In practice, we proceed as follows. We start an optimization run with a short Monte Carlo simulation with a large statistical error (e.g., $0.02$ Ha for the C$_2$ molecule), and we decrease progressively the statistical error at each iteration until the energy is converged to $10^{-3}$ Ha for three consecutive iterations. We choose the optimal parameters to be those from the iteration with the smallest value of $E_{\ensuremath{\text{VMC}}}$ plus three times the statistical error of $E_{\ensuremath{\text{VMC}}}$, which is often but not always the last iteration. A typical example of the convergence of the total VMC energy and of the standard deviation $\sigma_{\ensuremath{\text{VMC}}}$ is shown in Fig. \[fig:c2\_ktt\_cas87j\_vmc\_emin\_csfoj\_opt3\_lin\_new2\] for the simultaneous optimization of the Jastrow, CSF and orbital parameters in a wave function composed of a CAS(8,7) determinantal part multiplied by a Jastrow factor. In this case, the energy converges in 4 or 5 iterations. The standard deviation typically converges a little slower than the energy since we are optimizing just the energy here. A faster convergence, to a somewhat smaller value of the standard deviation, can be achieved by optimizing a linear combination of the energy and variance as in Ref. [@UmrFil-PRL-05].
Another possibility is to *alternate* between the optimization of the different kinds of parameters until global convergence. This has the advantage of allowing one to use different optimization methods for the various parameters, e.g. optimization of the Jastrow factor and the CSF coefficients with the Newton or linear method and optimization of the orbitals with the less expensive but still very efficient perturbative method. Fig. \[fig:c2\_ktt\_hfj\_vmc\_emin\_oj\_lin\_alter\] shows the convergence of the total VMC energy and of the standard deviation during the alternated optimization of the Jastrow parameters and of the orbital parameters in a wave function composed of a single Slater determinant multiplied by a Jastrow factor for the all-electron C$_2$ molecule. The convergence of the energy is surprisingly very slow, the convergence of the standard deviation is even worse. This is an indication of the presence of a strong coupling between some Jastrow and orbital parameters. This situation is in sharp contrast with the case where a pseudopotential is used to remove the core electrons. Fig. \[fig:c2\_ps\_hfj\_vmc\_emin\_oj\_lin\_alter\] shows the convergence of the total VMC energy during the alternated optimization of the Jastrow parameters and of the orbital parameters in a wave function composed of a single Slater determinant multiplied by a Jastrow factor for the C$_2$ molecule with a Hartree-Fock pseudopotential [@Shi-UNP-XX] and an adequate Gaussian one-electron basis set. The convergence is very fast, the energy being essentially converged within the statistical error in $1$ macroiteration. This favorable behavior has already been observed in other systems with pseudopotentials [@Fil-PRIV-XX], but we have also found pseudopotential systems for which the convergence is not as fast.
For the all-electron case, it thus seems that simultaneous optimization of the parameters is much preferable. The coupling between the different parameters seems to be too strong to allow an efficient alternated optimization. For large systems most of the wave function parameters are orbital and CSF parameters for which the perturbative method works well. It seems then promising to simultaneously optimize all the parameters with the Newton or the linear methods, using for the part of the Hessian or the Hamiltonian matrices involving the CSF and orbital coefficients rough approximations inspired by the perturbative method [@FilTouUmr-JJJ-XX].
Systematic improvement by wave function optimization {#sec:systematic_improvement}
----------------------------------------------------
$E_{{\ensuremath{\text{VMC}}}}$ $ E_{{\ensuremath{\text{DMC}}}}$
--------------------------------- ------------------------------------------- --------------------------------- ---------------------------------- -----
Jastrow $\times$ determinant Jastrow (24) -75.7648(5) -75.8570(5) 1.4
Jastrow (24) + orbitals (44) -75.8069(5) -75.8682(5) 1.1
Jastrow $\times$ CAS(8,5) Jastrow (24) -75.8045(3) -75.8750(5) 1.3
Jastrow (24) + CSFs (6) -75.8094(5) -75.8807(5) 1.3
Jastrow (24) + CSFs (6)+ orbitals (52) -75.8374(5) -75.8882(5) 1.0
Jastrow $\times$ CAS(8,7) Jastrow (24) -75.8469(5) -75.8973(5) 1.2
Jastrow (24) + CSFs (49) -75.8546(5) -75.9032(5) 1.2
Jastrow (24) + CSFs (49) + orbitals (64) -75.8769(5) -75.9092(5) 0.9
Jastrow $\times$ CAS(8,8) Jastrow (24) -75.8462(5) -75.8999(6) 1.1
Jastrow (24) + CSFs (164) -75.8562(5) -75.9050(6) 1.1
Jastrow (24) + CSFs (164) + orbitals (70) -75.8801(6) -75.9099(5) 0.9
Jastrow $\times$ RAS-SDTQ(8,26) Jastrow + CSFs + orbitals (extrapolation) -75.9016(5) -75.9191(5) –
-75.9265(8)
Table \[tab:energies\] reports the total VMC and DMC energies, $E_{\ensuremath{\text{VMC}}}$ and $E_{\ensuremath{\text{DMC}}}$, and the VMC standard deviation of the local energy $\sigma_{\ensuremath{\text{VMC}}}= \sqrt{\langle E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}})^2\rangle - \langle
E_{\ensuremath{\text{L}}}({\ensuremath{\mathbf{R}}})\rangle^2}$ for the different trial wave functions considered. For the single-determinant, CAS(8,5), CAS(8,7) and CAS(8,8) wave functions, we present the results for three levels of optimization. At the first level, only the Jastrow factor is optimized. At the second level, the Jastrow factor and the CSF coefficients are optimized together. At the third level, the Jastrow factor, the CSF coefficients and the orbitals are all optimized together. Going from one level to the next one improves the accuracy of the wave function but also increases the computational cost of the optimization. We note that it is important to reoptimize the determinantal (CSF and orbital) parameters, along with the Jastrow parameters, rather than keeping them fixed at the values obtained from the MCSCF wave functions. For each wave function, the effect of reoptimizing the determinantal part is to lower the VMC energy by about $0.03$ to $0.04$ Ha, and the standard deviation of the energy by about $0.2$ to $0.3$ Ha. More remarkably, even though the optimization is performed at the VMC level, the DMC energy also goes down by about $0.01$ Ha, implying that the nodal hypersurface of the trial wave function also improves. In addition, one observes a systematic improvement of the VMC and DMC energies when the size of the CAS increases, provided that at least the CSF coefficients are reoptimized with the Jastrow factor.
Including all the $110481$ CSFs of the RAS-SDTQ(8,26) wave function is too costly in quantum Monte Carlo but one can use a series of truncated wave functions obtained by retaining only small numbers of CSFs with coefficients larger in absolute value than a variable cutoff, and then estimate the energy by extrapolation to the limit that all the CSFs are kept. Fig. \[fig:c2\_6s4p1d\_csfj\_cusp\_emin\_csfoj\] shows the VMC and DMC energies obtained with these truncated, fully reoptimized multi-determinantal wave functions with respect to the sum of the squares of the MCSCF CSF coefficients retained, $\sum_{i=1}^{N_{\ensuremath{\text{CSF}}}} (c_i^{\text{MCSCF}})^2$. Since the RAS-SDTQ(8,26) wave function is normalized, the latter quantity is equal to $1$ in the limit where all the CSFs are kept in the wave function. Experience shows that the energies are well extrapolated by quadratic fits. The extrapolated DMC energy is $-75.9191(5)$ which amounts for $98.6\%$ of the correlation energy (using the HF energy of -75.40620 Ha calculated in Ref. ).
On the other hand, to calculate accurate well depths (dissociation energy + zero-point energy) it is often sufficient to rely on some partial cancellation of error between the atom and the molecule by employing atomic and molecular wave functions that are consistent with each other. For example, using the DMC energy of the C$_2$ molecule given by the Jastrow-Slater full-valence CAS(8,8) wave function and the DMC energy of the C atom given by the consistent Jastrow-Slater full-valence CAS(4,4) wave function with the same one-electron basis leads to a well depth of 6.46(1) eV, in perfect agreement within the uncertainty with the exact, nonrelativistic well depth estimated at 6.44(2) eV [@BytRue-JCP-05; @BytRue-JCP-05-note]. In contrast, the well depth calculated from MCSCF with the molecular CAS(8,8) and atomic CAS(4,4) wave functions (without Jastrow factor) is 5.62 eV, in poor agreement with the exact value.
Conclusions {#sec:conclusion}
===========
We have studied three wave function optimization methods based on energy minimization in a VMC context: the Newton, linear and perturbative methods. These general methods have been applied here to the optimization of wave functions consisting of a multiconfiguration expansion multiplied by a Jastrow factor for the all-electron C$_2$ molecule. The Newton and linear methods are both very efficient for the optimization of the Jastrow, CSF and orbital parameters, the linear method being generally more stable. The less computationally expensive perturbative method is efficient only for the CSF and orbital parameters. We have used the linear method to simultaneously optimize the Jastrow, CSF and orbital parameters, a much more efficient procedure than alternating between optimizing the different kinds of parameters. The linear method is capable of yielding not only ground state energies but excited state energies as well [@NigMel-PRL-01].
Although the optimization is performed at the VMC level, we have observed for the C$_2$ molecule studied here, as well as for other systems not discussed in the present paper, that as more parameters are optimized the DMC energies decrease monotonically, implying that the nodal hypersurface also improves monotonically. In fact, a sequence of trial wave functions consisting of multiconfiguration expansions of increasing sizes multiplied by a Jastrow factor, with all the Jastrow, CSF and orbital parameters optimized together allows one to systematically reduce the fixed-node error of DMC calculations for the systems studied.
Future directions for this work include optimization of the exponents of the one-electron basis functions (either Slater or Gaussian functions), direct optimization of the DMC energy and optimization of the geometry.
We thank Claudia Filippi, Peter Nightingale, Sandro Sorella, Richard Hennig, Roland Assaraf, Andreas Savin, Anthony Scemama, Wissam Al-Saidi and Paola Gori-Giorgi for stimulating discussions and useful comments on the manuscript, and Eric Shirley for having provided us with the code for generating Hartree-Fock pseudopotentials. This work was supported in part by the National Science Foundation (DMR-0205328, EAR-0530301), Sandia National Laboratory, a Marie Curie Outgoing International Fellowship (039750-QMC-DFT), and DOE-CMSN. The calculations were performed at the Cornell Nanoscale Facility and the Theory Center.
|
---
abstract: |
By exploiting differences in muon lifetimes it is possible to distinguish $\nu_{\mu}$ from $\overline{\nu_{\mu}}$ charged current interactions in underground neutrino detectors. Such observations would be a useful tool in understanding the source of the atmospheric neutrino anomaly.\
Subject headings: Cosmic Rays — Elementary Particles — Neutrino Oscillations\
address: 'University of Notre Dame, Notre Dame, Indiana 46556'
author:
- 'J.M. LoSecco'
date: 'June 3, 1998'
title: 'Measuring the $\nu_{\mu}$ to $\overline{\nu_{\mu}}$ Ratio in a High Statistics Atmospheric Neutrino Experiment'
---
Introduction
============
The atmospheric neutrino anomaly [@haines; @kamioka; @imbo] is the discrepancy between the observed and expected rate of electron and muon neutrino interactions in underground detectors. In general it is believed that these neutrinos originate in the Earth’s atmosphere as a consequence of the decay of short lived particles created by cosmic ray interactions.
The most popular explanation for the source of the anomaly is the oscillation of muon neutrinos. A number of hypothetical solutions have suggested a new form of interaction. For example Ma and Roy [@Ma] point out that a new diagonal neutral current interaction for the $\nu_{\tau}$ could produce a coherent picture for all current neutrino problems (solar[@solar], atmospheric and LSND[@lsnd]). Such new interactions would have different effects on neutrinos and antineutrinos so there is some interest in distinguishing $\nu$ from $\overline{\nu}$. In general a charged current neutrino interaction produces a charged muon or electron. The sign of the charge can be used to infer the particle/antiparticle nature of the interacting neutrino.
Morphological methods have been employed to distinguish charged current muon and electron events. But the effect was initially recognized when the fraction of event containing a muon decay signature was considerably below expectations[@haines].
This paper points out that CPT violating differences in the detector itself make it possible to distinguish on a statistical basis between $\nu_{\mu}$ and $\overline{\nu_{\mu}}$ induced interactions.
The Method
==========
Due to the possibility of muon capture [@muonO] the $\mu^{-}$ has a larger decay width than $\mu^{+}$ when stopped in normal matter. $$\Gamma_{\mu^{+}} = \Gamma_{\mu} = 1/\tau_{+}$$ $$\Gamma_{\mu^{-}} = \Gamma_{\mu} + \Gamma_{Capture} = 1/\tau_{-}$$
This leads to a shorter lifetime for $\mu^{-}$ than for $\mu^{+}$ when they decay in ordinary matter. The effect may not be large. It is about 18% for muons in water but increases with $Z$ so is more pronounced in heavier materials.
The observed time distribution for muon decays is the weighted sum of the two exponential decay distributions.
$$f_{-} e^{-t/\tau_{-}} + (1-f_{-}) e^{-t/\tau_{+}}$$ where $f_{-}$ is the fraction of decays due to a $\mu^{-}$ and $\tau_{-}$ and $\tau_{+}$ are the [*known*]{} decay lifetimes for the $\mu^{-}$ and $\mu^{+}$ respectively in the detector environment.
The mean value of the measured lifetime of a mixture of $\mu^{-}$ and $\mu^{+}$ is then: $$\tau_{Observed} = f_{-} \tau_{-} + (1-f_{-}) \tau_{+}
= \tau_{+} - f_{-} (\tau_{+} - \tau_{-})$$ or $$f_{-}=\frac{\tau_{+}-\tau_{Observed}}{\tau_{+} - \tau_{-}}$$ Where $\tau_{Observed}$ is the measured value for the mean muon decay time in the muon neutrino sample.
In general detectors only sample the muon decay rate in a time window following the interaction so that there is a correction to this expression for truncation of the interval. For a data sample restricted to the time range $t_{1}<t<t_{2}$, $\tau_{\pm}$ in the expression above is modified to: $$\tau_{\pm} \rightarrow \tau_{\pm}
\frac{e^{(t2-t1)/\tau_{\pm}}(1+\frac{t_1}{\tau_{\pm}})
-(1+\frac{t_2}{\tau_{\pm}})}
{e^{(t2-t1)/\tau_{\pm}}-1}$$
A cleaner result might be obtained by fitting the observed time distribution to extract $f_{-}$ (and confirm the values of $\tau_{+}$ and $\tau_{-}$). A number of consistency checks are possible. The fraction of decays attributable to $\mu^{-}$ decreases more rapidly than for $\mu^{+}$ so one may get a more accurate measurement by using a delayed sample of decays. All such temporal subsamples must yield a consistent value for $f_{-}$.
With sufficient statistics this method can be exploited in bins of neutrino energy or flight distance which are the relevant observables for the oscillation hypothesis. Vacuum oscillations should show no difference in the $\mu^{+}$ to $\mu^{-}$ fraction so differences would be a clear indication of new physics.
Complications
=============
The value of $f_{-}$ is a measurement of the $\mu^{-}$ fraction of the muon decay sample. It is at best an indirect measurement of the $\nu_{\mu}$ to $\overline{\nu_{\mu}}$ flux ratio. The cross sections for interaction of these two neutrino types are quite different so the observed value of $f_{-}$ must be corrected.
The triggering efficiency and reconstruction efficiency for $\nu_{\mu}$ and $\overline{\nu_{\mu}}$ induced reactions may be different and must be corrected.
Muon polarization effects may make the detection efficiency for the two muon charges different. Some accounting for a lower efficiency for observing a signal from muon capture may be needed.
Subthreshold pion decays that give rise to the decay sequence $\pi \rightarrow \mu \rightarrow e$ may also populate the post interaction time distribution. Subthreshold pions are additional tracks in the initial neutrino interaction that escape detection but will subsequently decay. These can be studied in several ways. The muon decay time distribution observed in $\nu_{e}$ interactions can be subtracted from that observed in muon type reactions. Events with multiple muon decays can be studied to understand the rate for which subthreshold pion decays occur. With sufficiently high detection efficiency the problem could be eliminated by removing events in which more than one muon decay is observed.
A spatial cut might be possible in that muon decays occurring near the primary interaction vertex are removed since this is where such subthreshold pion decays would be found.
Conclusions
===========
The atmospheric neutrino anomaly has been firmly established. More information about the nature of the interactions is necessary to fully understand the physical mechanism responsible for the effect. By exploiting lifetime differences in muon decay one has access to the $\nu_{\mu}$ and $\overline{\nu_{\mu}}$ fractions of events. A sufficiently large sample would permit the study of the $\nu_{\mu}$ and $\overline{\nu_{\mu}}$ content as a function of energy and distance. The absence of any variation of this fraction with flight path would strengthen the case for neutrino oscillations. A variation would point to some new physics.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank my BaBar colleagues for discussions about CP and CPT violating observables.
T.J. Haines [*et al.*]{}, Phys. Rev. Lett. [**57**]{}, 1986 (1986). K.S. Hirata [*et al.*]{}, Phys. Lett. [**B205**]{}, 416 (1988). R. Becker-Szendy [*et al.*]{}, Phys. Rev. [**D46**]{}, 3720 (1992). D. Casper [*et al.*]{}, Phys. Rev. Lett. [**66**]{}, 2561 (1991). E. Ma and P. Roy, Phys. Rev. Lett. [**80**]{}, 4637 (1998). J.H. Bahcall and M.H. Pinsonneault, Rev. Mod. Phys. [**67**]{}, 1 (1995). C. Athanassopoulos [*et al.*]{}, Phys. Rev. Lett. [**75**]{}, 12650 (1995). I.S. Towner, Annu. Rev. Nucl. Part. Sci., [**36**]{}, 115 (1986).\
W.C. Haxton and C. Johnson, Phys. Rev. Lett. [**65**]{}, 1325 (1990).
|
---
abstract: 'We consider polariton condensation in a generalized Dicke model, describing a single-mode cavity containing quantum dots, and extend our previous mean-field theory to allow for finite-size fluctuations. Within the fluctuation-dominated regime the correlation functions differ from their (trivial) mean-field values. We argue that the low-energy physics of the model, which determines the photon statistics in this fluctuation-dominated crossover regime, is that of the (quantum) anharmonic oscillator. The photon statistics at the crossover are different in the high- and low- temperature limits. When the temperature is high enough for quantum effects to be neglected we recover behavior similar to that of a conventional laser. At low enough temperatures, however, we find qualitatively different behavior due to quantum effects.'
author:
- 'P. R. Eastham and P. B. Littlewood'
title: 'Finite-size fluctuations and photon statistics near the polariton condensation transition in a single-mode microcavity'
---
Introduction {#sec:introsec}
============
Microcavity polaritons[@hopfield58; @weisbuch92; @skolnick98; @savonapol98; @kavokinbook] are quasiparticles which form in wavelength-scale optical cavities containing dielectrics. They are mixed modes formed from cavity photons and dielectric excitations such as excitons. Since they are part photon polaritons are bosons, and are thus candidates for Bose condensation.
Experimental results on pumped microcavities continue to be linked to polariton condensation[@deng03; @richard05; @dang98; @senellart99]. The basic result is a threshold behavior of the luminescence intensity from a driven microcavity, while other features seen include non-thermal correlation functions for this luminescence, along with spatially and spectrally localized emission.
The central characteristic of Bose condensation is the generation of many-particle coherences which, in polariton condensation, appear in the electromagnetic field. The existence of polariton condensation remains controversial because polariton condensation is not the only phenomenon we can associate with coherent photons in microcavities. Most straightforwardly, if the cavity is driven into the weak-coupling regime one expects conventional lasing, and this is thought to be the correct interpretation of early claims for polariton condensation[@pau96; @fan97]. While the more recent experiments cannot be straightforwardly attributed to conventional lasing, more exotic alternatives to polariton condensation, such as polaritonic lasing [@laussy04], remain.
While there may appear to be many different routes to optical coherence in microcavities the relationships amongst these routes are not clear. Polariton condensation, polariton lasing, and conventional lasing are often assumed to be fundamentally distinct, but could equally well be related phenomena in different parameter regimes. This view is supported by recent work showing that adding decoherence processes to a mean-field theory of polariton condensation[@eastham00; @eastham01; @easthamthesis] leads to a crossover from condensation to conventional lasing[@szymanska02; @szymanska03]. It is also suggested by the fundamental connections between equilibrium and non-equilibrium phase transitions. The best known of these connections is between laser theory and the Landau theory of second-order phase transitions[@haken75; @risken70], but we note also recent work connecting the critical behavior of parametric oscillators and ferromagnets[@drummond05], and a treatment of the ideal Bose gas along the lines of laser theory[@scully99].
In this paper we investigate fluctuations close to polariton condensation, and how they affect the photon statistics. In general there are different regimes for the dominance of fluctuations: in a very large system at low density of excitation the thermal equilibrium transition is of the BEC variety,[@keeling04; @keeling05] so that spatial fluctuations are important. But since the polariton mass is very small (because of the large ratio between the wavelength of light and the typical exciton radius and exciton separation), at modest densities a BCS-like mean-field regime occurs and spatial fluctuations are small. Here the dominant fluctuations may be due to the finite size – or generically the finite population – of the system. This is very often also the case for conventional lasers[@haken75] (for similar reasons).
Here we consider finite-size fluctuations in isolation by studying a single-mode model microcavity. We obtain an approximate form for the free energy of the model by neglecting quantum effects. This form can be interpreted as the classical probability distribution for the intensity of the cavity field, and used to obtain all the static correlation functions of the cavity photons. It is identical to the accepted form for the intensity distribution near the onset of lasing, so we argue that within our approximations polariton condensation and lasing are not distinguished by the qualitative behavior of the static correlation functions. However the parameters in the intensity distribution are associated with different physics in the two theories, so there remains room for quantitative distinctions between them.
While neglecting quantum effects in a theory of condensation leads to the same intensity distribution predicted by classical laser theory, this approximation fails for a condensate at low temperatures. When the temperature becomes comparable with or less than the interaction energy of two photons the finite level spacing affects the correlation functions. We shall see that this leads to behavior for the correlation functions of a condensate at low temperature which is qualitatively different from the predictions of standard laser theory. This difference is not surprising because standard laser theory is a classical approximation, controlled by the photon number at threshold.
Our analysis concerns the thermal equilibrium of a simplified model of a microcavity. Although in some parameter regimes current experiments are far from thermal equilibrium, there are several reasons to study the equilibrium behavior. The above-threshold luminescence in some recent experiments[@deng03] is suggestive of thermal equilibrium, and as microcavities continue to develop[@tawara04; @pawlis04] experiments can be expected to reach states closer to thermal equilibrium. Furthermore, the behavior close to equilibrium is expected to be similar to that in equilibrium, and an understanding of the equilibrium physics provides the basis for developing non-equilibrium theories. Finally, the qualitative behavior we exhibit here, in particular the functional forms of the correlation functions, derives from the structure of the effective theory describing the collective variables. This structure may be independent of whether the theory describes an equilibrium system such as a conventional condensate or a non-equilibrium system such as a conventional laser.
The remainder of this paper is organized as follows. We begin with an outline of the main results in section \[sec:outline\], which compares the classical laser, the classical and quantum critical fluctuations of the polariton condensate in terms of the anharmonic oscillator. The rest of the paper provides a derivation of these results, and a deeper quantitative analysis of the model.
In section \[sec:model\] we present the model we consider for both lasing and condensation, and give some necessary background on the mean-field theory of polariton condensation. Section \[sec:freeenergy\] contains the general analysis resulting in the free energy in the transition region, including a discussion of the regime of applicability of the classical approximation. In section \[sec:corrfuns\] we use these general results to analyze the phase diagram and photon statistics of the condensate. In section \[sec:laser\] we briefly review the standard calculations of the intensity distribution in a laser near threshold, and compare with our results for the polariton condensate. In doing so, we note that conventional laser theory and its approximation to a phase transition relies on a “large-N” justification that is not usually exposed. Section \[sec:numbers\] contains numerical estimates of the size of the fluctuation-dominated and quantum regimes in current condensation experiments. In section \[sec:discussion\] we discuss prospects for systems with large quantum regimes, and the role of spatial fluctuations. Finally, section \[sec:conclusions\] summarizes our conclusions.
Outline of main results: the anharmonic oscillator {#sec:outline}
==================================================
We shall find that the classical laser and quantum/classical condensate lie in the same universality class as the anharmonic oscillator
$$H= \alpha \phi^\dagger \phi +\gamma \phi^\dagger
\phi^\dagger \phi \phi. \label{eq:anharmosc}$$
We choose $\phi$ to be normalized to obey the canonical Bose commutation relation $[\phi,\phi^{\dagger}] = 1$.
The eigenstates of (\[eq:anharmosc\]) are just the number states, $\lvert n\rangle$, with energies $E(n)=(\alpha-\gamma) n+\gamma n^2 \approx
\alpha n+\gamma n^2$. Thus the partition function is $$Z=\sum_{n=0}^\infty e^{-\beta (\alpha n + \gamma
n^2)},\label{eq:anharmz}$$ with $\beta = 1/k_BT$.
The parameter $\alpha$ is the tuning parameter through the transition. At the mean-field level, minimization of the exponent in (\[eq:anharmz\]) leads to $$\begin{aligned}
n_{\mathrm{min}} &= 0 \;\;\; &{\rm for } \;\; \alpha > 0 \nonumber \\ \\
n_{\mathrm{min}} &= \frac{|\alpha|}{2 \gamma} \;\;\; & {\rm for } \;\; \alpha <
0. \nonumber\end{aligned}$$ Expanding in terms of fluctuations, $\delta n = n- n_{\mathrm{min}}$, one obtains (on the “condensed” side $\alpha < 0$) quadratic number fluctuations controlled by $\gamma (\delta n)^2$. We shall choose the parameter $\gamma \propto 1/N$, with $N$ growing with the system size; the limit $N \to \infty$ gives the mean-field result. We show below that the Dicke model indeed gives rise to a partition function of the form of (\[eq:anharmz\]) after truncating higher order terms in the exponent that are systematically smaller in powers of $1/N$ (where in this case $N$ is the number of quantum dots in the cavity).
The summation in the partition function is of course still over discrete quantum states $\lvert n\rangle$, but if the temperature is high enough the discreteness is irrelevant and the sum can be replaced by an integral. We now use fields $\psi = \phi / \sqrt N$ rescaled by system size so that $\psi$ appropriately describes the classical electromagnetic field intensity. Then one may write the partition function as $$\label{eq:psi4}
Z \approx \int d\psi d \psi^* e^{-\beta N [ \alpha |\psi|^2 + \gamma N |\psi|^4]}$$ (remembering that $\gamma N = O(1)$). We derive this form explicitly for the Dicke model in section \[sec:freeenergy\]. We now remark that the action in (\[eq:psi4\]) is consistent with the steady-state distribution from a Fokker-Planck equation for diffusion in a quartic potential (see section \[sec:laser\]), which is the conventional approach to laser theory. Of course the “temperature” there is a fiction that is generated by couplings to (Markov) baths representing the outside world. Furthermore there is no quantum limit for that type of laser theory. In the classical limit the steady state laser and the polariton condensate have the same scaling form.
The anharmonic oscillator thus encapsulates the basic results of the two different models when we focus on the low energy physics. One signature of the transition is the intensity-intensity correlation function $g^{(2)}(0)$, which is straightforward to calculate for the anhmarmonic oscillator:
$$g^{(2)}(0)=\frac{\sum_n
n(n-1)e^{-\beta(\alpha n + \gamma n(n-1))}}{(\sum_n ne^{-\beta(\alpha
n + \gamma n(n-1))})^2} .$$
![$g^{(2)}(0)$ as a function of $\alpha$ for the thermal equilibrium of the anharmonic oscillator (\[eq:anharmosc\]). The top panel shows the predictions of the classical approximation, while the bottom panel is the result of numerically evaluating the partition function. $\gamma=1/40$, and $T=2,1/2,1/5,1/15$, and $1/50$ for the solid, dashed, dotted, dot-dashed, and long-dashed curves, respectively. The upper family of lines in the top panel are at $ 4(\gamma\beta)=(\alpha\beta)^2$, indicating the boundary of the classical fluctuation-dominated regime. The lower family are $\alpha=T$, indicating the range of validity of the classical approximation in the normal regime. The circles on the lower panel are the (discrete) values taken by $g^{(2)}(0)$ at $T=0$. []{data-label="fig:quantumgtwo"}](fig1.eps){width="3in"}
In Fig \[fig:quantumgtwo\] we plot $g^{(2)}(0)$ calculated numerically for (\[eq:anharmosc\]) as a function of $\alpha$, at several temperatures. The tuning parameter $\alpha$ is proportional to density (at fixed temperature) for the polariton problem, and generically is the pump rate in the laser model. The mean-field ordered phase is to the right, where we recover asymptotically $g^{(2)}(0) \to 1$ as expected for a classical macroscopic field. In the top panel the classical (high-temperature) result is shown, with different curves corresponding to different temperatures. As temperature is lowered the result approaches more and more closely the mean-field theory, which is of course a step function. For specificity we have chosen $\gamma = 1/N = 1/40$ for this demonstration, though the qualitative evolution will be the same for different values of $\gamma$.
While at high temperatures($\gg \gamma$) $g^{(2)}(0)$ resembles the classical form, it is very different at low temperatures, $\lesssim
\gamma$. Well below the transition the dominant contribution to $g^{(2)}(0)$ describes the thermal excitations of one and two photon states, so $$g^{(2)}(0) \approx 2e^{-2\beta\gamma}.$$ Although in this regime the statistics are strongly sub-Poissonian, the intensity is also very low, $\approx e^{-\beta
\alpha}$. On crossing the transition the intensity increases and $g^{(2)}(0)$ approaches $1$. At the lowest temperature $g^{(2)}(0)$ is a small constant below the transition, which then approximately follows the zero-temperature result above the transition: $g^{(2)}(0)=1-1/n$ with $n$ the nearest integer to $-\alpha/(2\gamma)-1/2$. At higher temperatures the quantum corrections decrease, and the form of $g^{(2)}(0)$ is intermediate between the classical result and the low-temperature one.
While the results are straightforwardly exposed in terms of the anharmonic oscillator, the mapping of the polariton condensate to the parameters $\alpha$ and $\gamma$ is more complex. The essential details are exposed in the phase diagram of Figure \[fig:critregions\].
![Fluctuation-dominated regimes in terms of density and temperature for the polariton condensation transition in the model (\[eq:dickeham\]), with $E(i)=\omega$ so that all excitons are resonant with the cavity mode. The solid line is the mean-field boundary, and dashed(dotted) lines mark the boundaries of the fluctuation-dominated regions for $N=10$(40). Curves correspond to those in Fig. \[fig:critregionsmu\]. The shading indicates the regions in which the static approximation holds for $N=10$. []{data-label="fig:critregions"}](fig2.eps){width="3in"}
The solid line is the mean-field phase boundary between the uncondensed (to the left) and condensed phases. Here the polariton density is measured per site, which explains why the mean-field boundary asymptotes to $\rho_{\mathrm{ex}} = 1/2$ (the boundary of inversion) at large temperatures. The dashed and dotted lines mark regimes on either side of the mean-field transition where we estimate fluctuations to be substantial. These lines correspond to the upper panel of lines in Figure \[fig:quantumgtwo\]. The strong asymmetry of the fluctuation regime about the mean-field line arises because we have chosen to expose the results with density as a parameter, rather than an external tuning parameter (here, naturally, the chemical potential). On the condensed side of the transition the number fluctuations are large because the chemical potential for polaritons becomes nearly clamped. The shading marks the estimated regime of validity of the classical approximation (akin to the lower panel of lines in Figure \[fig:quantumgtwo\]). Over most of the figure the shaded regime encloses the fluctuation regime, and the classical approximation (which guarantees that $g^{(2)}(0) > 1$) holds. For small densities and temperatures, however, we see that there is a strongly-interacting quantum regime.
Model {#sec:model}
=====
Mean-field theories of polariton condensation have now been developed from models of a range of systems, including those with propagating photons[@keeling05; @keeling04]. Here we consider the simplest model which leads to a theory of polariton condensation, the generalized Dicke[@dicke54] model $$\label{eq:dickeham} H=\omega \psi^\dagger \psi +
\sum_i \frac{E(i)}{2} S^{z}_i + \frac{g}{\sqrt{N}}\sum_i (S^{+}_i \psi
+ \psi^\dagger S^{-}_i).$$ This is the basis of standard laser theory, as well as of our earliest mean-field theory[@easthamthesis; @eastham01; @eastham00] of polariton condensation. It directly describes a system of localized excitons, for example in quantum dots, embedded in a three-dimensional microcavity. There is a single cavity mode, with annihilation operator $\psi$ and energy $\omega$, dipole coupled to a set of $N$ quantum dots or atoms of the gain medium. The state $S_i^z=+(-)1$ corresponds to the presence (absence) of an exciton on site $i$, or to an atom in the upper (lower) of the states coupled by the lasing transition.
We have included a factor of $1/\sqrt{N}$ explicitly in the light-matter coupling in (\[eq:dickeham\]), so that for a cavity mode coupled to $N$ dots $g$ is related to the observed Rabi splitting and not to the single-dot Rabi splitting. This scaling is formally necessary because we will be concerned with the properties of the model (\[eq:dickeham\]) for large $N$, so need the model to be well behaved in the limit $N\to\infty$. But we stress that with our convention changing the number of dots in a cavity of fixed volume corresponds to changing both $N$ and $g$.
To construct a theory of Bose condensation of polaritons from (\[eq:dickeham\]) one studies its thermodynamics, fixing the total number of excitons and photons $$N_{\mathrm{ex}}=\psi^\dagger\psi+\frac{1}{2}\sum_i
\left(S_i^z+1\right). \label{eq:nexdefn}$$ Thus we consider the free energy, which can be written as the functional integral $$\label{eq:gpf} \frac{F}{kT}=f=\ln \int \mathcal{D}\psi\mathcal{D}\bar{\psi}
e^{-NS_{\mathrm{eff}}},$$ where $$S_{\mathrm{eff}}=\int_0^\beta d\tau
\bar{\psi}(\partial_{\tau}+\tilde{\omega}_{c})\psi - \ln \det P. \label{eq:seff}$$ Here $\psi$ is related to the real electromagnetic field amplitude $\psi_0$ by $\psi_0=\psi\sqrt{N}$, while $\ln \det P$ corresponds to the free energy of the quantum dots in the field $\psi(\tau)$.
Eq. (\[eq:gpf\]) is the free energy in the grand-canonical ensemble. The constraint on $N_{\mathrm{ex}}$ is dealt with on average, by introducing a chemical potential $\mu$ which appears in (\[eq:seff\]) as a shift of the photon and exciton energies: $\omega
\to \tilde\omega=(\omega-\mu)$ and $E(i) \to \tilde E(i)=(E(i)-\mu)=2
\tilde\varepsilon_i$. In the limit $N\to\infty$ the relative fluctuations of $N_{\mathrm{ex}}$ which occur in the grand-canonical ensemble are negligible, and the grand-canonical and canonical ensembles are expected to be equivalent. However, we are now considering fluctuations in a finite system, for which the predictions of the canonical and grand-canonical ensembles could differ. The correct ensemble then depends on how the polariton number is constrained experimentally. We consider an idealized limiting case in which this is done by coupling to a reservoir, so that the grand-canonical ensemble is appropriate.
We can develop an asymptotic expansion of the free energy for large $N$ by expanding $S_{\mathrm{eff}}$ around its static saddle points $\psi(\tau)=\psi$. The saddle-point equation gives the mean-field estimate of the phase boundary between the normal state ($\psi=0$) and the condensed state ($\psi$ finite). Evaluating (\[eq:gpf\]) on the stable saddle point leads to the mean-field estimate of the free energy; this is the only extensive contribution to the free energy, so that the mean-field theory gives the exact asymptotic form for the free energy in the thermodynamic limit $N\to\infty$.
Polariton condensation in the model (\[eq:dickeham\]) can be viewed as a generalization of the ferroelectric transition discovered by Hepp and Lieb[@hepp73; @eastham01]. There has been recent interest in the physics associated with that ferroelectric transition in a finite model at zero temperature[@emary03; @vidal05]. This differs from the present problem in that there is no constraint on the polariton number, *i.e.* $\mu=0$, and as a consequence the rotating-wave approximation implicit in (\[eq:dickeham\]) cannot be made. This leads to qualitative differences[@emary03] in the dynamics of the unconstrained models at $T=0$.
General form of the Free energy near the transition {#sec:freeenergy}
===================================================
Away from the mean-field phase transition the saddle-point expansion provides a systematic approximation scheme for the free energy of the finite system. However, this scheme fails in the vicinity of the mean-field transition due to the diverging occupations of the soft fluctuations. In this section, we shall calculate the general form of the free energy of a large but finite system near the mean-field transition. To simplify the notation we shall take all the excitons to have the same energy, $E(i)=E_g$, but the resulting forms may be straightforwardly generalized to allow for a distribution of exciton energies.
We consider the action obtained by expanding (\[eq:seff\]) to fourth order in $\psi$, $$\begin{aligned}
S_{\mathrm{eff}}&=&S_0+\beta
\sum_{\omega}\bar{\psi}(\omega)\psi(\omega) \frac{(i\omega +
E_+)(i\omega + E_-)}{(i\omega+2\tilde{\varepsilon})}
\nonumber \\ && +
\frac{g^4}{2} \sum_{\omega_1+\omega_2=\omega_3+\omega_4}
\bar\psi_{\omega_1}\bar\psi_{\omega_2}\psi_{\omega_3}\psi_{\omega_4}
V_{\omega_1\omega_2\omega_3\omega_4} \nonumber \\ && \quad +
\ldots. \label{eq:quarticaction}\end{aligned}$$ $S_0$ is the action of the two-level systems in the absence of photons, while the remaining part describes photons in the medium of the two-level systems. There are resonances at $E_{\pm}$, which are the polariton energies relative to the chemical potential $$\label{eq:polenergies} E_\pm = \frac{1}{2} (\tilde\omega_c +
2\tilde\varepsilon \pm
\sqrt{(\tilde\omega_c-2\tilde\varepsilon)^2+4g^2 \tanh \left(\beta
\tilde\varepsilon\right)}),$$ while $V_{1234}$ is the photon-photon interaction mediated by the two-level systems, $$\begin{gathered}
V_{1234}=\sum_{\omega_f} \{
(i\omega_f-\tilde\varepsilon)^{-1}[i(\omega_f+\omega_3)+\tilde\varepsilon]^{-1} \\
\times [i(\omega_f+\omega_3-\omega_2)-\tilde\varepsilon]^{-1} \\ \times
[i(\omega_f+\omega_4+\omega_3-\omega_2)+\tilde\varepsilon]^{-1}\}.\label{eq:intsum}\end{gathered}$$ $\omega_{1}$, etc. are bosonic Matsubara frequencies, while $\omega_{f}=(n+3/4)2\pi T$ is a fermionic frequency, shifted to take account of the two-level constraint[@popov88; @keeling05].
Static free energy
------------------
As we approach the transition from the normal side one of the polariton energies $E_{\pm}$ is approaching zero, and perturbation theory fails. Over most of this fluctuation regime, however, $\omega_1=2\pi T$ is large compared with the energy of the soft mode. The dominant contribution to the free energy for a large system close to the transition then comes from the static (classical) paths. Retaining only these dominant contributions gives an approximation for (\[eq:gpf\]) near the transition, $$\begin{aligned}
\label{eq:staticfullcfenergy}f&\approx& \ln \int d\psi d\psi^*
e^{-N(\beta
\tilde\omega_c|\psi|^2-\ln\cosh\beta\sqrt{\tilde{\varepsilon}^2+g^2|\psi|^2})}
\\ &=& f_0+\ln \int d\psi d\psi^\ast e^{-N(a |\psi|^2+b |\psi|^4 + c
|\psi|^6 + \ldots)}. \label{eq:staticfenergy}\end{aligned}$$ Here $a=\beta E_+E_-/(2\tilde\varepsilon)$ is the static part of the Gaussian kernel in (\[eq:quarticaction\]), measuring the distance to the transition, $b$ is the static part of the interaction term, and $c, d,
\ldots$ are higher-order interaction strengths which do not depend on the system size $N$. $f_0=N \ln \cosh \beta\tilde\varepsilon$ is the free energy of the two-level systems in zero field. The explicit form for $b$, obtained either from (\[eq:intsum\]) or by expanding the exponent in (\[eq:staticfullcfenergy\]), is $$b=g^4 \beta \frac{\tanh(\beta
\tilde\varepsilon)-\beta\tilde\varepsilon\operatorname{sech}^2(\beta\tilde\varepsilon)}{8\tilde\varepsilon^3}. \label{eq:staticintstrength}$$
Reduction to an oscillator {#sec:freeenergy-osc}
--------------------------
At low temperatures the non-perturbative regime may lie outside the regime of validity of the static approximation, so that time-dependent paths must be considered. The divergences of perturbation theory are still at small frequencies, so for a large system close to the transition we can replace the action with its low frequency form. Considering for definiteness the region near the transition where $E_-$ vanishes, we have $\omega, E_-$ as small parameters, while generically $E_+$ and $\tilde\varepsilon$ are finite. The Gaussian term in (\[eq:quarticaction\]) can then be straightforwardly approximated as $$\frac{\beta E_+}{2\tilde\varepsilon}
\sum_{\omega}\bar{\psi}(\omega)\psi(\omega)
(i\omega+E_-), \label{eq:gausslowfreq}$$ with corrections proportional to the ratios of small to finite parameters. Approximating the interaction term is more involved, because the result of the summation (\[eq:intsum\]) takes different forms depending on how many of the external frequencies coincide. This problem can be avoided by restricting our attention to the low-temperature regime $T \ll
\tilde \varepsilon$, in which all the forms lead to the same low-frequency approximation $$\frac{g^4 \beta}{8\tilde\varepsilon^3}
\sum_{\omega_1+\omega_2=\omega_3+\omega_4}
\bar\psi_1\bar\psi_2\psi_3\psi_4.
\label{eq:quartlowfreq}$$ The corrections to (\[eq:quartlowfreq\]) are again small in terms of the ratios of small to large parameters, *e.g.* $\omega/\tilde\varepsilon$ and $T/\tilde\varepsilon$. Since the coefficient in (\[eq:quartlowfreq\]) is the low-temperature limit of $b$, it is convenient to replace it with $b$, leading to a low-energy approximation to the original theory which is valid at the quantum level for low temperatures, and at the classical level at higher temperatures.
(\[eq:gausslowfreq\]) and (\[eq:quartlowfreq\]) become, after rescaling the fields $\psi\to\psi\sqrt{\frac{2\tilde\varepsilon}{E_+
N}}$, the action for a quantum anharmonic oscillator (\[eq:anharmosc\]). The oscillator frequency $\alpha=E_-$, and the interaction strength is $\frac{b}{\beta N}\left(\frac{2\tilde\varepsilon}{E_+}\right)^2$, or $\frac{g^4}{2\tilde\varepsilon N E_+^2}$ at low temperatures.
Validity of the static approximation {#sec:freeenergy-stat}
------------------------------------
The static approximation leading to (\[eq:staticfullcfenergy\]) holds because the energy of fluctuations is much less than temperature. On the normal side of the transition this only occurs in a region close to the transition. However, the behavior of the excitation spectrum on the condensed side is different. This spectrum[@eastham01; @eastham00; @easthamthesis] comprises a mode which is always at zero frequency relative to $\mu$, and two modes which at the transition are at positive and negative of the non-vanishing polariton energy. Thus there appears to be no mode which becomes soft at the transition, and so the classical approximation does not appear to be controlled by the distance from the transition.
This puzzle is resolved by inspecting the anharmonic oscillator (\[eq:anharmz\]), which we introduced earlier in section \[sec:outline\]. On the normal side of the transition the nonlinear terms in (\[eq:anharmz\]) are irrelevant, and the parameter controlling the classical approximation is $\beta \alpha$. On the condensed side the exponent in (\[eq:anharmz\]) can be rewritten in terms of the number fluctuations $\delta n=n -
n_{min}=n-(-\alpha/(2\gamma))$ as $-\beta \gamma (\delta n)^2$. Thus the parameter controlling the static approximation on the condensed side is $\beta \gamma$. Returning to the polariton model close to the transition where $E_-$ vanishes, we see that the classical approximation holds for the low-energy fluctuations when $$\frac{b}{N}\left(\frac{2\tilde\varepsilon}{E_+}\right)^2 \ll 1,
\label{eq:cclassin}$$ provided the temperature is small compared with the Rabi splitting and $\tilde\varepsilon$. At higher temperatures we expect an inequality of the same general form, $TN/g\gtrsim 1$, but with numerical differences due to the renormalization of the effective interaction by the occupation of the high-energy polariton and the frequency dependence of the interaction.
Our previous computations[@eastham01; @eastham00; @easthamthesis] of the fluctuation spectrum were done at a Gaussian level, *i.e.* approximating the partition function of the fluctuations with that of a harmonic oscillator. This approximation predicts that the characteristic frequency of fluctuations about the condensate is zero because there is no linear term in $\delta n$ when the exponent of (\[eq:anharmz\]) is expanded around a finite $n$. To obtain a finite level spacing one must go beyond the Gaussian approximation and include interactions, which generate a finite level spacing $\sim
1/N$. This may be contrasted with the normal state, where the harmonic oscillator part of (\[eq:anharmz\]) is enough to generate a finite level spacing $\alpha$.
Correlation functions and phase diagram {#sec:corrfuns}
=======================================
The correlation functions of the cavity field can be obtained by differentiating (\[eq:staticfullcfenergy\]) with respect to $\tilde\omega_c$ or (\[eq:staticfenergy\]) with respect to $a$. Note that the field is classical: the integrand of (\[eq:staticfullcfenergy\]) can be interpreted as a probability distribution for the intensity of the cavity field. Non-classical fields are associated with time-dependent paths, which give complex integrands in the path integral (\[eq:gpf\]) that cannot be interpreted as classical probabilities.
The integral (\[eq:staticfullcfenergy\]) is only tractable numerically. However, the interactions $c, d, \ldots$ do not affect the asymptotic behavior of the correlation functions as $N\to\infty$ if $a\geq 0$ and $b>0$, because the quartic nonlinearity restricts fluctuations of the field to $|\psi|\lesssim
N^{-1/4}b^{-1/4}$. Therefore we can obtain the asymptotic forms of the correlation functions on the normal side of the mean-field phase boundary from (\[eq:staticfenergy\]) with $c, d, \ldots=0$. The corresponding free energy is, discarding additive constants, $$\begin{aligned}
f&=&\ln \int d\psi d\psi^\ast
e^{-N(a|\psi|^2+b|\psi|^4)}\label{eq:fquadintgl} \\ &=&\ln
\frac{e^{\frac{Na^2}{4b}}\operatorname{erfc}\left(\frac{a}{2}\sqrt{\frac{N}{b}}\right)}{\sqrt{bN}}.
\label{eq:fquadexplicit} \end{aligned}$$ On the condensed side where $a<0$ we may still use the approximation (\[eq:fquadintgl\]) so long as the order parameter $|\psi|$ is small. It is a weaker approximation than on the normal side, because the minimum of the exponent in (\[eq:staticfenergy\]) occurs for $|\psi|^2\sim 1$, so that truncation to a quartic theory produces errors in the leading asymptotics of the correlation functions. However, these errors are proportional to the order parameter, and so are numerically small close enough to the transition, even if they are not asymptotically small in $N$.
We now present explicit results for the behavior of the cavity field in the model (\[eq:gpf\]) when $N$ is finite but large. For orientation, the mean-field phase boundaries $E_{\pm}=0$ are shown as the solid lines in Fig. \[fig:critregions\]. $\rho_{ex}$ is the number of polaritons per site, $\langle N_{\mathrm{ex}}/N \rangle$ with $N_{\mathrm{ex}}$ given by (\[eq:nexdefn\]); note that this differs from the definition of $\rho_{ex}$ used in Refs. by a shift of $0.5$. Temperature is expressed in units of $g$, which is one-half of the collective Rabi splitting at resonance(see Eq. \[eq:polenergies\]).
For completeness we begin by considering the region far from the transition, where the correlation functions can be obtained using the saddle-point expansion. On the normal side of the mean-field transition there is no saddle-point contribution to the photon density $\langle \psi^\dagger \psi \rangle$, so the first non-vanishing contribution is at order $1/N$. This term is just the expectation value of the photon density from the Gaussian part of (\[eq:quarticaction\]), *i.e.* the photon density in a population of non-interacting polaritons with energies $E_{\pm}$: $$\begin{aligned}
\label{eq:photongaussian} \langle \psi^\dagger \psi \rangle &=& \frac{(E_+ -
2\tilde{\varepsilon})n_{\mathrm{B}}(E_+) + (2\tilde{\varepsilon} -
E_-)n_{\mathrm{B}}(E_-)}{N(E_+-E_-)}, \\ n_\mathrm{B}(x)& =&
\frac{1}{e^{\beta x}-1}.\end{aligned}$$ Since at this order in $N$ we have non-interacting particles the many-photon correlation functions are related to the photon number by Wick’s theorem. In particular, for the static part of the correlation function measured by Deng *et al.* we have $g^{(2)}(0)=\langle\psi^\dagger\psi^\dagger\psi\psi\rangle/\langle\psi^\dagger\psi\rangle^2=2+O(1/N)$.
On the condensed side of the mean-field transition the leading-order contributions to the correlation functions come from the saddle point. Thus $\langle \psi^\dagger \psi \rangle \sim 1$, and we expect $g^{(2)}(0)=1+O(1/N)$. Calculations of the higher-order terms are complicated due to the presence of the zero mode, and so we shall not pursue them here.
In the region close to the mean-field transition the saddle-point expansion must break down, to allow the correlation functions to smoothly interpolate between their forms in the two states. We can estimate the boundaries of this crossover region by equating the magnitudes of successive terms in the large-N expansion of the photon density, $\langle \psi^\dagger \psi \rangle$. For the approximate form (\[eq:fquadexplicit\]) we see that the crossover regime obeys $Na^2\lesssim 4 b$, or $$\label{eq:critinequality}\frac{2g^4}{\beta N
\tilde\varepsilon
E_+^2E_-^2}\left(\tanh(\beta\tilde\varepsilon)-\beta\tilde\varepsilon\operatorname{sech}^2(\beta\tilde\varepsilon)\right)
\gtrsim 1.$$ Note that the numerical factor in this expression is not meaningful, since it depends on the precise definition of the boundaries of the crossover regime.
![Fluctuation-dominated regimes for the polariton condensation transition with $\Delta=0$. Solid lines are the mean-field boundaries. Dashed (dotted) lines mark the boundaries of the fluctuation-dominated regions for $N=10$(40).[]{data-label="fig:critregionsmu"}](fig3.eps){width="3in"}
In Fig. \[fig:critregionsmu\] we show the phase diagram of the system as a function of $\mu$ and $T$, for $\Delta
g=\omega-E_g=0$. The solid lines are the phase boundary of the infinite system, $E_+E_-=0$, while the remaining lines are the boundaries of the crossover regions (\[eq:critinequality\]) for systems of 10 and 40 quantum dots. One general feature which can be seen in this figure is that the size of the fluctuation regions scales as $1/\sqrt{N}$, as can be deduced from (\[eq:fquadexplicit\]). The particular shapes of the fluctuation regions regions come from the interplay between the thermal occupations and the strength of the interaction, which is temperature and density dependent. In particular, the present model has the unusual feature that the fluctuation regions vanish in the high-temperature limit, where the limits of the inequality (\[eq:critinequality\]) are approximately $$4\tilde\varepsilon=g^2\beta-2\Delta g \pm
\frac{2\beta g^2}{\sqrt{3N}}.$$ Physically, this §occurs because for large $T$ the number of fluctuations at fixed chemical potential behaves as $n_{\mathrm{fluc}}=n_{\mathrm{B}}(E_+)+n_{\mathrm{B}}(E_-) \sim
T$ while the interaction strength $b$ vanishes like $1/T^4$. Thus the mean interaction energy $\langle n_{\mathrm{fluc}}^2
b \rangle$ vanishes in the high temperature limit. The occupation of the fluctuations diverges only as $T$ because they are confined to a finite energy range, so that increasing temperature does not increase the number of relevant fluctuation modes, as it would in the more familiar case of a semi-infinite band of states. The interaction vanishes because at high temperatures the field does not affect the occupation of the two-level systems: the free energy of a two-level system, which appears in the exponent of (\[eq:staticfullcfenergy\]), becomes independent of field $|\psi|$ as $T\to \infty$.
Fig. \[fig:critregions\] shows the same phase diagram in terms of density and temperature, obtained by relating chemical potential to density using the mean-field results. For the normal state we use $$\rho_{\mathrm{ex}}=\rho_{\mathrm{excitons}}=(1-\tanh(\beta\tilde\varepsilon))/2,
\label{eq:mfdenseq}$$ while for the condensate the quartic theory gives $$\rho_{\mathrm{ex}}=\rho_{\mathrm{excitons}}+kT\frac{\partial}{\partial
\mu}\left(\frac{a^2}{4b}\right).\label{eq:mfdenseqcond}$$ These relations introduce qualitative differences between the fluctuation-dominated regions in this figure and those in Fig. \[fig:critregionsmu\]. In particular, the fluctuation-dominated region becomes much larger on the condensed side, because the chemical potential is only weakly dependent on density in the condensate.
The shading on Fig. \[fig:critregions\] indicates the regimes of validity of the classical approximation, combining the analysis of sections \[sec:freeenergy-osc\] and \[sec:freeenergy-stat\] with Eqs. \[eq:mfdenseq\] and \[eq:mfdenseqcond\]. On the normal side of the mean-field transition we shade the region $\beta E_-<1$. On the condensed side the analogous inequality is (\[eq:cclassin\]). However, the numerical factors in this result are only correct close to the phase boundary. Since this only occurs at low densities we have plotted the inequality $T/g>1/(4N)$, corresponding to the low-density limit of (\[eq:cclassin\]).
In the upper panel of Fig. \[fig:photofmu\] we plot the prediction of (\[eq:fquadexplicit\]) for the number of photons in the the cavity $$N \langle \psi^\dagger\psi\rangle =
-\frac{\partial f}{\partial a},$$ as a function of the deviation of $\mu$ from its mean-field critical value $\mu_c$. We take $\Delta=0$, and plot curves for $T/g=0.25, 0.75$ and $1.5$, and for $N=10$ and $40$.
![Photon number $N\langle \psi^\dagger \psi \rangle$ (top panel) and density $\langle \psi^\dagger \psi \rangle$ (bottom panel) as functions of $(\mu-\mu_c)/(2g)$ for $\Delta=0$. $T/g=0.25$ (solid lines), 0.75 (dashed lines), and 1.5 (dotted lines). $N=10$ (lower line of each pair in the top panel, upper in the bottom panel) and $N=40$ (upper line in top panel, lower in bottom panel). Shading marks the condensed region of the mean-field theory.[]{data-label="fig:photofmu"}](fig4.eps){width="3in"}
In general each of the $N$ two-level systems makes a contribution of order $1/N$ to the number of cavity photons, due to the scaling of the coupling constant. Well below the transition we could neglect the interactions between the photons and approximate the integrand of (\[eq:staticfenergy\]) by a Gaussian. This gives $N \langle
\psi^\dagger \psi \rangle\approx 1/a$. In the Gaussian regime each two-level system contributes incoherently to the cavity field, so the total photon number is of order $1$; such scaling is demonstrated by the collapse of the curves corresponding to different system sizes in the left side of the top panel of Fig. \[fig:photofmu\].
As we increase the chemical potential through the mean-field transition the occupancy of the cavity field increases, and the interactions begin to generate coherence between the contributions of the different two-level systems. Far in the condensed state this coherence is complete: all the $\sim N$ two-level systems contribute coherently to the cavity field, giving a photon number which scales as $N$. This scaling can be seen on the right of the lower panel of Fig.\[fig:photofmu\], which shows the photon number per two-level system. The order of $N$ contribution to the photon number comes from the saddle point of (\[eq:fquadintgl\]), and is $-a/(2b)$. This saddle-point contribution is the exponential in (\[eq:fquadexplicit\]); it survives in the condensed state, but is canceled by the asymptotic expansion of the error function well into the normal state.
In the region near the transition neither the Gaussian nor saddle-point approximations are appropriate, and the full form (\[eq:fquadexplicit\]) must be used. There is partial coherence amongst the two-level systems, leading to a photon number which scales as $\sqrt{N}$. Explicitly, we find $N\langle\psi^\dagger\psi\rangle=\sqrt{N/(\pi b)}$ for the photon number at the transition. The scaling can be shown from a more general argument by noting that the $a$ dependent part of (\[eq:fquadintgl\]) is a function of $\sqrt{N}a$ and $b$, so at the transition $$\begin{aligned}
N\langle \psi^\dagger \psi \rangle = -\left.\frac{\partial
f(\sqrt{N}a,b)}{\partial a}\right|_{a=0} \nonumber \\ =
-\sqrt{N}g(0,b). \end{aligned}$$
Fig. \[fig:iicorrfun\] illustrates the results obtained from (\[eq:fquadexplicit\]) for the static intensity-intensity correlation function $$g^{(2)}(\tau=0)=1+\frac{\frac{\partial^2f}{\partial
a^2}}{\left(\frac{\partial f}{\partial a}\right)^2}.$$ Well below the transition $g^{(2)}(0)$ approaches the value of $2$ associated with a thermal mixture of non-interacting photons, while well above it $g^{(2)}(0)$ approaches $1$, corresponding to the bunched photons of *e.g.* a coherent state or large-amplitude number state. The crossover occurs over a range of chemical potentials which scales as $1/\sqrt{N}$.
![Intensity-intensity correlation function $g^{(2)}(0)$ of the cavity photons as a function of chemical potential $(\mu-\mu_c)/(2g)$ for $\Delta=0$. $T/g=0.25$ (solid lines), 0.75 (dashed lines), and 1.5 (dotted lines). $N=10$ (lower line of each pair in the normal region) and $N=40$ (upper line). Shading marks the condensed region of the mean-field theory. Circles correspond to the lines marking the fluctuation-dominated regimes of the normal state in Fig. \[fig:critregions\]. The vertical line corresponds to the limit of the shading in the normal state on Fig. \[fig:critregions\] for $T/g=0.25$; the analogous lines for $T/g=0.75$ and $1.5$ are off the scale.[]{data-label="fig:iicorrfun"}](fig5.eps){width="3in"}
Laser fluctuations {#sec:laser}
==================
In this section, we briefly review the standard theory of fluctuations in a single-mode laser close to threshold, following the work of Risken[@risken70]. We shall see that the photon statistics predicted by this theory are similar to those predicted by the theory of polariton condensation. This similarity holds in spite of the fact that the theory of condensation describes a system in thermal equilibrium while laser theory describes one which is not in thermal equilibrium.
The canonical laser theories are based on the dynamics of models such as (\[eq:dickeham\]) in the presence of dissipation processes. Such dissipation processes can be formally generated by coupling each dot, and the photon mode, to an oscillator bath. Thus one considers Hamiltonians of the general form $$H_{T}=H+H_{\mathrm{res}}+H_{\mathrm{ph-res}}+H_{\mathrm{s-res}}, \label{eq:laserham}$$ where $H=H_{\mathrm{ph}}+H_{\mathrm{s}}+H_{\mathrm{s-ph}}$ is given by Eq. (\[eq:dickeham\]), $H_{\mathrm{res}}$ describes the harmonic oscillator baths, and the remaining two terms couple the system and baths. The standard model is to couple the cavity mode to its bath with Hamiltonian $$H_{\mathrm{ph-res}}= \sum_p g_p(\psi^\dagger d_p + d^\dagger_p
\psi),$$ and the dots to their baths with Hamiltonian $$H_{\mathrm{s-res}}= \sum_i \sum_p g^{\bot}_{pi}(d_{pi}^\dagger
S_i^-+S_i^+ d_{pi}) + g^{\|}_{ip}d_{pi}^\dagger d_{pi}
S_z.$$
To develop dynamical equations for the system variables alone one takes the Liouville equation for the evolution of the density matrix and averages over the reservoir variables. The interaction between the system and reservoir variables is treated using second-order perturbation theory, leading to an integro-differential equation for the time evolution of the reduced density matrix $\rho$. This equation is then approximated by a first-order differential equation, implicitly neglecting memory effects, formally by having Markovian baths whose spectra are completely white.
Finally, one integrates the atomic degrees of freedom out of the density matrix for the system, assuming that the atomic system is very strongly damped. This leads to an equation of motion for the $P$ representation of the field density matrix, $W(\psi,\psi^\ast,t)$. Neglecting terms associated with the quantum nature of the field[@risken70] this equation becomes the Fokker-Planck equation describing diffusion in a quartic potential: $$\begin{aligned}
\frac{\partial W}{\partial t}+\zeta
\frac{\partial}{\partial\psi} (d-|\psi|^2)\psi W + \zeta
\frac{\partial}{\partial\psi^\ast} (d-|\psi|^2)\psi^\ast W \nonumber
\\ = 4q\frac{\partial^2 W}{\partial\psi\partial\psi^\ast}.
\label{eq:laserfpunscaled}\end{aligned}$$ The coefficients in this form are the strength of the nonlinearity $\zeta$(denoted $\beta$ in Ref. ), the linear gain or loss $\zeta d$, and a diffusion constant $q$.
To compare with the theory of fluctuations near polariton condensation we make the same rescaling of the field as we did there, $\psi \to \psi\sqrt{N}$. In terms of this rescaled field the steady-state probability distribution obtained from (\[eq:laserfpunscaled\]) is proportional to the integrand of (\[eq:fquadintgl\]). The parameters are related to the laser parameters as $a=-\zeta d/(2q)$ and $b=\zeta N/(4q)$. The scaling of these terms follows from the microscopic expressions for $\zeta, q$ and $d$ given in Eq. (2.17, 2.27) of Ref. . Substituting $g\to g/\sqrt{N}$ so that $g$ is as defined in Eq. \[eq:dickeham\], and noting that the total inversion $\sigma$ scales as $N$, we find $q \sim N^0$, $\zeta \sim N^{-1}$, $\zeta d \sim N^0$. Therefore, as in the polariton condensate, the parameters $a$ and $b$ are independent of $N$.
Thus in the classical approximation the photon statistics of a laser and a condensate differ only due to the dependence of $a$ and $b$ on the microscopic parameters of the system. This is similar to the usual universality which occurs close to a second-order phase transition, where the singular parts of the observables are given by universal scaling functions, with non-universal relations between the scaling parameters and physical parameters such as temperature. In the finite systems considered here the observables are not singular, but for large $N$ the correlation functions in the two problems are identical functions of a parameter describing the distance from the transition. This arises because the collective behavior of the electromagnetic field in both theories is described by a quartic “free-energy” , and the system size $N$ enters these free energies in the same way.
Within our approximations, then, the only scope for making a distinction between polariton condensation and lasing arises from the forms of $a$ and $b$ in the two systems. These forms are model-dependent, and expected to change in more realistic models of either system. However, we note that there are differences in the physics of these forms in the laser and the condensate, which could be expected to be robust. For example, in the laser the noise strength $q$ is the spontaneous emission rate into the lasing mode, while in the condensate the analogous noise strength is temperature. It appears that the laser amplifies spontaneous emission noise, while the condensate amplifies thermal noise. The parameters $a$ and $b$ also contain different physics: in the laser $a$ is associated with gain or loss and $b$ with gain depletion, while in the condensate $a$ is a single-particle energy and $b$ an interaction strength. Thus the size of the threshold regime in absolute units is controlled by different physics in the two theories.
The analogy between the laser threshold and a second-order phase transition is well-known, and previous authors have pointed out that this analogy extends beyond the thermodynamic limit[@rice94; @haken75]: in a finite system the singularities associated with a phase transition are rounded in the same way that the sharp singularities of the lasing transition in a rate-equation treatment are rounded by noise. We see that for the laser model (\[eq:laserham\]) the parameter controlling this rounding is $N$, the system size, *exactly* as in the usual thermodynamic case. This appears to differ from the picture obtained from classical stochastic models[@rice94], in which the corresponding parameter is the fraction of the spontaneous emission directed into the lasing mode.
Quantitative estimates for current experiments {#sec:numbers}
==============================================
We can use our model to obtain some indication of the scale of the fluctuation-dominated region in current experimental systems. Such systems are generally planar microcavities, rather than the three-dimensional cavity of (\[eq:dickeham\]). However the luminescence which may be evidence for condensation is localized, suggesting that the condensate itself is localized, presumably due to some combination of kinetics, self-trapping, and disorder. In a condensate localized on scale $L$ we expect there to be an energy cost $\sim (L^2 m_p^2)^{-1}$ associated with spatial variations, where $m_p
\sim 10^{-5} m_e$ is the in-plane mass of the polaritons. If this energy cost is large compared with temperature we expect spatial fluctuations to be unimportant, and the present theory to apply.
Let us consider in particular the data discussed by Weihs, Deng, Snoke, and Yamamoto in Ref. . These experiments are done on a GaAs planar microcavity with twelve quantum wells and a Rabi splitting of $14.9 \mathrm{meV}$, so $g\approx 86\mathrm{K}$. Their condensate is typically $5 {\mathrm{\mu m}}$ in diameter, giving a temperature scale for spatial fluctuations of $60\mathrm{K}$. We take the Mott density, $10^{12} \mathrm{cm^{-2}}$, to be an upper bound on the density of available exciton sites, so the condensate of diameter $5 {\mathrm{\mu m}}$ corresponds to $N\sim 10^5$.
The temperature in these systems is extremely low relative to the Rabi splitting: $T=5K$ corresponds to $T/g=0.06$. At these low temperatures the critical density is not associated with the single-mode phase transition studied here but with spatial fluctuations, whose effect on the mean-field theory is analyzed in Refs. and . Thus our theory cannot describe the transition that is being crossed in the experiments. However, once the condensate has formed we expect it to apply, since the condensate which forms is apparently localized, and the spatial degrees of freedom are frozen out. We estimate that the achieved condensate densities of $10^{10}\mathrm{cm^{-2}}$ correspond to $\rho_{\mathrm{ex}} \approx
10^{10}/{n_{\mathrm{Mott}}}=10^{-2}$. For this density the transition temperature of our theory is $T/g=0.2 \approx 20\mathrm{K}$. For $N=10^{5}$ the fluctuation-dominated region extends over around $20\mathrm{mK}$ on the normal side of the transition and $1\mathrm{K}$ on the condensed side. The experimental temperature is far outside this region, so the present theory suggests that the fluctuations of the condensate are negligible and the light from the condensate area is almost perfectly coherent. If raising the temperature did not change the localization then this would allow the sharp crossover predicted here to be observed. This complication would be absent for systems with external in-plane confinement of the polaritons, such as pillar microcavities[@vahala03].
For a system in thermal equilibrium, we have argued that the quantum regime occurs when the temperature is smaller than the characteristic level spacing produced by the photon-photon interaction. For the particular model used here this is when $TN \sim g$. We estimate that this gives a scale of $1\mathrm{mK}$ for the systems of Ref. , which is far below the achieved temperatures. Furthermore, these are open systems coupled to baths. In such a system, the effect of dissipation is loosely to broaden the level spectrum through the decay process. To have quantum effects that dominate for the longest times one then requires linewidths that are narrow in comparison to the level spacing. This is not achieved, since a temperature scale of $1\mathrm{mK}$ corresponds to a linewidth on the order of $10 \mu \mathrm{eV}$, far below the linewidth associated with the decay of the cavity mode.
Discussion {#sec:discussion}
==========
Recently, three groups[@yoshie04; @reithmaier04; @peter05] have achieved strong-coupling of single quantum dots in microcavities, and are thus beginning to approach the regime, already achieved in atom cavity optics[@thompson92], of a single two-level system strongly coupled to a cavity mode. Clearly the nonlinear regime[@mckeever03] of such a system is dominated by quantum effects. However, we note that our model suggests that this single-atom limit is not the only way to see strong quantum effects. For a cavity of fixed volume one should replace $g/\sqrt{N}$ with $g$ in (\[eq:dickeham\]). The effective photon-photon interaction then scales as $N$(whereas it scales as $N^{-1}$ at fixed dot density), so that quantum effects survive to higher temperatures when the number of dots increases in a cavity of fixed volume: a situation opposite to the normal thermodynamic limit we consider elsewhere, in which quantum effects move to lower temperatures as the system size increases. In this context it is interesting to note the experiments of Ref. , in which the quantum statistical properties of light emitted from a weakly driven atom-cavity system are measured. The observed $g^{(2)}(0)$ was below the classical limit of 1, and furthermore its value was apparently independent of the number of atoms in the cavity. This would naturally arise if the noise increased as $N$, as it does in standard laser theory, canceling the increase in the level spacing.
The behavior of a finite system in which spatial fluctuations are allowed could be analyzed using renormalization group arguments[@cardybook]. We expect it to depend on the interplay between system size and an effective co-ordination number or interaction range: a small system with a large co-ordination (long-range interactions) will be dominated by finite-size fluctuations over a larger region of the phase diagram than it is dominated by spatial fluctuations, whereas the reverse will occur for a large system with a small co-ordination (short-range interactions). In the former case we can view the finite-size fluctuations as corrections to mean-field theory, as here. This cannot be true in the latter case, however, since the theory of the finite system should then involve critical exponents different from those of mean-field theory.
The phase diagram of the infinite two-dimensional system, allowing for spatial fluctuations, has been studied in Ref. . The deviations between the phase boundary there and that of mean-field theory indicate the regions in which spatial fluctuations of an infinite system are significant. Since the photon mass is very small, effectively providing a long-range interaction between excitons, these deviations are only significant at very low densities, where even a long-range interaction gives a small co-ordination number. Hence, except at low densities, we expect even a relatively large system to have mean-field like finite-size behavior.
Conclusions {#sec:conclusions}
===========
In this paper we have analyzed the behavior of a model of a finite polariton condensate close to the mean-field transition, and shown that it is formally similar to the theory describing a laser close to the laser threshold. This similarity fails when the polariton condensate reaches a low-temperature quantum regime, since the laser theory is classical – notwithstanding that the noise in the laser theory might ultimately have its origins in quantum effects.
In the classical regime we find that the intensity distribution for the photons in a finite polariton condensate is of the same form as that obtained from conventional laser theory. Thus the photon statistics are not expected to reveal any fundamental difference between a condensate and a laser, but of course the parametrization is different and the scales have different meanings. In particular, for a condensate in thermal equilibrium the sharpness of the crossover is controlled by temperature, whereas in a laser it is controlled by the noise introduced by coupling to external baths.
Quantum effects dominate in general when the level spacing exceeds the noise strength. We have seen that for the polariton condensate in equilibrium the relevant level spacing is the photon-photon interaction, and the relevant noise strength is temperature. In the conventional thermodynamic limit of our model the temperature scale for quantum effects decreases with system size, but the opposite occurs when increasing the dot number with fixed cavity volume. Hence it is not always the smallest systems which have the most strongly quantum-mechanical collective behavior.
We thank Simon Kos, Vikram Tripathi, Jonathan Keeling, and Mike Gunn for helpful discussions of this work, and acknowledge support from the EPSRC, Sidney Sussex College, Cambridge, and the EU RTN Project No. HPRN-2002-00298.
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---
abstract: 'This paper proposes a novel two-layer Volt/VAR control (VVC) framework to regulate the voltage profiles across an unbalanced active distribution system, which achieves both the efficient open-loop optimization and accurate closed-loop tracking. In the upper layer, the conventional voltage regulation devices with discrete and slow-response characteristics are optimally scheduled to regulate voltage profiles in an hourly timescale while improving economic operations based on the receding horizon optimization (RHO) in a centralized manner. A generalized linearized branch flow model (G-LBFM) is developed to incorporate tap changers into branches, which significantly reduces the computational complexity compared to the original mixed-integer non-convex case. In the lower layer, we develop an integral-like control algorithm rather than resorting to the droop-based rules for real-time reactive power dispatch of distributed energy resources (DERs) to achieve accurate voltage tracking and mitigate fast voltage fluctuations in a decentralized (purely local) fashion. Further, a sufficient stability condition of the integral rule is presented to guarantee the closed-loop stability. Case studies are carried out on the unbalanced IEEE 123-Node Test Feeder to validate the effectiveness of the proposed method.'
author:
- |
Yifei Guo, Qianzhi Zhang, Zhaoyu Wang, \
Fankun Bu, and Yuxuan Yuan, [^1] [^2]
title: |
Two-Layer Volt/VAR Control in Unbalanced Active Distribution Systems: Efficient Optimization\
and Accurate Tracking
---
[Shell : Bare Demo of IEEEtran.cls for Journals]{}
unbalanced active distribution system, distributed energy resource (DER), receding horizon optimization (RHO), two-layer control, Volt/VAR control (VVC).
Introduction
============
/reactive power (VAR) control (VVC) is an essential task to ensure the secure operation of distribution systems. Generally, it is performed by using a predefined set of rules to schedule the various conventional voltage regulation devices, including capacitor banks (CBs), step-voltage regulators (SVRs) and on-load tap changing transformers (OLTCs), which might not able to tackle fast voltage issues caused by the high variability of loads and rapidly developed distributed energy resources (DERs), e.g., distributed wind and photovoltaic (PV) generation, due to their slow response and limited operation times. Therefore, the inverter-based DERs with fast and continuous VAR capability have been encouraged to provide necessary Volt/VAR support for distribution systems [@IEEE1547_2018]. To optimally coordinate overall Volt/VAR regulation resources and achieve specific control goals, such as loss reduction and voltage deviation mitigation, the centralized or distributed optimization methods have been widely investigated [@Pal]–[@Feng]. See [@OV1] and [@OV3] for related surveys. Given such optimization-based methods are mostly designed in open-loop fashion, potential model errors, prediction errors and communication delays/noises would deteriorate their performance for real-time implementations [@haozhu]. Besides, the *single-layer* optimization/control might not be able to exploit different response capabilities of various VVC devices in the best way.
In this context, the idea of *multi-layer* (also termed as *multi-level* or *multi-stage*) control has attracted a lot of attention, which aims to coordinate different devices in separable timescales. In [@LM_two_stage]–[@JYP_two_stage], the coordination between OLTC and CBs in traditional distribution systems were coordinated using two-stage frameworks. In [@DJ_multi_time], a two-timescale Volt/VAR optimization method was proposed wherein the OLTCs and CBs are dispatched in hourly timescale while the DERs are dispatched in 15-min timescale explicitly considering their uncertainties. Similarly in [@Yifei2], a two-stage open-loop optimization model based on model predictive control was established to coordinate multiple devices. In [@Yu], a bi-level voltage management scheme was proposed, including the centralized coordination of OLTCs and CBs and a distributed consensus-based algorithm to dispatch electric springs working on a faster timescale as supplementary control for critical loads. In [@YWang], a distributed two-layer VVC scheme was developed to dispatch the grouped PV inverters, including a distributed optimization-based 15-min dispatching and droop-based real-time control. The authors in [@HSB]–[@THong_droop] proposed to schedule the parameters of droop controllers by upper-layer optimization. In [@Czhang_multi_time], the robust VVC was specially addressed.
It can be concluded that a preferable design of the multi-layer VVC is the combination of open-loop optimization to schedule the discrete devices with slow response and an easy-to-implement control law with minimal computation and communication requirements in real-time layer to dispatch the flexible DERs. However, due to existence of discrete devices and nonlinearity of ac power flow, the optimization problems are essentially non-convex mixed-integer nonlinear programming problems which become intractable for large systems. To tackle this complexity issue, some heuristics methods such as particle swarm optimization [@LM_two_stage; @DJ_multi_time] and differential evolution [@Yu], are used to solve the problems or some standard nonlinear programming algorithms are adopted after relaxing the discrete variables to continuous ones [@Pal; @Anna; @BAR_dis_tap]. However, these methods are sometimes time-consuming and often suffer the sub-optimality [@Yifei2]. The three-phase unbalanced cases may further complicate the optimization problems, which was not addressed in most previous multi-layer architectures. For the real-time layer, as advocated by [@IEEE1547_2018], a popular choice is the linear (piece-wise) droop control [@YWang]–[@THong_droop], which, however, may be challenged by its vulnerability to closed-loop instability due to improper control parameter selection and inaccurate voltage tracking due to its inherent proportional rule [@OV1]. In this paper, we offer a two-layer VVC framework to coordinate the optimal settings of OLTC, SVRs and CBs and the real-time VAR adjustment of inverter-based DERs in separable timescales. The upper-layer control is developed based on the centralized receding horizon optimization (RHO) with the help of rolling predictions of DER generation and load consumption, which is established using the three-phase generalized linearized branch flow model (G-LBFM) that incorporates a tap changer over a branch. In the lower layer, we propose an integral-like algorithm with the aim of tracking the voltage references scheduled by the upper layer control, instead of resorting to the droop control to adjust the VAR outputs of DER inverters. Compared with existing methods, the advantages of the proposed method can be summarized as follows:
- With the G-LBFM, the nonlinearity of three-phase power flow caused by tap changers can be approximated by a linear model with a sufficient accuracy. In this way, the RHO model becomes a standard mix-integer quadratic programming (MIQP) problem that can be efficiently handled by solvers.
- Compared with the droop control, the proposed integral-like control can achieve more accurate voltage tracking by fully exploiting the VAR capabilities of inverters. Due to closed-loop nature of proposed method, better tracking performance has been shown compared with the optimization method based on system-wide information.
- A sufficient condition with rigorous proof is provided to avoid Volt/VAR hunting/oscillation and ensure the closed-loop stability of the lower-layer control.
The reminder of the paper is organized as follows. Section \[sec:overview\] gives a overview of the proposed two-layer method. Section \[sec:model\] presents the generalized branch flow and voltage regulation device models. Section \[sec:stage1\] present the open-loop upper-layer control. Section \[sec:stage2\] presents the decentralized real-time VAR control algorithm of DERs in the lower layer. Simulation results are given in Section \[sec:case\], followed by conclusions.
Overview of Two-Layer Volt/VAR Control {#sec:overview}
======================================
The basic idea of the two-layer VVC is to deal with different voltage issues in separable timescales (voltage deviations in the timescale of a few hours and voltage fluctuations in the timescales ranging from a few seconds to minutes) by exploiting multiple VVC resources with different response time.
The upper layer aims at optimally scheduling the discrete devices OLTC, SVRs and CBs based on the receding horizon optimization to improve the economic operation of the distribution systems while regulating the voltage profile within the predefined range $[0.95,1.05]$ p.u.. The RHO method is used here to address the operation limits of switchable devices over a given period and more accurate prediction of load and DER generation (wind/solar). It is inherently an open-loop method that requires some prior information (e.g., network topology and parameters) and some online operation information (e.g., wind, solar and load prediction and operation status of OLTC, SVRs and CBs). Note that, the potential VAR capabilities of DERs are considered in the optimization rather than being neglected to avoid overuse of other devices The corresponding optimal voltage solution of the RHO problem will be used as references for the lower layer control, resulting in a hierarchical control structure.
The lower layer aims to mitigate the voltage fluctuations in the timescales ranging from seconds to minutes by commanding the VAR outputs of DER inverters. An integral-like real-time control algorithm is developed to track the voltage references scheduled by the upper layer. Each DER inverter will update its VAR output according to its local voltage measurements. Given the algorithm requires very little computation and local information, it can be carried out in the time scale of a few seconds in a decentralized manner.The closed-loop nature of lower-layer control can largely reduce the risk of voltage violations, which might be caused by the model or prediction errors in the upper-layer optimization.
System Models {#sec:model}
=============
Generalized Branch Flow Model
-----------------------------
In this subsection, we propose a *generalized branch flow model* to take into account the existence of SVRs, which is not addressed in the traditional BFM. Additionally, it will be extended to unbalanced three-phase systems.
### Single-Phase Model
A distribution system is typically operated with tree topology. Consider a radial network comprising $N+1$ buses denoted by set $\mathcal{N}\bigcup\{0\}$, $\mathcal{N}:=\left\{1,\ldots,N\right\}$ and $N$ branches denoted by $\mathcal{E}$. Bus 0 denotes the slack bus (high-voltage side of main transformer). For each bus $j\in\mathcal{N}$, $p_{{\rm c},j}$ and $q_{{\rm c},j}$ are the real and reactive power consumptions; $p_{{\rm inv},j}$ and $q_{{\rm inv},j}$ are the real and reactive power injections by the DER inverter; $q_{{\rm cap},j}$ denotes the VAR injections from capacitor banks; $v_{j}:=|V_{j}|^2$ represents the squared voltage magnitude; $\mathcal{N}_{j}$ denotes the set of children buses. For any branch $({ i,j})\in\mathcal{E}$, $r_{ij}$ and $x_{ij}$ are the branch resistance and reactance while $P_{ij}$ and $Q_{ij}$ denote the real and reactive power flow from bus $i$ to bus $j$, respectively.
![Generalized single-phase branch model.[]{data-label="1ph_branch"}](1ph_branch.png "fig:"){width="2.5in"}\
A generalized single-phase branch model with a tap changer is shown in Fig. \[1ph\_branch\]. Upon this, the LBFM is generalized to, $\forall (i,j)\in\mathcal{E}$,
\[BFM\_alpha\] $$\begin{aligned}
P_{ij}&=\sum_{k\in\mathcal{N}_{j}} P_{jk}+p_{{\rm c},j}-p_{{\rm inv},j}\\
Q_{ij}&=\sum_{k\in\mathcal{N}_j} Q_{jk}+q_{{\rm c},j}-q_{{\rm inv},j}-q_{{\rm cap},j}\\
t_{ij}^2\cdot v_{i}-v_{j}&=2(r_{ij}P_{ij}+x_{ij}Q_{ij})\\[1mm]
t_{ij}&=1+n_{{\rm tap},ij}\cdot\Delta{tap}_{ij}\end{aligned}$$
where $n_{{\rm tap},ij}$ and $\Delta{tap}_{ij}$ denote the tap position and step, respectively. Generally, the series impedance and shunt admittance of SVR (auto-transformer) are small in per unit, so that they can be neglected here [@Kersting]. Obviously, the existence of $t_{ij}^2v_i$ makes the optimization-based VVC problems become nonlinear nonconvex mixed-integer problems. As known, the constraints with integer variables can be handled by mixed-integer solvers using the algorithms such as branch-and-bound, provided its relaxation is convex. However, this is not the case with (\[BFM\_alpha\]c) because of the multiplication of $t_{ij}^2v_i$ and the quadratic term $t_{ij}^2$. That is, even after relaxing the integer variables, the constraints still hold a nonlinear nature, making the problem hard to solve. To tackle with this, a linear approximation of $t_{ij}^2\cdot v_{i}$ is derived as, $$\begin{aligned}
\label{linearization}
\nonumber t_{ij}^2\cdot v_{i}&=\left(1+2n_{{\rm tap},ij}\cdot\Delta{tap}_{ij}+n_{{\rm tap},ij}^2\cdot\left(\Delta{tap}_{ij}\right)^2\right)\cdot v_{i}\\
\nonumber &\approx v_{i}+2n_{{\rm tap},ij}\cdot\Delta{tap}_{ij}\cdot v_{i}\\
&\approx v_{i}+2n_{{\rm tap},ij}\cdot\Delta{tap}_{ij}\cdot v_{{\rm nom}}.\end{aligned}$$ Such an approximation is believed to hold because the term $n_{{\rm tap},ij}\Delta tap_{ij}<<1$ and $v_{i}\approx v_{\rm nom}$ during normal operation. The model errors of this linear approximation are presented in Fig. \[error\] where the tap position ranges from $-16$ to $+16$. It can be seen that the model errors are no worse than $\pm 1$ tap position under a wide operation range and are no worse than $\pm 2$ tap position except some extreme cases that seldom happen in real operation, implying a sufficient accuracy used in our VVC problem. Note that, as discussed before, the impact of model errors can be compensated by the closed-loop nature of the lower-layer control to avoid severe voltage deviations.
![Model error of a linear approximation of term $t_{ij}^2\cdot v_{i}$.[]{data-label="error"}](taperror.png "fig:"){width="2.4in"}\
Then, substituting (\[linearization\]) into (\[BFM\_alpha\]c), one can obtain, $\forall(i,j)\in\mathcal{E}$, $$\begin{aligned}
\label{LGBFM}
\hspace{-3mm}v_{i}-v_{j}&=2(r_{ij}P_{ij}+x_{ij}Q_{ij})-2n_{{\rm tap},ij}\cdot\Delta{tap}_{ij}\cdot v_{{\rm nom}}.\end{aligned}$$ Hence, the G-LBFM can be expressed by combining (\[BFM\_alpha\]a)-(\[BFM\_alpha\]b) and (\[LGBFM\]).
### Extension to Unbalanced Three-Phase Systems
![Generalized three-phase branch model.[]{data-label="3ph_branch"}](3ph_branch.png "fig:"){width="2in"}\
The G-LBFM is extended to unbalanced three-phase systems (see Fig. \[3ph\_branch\]) where $z_{ij}$ is extended to ${\bm z}_{ij}:={\bm r}_{ij}+{\rm j}{\bm x}_{ij}\in\mathbb{C}^{3\times3}$, $n_{{\rm tap},ij}$ is extended to ${\bm n}_{{\rm tap},ij}:=[n_{{\rm tap},ij}^{\rm a},n_{{\rm tap},ij}^{\rm b},n_{{\rm tap},ij}^{\rm c}]^T$ and let $\Delta{\bm{tap}}_{ij}:=[\Delta{tap}_{ij}^{\rm a},\Delta{tap}_{ij}^{\rm b},\Delta{tap}_{ij}^{\rm c}]^T$. Correspondingly, the three-phase voltages, power consumption, power injections and line power flows in (\[BFM\_alpha\]) are extended to a three-phase form ${\bm v}_{i}$, ${\bm p}_{{\rm c},j}$, ${\bm q}_{{\rm c},j}$, ${\bm p}_{{\rm inv},j}$, ${\bm q}_{{\rm inv},j}$, ${\bm q}_{{\rm cap},j}$, ${\bm P}_{ij}$ and ${\bm Q}_{ij}$, respectively.
Assume the system unbalance is not too severe [@qianzhi1], the G-LBFM is extended to a three-phase case as, $\forall (i,j)\in\mathcal{E}$,
\[BFM\_3ph\] $$\begin{aligned}
{\bm P}_{ij}&=\sum_{k\in\mathcal{N}_{j}}{\bm P}_{jk}+{\bm p}_{{\rm c},j}-{\bm p}_{{\rm inv},j}\\
{\bm Q}_{ij}&=\sum_{k\in\mathcal{N}_{j}}{\bm Q}_{jk}-{\bm q}_{{\rm c},j}-{\bm q}_{{\rm inv},j}-{\bm q}_{{\rm cap},j}\\
{\bm v}_{i}-{\bm v}_{j}&=2\left(\bar{\bm r}_{ij}{\bm P}_{ ij}+\bar{\bm x}_{ij}{\bm Q}_{ij}\right)+2{v}_{\rm nom}\cdot{\bm n}_{{\rm tap},ij}\odot\Delta{\bm{tap}}_{ij}\end{aligned}$$
with $$\begin{aligned}
\bar{\bm r}_{ij}&={\rm Re}\{\bm{\alpha\alpha}^H\}\odot{\bm r}_{ij}+{\rm Im}\{\bm{\alpha\alpha}^H\}\odot{\bm x}_{ij},\\
\bar{\bm x}_{ij}&={\rm Re}\{\bm{\alpha\alpha}^H\}\odot{\bm x}_{ij}-{\rm Im}\{\bm{\alpha\alpha}^H\}\odot{\bm r}_{ij},\end{aligned}$$ where ${\bm\alpha}:=[1, e^{\rm-j2\pi/3}, e^{\rm j2\pi/3}]^T$ and $\odot$ denotes the element-wise multiplication.
### Compact Representation
To better present the algorithm in the lower layer and some related analyses, we further rewrite the G-LBFM in a compact form. Firstly, similar as the compact single-phase model in [@haozhu], the three-phase voltages (except bus 0), real/reactive power load and injections at bus and active/reactive power flows and tap position over each branch are compactly represented by column vectors ${\bm v}, {\bm p}_{\rm c}, {\bm p}_{\rm inv},{\bm q}_{\rm inv}, {\bm q}_{\rm cap}, {\bm q}_{\rm c}, \bm P, \bm Q$ and $\bm n$, respectively. Branch resistances and reactances are compactly represented by block-diagonal matrices $\bm R$ and $\bm X$, respectively. To be noticed, we only consider the buses and branches that actually exist in real systems instead of considering a completed three-phase system by adding some virtual buses and branches. Let $\varphi_i$ and $\varphi_{ij}$ be the actual phase sets of bus $i$ and branch $(i,j)$. Let $\bar{\bm G}:=[{\bm g}_0\,\,{\bm G}^T]^T\in\mathbb{R}^{(N+1)\times N}$ be the (single-phase) graph incidence matrix of the distribution network where ${\bm g}_0^T$ denotes the first row of $\bar{\bm G}$ [@incidencematrix]. The extended incidence matrix of a three-phase system $\bm{\bar{A}}:=[\bm{\bar{A}}_{ij}]_{\sum|\varphi_{ij}|\times\sum|\varphi_{ij}|}$ is defined as, $$\begin{aligned}
\bar{\bm A}_{ij}=\bar{G}_{ij}\otimes{\bm I}_{|\varphi_{ij}|} \end{aligned}$$ where ${\bm I}_{|\varphi_{ij}|}$ is the $\varphi_i\times\varphi_j$ identity matrix; $j$ is the number index of branch $(i,j)$, i.e., $j{\rm th}$ column of $\bar{\bm G}$ corresponds to branch $(i,j)$. Then, the compact G-LBFM of the whole system can be expressed as,
\[CBFM\] $$\begin{aligned}
\hspace{-3mm}-\bm{AP}&=-{\bm p}_{\rm inv}+{\bm p}_{\rm c}\\
\hspace{-3mm} -\bm{AQ}&=-{\bm q}_{\rm inv}-{\bm q}_{\rm cap}+{\bm q}_{\rm c}\\
\hspace{-3mm} \begin{bmatrix}{\bm a}_0\,\,{\bm A}^T\end{bmatrix}\begin{bmatrix}{\bm v}_0\\{\bm v}\end{bmatrix}&=2(\bm{RP+XQ})+2{v}_{\rm nom}\cdot{\bm n}\odot\Delta\bm{tap}\end{aligned}$$
where ${\bm a}_0^T$ denotes the submatrix of $\bar{\bm A}$ corresponding to bus $0$ while $\bm A$ denotes the remaining submatrix.
By substituting (\[CBFM\]a) and (\[CBFM\]b) into (\[CBFM\]c), we have $$\begin{aligned}
\label{volt}
{\bm v}=\bm{Mq}_{\rm inv}+\bm\mu\end{aligned}$$ where $$\begin{aligned}
&\hspace{25mm}{\bm M}=2{\bm A}^{-T}{\bm X}{\bm A}^{-1}\\
{\bm\mu}=&{\bm A}^{-T}\Big(-{\bm a}_0{\bm v}_0+2{\bm R}{\bm A}^{-1}({\bm p}_{\rm inv}-{\bm p}_{\rm c})\\
&\hspace{8mm}+2{\bm X}{\bm A}^{-1}\left({\bm q}_{\rm cap}-{\bm q}_{\rm c}\right)+2{v}_{\rm nom}\cdot{\bm n}\odot\Delta\bm{tap}\Big).\end{aligned}$$ Clearly, $\bm\mu$ can be considered as the component contributed by all other factors except the VAR injections ${\bm q}_{\rm inv}$.
Voltage Regulation Device Models
--------------------------------
### Step-Voltage Regulator
The SVRs can be modelled as an ideal transformer with a tap changer due to their small impedance. For single-phase SVRs, each phase has its own compensator circuit, so that the taps can be changed separately. For three-phase SVRs with only one compensator circuit, the taps on all windings change the same [@Kersting]. Here, both single-phase SVRs and three-phase SVRs are applied.
### OLTC Transformer
For the substation OLTC transformer, by neglecting the shunt admittance, it can be modelled as a series impedance with an ideal three-phase voltage regulator, which can be also modelled by the G-LBFM.
### Capacitor Bank
The switchable CBs can be approximately modelled by constant reactive power sources with discrete characteristics. Similarly, they are connected in a single-phase way, which enables separable VAR compensations on each phase. The operation limits of SVRs can be expressed as, for any $\rm(i,j)\in\mathcal{E}_{SVR}$ and $\phi\in\{\rm a,b,c\}$, $$\begin{aligned}
\underline{n}_{\rm ij}^\phi\leq n_{\rm ij}^\phi\leq \overline{n}_{\rm ij}^\phi,\,\,n^\phi_{\rm ij}\in\mathbb{Z}\end{aligned}$$$$\begin{aligned}
q_{\rm CB}^{\phi}=n_{\rm CB}^{\phi}\Delta q_{\rm CB}^{\phi}\end{aligned}$$ where $\underline{n}_{\rm ij}^\phi$ and $\overline{n}_{\rm ij}^\phi$ denotes the lower and upper bounds of tap position.
### DER Inverter
The voltage source inverter-based DER has a typical cascading control structure, i.e., inner current control loop and outer control loop, which enables the decoupled active and reactive power control. The total power outputs are limited by the inverter capacity and it is assumed that the active power has higher priority than reactive power when the total output power reaches to the capacity limit. Besides, in LV distribution systems, DER inverters are generally connected to a single phase, enabling separable VAR support on each phase.
Receding Horizon Optimization for Optimal Scheduling of OLTC, SVRs and CBs {#sec:stage1}
==========================================================================
In the upper-layer, a RHO-based VVC strategy is developed to deal with hourly voltage issues while improving the economic operation of distribution systems by optimally coordinating the operation of voltage regulation devices. Each term in the objective is formulated as follows,
- Network power losses $$\begin{aligned}
\label{losses}
\hspace{-5mm}J_{\rm loss}(t):=\sum_{(i,j)\in\mathcal{E}}\sum_{\phi\in\varphi_{ij}}r_{ij}^{\phi\phi}\cdot\frac{\big(P_{ij}^\phi(t)\big)^2+\big(Q_{ij}^\phi(t)\big)^2}{v_{\rm nom}}.
\end{aligned}$$
- Operation costs of SVRs $$\begin{aligned}
\label{SVRs_change}
\hspace{-5mm}J_{\rm tap}(t)&:=\sum_{(i,j)\in\mathcal{E}}\sum_{\phi\in\varphi_{ij}}\left(n_{{\rm tap},ij}^{\phi}(t)-n_{{\rm tap},ij}^{\phi}(t-1)\right)^2.\end{aligned}$$
- Operation costs of CBs $$\begin{aligned}
\label{CBs_change}
\hspace{-5mm}J_{\rm cap}(t)&:=\sum_{i\in\mathcal{N}}\sum_{\phi\in\varphi_i}\left(N_{{\rm cap},i}^{\phi}(t)-N_{{\rm cap},i}^{\phi}(t-1)\right)^2.\end{aligned}$$
Thus, the objective of the RHO-based VVC problem is formulated over a prediction length $T_H$ with the penalty factors $C_{\rm loss},C_{\rm loss},C_{\rm cap}$ and $C_\delta$ assigned to terms (\[losses\])–(\[CBs\_change\]) and the voltage violation term.
\[VVC1\] $$\begin{aligned}
\nonumber&\text{minimize}\,\,\,\sum_{t=1}^{T_H}\Big(C_{\rm loss}\cdot J_{\rm loss}(t)+C_{\rm tap}\cdot J_{\rm tap}(t)\\
&\hspace{16mm}+C_{\rm cap}\cdot J_{\rm cap}(t)\Big)+C_\delta\cdot\sum_{i\in\mathcal{N}}\sum_{\phi\in\varphi_i}\big(\delta_i^\phi\big)^2\\
\nonumber&\text{over}\,\,\,n_{{\rm tap},ij}^{\phi}(t),N^{\phi}_{{\rm cap},i}(t),q_{{\rm inv},i}^{\phi}(t),P_{ij}^{\phi}(t),Q_{ij}^\phi(t),v_{i}^{\phi}(t),\delta_i^\phi,\\
\nonumber&\forall\,t\in\{1,\ldots,T_H\},\,\forall\,({i,j})\in\mathcal{E},\,\forall\, i,j\in\mathcal{N},\,\forall\,\phi\in\{\rm a,b,c\}\\
&\nonumber\text{s}\text{ubject}\,\,\text{to}\\
&P_{ij}^{\phi}(t)=\sum_{k\in\mathcal{N}_{j}}P_{jk}^{\phi}(t)+{\hat p}_{{\rm c},j}^{\phi}(t)-{\hat p}_{{\rm inv}, j}^{\phi}(t)\\
&Q_{ij}^{\phi}(t)=\sum_{k\in\mathcal{N}_{j}}Q_{jk}^{\phi}(t)+{\hat q}_{{\rm c},j}^{\phi}(t)-q_{{\rm inv}, j}^{\phi}(t)-q_{{\rm cap},j}^\phi(t)\\
&\nonumber v_{i}^{\phi}(t)-v^{\phi}_{j}(t)=\sum_{\phi^\prime\in\varphi_{ij}}2\left(\overline{r}^{\phi\phi^\prime}_{ij}P^{\phi}_{ij}(t)+\overline{x}^{\phi\phi^\prime}_{ij}Q^{\phi}_{ij}(t)\right)\\
&\hspace{25mm}+2{v}_{\rm nom}\cdot n_{{\rm tap},ij}^\phi(t)\cdot\Delta{tap}^\phi_{ij}\\
&\underline{n}_{{\rm tap},ij}^{\phi}\leq{n}_{{\rm tap},ij}^{\phi}(t)\leq\overline{n}_{{\rm tap},ij}^{\phi},\,\,{n}_{{\rm tap},ij}^{\phi}(t)\in\mathbb{Z}\\
&\left|{n}_{{\rm tap},ij}^{\phi}(t)-{n}_{{\rm tap}, ij}^{\phi}(t-1)\right|\leq\Delta\overline{n}_{{\rm tap},ij}^{\phi}\\
&\sum_{i=1}^T\left|{n}_{{\rm tap},ij}^{\phi}(t)-{n}_{{\rm tap},ij}^{\phi}(t-1)\right|\leq\Delta_\Sigma\overline{n}_{{\rm tap},ij}^{\phi}\\
&q_{{\rm cap},i}^\phi(t)=N_{{\rm cap},i}^{\phi}(t)\cdot\Delta q_{{\rm cap},i}^{\phi},\,\,{N}_{{\rm cap},i}^{\phi}(t)\in\mathbb{Z}\\
&0\leq{N}_{{\rm cap},i}^{\phi}(t)\leq\overline{N}_{{\rm cap},i}^{\phi}\\
&\left|{N}_{{\rm cap},i}^{\phi}(t)-{N}_{{\rm cap},i}^{\phi}(t-1)\right|\leq\Delta\overline{N}_{{\rm cap},i}^{\phi}\\
&\sum_{t=1}^{T}\left|{N}_{{\rm cap},i}^{\phi}(t)-{N}_{{\rm cap},i}^{\phi}(t-1)\right|\leq\Delta_\Sigma\overline{N}_{{\rm cap},i}^{\phi}\\
&\underline{q}_{{\rm inv},i}^\phi(t)\leq q_{{\rm inv},i}^\phi(t)\leq\overline{q}_{{\rm inv},i}^\phi(t)\\
&\hspace{-1mm}-\underline{q}_{{\rm inv},i}^{\phi}(t)=\overline{q}_{{\rm inv},i}^{\phi}(t)=\eta\sqrt{(s_{{\rm inv},i}^{\phi})^2-(p_{{\rm inv},i}^\phi(t))^2}\\
&\underline{v}_{i}^{\phi}-\delta_i^\phi\leq v_{i}^{\phi}(t)\leq\overline{v}_{i}^{\phi}+\delta_i^\phi.\end{aligned}$$
The constraints (\[VVC1\]b)–(\[VVC1\]d) represent the power flow constraints where ${\hat p}_{{\rm c},j}^{\phi}, {\hat p}_{{\rm inv},j}^\phi$ and ${\hat q}_{{\rm inv},j}^\phi$ can be obtained from real-time prediction. (\[VVC1\]e)–(\[VVC1\]k) denote the operation limits of SVRs and CBs, respectively.[^3] (\[VVC1\]l)-(\[VVC1\]m) denote the operation constraints of DER inverters. In this layer, we reserve partial VAR capabilities of inverters in the optimization by introducing a scalar $0<\eta<1$, in which way, the effects of prediction errors of DERs can be effectively reduced. Otherwise, it might result in conservative operations of other discrete devices and, consequently, the lower-layer control might fail to regulate the voltages within the predefined range. (\[VVC1\]n) denotes the voltage constraints. To avoid infeasible cases caused by the voltage limit, the slack term $\delta^{\phi}_i$ is introduced which will be significantly punished in the objective function as the last term. For the branch $(i,j)$ with a three-phase SVR with only one compensator circuit, an additional constraint on tap changers should be added into (\[VVC1\]), which is as follows, $$\begin{aligned}
n_{{\rm tap},ij}^a(t)=n_{{\rm tap},ij}^b(t)=n_{{\rm tap},ij}^c(t),\,\,\forall t.\end{aligned}$$ To be noticed, given the constraints in (\[VVC1\]g) and (\[VVC1\]k) with a sum of absolute terms cannot be directly handled by solvers, auxiliary variables $n_{{\rm tap},ij,+}^{\phi}(t),n_{{\rm tap},ij,-}^{\phi}(t), N_{{\rm cap},ij,+}^{\phi}(t)$ and $ N_{{\rm cap},ij,-}^{\phi}(t)$ are introduced to transform the constraints into an equivalent linear form. Constraint (\[VVC1\]g) is equivalently expressed by,
$$\begin{aligned}
&\hspace{-2mm}\sum_{t=1}^{T}\left(n_{{\rm tap},ij,+}^{\phi}(t)+n_{{\rm tap},ij,-}^{\phi}(t)\right)\leq\Delta_\Sigma\overline{n}_{{\rm tap},ij}^{\phi}\\
&\hspace{-2mm}n_{{\rm tap},ij,+}^{\phi}(t)-n_{{\rm tap},ij,-}^{\phi}(t)=n_{{\rm tap},ij}^{\phi}(t)-n_{{\rm tap},ij}^{\phi}(t-1)\\[1mm]
&\hspace{-2mm}n_{{\rm tap},ij,+}^{\phi}(t)\geq0,\,\, n_{{\rm tap},ij,-}^{\phi}(t)\geq0\end{aligned}$$
and, similarly, (\[VVC1\]k) becomes
\[CBreformulate\] $$\begin{aligned}
&\hspace{-1mm}\sum_{t=1}^{T}\left(N_{{\rm cap},i,+}^{\phi}(t)+N_{{\rm cap},i,-}^{\phi}(t)\right)\leq\Delta_\Sigma\overline{N}_{{\rm cap},i}^{\phi}\\
&\hspace{-1mm}N_{{\rm cap},i,+}^{\phi}(t)-N_{{\rm cap},i,-}^{\phi}(t)=N_{{\rm cap},i}^{\phi}(t)-N_{{\rm cap},i}^{\phi}(t-1)\\[1mm]
&\hspace{-1mm}N_{{\rm cap},i,+}^{\phi}(t)\geq0, N_{{\rm cap},i,-}^{\phi}(t)\geq0.\end{aligned}$$
After such reformulations, the problem (\[VVC1\]) becomes a (convex) mixed-integer quadratic programming (MIQP) problem and can thus be efficiently solved by MIQP solvers. Only the first-step optimal solution will be used to schedule the OLTC transformer, SVRs and CBs while the corresponding voltage solution will be used in the lower-layer control.
Decentralized Integral-Like Control for Reactive Power Dispatch of DER Inverters {#sec:stage2}
================================================================================
In this layer, a decentralized control scheme for VAR dispatch of DER inverters is developed to track the voltage references scheduled by the upper-layer control and mitigate the voltage fluctuations in the timescale of seconds.
Compact and Reduced Generalized Branch Flow Model
-------------------------------------------------
In Section III, a generalized BFM was given. To better present the control algorithm and related analyses in this section, we further rewrite the generalized BFM in a compact form. Firstly, three-phase voltages except bus 0, active/reactive power loads and injections at each node and active/reactive power flows and tap position over each branch are compactly represented by column vectors ${\bf v}, {\bf p}_c, {\bf p}_{\rm inv},{\bf q}_{\rm inv}, {\bf q}_{\rm cap}, {\bf q}_c, \bf P,Q$ and $\bf n$, respectively. Branch resistances and reactances are compactly represented by block-diagonal matrices $\bf R$ and $\bf X$, respectively. Secondly, we have to remove the elements corresponding to the phase(s) which does not actually exist in real systems though it is uniformly represented in model (\[BFM\_3ph\]), so that some matrix properties will hold in the following analysis. For example, let $\varphi_{\rm ij}\subseteq\{\rm a,b,c\}$ be the phase set of branch $\rm (i,j)$. If ${\rm a}\notin\varphi_{\rm ij}$, then $P_{\rm ij}^{\rm a}, Q_{\rm ij}^{\rm a}$ should be removed from $\bf P, Q$ and the corresponding impedance entries from $\bf R$ and $\bf X$.
Let $\bar{\bf G}:=[{\bf g}_0\,\,{\bf G}^T]^T\in\mathbb{R}^{(N+1)\times N}$ be the original (single-phase) graph incidence of the distribution network where ${\bf g}_0^T$ denotes the first row of $\bar{\bf G}$ [@incidencematrix]. The generalized graph incidence matrix of the distribution system considering multi-phase connection can be defined as, $$\begin{aligned}
\bar{\bf A}_{\rm il}=\bar{G}_{\rm il}\otimes{\bf I}_{|\varphi_{\rm ij}|}\end{aligned}$$ where $l$ is the number index of branch $\rm(i,j)$, i.e., the $l\rm th$ column of $\bf\bar{G}$ corresponds to branch $\rm(i,j)$. Then, the compact and reduced BFM for a three-phase unbalanced system can be expressed as,
\[CBFM\] $$\begin{aligned}
\hspace{-3mm}-{\bf AP}&=-{\bf p}_{\rm inv}+{\bf p}_{c}\\
\hspace{-3mm} -{\bf AQ}&=-{\bf q}_{\rm inv}-{\bf q}_{\rm cap}+{\bf q}_{c}\\
\hspace{-3mm} \begin{bmatrix}{\bf a}_0\,\,{\bf A}^T\end{bmatrix}\begin{bmatrix}{\bf v}_0\\{\bf v}\end{bmatrix}&=2({\bf RP+{X}Q})+2{v}_{\rm nom}\cdot{\bf n}\odot\Delta{\bf Tap}\end{aligned}$$
where ${\bm a}_0^T$ denotes the submatrix of $\bar{\bf A}$ corresponding to bus $0$ while $\bm A$ denotes the remaining submatrix.
By substituting (\[CBFM\]a) and (\[CBFM\]b) into (\[CBFM\]c), we have $$\begin{aligned}
\label{volt}
{\bf v}={\bf Mq}_{\rm inv}+\bm\mu\end{aligned}$$ where $$\begin{aligned}
&\hspace{25mm}{\bf M}=2{\bf A}^{-T}{\bf X}{\bf A}^{-1}\\
{\bm\mu}=&{\bf A}^{-T}\Big(-{\bf a}_0{\bf v}_0+2{\bf R}{\bf A}^{-1}({\bf p}_{\rm inv}-{\bf p}_c)\\
&\hspace{6mm}+2{\bf X}{\bf A}^{-1}\left({\bf q}_{\rm cap}-{\bf q}_c\right)+2{v}_{\rm nom}\cdot{\bf n}\odot\Delta{\bf Tap}\Big).\end{aligned}$$ Clearly, $\bm\mu$ can be considered as components of voltages contributed by all other factors except the VAR injections ${\bf q}_{\rm inv}$.
*Proposition 1:* Matrix $\bf M$ is positive definite.
*Proof:* Since the distribution network is essentially a connected tree graph, it can be found that if we properly arrange the column sequence of $\bf A$ according to the negative end node of each branch, $\bf A$ will be a upper triangular matrix with all diagonal entries being equal to $-1$, implying it is non-singular.
For single-phase branches, it naturally has $$\begin{aligned}
\bar{\bf x}_{\rm ij}=\begin{bmatrix}x_{\rm ij}^\phi\end{bmatrix}\succ0,\,\, \phi\in\varphi_{\rm ij}.\end{aligned}$$
For two-phase branches, without loss of generality, suppose $\varphi_{\rm ij}=\{\rm a,b\}$ here. For any non-zero vector ${\bf y}\in\mathbb{R}^2$, we have $$\begin{aligned}
\nonumber{\bf y}^T\bar{\bf x}_{\rm ij}{\bf y}=&\begin{bmatrix}y_1\\y_2\end{bmatrix}^T\begin{bmatrix}x^{\rm aa}_{\rm ij}&-\dfrac{1}{2}x^{\rm ab}_{\rm ij}-\dfrac{\sqrt{3}}{2}r^{\rm ab}_{\rm ij}\\-\dfrac{1}{2}x^{\rm ba}_{\rm ij}+\dfrac{\sqrt{3}}{2}r^{\rm ba}_{\rm ij}&x^{\rm bb}_{\rm ij}
\end{bmatrix}\begin{bmatrix}y_1\\y_2\end{bmatrix}\\[2mm]
\nonumber=&\frac{x_{\rm ij}^{\rm ab}+x_{\rm ij}^{\rm ba}}{4}\cdot\left(y_1-y_2\right)^2+\left(x_{\rm ij}^{\rm aa}-\frac{x_{\rm ij}^{\rm ab}+x_{\rm ij}^{\rm ba}}{4}\right)\cdot y_1^2\\&+\left(x_{\rm ij}^{\rm bb}-\frac{x_{\rm ij}^{\rm ab}+x_{\rm ij}^{\rm ba}}{4}\right)\cdot y_2^2>0\end{aligned}$$ since $x_{\rm ij}^{\rm aa}\approx x_{\rm ij}^{\rm bb}>x_{\rm ij}^{\rm ab}\approx x_{\rm ij}^{\rm ba}>0$, which is widely believed to hold in real distribution systems. Therefore, $\bar{\bf x}_{\rm ij}\succ0$.
For any three-phase branch, it can be also easily proven that $\bar{\bf x}_{\rm ij}\succ0$ using the similar method as above.
Then, as defined before, we have $$\begin{aligned}
{\bf X}={\rm blkdiag}\{\bar{\bf x}_{\rm ij}\}\succ0.\end{aligned}$$ Accordingly, for any non-zero vector $\bf y\in\mathbb{R}^{\sum|\varphi_{\rm ij}|}$, $$\begin{aligned}
{\bf y}^T{\bf M}{\bf y}=2{\bf y}^T{\bf A}^{-T}{\bf X}{\bf A}^{-1}{\bf y}=2({\bf A}^{-1}{\bf y})^T{\bf X}({\bf A}^{-1}{\bf y})>0\end{aligned}$$ always holds, indicating $\bf M\succ0$.
Problem Formulation
-------------------
The VVC-II aims to track the projected voltage references $[\bf v_{\rm ref}]^{\overline{\bf v}}_{\underline{\bf v}}$ scheduled by VVC-I while reducing the inverter losses, of which the problem formulation is compactly presented as,
\[VVC-II\] $$\begin{aligned}
\text{minimize}\hspace{3mm}&f:=\frac{1}{2}\left|\left|{{\bf v}}-{\bf v}_{\rm ref}\right|\right|_{{\bm\Phi}_v}^2+\frac{1}{2}\left|\left|{\bf q}_{\rm inv}\right|\right|_{{\bm\Phi}_{q}}^2\\
\nonumber\text{over}\hspace{3mm}&{\bf v}, {\bf q}_{\rm inv}\\
\text{subject to}\hspace{3mm}&{\bf v}={{\bf Mq}_{\rm inv}}+\bm\mu\\
&\underline{\bf q}_{\rm inv}\leq{\bf q}_{\rm inv}\leq \overline{\bf q}_{\rm inv}\end{aligned}$$
where ${\bf\Phi}_v\succ0$ and ${\bf\Phi}_q\succ0$ are the weighting matrices; $\underline{\bf q}_{\rm inv}$ and $\overline{\bf q}_{\rm inv}$ are the reactive power min/max limits of DER inverters at each node. Notice that, for those buses/phases without DER integration, the corresponding elements in $\underline{\bf q}_{\rm inv}$ and $\overline{\bf q}_{\rm inv}$ are always equal to zero, which does not affect the coming control algorithm design and analyses.
Integral-Like Algorithm
-----------------------
Firstly, the (squared) voltage reference at bus $(i,\phi)$ is scheduled by[^4], $$\begin{aligned}
\label{vref}
{v}_{{\rm ref},{i}}^\phi(t)=\left[\nu_i^{\phi}\right]^{\overline{v}_{i}^{\phi}}_{\underline{v}_{i}^{\phi}},\, \phi\in\varphi_i\end{aligned}$$ where $\nu_i^{\phi}$ denotes the first-step optimal voltage solution at bus $(i,\phi)$ in the RHO-based upper-layer control; $[\ast]^{\overline{v}_{i}^{\phi}}_{\underline{v}_{i}^{\phi}}$ denotes the projection operator onto the constraint set $[{\underline{v}_{i}^{\phi}},{\overline{v}_{i}^{\phi}}]$. The projection is necessary here because we introduce a slack variable in the voltage constraint in the upper-layer optimization, which implies there might be voltage solution on some buses violating the predefined limit. The voltage reference (\[vref\]) will be maintained unchanged during every upper-layer control period.
Each DER inverter connected to bus $(i,\phi)$ updates its VAR output according to the integral-like control law, $$\begin{aligned}
\label{DeVVC}
\hspace{-2mm}{q}_{{\rm inv},i}^{\phi}(t+1)=\left[{q}_{{\rm inv},i}^{\phi}(t)-\gamma\left({v}_{i}^{\phi}(t)-{v}_{{\rm ref},{i}}^\phi(t)\right)\right]^{\overline{q}_{{\rm inv},i}^\phi(t)}_{\underline{q}_{{\rm inv},i}^\phi(t)}\end{aligned}$$ or compactly, $$\begin{aligned}
\hspace{-2mm}{\bm q}_{{\rm inv}}(t+1)=\left[{\bm q}_{{\rm inv}}(t)-\gamma\left({\bm v}(t)-{\bm v}_{{\rm ref}(t)}\right)\right]^{\overline{\bm q}_{{\rm inv}}(t)}_{\underline{\bm q}_{{\rm inv}}(t)}\end{aligned}$$ where $\gamma>0$ denotes the step size; ${\overline{q}_{{\rm inv},i}^\phi(t)}=-{\underline{q}_{{\rm inv},i}^\phi(t)}=\sqrt{(s_{{\rm inv},i}^{\phi})^2-(p_{{\rm inv},i}^\phi(t))^2}$. Accordingly, each DER inverter updates its VAR output according to its local instantaneous phase voltage measurement.
Optimality Analysis
-------------------
The optimality of the presented control algorithm under a fixed-point problem is given as follows.
*Proposition 2*: Under a fixed point $({\bm\mu},\underline{\bf q}_{\rm inv} ,\overline{\bf q}_{\rm inv})$, suppose (\[volt\]) holds, if the sequence $\{{\bf q}_{\rm inv}(t)\}$ generated by the DeGP algorithm (\[DeGP\]) converges to a limit point, this limit point must be the global optima of the VVC-II problem (\[VVC-II\]) over the constraint set $[\underline{\bf q}_{\rm inv},\overline{\bf q}_{\rm inv}]$.
*Proof*: Since ${\bm\Phi}_v={\bf M}^{-T}\succ0$ and ${\bm\Phi}_q\succ0$, problem (\[VVC-II\]) is essentially a strongly convex QP problem and thus there must exist a unique global optima.
Let ${\bf q}_{\rm inv}^{\star}$ be the limit point of the sequence generated by the DeGP algorithm (\[DeGP\]). Accordingly, we have $$\begin{aligned}
{\bf q}^{\star}_{\rm inv}=\left[{\bf q}_{\rm inv}^\star-\gamma\nabla f({\bf q}_{\rm inv}^{\star})\right]^{\overline{\bf q}_{\rm inv}}_{\underline{\bf q}_{\rm inv}}. \end{aligned}$$ According to *Projection Theorem* [@NP Prop. B.11], it follows that $$\begin{aligned}
\nonumber ({\bf q}_{\rm inv}^\star-\gamma\nabla f({\bf q}_{\rm inv}^{\star})-{\bf q}_{\rm inv}^\star)^T&\left({\bf q}_{\rm inv}-{\bf q}_{\rm inv}^\star\right)\leq0,\,\\
&\hspace{1mm}\forall\, {\bf q}_{\rm inv}\in[\underline{\bf q}_{\rm inv},\overline{\bf q}_{\rm inv}].\end{aligned}$$ Consequently, we have $$\begin{aligned}
\nabla f({\bf q}_{\rm inv}^\star)^T\left({\bf q}_{\rm inv}-{\bf q}_{\rm inv}^\star\right)\geq0,\,\forall\,{\bf q}_{\rm inv}\in[\underline{\bf q}_{\rm inv},\overline{\bf q}_{\rm inv}]\end{aligned}$$ which indicates ${\bf q}_{\rm inv}^\star$ must be the global optima.
Convergence/Stability Analysis
------------------------------
The convergence of DeGP is essentially approximately equivalent to the closed-loop stability of the dynamic system. A sufficient condition is presented as follows.
*Proposition 3*: Under a fixed point $({\bm\mu},\underline{\bf q}_{\rm inv} ,\overline{\bf q}_{\rm inv})$, suppose (\[volt\]) holds, if $\gamma<{2}/({\varepsilon\left\|{\bf M}+{\bm\Phi}_q\right\|})$, the sequence generated by the DeGP algorithm (\[DeGP\]) is guaranteed to converge to the stationary point ${\bf q}^{\star}_{\rm inv}$.
Stability Analysis
------------------
The closed-loop system stability (the distribution network with the decentralized VAR controllers) under a fixed point $(\bm v_{\rm ref}, \underline{\bm q}_{\rm inv}, \overline{\bm q}_{\rm inv})$ is investigated. Let $({\bm q}_{\rm inv}^\ast,{\bm v}^{\ast})$ be the stationary point of closed-loop system. According to ($\ref{DeVVC}$), we have $$\begin{aligned}
\nonumber&\left\|{\bm q}_{\rm inv}(t+1)-{\bm q}_{\rm inv}^\ast\right\|\\
\nonumber&\hspace{0mm}=\left\|\left[{\bm q}_{\rm inv}(t)-\gamma\left({\bm v}-{\bm v}_{\rm ref}\right)\right]^{\overline{\bm q}_{\rm inv}}_{\underline{\bm q}_{\rm inv}}-\left[{\bm q}_{\rm inv}^\ast-\gamma\left({\bm v}^\ast-{\bm v}_{\rm ref}\right)\right]^{\overline{\bm q}_{\rm inv}}_{\underline{\bm q}_{\rm inv}}\right\|\\
\nonumber&\hspace{0mm}\leq\left\|{\bm q}_{\rm inv}(t)-\gamma\left({\bm v}(t)-{\bm v}_{\rm ref}\right)-{\bm q}_{\rm inv}^\ast+\gamma\left({\bm v}^\ast-{\bm v}_{\rm ref}\right)\right\|\\
\nonumber&\hspace{0mm}=\left\|{\bm q}_{\rm inv}(t)-\gamma\left({\bm M}\bm q_{\rm inv}(t)+\bm\mu-{\bm v}_{\rm ref}\right)\right.\\
\nonumber&\hspace{38mm}\left.\ -{\bm q}_{\rm inv}^\ast+\gamma\left({\bm M}\bm q_{\rm inv}^\ast+\bm\mu-{\bm v}_{\rm ref}\right)\right\|\\
&\hspace{0mm}\leq\left\|{\bm I}-\gamma{\bm M}\right\|\left\|{\bm q}_{\rm inv}(t)-{\bm q}_{\rm inv}^\ast\right\|\end{aligned}$$ where the first inequality holds because the projection operation is a non-expensive mapping. Accordingly, a sufficient stability condition that guarantees $\left\|{\bm q}_{\rm inv}(t)-{\bm q}_{\rm inv}^\ast\right\|$ contracting at each iteration is $$\begin{aligned}
\label{stabilitycond}
\left\|\bm I-\gamma\bm M\right\|<1\end{aligned}$$ which provides a upper bound of $\gamma$ to avoid hunting between independent VAR control loops of DERs. Obviously, $f({\bf q}_{\rm inv}(t))\geq0$ always holds. And according to ($\ref{VVC-II}$), it is clear that $f$ satisfies the Lipschitz condition $$\begin{aligned}
\hspace{-1mm} \left\|\nabla f({\bf q}_{{\rm inv}}^{(1)})-\nabla f({\bf q}_{{\rm inv}}^{(2)})\right\|\leq\big\|{\bf M}+{\bm\Phi}_q\big\|\left\|{\bf q}_{{\rm inv}}^{(1)}-{\bf q}_{{\rm inv}}^{(2)}\right\|. \end{aligned}$$ for any ${\bf q}_{\rm inv}^{(1)},{\bf q}_{\rm inv}^{(2)}\in[\underline{\bf q}_{\rm inv},\overline{\bf q}_{\rm inv}]$.
Let $\Omega({\bf q}_{\rm inv}):=\left[{\bf q}_{\rm inv}-\gamma\nabla f({\bf q}_{\rm inv})\right]^{\overline{\bf q}_{\rm inv}}_{\underline{\bf q}_{\rm inv}}$ be the mapping defined in (\[CGP\]). According to *Descent Lemma* [@NP], it follows that, $$\begin{aligned}
\nonumber f&({\bf q}_{{\rm inv}}(t+1))\\
\nonumber&\leq f({\bf q}_{{\rm inv}}(t))+\varepsilon\big(\Omega({\bf q}_{\rm inv}(t))-{\bf q}_{\rm inv}(t)\big)^T\nabla f({\bf q}_{{\rm inv}}(t))\\
&\nonumber\hspace{3mm}+\frac{\varepsilon^2\big\|{\bf M}+{\bm\Phi}_q\big\|}{2}\big\|\Omega({\bf q}_{\rm inv}(t))-{\bf q}(t)\big\|^2\\
&\nonumber\leq f({\bf q}_{{\rm inv}}(t))\\
&\hspace{3mm}-\left(\frac{\varepsilon}{\gamma}-\frac{\varepsilon^2\big\|{\bf M}+{\bm\Phi}_q\big\|}{2}\right)\big\|\Omega({\bf q}_{\rm inv}(t))-{\bf q}_{\rm inv}(t)\big\|^2\end{aligned}$$ where the second inequality can be obtained from *Projection Theorem* [@NP Prop. B.11].
Thus, a sufficient condition to guarantee the convergence (closed-loop stability) is given by $$\begin{aligned}
\label{suff}
\gamma<\frac{2}{\varepsilon\big\|{\bf M}+{\bm\Phi}_q\big\|}\end{aligned}$$ which completes the proof.
Asynchronous Implementation
---------------------------
Notice that, the above presented DeGP algorithm and corresponding analyses are based on the assumption of synchronous update of DER inverters, which requires a synchronized clock in real-life implementation. However, in distribution systems, DERs are owned by different customers. Hence, it might be costly to coordinate each DER. To allow for this, an asynchronous counterpart of DeGP is designed by, $$\begin{aligned}
q_{{\rm inv},i}^\phi(t+1)=q^\phi_{{\rm inv},i}(t)+\delta q_{{\rm inv},i}^\phi(t)\end{aligned}$$ where $\delta q_{{\rm inv},i}^\phi(t)$ is given by, $$\begin{aligned}
\delta q_{{\rm inv},i}^{\phi}(t):=&\left\{
\begin{matrix}{\varepsilon}\bigg(\left[{q}_{{\rm inv},i}^\phi(t)-\gamma\big({v}_{i}^\phi(t)-{v}^\phi_{{\rm ref},i}\right.\\
&\hspace{-48mm}\left.+{\Phi}_{q,i}^\phi{q}_{{\rm inv},i}^\phi(t)\big)\right]^{\overline{q}_{{\rm inv},i}^\phi}_{\underline{q}_{{\rm inv},i}^\phi}-q_{{\rm inv},i}^\phi\bigg),\,t\in\mathcal{T}_i\\
0,\,t\notin\mathcal{T}_i.
\end{matrix}
\right.\end{aligned}$$ We conduct the convergence analysis of asynchronous DeGP by imposing the concept of *Partial Asynchronism*:
i)
: *Bounded Update Delay*: Each DER inverter performs an update of reactive power at least once during any $B$ time units;
ii)
: *Bounded Information Delay*: The information feedback $(v_{i}^{\phi},q_{{\rm inv},i}^\phi)$ used by each DER controller is outdated by at most $B$ time units.
Interestingly, since DeGP only exploits the local instantaneous voltage and reactive power information from the physical system as feedback, which can always reflect the up-to-date information of other DERs. Therefore, it does not suffer from any information delay.
Similarly, according to *Descent Lemma*, it follows that $$\begin{aligned}
\nonumber &f({\bf q}_{{\rm inv}}(t+1))\\
\nonumber&\leq f({\bf q}_{{\rm inv}}(t))+\delta{\bf q}_{{\rm inv}}^T(t)\nabla f({\bf q}_{{\rm inv}}(t))+\frac{\big\|{\bf M}+{\bm\Phi}_q\big\|}{2}\big\|\delta{\bf q}_{{\rm inv}}(t)\big\|^2\\
& \leq f({\bf q}_{{\rm inv}}(t))-\left(\frac{1}{\varepsilon\cdot\gamma}-\frac{\big\|{\bf M}+{\bm\Phi}_q\big\|}{2}\right)\cdot\big\|\delta{\bf q}_{{\rm inv}}(t)\big\|^2\end{aligned}$$ which leads to the same result as (\[suff\]). Thus, the asynchronous DeGP has the same convergence boundary with respect to $\gamma$ due to the instantaneous local feedback mechanism.
Case Study {#sec:case}
==========
![IEEE 123-Node Test Feeder.[]{data-label="IEEE123"}](IEEE123.png "fig:"){width="3.42in"}\
In this section, the proposed VVC framework is verified on the unbalanced IEEE 123-Node Test Feeder [@testfeeder]. Multiple PV and full-inverter wind turbines (WTs), each with a rated capacity of 200 kW, are integrated into the system with the locations shown in Fig. \[IEEE123\]. We develop a simulation framework in MATLAB R2019b, which integrates YALMIP Toolbox [@yalmip] with IBM ILOG CPLEX 12.9 solver [@cplex] for optimization, and the open-source Open Distribution System Simulator (OpenDSS) [@opendss] for power flow analysis. The OpenDSS can be controlled from MATLAB through a component object model interface, allowing us to carry out the VVC algorithms, perform power flow calculations, and retrieve the results. The control mode is disabled in OpenDSS to ensure that the tap changers are not automatically adjusted according to local voltages during a simulation. The daily residential load and solar profiles are obtained from the historical data [@Holcomb] and the wind power time-series is generated from the wind datasets provided by National Renewable Energy Laboratory [@winddata].
Static Performance
------------------
### Upper-Layer Control
[Linear Model]{} [ Nonlinear Model]{}
----------- ------------------ ----------------------
Losses 24.14 kW 118.67 kW
\[0.3mm\]
: Solution Quality With Different Models[]{data-label="ULC_static1"}
$T_H$ [Linear Model]{} [ Nonlinear Model]{}
-------------- ------------------ ---------------------- -- --
1 0.5045 s 5.625 s
\[0.5mm\] 3 0.8666 s 15.11 s
\[0.5mm\] 5 1.294 s 760.5 s
\[0.5mm\] 10 15.51 s 151.4 s
\[0.5mm\] 12 181.5 s 382.5 s
\[0.5mm\]
: Computation Time Under Different Models[]{data-label="ULC_static2"}
The computation performance of the upper-layer control under a static case is shown here. We take $T_H=1$, $C_{\rm tap}=C_{\rm cap}=0$ and $\eta=0.8$ to compare the quality of the solutions. The optimization problem with the G-LBFM constraint can be directly solved by CPLEX 12.9 while the problem with original nonlinear GBFM constraint is solved by IPOPT 3.13. Besides, we further make a comparison with the continuous relaxation-based method [@BAR_dis_tap], wherein the relaxed discrete variables are rounded to the closest integer values. The solutions are used to schedule the VVC devices in the test system (in OpenDSS) and the resultant real losses are measured, which are shown in Table \[ULC\_static1\]. It can be seen that the proposed method with the linear approximation is able to compute a much better solution than the one obtained by continuous relaxation-based method. We further compare the computation efficiency of the two methods under different receding horizon, which is illustrated in Table \[ULC\_static2\]. Obviously, the proposed method shows much better computation efficiency, which is beneficial for the real-time control.
### Lower-Layer Control
The tracking performance of the lower-layer control algorithm under a static case is also tested here to illustrate the convergence performance. We compare our methods[^5] with the droop control[^6] and the open-loop optimization method[^7]. The sufficient stability condition calculated from (\[stabilitycond\]) is $\gamma<0.022$. Fig. \[static\_LLC\] shows the convergence performance of different methods with the metric $\|\bm v-\bm v_{\rm ref}\|^2$. It can be seen that the proposed integral-like method with $\gamma=0.01$ or $0.02$ can efficiently converge to a stationary point within 20 iterations, implying a good tracking capability. In comparison, the droop control converges to an equilibrium after several steps of oscillation which is remote to the voltage reference. This is because the droop control is essentially the difference control, however, the proposed integral-like method is the non-difference control. The final result of the proposed method (see the zoomed part) is even better than that of the optimization-based method due to the online feedback mechanism. Additionally, a larger $\gamma$ naturally implies faster response and thus better tracking capability. But as $\gamma$ increases, the closed-loop system tends to be oscillating and finally diverges. So, there will be a trade off between fast tracking and stability in real-life applications.
![Static performance with different methods for lower-layer control.[]{data-label="static_LLC"}](static_LLC.png "fig:"){width="3.2in"}\
Dynamic Simulation
------------------
![Daily residential load, wind and solar power profiles.[]{data-label="timeseries"}](loadPVwind_timeseries.png "fig:"){width="3.25in"}\
The dynamic simulation is also performed. The daily load, solar and wind power profile are illustrated in Fig. \[timeseries\]. In the proposed method, the upper-layer control period is 1 h and in the lower layer, VAR outputs of DER inverters are updated every 5 s. Besides, we set $T_H=3$ and $\gamma=0.02$. To better demonstrate the effectiveness of the proposed method, it is compared with cases with no control, with only upper-layer control, as well as with the continuous-relaxation-based control method. The operation of tap changers and CBs, as well as the VAR outputs of DERs over 24 hours are shown in Fig. \[CB\]–\[qinv\], respectively.
![Operation of CBs.[]{data-label="CB"}](CB.png "fig:"){width="3.2in"}\
![Operation of tap changers.[]{data-label="TAP"}](tap.png "fig:"){width="3.1in"}\
![Reactive power outputs of DER inverters.[]{data-label="qinv"}](qinv.png "fig:"){width="3.1in"}\
![Power losses.[]{data-label="loss"}](loss.png "fig:"){width="3.1in"}\
Fig. \[loss\] shows the total power losses of the system. It can be seen that the proposed method with two-layer control can effectively reduce the losses, especially during 16:00–24:00 with heavy load consumption and show much better performance than the continuous-based method. The voltage profiles with different control strategies are illustrated in Fig. \[Vprofile\]. Without Volt/VAR control, the voltages violate the range $[0.95,1.05]$ p.u. when DERs have high production e.g., 00:00–04:00 and 09:00–13:00 or when the load demand is heavy, e.g., 16:00–20:00. Only With the upper-layer control, in most time, the voltages can be regulated within the predefined range $[0.95,1.05]$ p.u. with the help of tap changers and CBs, which validates the effectiveness of the open-loop upper-layer control. However, there are still some bus voltages that exceed $1.05$ p.u. in some moments (highlighted in Fig. \[Vprofile\](b)) due to the high fluctuations of DERs and load. In contrast, the two-layer control method can effectively regulate the voltages within the feasible range all the time, and due to the VAR support from DERs, the fast voltage fluctuations are significantly reduced, which can also be clearly observed from the overall voltage performance shown in Fig. \[dV\] and the representative bus voltage at Node 114 (Phase a) shown in Fig. \[V114\], located in the end of the feeder. It can be observed that the voltage under the two-layer control can accurately track the reference, validating the good tracking capability of the integral-like control algorithm in lower layer. Besides, given the control parameter $\gamma$ is designed within the stability boundary, there is no Volt/VAR hunting phenomenon.
![Overall voltage deviations with different strategies.[]{data-label="dV"}](dV.png "fig:"){width="3.15in"}\
![Voltage of Phase a at Node 114 with and without lower-layer control.[]{data-label="V114"}](V114.png "fig:"){width="3.15in"}\
Conclusion {#sec:con}
==========
This paper proposed a two-layer VVC framework for unbalanced distribution systems with high-penetration of inverter-based DERs. The upper layer optimally coordinates the OLTC, SVRs and CBs to improve economic benefits while correcting the long-term voltage deviations based on the RHO method. A G-LBFM was proposed to model the three-phase system with tap changers so that the optimization problem can be efficiently solved in real time. A decentralized integral-like algorithm was developed in the lower layer to deal with the fast voltage fluctuations by exploiting the VAR capabilities of DER inverters. Compared with the cases of no control and only upper layer control, the simulation results show that the proposed two-layer VVC method can effectively reduce the system losses while regulating the voltages within a predefined range in different timescales, due to the combination of open-loop optimization and closed-loop voltage reference tracking. Compared with the continuous-relaxation-based method, the proposed optimization model in upper layer can provide a much better solution and show much better computation efficiency. Compared with the droop-based real-time control, the proposed integral-like control in lower layer can better reduce the voltage deviations by fully exploiting the VAR capabilities of inverter-based DERs. The lower layer control also shows better numerical results than the optimization-based method due to its closed-loop nature.
[99]{} “IEEE Standard for Interconnection and Interoperability of Distributed Energy Resources with Associated Electric Power Systems Interfaces,” *IEEE Std 1547-2018* , pp. 36–41, 2018. Y. P. Agalgaonkar, B. C. Pal, and R. A. Jabr, “Distribution voltage control considering the impact of PV generation on tap changers and autonomous regulators," *IEEE Trans. Power Syst.*, vol. 29, no. 1, pp. 182-192, Jan. 2014. P. Li, H. Ji, C. S. Wang, J. Zhao, G. Song, F. Ding, and J. Wu, “Coordinated control method of voltage and reactive power for active distribution networks based on soft open point,“ *IEEE Trans. Sustain. Energy*, vol. 8, no. 4, pp. 1430-1442, Oct. 2017. Q. Zhang, K. Dehghanpour and Z. Wang, ”Distributed CVR in unbalanced distribution systems with PV penetration," *IEEE Trans. Smart Grid*, vol. 10, no. 5, pp. 5308-5319, Sep. 2019. Y. Guo, Q. Wu, H. Gao, X. Chen, J. [Ø]{}stergaard, and H. Xin, “MPC-Based Coordinated Voltage Regulation for Distribution Networks With Distributed Generation and Energy Storage System," *IEEE Trans. Sustain. Energy*, vol. 10, no. 4, Oct. 2019. A. Kulmala, S. Repo, and P. Jarventausta, “Coordinated voltage control in distribution networks including several distributed energy resources," *IEEE Trans. Smart Grid*, vol. 5, no. 4, pp. 2010-2020, Jul. 2014. H. Sun, et al, “Review of challenges and research opportunities for voltage control in smart grids," *IEEE Trans. Power Syst.*, in press, 2019. K. E. Antoniadou-Plytaria, I. N. Kouveliotis-Lysikatos, P. S. Georgilakis, and N. D. Hatziargyriou, “Distributed and decentralized voltage control of smart distribution networks: models, methods, and future research,“ *IEEE Trans. Smart Grid*, vol. 8, no. 6, pp. 2999-3008, Nov. 2017. B. A. Robbins, H. Zhu and A. D. Domínguez-García, ”Optimal tap setting of voltage regulation transformers in unbalanced distribution systems," *IEEE Trans. Power Syst.*, vol. 28, no. 2, pp. 1470-1482, May 2013. C. Feng, Z. Li, M. Shahidehpour, F. Wen, W. Liu, and X. Wang, “Decentralized short-term voltage control in active power distribution systems," *IEEE Trans. Smart Grid*, vol. 9, no. 5, pp. 4566-4576, 2018. H. Zhu and H. Liu, “Fast local voltage control under limited reactive power: optimality and stability analysis," *IEEE Trans. Power Syst.*, vol. 31, no. 5, Sep. 2016.
L. Mokgonyana, J. Zhang, L. Zhang and X. Xia, “Coordinated two-stage volt/var management in distribution networks," *Elect. Power Syst. Research*, vol. 141, pp. 157-164, 2016. J. Y. Park, S. R. Nam, and J. K. Park, “Control of a ULTC considering the dispatch schedule of capacitors in a distribution system," *IEEE Trans. Power Syst.*, vol. 22, no. 2, pp. 755–761, May 2007. D. Jin, H. Chiang and P. Li, “Two-timescale multi-objective coordinated Volt/Var optimization for active distribution networks," *IEEE Trans. Power Syst.*, vol. 34, no. 6, pp. 4418-4428, Nov. 2019. Y. Guo, Q. Wu, H. Gao, S. Huang, B. Zhou and C. Li, “Double-time-scale coordinated voltage control in active distribution networks based on MPC," *IEEE Trans. Sustain. Energy*, vol. 11, no. 1, pp. 294-303, Jan. 2020. Y. Zheng, D. J. Hill, K. Meng and S. Y. Hui, “Critical bus voltage support in distribution systems with electric springs and responsibility sharing," *IEEE Trans. Power Syst.*, vol. 32, no. 5, pp. 3584-3593, Sep. 2017. Y. Wang, T. Zhao, C. Ju, Y. Xu and P. Wang, “Two-level distributed voltage/var control using aggregated PV inverters in distribution networks," *IEEE Trans. Power Del.*, in press. H. S. Bidgoli and T. Van Cutsem, “Combined local and centralized voltage control in active distribution networks," *IEEE Trans. Power Syst.*, vol. 33, no. 2, pp. 1374-1384, Mar. 2018. S. Weckx, C. Gonzalez and J. Driesen, “Combined central and local active and reactive power control of PV inverters," *IEEE Trans. Sustainable Energy*, vol. 5, no. 3, pp. 776-784, Jul. 2014. T. Hong, D. Zhao, Y. Zhang, B. Cui and Y. Tian, “Optimal voltage reference for droop-based DERs in distribution systems," *IEEE Trans. Smart Grid*, in press. C. Zhang, Y. Xu, Z. Dong and J. Ravishankar, “Three-stage robust inverter-based voltage/var control for distribution networks with high-level PV," *IEEE Trans. Smart Grid*, vol. 10, no. 1, pp. 782-793, Jan. 2019.
W. H. Kersting, “Distribution System Modeling and Analysis 3rd ed,” *CRC Press*, FL, 2012. R. B. Bapat, “Graphs and Matrices,” *Springer*, New York, 2010.
IEEE PES Distribution Systems Analysis Subcommittee Radial Test Feeders. \[Online\]. Available: http://www.ewh.ieee.org/soc/pes/dsacom/ testfeeders/index.html OpenDSS Simulation Tool \[Online\]. Available: http://smartgrid.epri. com/SimulationTool.aspx J. Löfberg,“YALMIP: A toolbox for modeling and optimization in MATLAB," in *Proc. CACSD Conf.*, Taipei, Taiwan, 2004. B. Christian, B. Pierre and A. Lodi,“Solving mixed-integer quadratic programming problems with IBM-CPLEX: a progress report," in *Proc. 26th RAMP Symposium*, Tokyo, Japan, Oct. 2014. C. Holcomb, “Pecan Street Inc.: A test-bed for NILM,” in *International Workshop on Non-Intrusive Load Monitoring*, 2012. \[Online\]. Available: https://www.pecanstreet.org/ Eastern Wind Data Set. \[Online\]. Available: https://www.nrel.gov/grid/ eastern-wind-data.html
[^1]: This work was supported by the U.S. Department of Energy Wind Energy Technology Office under DE-EE8956 and the National Science Foundation under ECCS 1929975. *(Corresponding author: Zhaoyu Wang)*
[^2]: The authors are with the Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011 USA. (email:yifeig@iastate.edu; qianzhi@iastate.edu;wzy@iastate.edu; fbu@iastate.edu; yuanyx@iastate.edu)
[^3]: Let $\overline{N}_{{\rm cap},i}^{\phi}=0$ for those buses/phases without CBs and $\underline{n}_{{\rm tap},ij}^{\phi}=\overline{n}_{{\rm tap},ij}^{\phi}=0$ for branches without SVRs.
[^4]: Here, we abuse the notation of time stamp $t$ in the upper and lower layers, which actually corresponds to different time intervals.
[^5]: Note that, in the test of our method (also for the later droop control), at each iteration, the real voltage measurements (from OpenDSS) are used as feedback to perform the update instead of using the computation results from the linearized model.
[^6]: We originally select the droop gain according to the widely-used principle [@Czhang_multi_time], but due to the large gain, the droop control fails to converge under our problem, i.e., the closed-loop system is unstable, which is not shown in Fig. \[static\_LLC\]. Therefore, we gradually reduce the gain until it converges and the corresponding result is illustrated in the figure.
[^7]: The optimization method is used to minimize $\left\|\bm v-\bm v_{\rm ref}\right\|^2$ over the constraints $\bm q\in[\underline{\bm q}_{\rm inv},\overline{\bm q}_{\rm inv}]$ and linearized BFM (so that the problem is convex) using the solver CPLEX 12.9.
|
---
abstract: |
In this paper we present elementary computations for some *Markov modulated* counting processes, also called counting processes with *regime switching*. Regime switching has become an increasingly popular concept in many branches of science. In finance, for instance, one could identify the background process with the ‘state of the economy’, to which asset prices react, or as an identification of the varying default rate of an obligor. The key feature of the counting processes in this paper is that their intensity processes are functions of a finite state Markov chain. This kind of processes can be used to model default events of some companies.
Many quantities of interest in this paper, like conditional characteristic functions, can all be derived from conditional probabilities, which can, in principle, be *analytically* computed. We will also study limit results for models with rapid switching, which occur when inflating the intensity matrix of the Markov chain by a factor tending to infinity. The paper is largely expository in nature, with a didactic flavor.\
[*Keywords:*]{} Counting process, Markov chain, Markov modulated process, Regime switching.\
[*AMS subject classification:*]{} 60G44, 60G55, 60J27.
author:
- Michel Mandjes
- Peter Spreij
title: Explicit computations for some Markov modulated counting processes
---
Introduction
============
In this paper we present some elementary computations concerning some *Markov modulated* (MM) counting processes, denoted $N$, also called counting processes with *regime switching*. Such processes fall into the class of *hybrid models* [@YinZhu] and are in fact Hidden Markov processes [@EAM]. Although in the present paper we restrict ourselves to certain counting processes, it is worth mentioning that owing to its various attractive features, regime switching has become an increasingly popular concept in many branches of science. In a broad spectrum of application domains it offers a natural framework for modeling situations in which the stochastic process under study reacts to an autonomously evolving environment. In finance, for instance, one could identify the background process with the ‘state of the economy’, to which asset prices react, or as an identification of the varying default rate of an obligor. In operations research, in particular in wireless networks, the concept can be used to model the channel conditions that vary in time, and to which users react. In the literature in the latter field there is a sizeable body of work on Markov-modulated queues, see e.g. [@ASM Ch. XI] and [@NEUTS], while Markov modulation has been intensively used in insurance and risk theory as well [@AA]. In the economics literature, the use of regime switching dates back to at least the late 1980s [@HAM]. Various specific models have been considered since then, see for instance [@ANG; @EMAM; @ESIU]. For other direct applications of models with regime switching in finance (hedging of claims, interest rate models, credit risk, application to pension funds) we refer to [@chen; @JP2008; @JP2012; @Yin2009; @ZhouYin2003] for recent results.
The key feature of the counting processes, commonly denoted $N$, in this paper is that their intensity processes are of the form $\lambda_t=\lambda(X_t,N_t)$, where $X$ is a finite state Markov chain whose jumps with probability one never coincide with the jumps of the counting process. For mathematical convenience we assume without loss of generality that $X$ takes its values in the set of $d$-dimensional basis vectors.
This kind of processes can be used to model default events of some companies. We restrict our treatment to models where the intensity is of a special form, leading to the MM one point process which can be used to model the default event of a single company, its extension to the situation of defaults of various companies and an MM Poisson process, which can be used to model defaults for a large pool of obligors whose individual intensities of default are all the same and small.
The intensities $\lambda_t=\lambda(X_t,N_t)$ that we use will be affine in $X_t$, i.e. $\lambda_t=\lambda^\top X_tf(N_t)$ for some $\lambda\in{\mathbb{R}}^d$ and some function $f$. It is possible to show that the joint process $(X,N)$ is Markov, in fact it is an affine process after a state transformation. This means that for many quantities of interest, like conditional characteristic functions, one can in principle use the full technical apparatus that has become available for affine process, see [@dfs]. However, as these quantities can all be derived from conditional probabilities (our processes are finite, or at most countably, valued), using these techniques is like making a detour since the conditional probabilities can be derived by more straightforward methods. Moreover these conditional probabilities give a *direct* insight into the probabilistic structure of the process and can in principle be *analytically* computed. Therefore, we circumvent the theory of affine processes and focus on direct computation of all conditional probabilities of interest.
We will also study limit results for models with rapid switching, which occur when inflating the intensity matrix of the Markov chain by a factor tending to infinity. Rapid switching between (macro) economic states is unrealistic, but in models for the profit and loss of trading positions, especially in high frequency trading, rapid switching may take place, see [@graziano]. We will see that the limit processes have intensities that are expectations under the invariant distribution of the chain. This is similar to what happens in the context of Markov modulated Ornstein-Uhlenbeck processes [@HuangMandjesSpreij], see also [@quintet], whereas comparable results under scaling in the operations research literature can be found in [@BMT] and [@BKMT].
The paper is largely expository in nature, with a didactic flavor. We do not claim novelty of all results below. Rather we emphasize the uniform approach that we follow, using martingale methods, that may also lead to alternative proofs of known results, e.g. those concerning transition probabilities by using ‘${\varepsilon}$-arguments’ as in [@NEUTS]. The organization of the paper is as follows. In Section \[section:mm\] we study Markov modulated model for the total number of defaults when there are $n$ obligors. As a primer, in Section \[section:mm0\] we extensively study the Markov modulated model for a single obligor, in particular its distributional properties. Then we switch to the more general situation of Section \[section:mmmultiple\], where our approach is inspired by the easier case of the previous section. All results are basically obtained by exploiting the Markovian nature of the joint process $(X,N)$. Section \[section:mmpoisson\] gives a few results for the Markov modulated Poisson process. Conditional probabilities of future values of the counting processes, when only its own past can be observed (and not the underlying Markov chain) can be computed using filtering theory, which is the topic of Section \[section:filtering\]. In Section \[section:rapid\] we obtain the limit results for processes where the Markov chain is rapidly switching.
The MM model for multiple obligors {#section:mm}
==================================
We assume throughout that a probability space ${(\Omega,\mathcal{F},{\mathbb{P}})}$ is given. Suppose we have $n$ obligors with default times $\tau^i$ for obligor $i$, $i=0,\ldots,n$. Let $Y^i_t={{\bf 1}}_{\{\tau^i\leq t\}}$, $t\in [0,\infty)$. Here we encounter the canonical set-up for the intensity based approach in credit risk modelling, see [@filipovic Chapter 12] or [@br Chapter 6] for further details on probabilistic aspects. We postulate for each $i\in\{1,\ldots,n\}$ $$\label{eq:yi}
{{\mathrm d} }Y^i_t= \lambda_t(1-Y^i_t)\,{{\mathrm d} }t + {{\mathrm d} }m^i_t,$$ for $\lambda_t$ a nonnegative process to be specified, but which is the same for each obligor $i$. Here each $m^i$ is a martingale w.r.t. to the filtration, call it ${\mathbb{F}}^i$, generated by $Y^i$ and the process $\lambda$. We impose that the $\tau_i$ are conditionally independent given $\lambda$. Hence, simultaneous defaults occur with probability zero, as the $\tau^i$ have a continuous distribution. By the conditional independence assumption, the $m^i$ are also martingales w.r.t. ${\mathbb{F}}=\vee_{i=1}^n{\mathbb{F}}^i$. The process $\lambda$ is assumed to be predictable w.r.t. ${\mathbb{F}}$. In all what follows in this section we take $N_t=\sum_{i=1}^nY^i_t$.
The MM one point process {#section:mm0}
------------------------
For a better understanding of what follows, we single out the special case $n=1$ and we write $\tau$ instead of $\tau^1$. There is some advantage in starting with a simpler case that allows for more explicit formulas, is more transparent, and that at the same time can serve as a warming up for the more general setting.
### The general one point process with intensity {#section:inhomo}
Let us consider the basic case, the random variable $\tau$ has an exponential distribution with parameter $\lambda$, and $Y_t={{\bf 1}}_{\{\tau\leq t\}}$, $t\in [0,\infty)$. Then $Y$ has semimartingale decomposition $$\label{eq:y1}
{{\mathrm d} }Y_t = \lambda(1-Y_t)\,{{\mathrm d} }t+{{\mathrm d} }m_t,$$ where $\lambda >0$ and $m$ a martingale w.r.t. the filtration generated by the process $Y$. As a matter of fact, the distributional property of $\tau$ is equivalent to the decomposition of $Y$ in . Clearly $Y_t$ is a Bernoulli random variable, so $y(t):={\mathbb{E}}Y_t={\mathbb{P}}(Y_t=1)={\mathbb{P}}(\tau\leq t)$. Alternatively, taking expectations, we get the ODE $$\dot{y}(t)=\lambda (1-y(t)),$$ which is, with $y(0)=0$, indeed solved by $$y(t)=1-\exp(-\lambda t).$$ Let $\Lambda^\tau$ be the compensator of $Y$, then $$\Lambda^\tau_t=\int_0^t\lambda(1-Y_s)\,{{\mathrm d} }s = \int_0^t\lambda{{\bf 1}}_{\{s<\tau\}}\,{{\mathrm d} }s = \int_0^{t\wedge \tau}\lambda \,{{\mathrm d} }s=\lambda (\tau\wedge t).$$ Note that $Y$ can be considered as $N^\tau$, the at $\tau$ stopped Poisson process with intensity $\lambda$. The compensator $\Lambda$ of $N$ stopped at $\tau$ indeed yields $\Lambda^\tau$.\
As a first generalization we change the above setup in the sense that we postulate $$\label{eq:y2}
{{\mathrm d} }Y_t = \lambda_t(1-Y_t)\,{{\mathrm d} }t+{{\mathrm d} }m_t,$$ where $\lambda$ is a nonnegative locally integrable Borel function, also known as the (time varying) hazard rate. As above one can show that $$y(t)=1-\exp(-\int_0^t\lambda_s\,{{\mathrm d} }s).$$ In a next generalization we suppose that $\lambda$ becomes a random process defined on an auxiliary probability space $(\Omega',{\mathcal{F}}',{\mathbb{P}}')$. We can look at the product probability space $(\Omega\times\Omega',{\mathcal{F}}\otimes{\mathcal{F}}',{\mathbb{P}}\otimes{\mathbb{P}}')$ and redefine in the obvious way $Y$, $\tau$ and $\lambda$ on this product space. The filtration we will use consists of the $\sigma$-algebras ${\mathcal{F}}^Y_t\otimes{\mathcal{F}}^\lambda_t$.
It is assumed that $\lambda$ is predictable and a.s. locally integrable w.r.t. Lebesgue measure. For a given trajectory $\lambda_t=\lambda_t(\omega')$ we define $Y$ on ${(\Omega,\mathcal{F},{\mathbb{P}})}$ as in . With ${\mathcal{F}}^\lambda$ the $\sigma$-algebra generated by the full process $\lambda$, we have that $${\mathbb{E}}[Y_t|{\mathcal{F}}^\lambda]=1-\exp(-\int_0^t\lambda_s\,{{\mathrm d} }s),$$ and hence $$y(t)={\mathbb{E}}Y_t=1-{\mathbb{E}}\exp(-\int_0^t\lambda_s\,{{\mathrm d} }s).$$ Alternatively, one can construct the point process $Y$ as follows. Let $(\Omega, {\mathcal{F}}, {\mathbb{Q}})$ be a probability space on which is defined a standard Poisson process $Y$ and *independently* of $Y$ the nonnegative predictable process $\lambda$. Put $L_t={\mathcal{E}}(\mu)_t$, the Doléans exponential of the ${\mathbb{Q}}$-local martingale $\mu$ given by $\mu_t=\int_0^t (\lambda_s{{\bf 1}}_{\{Y_{s-}=0\}}-1)\,{{\mathrm d} }(Y_s-s)$. Note that $L_0=1$. Let $\tau_k$ be the consecutive jump times of $Y$, $\tau_0=0$. Note that the differences $\tau_k-\tau_{k-1}$ have a standard exponential distribution under ${\mathbb{Q}}$. The assertion of the following lemma is a variation on Equation (4.23) in [@br].
The density process $L$ allows the following explicit expression, $$L_t=(\lambda_{\tau_1})^{Y_t}\exp(t-\int_0^{\tau_1\wedge t}\lambda_s\,{{\mathrm d} }s){{\bf 1}}_{\{Y_t\leq 1\}}.$$ If $\lambda$ is a bounded process, $L$ is a martingale, hence ${\mathbb{E}}L_t=L_0=1$.
By construction, $L$ is a local martingale. For bounded $\lambda$ we have ${\mathbb{E}}\int_0^t L_s^2\,{{\mathrm d} }s\leq C\exp(2t)$ for some constant $C$, which yields $L$ a square integrable martingale. The given expression for $L_t$ can be verified by an elementary, but slightly tedious computation.
Under the assumption that $L$ is a martingale (guaranteed for bounded $\lambda$), by Girsanov’s theorem, see [@bremaud Chapter VI, T3 and T4], we can define for every $T>0$ a probability ${\mathbb{P}}$ on $(\Omega,{\mathcal{F}}_T)$ such that $$m_t:=Y_t-t-\langle Y,\mu\rangle_t=Y_t-\int_0^t\lambda_s{{\bf 1}}_{\{Y_{s-}=0\}}\,{{\mathrm d} }s$$ is a local martingale under ${\mathbb{P}}$. Note that ${\mathbb{P}}(Y_T>1)={\mathbb{E}}_{\mathbb{Q}}{{\bf 1}}_{\{Y_T>1\}}L_T=0$. Hence, under ${\mathbb{P}}$ we have ${{\bf 1}}_{\{Y_s=0\}}=1-Y_s$ and the expression for $m_t$ coincides with for $t\leq T$. Note that $L$ cannot be uniformly integrable, since $L_\infty=0$, which follows from $L_{\tau_2}=0$. Hence it is not automatic that one can define a probability ${\mathbb{P}}$ on $(\Omega,{\mathcal{F}})$ such that $m$ is a martingale on $[0,\infty)$. Note that the laws under ${\mathbb{P}}$ and ${\mathbb{Q}}$ of $\lambda$ are the same.
### The one point process with MM intensity {#section:mm1}
In this section we consider , where we specify $\lambda_t$ as a function of a finite state Markov chain $X_t$, i.e. $\lambda_t=\lambda(X_t)$. We see that, trivial cases excluded, unlike the constant hazard rate $\lambda$ in , we now have a rate that assumes different values according to the states of the Markov chain. We thus have a rate that is subject to *regime switching*, one also says that we have a *Markov modulated* rate. In order to pose a precise mathematical model, we make some conventions. Let $d$ be the size of the state space of the Markov chain $X$. Then w.l.o.g. we may assume that $X$ takes its values in the set $\{e_1,\ldots,e_d\}$ of $d$-dimensional standard basis vectors. This implies that any function of $X_t$ can be written as a linear map of $X_t$, in particular $\lambda(X_t)=\lambda^\top X_t$, where on the right hand side $\lambda$ is a vector in ${\mathbb{R}}^d_+$.
Let $Q$ be the transition matrix of $X$, for which we use the convention that $Q_{ij}$ for $i\neq j$ is the intensity of a transition from state $j$ to state $i$. As a consequence the *column sums* of $Q$ are equal to zero. We then have $${{\mathrm d} }X_t=QX_t\,{{\mathrm d} }t+ {{\mathrm d} }M^X_t,$$ where $M^X$ a martingale with values in ${\mathbb{R}}^d$. We also assume that $Q$ is irreducible and we denote by $\pi$ the vector representing the invariant distribution.
Furthermore it will be throughout assumed that $Y$ and $X$ have no simultaneous jumps, hence the quadratic variation process $[X,Y]$ ($[X,Y]_t=\sum_{s\leq t}\Delta X_s\Delta Y_s$) is identically zero.\
For the single obligor case, we pose the following model with regime switching, $${{\mathrm d} }Y_t= \lambda^\top X_t(1-Y_t)\,{{\mathrm d} }t + {{\mathrm d} }m_t,$$ where $\lambda\in{\mathbb{R}}^d_+$.
One way of constructing this model is by realizing it on a product space with $\lambda_t=\lambda^\top X_t$ as in Section \[section:inhomo\]. Alternatively, one can realize $Y$ as standard Poisson process and independently of it, $X$ as a Markov chain on the auxiliary space under ${\mathbb{Q}}$. By independence, one has $[X,Y]=0$ under ${\mathbb{Q}}$ and as these brackets remain the same under an absolutely continuous change of measure using the ${\mathbb{Q}}$-martingale $\mu$ of the previous section, we are then guaranteed to have $[X,Y]=0$ under ${\mathbb{P}}$ as well. In this case it is possible to have ${\mathbb{P}}$ defined on $(\Omega,{\mathcal{F}})$ for ${\mathcal{F}}={\mathcal{F}}_\infty$, where we use the filtration generated by $Y$ and $X$. As a side remark we note that ${\mathbb{P}}$ will not be absolutely continuous w.r.t. ${\mathbb{Q}}$ on ${\mathcal{F}}_\infty$.\
In all what follows in this paper we adopt the following *Convention: we will use the generic notation $M$ for a martingale, possibly even of varying dimensions, whose precise form is not important.*\
An important role will be played by the matrices $Q_{k\lambda}:=Q-k\,{\mathrm{diag}}(\lambda)$ for $k\geq 0$. Here ${\mathrm{diag}}(\lambda)$ is the diagonal matrix with $ii$-element equal to $\lambda_i$. Here is a, possibly known, stability result for the matrix $Q_\lambda$ (we take $k=1$, but a similar result is obviously true for all positive $k$).
\[lemma:qlambda\] Let $\lambda_i>0$ for all $i$. Then the matrix $Q_\lambda$ is invertible and $\exp(Q_\lambda t)\to 0$ for $t\to\infty$.
That $Q_\lambda$ is invertible, can be seen as follows. Write $$Q_\lambda=-(I-Q{\mathrm{diag}}(\lambda)^{-1}){\mathrm{diag}}(\lambda)$$ and note that $Q{\mathrm{diag}}(\lambda)^{-1}$ is also the intensity matrix of a Markov chain, as its off-diagonal elements are positive and ${{\bf 1}}^\top Q{\mathrm{diag}}(\lambda)^{-1}=0$. Therefore $I-Q{\mathrm{diag}}(\lambda)^{-1}$ is invertible, and so is $Q_\lambda$.
In proving the limit result, we give a probabilistic argument.[^1] Consider the augmented matrix $$Q^a_\lambda =
\begin{pmatrix}
0 & -{{\bf 1}}^\top Q_\lambda \\
0 & Q_\lambda
\end{pmatrix},$$ which is the transition matrix of a Markov chain taking values in $\{e^a_0,\ldots,e^a_d\}$, labelled as the standard basis vectors of ${\mathbb{R}}^{d+1}$. Clearly, $0$ is an absorbing state, and the only one. Hence whatever initial state $x^a(0)$, we have that $\exp(Q^a_\lambda t)x^a(0)\to e^a_0$ for $t\to\infty$. Computing the exponential and taking $x^a(0)\neq e^a_0$, we find $$\exp(Q^a_\lambda t)x^a=
\begin{pmatrix}
1 & {{\bf 1}}^\top(I-\exp(Q_\lambda t)) \\
0 & \exp(Q_\lambda t)
\end{pmatrix}x^a(0) =
\begin{pmatrix}
{{\bf 1}}^\top(I-\exp(Q_\lambda t))x(0) \\
\exp(Q_\lambda t)x(0).
\end{pmatrix}$$ Hence $\exp(Q_\lambda t)\to 0$.
In a next section, see Remark \[remark:n=1\], we shall see how to compute ${\mathbb{P}}(Y_t=1)$. It turns out to be the case that $${\mathbb{P}}(Y_t=1)=1-{{\bf 1}}^\top \exp(Q_\lambda t)x(0).$$ We conclude in view of Lemma \[lemma:qlambda\] that ${\mathbb{P}}(Y_t=1)\to 1$ for $t\to\infty$. Hence, with probability one, the obligor eventually defaults, as expected.
The MM model for multiple obligors {#section:mmmultiple}
----------------------------------
In Section \[section:mm1\] we have seen results for default processes in the situation of a single obligor. In the present section we generalize those results, at the cost of considerably more complexity, to the situation of multiple obligors.
### Multiple obligors with time-varying intensity {#section:mcl}
Recall . Let’s first look at the constant intensity case, $\lambda_t=\lambda>0$. Then $N_t=\sum_{i=1}^nY^i_t$ satisfies $$\label{eq:bin}
{{\mathrm d} }N_t=\lambda(n-N_t)\,{{\mathrm d} }t+ {{\mathrm d} }m_t,$$ where $m=\sum_{i=1}^nm^i$. By the independence of the default times, $m$ is a martingale w.r.t. ${\mathbb{F}}$ and $N_t$ has the $\mathrm{Bin}(n,1-\exp(-\lambda t))$ distribution. Moreover, given $N_u$, $u\leq s$, $N_t-N_s$ has for $t>s$ the $\mathrm{Bin}(n-N_s,1-\exp(-\lambda (t-s)))$ distribution. This model has long ago been used in software reliability going back to [@jm], with various refinements, like in a Bayesian set up the parameters $n$ and $\lambda$ being random, see [@ks; @littlewood] or with time varying but deterministic intensity function $\lambda(t)$, see [@go].\
Next we look at the case of time varying, possibly random, $\lambda$. By the assumed conditional independence of the $\tau^i$ given $\lambda$ we have, similar to the constant $\lambda$ case, that $N_t$, conditional on the process $\lambda$, has a $\mathrm{Bin}(n, 1-\exp(-\Lambda_t))$ distribution with $\Lambda_t=\int_0^t\lambda_s\,{{\mathrm d} }s$.
Let $p^k(t)={\mathbb{P}}(N_t=k|{\mathcal{F}}^\lambda)$, put $$p(t)=\begin{pmatrix} p^{0}(t) \\ \vdots \\ p^{n}(t)
\end{pmatrix}$$ and $$\label{eq:A}
A=\begin{pmatrix}
-n & 0 & \cdots & \cdots & \cdots & 0 \\
n & -(n-1) & 0 & \cdots & \cdots & 0 \\
0 & n-1 & -(n-2) & 0 & \cdots & 0 \\
\vdots & \ddots & \ddots & \ddots & \ddots & \vdots \\
\vdots & & \ddots & \ddots & -1 & 0 \\
0 & \cdots & \cdots & 0 & 1 & 0
\end{pmatrix}.$$ Then we have for $p(t)$ the system of differential equations $$\dot{p}(t)=\lambda_tAp(t),$$ which has solution (here we use that $\lambda$ is real-valued) $$p(t)=\exp(\Lambda_t A)e_0,$$ where $\Lambda_t=\int_0^t\lambda_s\,{{\mathrm d} }s$ and $e_0$ is the first standard basis vector of ${\mathbb{R}}^{n+1}$. For the vector whose elements are the unconditional probabilities ${\mathbb{P}}(N_t=k)$ one has to take the expectation and it depends on the specification of $\lambda$ whether this results in analytic expressions. We will see that this happens in case of a Markov modulated rate process.
### The MM case {#section:mmmo}
We assume to have a finite state Markov process as in Section \[section:mm1\] and let $\lambda_t=\lambda^\top X_{t-}$. For $N_t$ one now has its submartingale decomposition $${{\mathrm d} }N_t=\lambda^\top X_t(n-N_t)\,{{\mathrm d} }t+ {{\mathrm d} }m_t.$$ This is the model of Section \[section:mm1\] extended to more obligors. The default rate for each obligor has become random ($\lambda^\top X_t$), but is taken the same for all of them.
Let $\nu^k_t={{\bf 1}}_{\{N_t=k\}}$, $k=0,\ldots,n$. For notational convenience we set $\nu^{-1}_t=0$. It follows that $\Delta\nu^k_{t}=1$ iff $N_t$ jumps from $k-1$ to $k$ at $t$, and $\Delta\nu^k_{t}=-1$ iff $N_t$ jumps from $k$ to $k+1$. This can be summarized by $${{\mathrm d} }\nu^k_{t}=(\nu^{k-1}_{t-}-\nu^k_{t-})\,{{\mathrm d} }N_t.$$ In vector form this becomes $$\label{eq:nu0}
{{\mathrm d} }\nu_t=(J-I)\nu_{t-}\,{{\mathrm d} }N_t,$$ where $$J=\begin{pmatrix}
0 & & & & \\
1 & 0 & & & \\
0 & 1 & \ddots & & \\
\vdots & \ddots & \ddots & \ddots & 0 \\
0 & \cdots & 0 & 1 & 0
\end{pmatrix}.$$ Using the dynamics for $N$, we get $$\begin{aligned}
{{\mathrm d} }\nu^k_{t} & =(\nu^{k-1}_{t-}-\nu^k_{t-})(\lambda^\top X_{t-}(n-N_t)\,{{\mathrm d} }t+ {{\mathrm d} }m_t) \\
& = \lambda^\top X_t ((n-k+1)\nu^{k-1}_t-(n-k)\nu^k_t)\,{{\mathrm d} }t +{{\mathrm d} }M_t.\end{aligned}$$ Letting $\nu_t=\begin{pmatrix}
\nu^0_t \\
\vdots \\
\nu^n_t
\end{pmatrix}$, we get from the above display $$\label{eq:nu}
{{\mathrm d} }\nu_t=\lambda^\top X_t A\nu_t\,{{\mathrm d} }t+ {{\mathrm d} }M_t,$$ where $A$ is as in . This equation for $\nu$ is a main ingredient in the next result.
\[prop:zeta\] Let $\zeta_t=\nu_t\otimes X_t$. The process $\zeta$ is Markov with transition matrix $\mathbf{Q}$, where $\mathbf{Q}=(A\otimes{\mathrm{diag}}(\lambda)+I\otimes Q)$. It follows that ${\mathbb{E}}[\zeta_t|{\mathcal{F}}_s]=\exp({\mathbf{Q}}(t-s))\zeta_s$.
We will use equation together with the dynamics of $X$. Using the product rule and the fact that $N$ and $X$ do not jump at the same time and summarizing again all martingale terms again as $M$, we get (recall the multiplication rule $(A\otimes B)(C\otimes D)=(AC)\otimes (BD)$) $$\begin{aligned}
{{\mathrm d} }(\nu_t\otimes X_t) & = \left((A\nu_t\lambda^\top X_t)\otimes X_t+\nu_t \otimes (QX_t)\right)\,{{\mathrm d} }t + {{\mathrm d} }M_t \\
& = \left((A\nu_t)\otimes (X_t\lambda^\top X_t)+\nu_t \otimes (QX_t)\right)\,{{\mathrm d} }t + {{\mathrm d} }M_t \\
& = \left((A\nu_t)\otimes ({\mathrm{diag}}(\lambda) X_t)+I\nu_t \otimes (QX_t)\right)\,{{\mathrm d} }t + {{\mathrm d} }M_t \\
& = (A\otimes{\mathrm{diag}}(\lambda)+I\otimes Q)(\nu_t \otimes X_t)\,{{\mathrm d} }t + {{\mathrm d} }M_t \\
& = \mathbf{Q}(\nu_t \otimes X_t)\,{{\mathrm d} }t+ {{\mathrm d} }M_t.\end{aligned}$$ Note that $\zeta_t$ by construction consists of the indicators of the values of the joint process $(\nu,X)$. Hence the equation ${{\mathrm d} }\zeta_t={\mathbf{Q}}\zeta_t\,{{\mathrm d} }t + {{\mathrm d} }M_t$ reveals, cf. Lemma 1.1 in Appendix B of [@EAM], that $\zeta$ (and hence $(\nu,X)$) is Markov.
An explicit computation shows $$\label{eq:bq}
{\mathbf{Q}}=\begin{pmatrix}
Q_{n\lambda} & 0 & \cdots & \cdots & \cdots & 0 \\
n\,{\mathrm{diag}}(\lambda) & Q_{(n-1)\lambda} & 0 & \cdots & \cdots & 0 \\
0 & (n-1)\,{\mathrm{diag}}(\lambda) & Q_{(n-2)\lambda} & 0 & \cdots & 0 \\
\vdots & \ddots & \ddots & \ddots & \ddots & \vdots \\
\vdots & & \ddots & \ddots & Q_\lambda & 0 \\
0 & \cdots & \cdots & 0 & {\mathrm{diag}}(\lambda) & Q
\end{pmatrix},$$ where for $k\in{\mathbb{N}}$ we have $Q_{k\lambda}=Q-k\,{\mathrm{diag}}(\lambda)$.
The original dynamic equations for $X_t$ and $N_t$ can be retrieved from Proposition \[prop:zeta\]. Realizing the relations $X_t=({{\bf 1}}^\top \otimes I)\zeta_t$ and $({{\bf 1}}^\top \otimes I)\mathbf{Q}={{\bf 1}}^\top\otimes Q$, and ${{\bf 1}}^\top A=0$, we obtain from Proposition \[prop:zeta\] $$\begin{aligned}
{{\mathrm d} }X_t & =({{\bf 1}}^\top \otimes I)\left(\mathbf{Q}(\nu_t \otimes X_t)\right)\,{{\mathrm d} }t+ {{\mathrm d} }M_t \\
& = ({{\bf 1}}^\top\otimes Q)(\nu_t\otimes X_t)\,{{\mathrm d} }t+ {{\mathrm d} }M_t \\
& = QX_t\,{{\mathrm d} }t+ {{\mathrm d} }M_t.\end{aligned}$$ Similarly, we get from $\nu_t=(I\otimes {{\bf 1}}^\top)\zeta_t$, $$\begin{aligned}
{{\mathrm d} }\nu_t & =(I\otimes {{\bf 1}}^\top)\left(\mathbf{Q}(\nu_t \otimes X_t)\right)\,{{\mathrm d} }t+ {{\mathrm d} }M_t \\
& = (A\otimes \lambda^\top)(\nu_t\otimes X_t)\,{{\mathrm d} }t+ {{\mathrm d} }M_t \\
& = A\nu_t\lambda^\top X_t\,{{\mathrm d} }t+ {{\mathrm d} }M_t.\end{aligned}$$ Using $\begin{pmatrix} 0 & 1 & \cdots & n\end{pmatrix}A\nu_t=\begin{pmatrix} n & \cdots & 1 & 0\end{pmatrix}\nu_t=n-N_t$, we get from the last display the decomposition ${{\mathrm d} }N_t=(n-N_t)\lambda^\top X_t \,{{\mathrm d} }t+ {{\mathrm d} }m_t$ back.
Letting $\pi(t)={\mathbb{E}}\zeta_t$, we obtain from Proposition \[prop:zeta\] the ODE $$\label{eq:pi}
\dot{\pi}(t) ={\mathbf{Q}}\pi(t)$$ with the initial condition $\pi(0)=e_0\otimes x(0)$, where $e_0$ has $1$ as its first element, all other elements being zero. We will give a rather explicit expression for $\pi(t)=\exp({\mathbf{Q}}t)\pi(0)$, for which we need some additional results.
The differential equation for $\pi$ is the following type of forward equation, $$\dot{F}=\mathbf{Q}F.$$ Here $F$ can be any matrix valued function of appropriate dimensions. We will block-diagonalize the matrix $\mathbf{Q}$. The transformation that is needed for that is given by the matrix $V$ whose $ij$-block ($i,j=0,\ldots,n$) is $$V_{ij}={n-j \choose n-i}(-1)^{i-j}I.$$ Note that $V_{ij}=0$ for $i<j$, $V$ is block lower-triangular. The inverse matrix is also block lower-triangular with blocks $$V^{-1}_{ij}={n-j \choose n-i}I.$$ One may check by direct computation that indeed $VV^{-1}=I$. It is straightforward to verify that $\mathbf{Q}^V:=V^{-1}\mathbf{Q}V$ is block-diagonal with $i$-th block ($i=0,\ldots,n$) equal to $$\mathbf{Q}^V_i=Q_{(n-i)\lambda}.$$ Putting $G=V^{-1}F$ we obtain $$\dot{G}=\mathbf{Q}^V\,G,$$ whose solution satisfying $G(0)=I$ is block diagonal with $i$-th block $G_i(t)=\exp(Q_{(n-i)\lambda}t)$. We thus obtain the following lemma.
\[lemma:diag\] The solution to the forward ODE $\dot{F}=\mathbf{Q}F$ with initial condition $F(0)$ is given by $F(t)=\exp({\mathbf{Q}}t)F(0)$, where $$\exp({\mathbf{Q}}t)=V
\begin{pmatrix}
\exp(Q_{n\lambda}t) & & \\
& \ddots & \\
& & \exp(Qt)
\end{pmatrix}
V^{-1}.$$ If $F(t)=\exp({\mathbf{Q}}t)$, its blocks $F_{ij}(t)$ can be explicitly computed. One has $F_{ij}(t)=0$ if $i<j$, and for $i\geq j$ it holds that $$F_{ij}(t)={n-j \choose n-i} \sum_{k=j}^i(-1)^{i-k} {i-j\choose i-k}\exp(Q_{(n-k)\lambda}t).$$
We use the block triangular structure of $V$ and $V^{-1}$ together with the block diagonal structure of ${\mathbf{Q}}^V$ to compute $$\begin{aligned}
F_{ij}(t) & = \sum_{k=j}^{i}V_{ik}\exp(Q_{(n-k)\lambda}t)V_{kj} \\
& = \sum_{k=j}^{i}{n-k \choose n-i}(-1)^{i-k}\exp(Q_{(n-k)\lambda}t){n-j \choose n-k} \\
& = {n-j \choose n-i}\sum_{k=j}^{i}(-1)^{i-k}{i-j \choose i-k}\exp(Q_{(n-k)\lambda}t),\end{aligned}$$ as stated.
\[proposition:pink\] The solution $\pi(t)$ to the system of ODEs under the initial condition $\pi(0)=e_0\otimes x(0)$ has components $\pi^i(t)\in{\mathbb{R}}^d$ given by $$\label{eq:pisol}
\pi^i(t)={n \choose i}\sum_{k=0}^i(-1)^{i-k}{i\choose k}\exp(Q_{(n-k)\lambda}t)x(0).$$
We use Lemma \[lemma:diag\] and recall the specific form of the initial condition $\pi(0)$. We have to compute $\exp({\mathbf{Q}}t)\pi(0)$ and obtain from Lemma \[lemma:diag\] with $j=0$ for $\pi^i(t)=F_{i0}(t)$ $$\begin{aligned}
\pi^i(t) & ={n \choose n-i} \sum_{k=0}^i(-1)^{i-k} {i\choose i-k}\exp(Q_{(n-k)\lambda}t)x(0) \\
& ={n \choose i} \sum_{k=0}^i(-1)^{i-k} {i\choose k}\exp(Q_{(n-k)\lambda}t)x(0). \end{aligned}$$
\[remark:n=1\] Let us look at a special case, $n=1$. Then we can write $N_t=Y_t$ and it is sufficient to compute $$\label{eq:zt}
\pi^1(t)={\mathbb{E}}(Y_tX_t)=\left(\exp(Qt)-\exp(Q_\lambda t)\right)x(0).$$ As a consequence we are able to compute ${\mathbb{P}}(Y_t=1)={{\bf 1}}^\top{\mathbb{E}}(Y_tX_t)$, $${\mathbb{P}}(Y_t=1)=1-{{\bf 1}}^\top \exp(Q_\lambda t)x(0),$$ since ${{\bf 1}}^\top\exp(Q_t)={{\bf 1}}^\top$. As $\exp(Qt)\to \pi{{\bf 1}}^\top$, we conclude in view of Lemma \[lemma:qlambda\] from that $\pi^1(t)\to\pi$ for $t\to\infty$. This result should be obvious, as $Y_t$ eventually becomes 1 and $X_t$ converges in distribution to its invariant law.
For the case $n>1$ the expressions for $\pi^i(t)$ are a bit complicated, but their asymptotic values for $t\to\infty$, are as expected, $\pi^i(t)\to 0$ for $i<n$, whereas $\pi^n(t)\to \pi$. This again follows from Lemma \[lemma:qlambda\].
Proposition \[proposition:pink\] has the following corollary.
\[cor:multiphi\] Let $\phi(t,u)={\mathbb{E}}\exp({\mathrm{i}}u N_t)X_t$. It holds that $$\phi(t,u)=\sum_{k=0}^n{n\choose k}\exp({\mathrm{i}}uk)(1-\exp({\mathrm{i}}u))^{n-k}\exp(Q_{(n-k)\lambda}t)x(0).$$
We shall use the elementary identity $$\sum_{k=j}^n\beta^k{n\choose k}{k\choose j}={n\choose j}\beta^j(1+\beta)^{n-j}$$ for $\beta=-e^{-{\mathrm{i}}u}$ in the last step in the chain of equalities below. From Proposition \[proposition:pink\] we obtain $$\begin{aligned}
{\mathbb{E}}\exp({\mathrm{i}}u N_t)X_t & = \sum_{k=0}^ne^{{\mathrm{i}}u k}\pi^k(t) \\
& = \sum_{k=0}^ne^{{\mathrm{i}}u k}{n \choose k}\sum_{j=0}^k(-1)^{k-j}{k\choose j}\exp(Q_{(n-j)\lambda}t)x(0) \\
& = \sum_{j=0}^n\sum_{k=j}^n(-e^{{\mathrm{i}}u})^k{n \choose k}{k\choose j}(-1)^{j}\exp(Q_{(n-j)\lambda}t)x(0) \\
& = \sum_{j=0}^n {n\choose j}e^{{\mathrm{i}}ju}(1-e^{{\mathrm{i}}u})^{n-j}\exp(Q_{(n-j)\lambda}t)x(0).\end{aligned}$$
\[remark:bin\] Alternatively, one can compute a moment generating function $\psi(t,v)={\mathbb{E}}\exp(-vN_t)X_t$ for $v\geq 0$. Let $B$ have a binomial distribution with parameters $n$ and $p=1-\exp(-v)$. Then we have for $\psi(t,v)$ the compact expression $\psi(t,v)={\mathbb{E}}\exp((Q-B{\mathrm{diag}}(\lambda))t)x(0)={\mathbb{E}}\exp(Q_{\lambda B}t)x(0)$.
There appears to be no simpler representation for $\phi(t,u)$. We note that this function also satisfies the PDE $$\dot{\phi}(t,u)=(Q+n(e^{{\mathrm{i}}u}-1){\mathrm{diag}}(\lambda))\phi(t,u)+{\mathrm{i}}(e^{{\mathrm{i}}u}-1){\mathrm{diag}}(\lambda)\frac{\partial \phi(t,u)}{\partial u}.$$ Just by computing the partial derivatives, one verifies that this equation holds. Alternatively, one can apply the Itô formula to $\exp({\mathrm{i}}u N_t)X_t$ followed by taking expectations.
### Conditional probabilities {#section:cpf}
The vehicle we use is the process $\zeta$, recall $\zeta_t=\nu_t\otimes X_t$. Our aim is to find expressions for $\zeta_{t|s}={\mathbb{E}}[\zeta_t|{\mathcal{F}}_s]$ for $t>s$, from which one can deduce the conditional probabilities ${\mathbb{E}}[\nu_t|{\mathcal{F}}_s]$ and ${\mathbb{E}}[N_t|{\mathcal{F}}_s]$. By the Markov property, Proposition \[prop:zeta\], we have ${\mathbb{E}}[\zeta_t|{\mathcal{F}}_s]=\exp({\mathbf{Q}}(t-s))\zeta_s$. Let $\zeta_{t|s}={\mathbb{E}}[\zeta_t|{\mathcal{F}}_s]$ and $\zeta^k_{t|s}={\mathbb{E}}[{{\bf 1}}_{\{N_t=k\}}X_t|{\mathcal{F}}_s]$. We aim at a more explicit representation of the conditional probabilities $\zeta^k_{t|s}$ for $k\geq 0$. Note that $\zeta^k_{t|s}=(e_k^\top\otimes I)\zeta_{t|s}$. Hence $\zeta^k_{t|s}=(e_k^\top\otimes I)\exp(\mathbf{Q}(t-s))\zeta_s$. Using Lemma \[lemma:diag\], we have $$\zeta^k_{t|s}=(e_k^\top\otimes I)V
\begin{pmatrix}
\exp(Q_{n\lambda}(t-s)) & & \\
& \ddots & \\
& & \exp(Q(t-s))
\end{pmatrix}
V^{-1}\zeta_s.$$ By matrix computations as before this leads to the following result.
\[prop:zetak\] It holds that $$\zeta^k_{t|s}=\sum_{j=0}^k{n-j \choose k-j}\sum_{i=0}^k(-1)^{k-i}{k-j\choose k-i}\exp(Q_{(n-i)\lambda}(t-s))\zeta^j_s.$$
Note that in the formula of this proposition, only one of the $\zeta^j_s$ is different from zero and then equal to $X_s$. Effectively, the sum over $j$ thus reduces to one term only. The conditional probabilities $\nu^k_{t|s}={\mathbb{P}}(N_t=k|{\mathcal{F}}_s)$ can now simply be computed as ${{\bf 1}}^\top\zeta^k_{t|s}$. Note that these still depend on $X_s$, and one has the explicit expression $${\mathbb{E}}[\nu^k_t|{\mathcal{F}}_s]=
\sum_{j=0}^n{n-j \choose n-k} \sum_{i=j}^k(-1)^{k-i} {k-j\choose k-i}{{\bf 1}}^\top\exp(Q_{(n-i)\lambda}(t-s))X_s\nu^j_s.$$
Consider the special case $n=1$ and let $Z_t=Y_tX_t$, $Y_t$ as in Section \[section:mm1\]. This amounts to taking $k=n=1$ in Proposition \[prop:zetak\] and one gets for $Z_{t|s}={\mathbb{E}}[Z_t|{\mathcal{F}}^Y_s]$ the simpler expression $$Z_{t|s} =
\exp(Q_\lambda (t-s))Z_s+\big(\exp(Q(t-s))-\exp(Q_\lambda (t-s))\big)X_s. \label{eq:zts}$$
The next purpose is to compute ${\mathbb{E}}[e^{{\mathrm{i}}uN_t}X_t|{\mathcal{F}}_s]$ and from that one ${\mathbb{E}}[e^{{\mathrm{i}}uN_t}|{\mathcal{F}}_s]={{\bf 1}}^\top{\mathbb{E}}[e^{{\mathrm{i}}uN_t}X_t|{\mathcal{F}}_s]$.
The following hold. $$\begin{aligned}
{\mathbb{E}}[e^{{\mathrm{i}}uN_t}X_t|{\mathcal{F}}_s] &=\sum_{k=0}^n\sum_{j=k}^n {n-k \choose j-k}(1-e^{{\mathrm{i}}u})^{n-j}e^{{\mathrm{i}}uj}\exp(Q_{(n-j)\lambda} (t-s))\zeta^k_s, \nonumber\\
{\mathbb{E}}[e^{{\mathrm{i}}uN_t}|{\mathcal{F}}_s]&=\sum_{k=0}^n\sum_{j=k}^n {n-k \choose j-k}(1-e^{{\mathrm{i}}u})^{n-j}e^{{\mathrm{i}}uj}{{\bf 1}}^\top\exp(Q_{(n-j)\lambda} (t-s))\zeta^k_s.\label{eq:cfmmn}\end{aligned}$$
We start from the identity $e^{iuN_t}X_t=\mathbf{F}\zeta_t$, with $
\mathbf{F}=e(u)\otimes I,
$ where $e(u)=\begin{pmatrix} 1 & e^{{\mathrm{i}}u} & \cdots & e^{n{\mathrm{i}}u} \end{pmatrix}$. Hence we have $${\mathbb{E}}[e^{{\mathrm{i}}uN_t}X_t|{\mathcal{F}}_s]=(e(u)\otimes I)\exp(\mathbf{Q}(t-s))\zeta_s.$$ This can be put into the asserted more explicit representation, involving the matrices $Q_{k\lambda}$ by application of Proposition \[prop:zetak\]. The second assertion is a trivial consequence.
It is conceivable that only $N$ is observed, and not the background process $X$. In such a case one is only able to compute conditional expectation of quantities as above conditioned on ${\mathcal{F}}^N_s$ instead of ${\mathcal{F}}_s$. See Section \[section:filter2\] for results.
The Markov Modulated Poisson process {#section:mmpoisson}
====================================
In this section we study MM Poisson processes. These have an intensity process $\lambda_t=\lambda^\top X_t$, using the same notation as before. In terms of defaultable obligors, such processes occur as limits of the total number of defaults $N_t$ as in Section \[section:mmmultiple\] where $n\to\infty$ and the vector $\lambda$ is scaled to become $\lambda/n$, as we shall see later. So we can use this to approximate the total number of defaults in a market with a large number of obligors, where each of them has small default rate.
The model
---------
The point of departure is to postulate the dynamics of the counting process $N$ as $${{\mathrm d} }N_t=\lambda^\top X_t\,{{\mathrm d} }t+ {{\mathrm d} }m_t.$$ We follow the same approach as before. So we use that conditionally on ${\mathcal{F}}^X$ we have that $N_t$ has a $\mathrm{Poisson}(\Lambda_t)$ distribution with $\Lambda_t=\int_0^t\lambda^\top X_s\,{{\mathrm d} }s$. It follows that $${\mathbb{E}}[{{\bf 1}}_{\{N_t=k\}}X_t|{\mathcal{F}}^X]=\frac{1}{k!}\Lambda_t^k\exp(- \Lambda_t)X_t=:p^{k}(t)X_t,$$ and $$\frac{{{\mathrm d} }}{{{\mathrm d} }t} p^{k}(t)=p^{k-1}(t)-p^{k}(t)\lambda^\top X_t.$$ Then we obtain $${{\mathrm d} }{\mathbb{E}}[{{\bf 1}}_{\{N_t=k\}}X_t|{\mathcal{F}}^X]=\big(p^{k-1}(t)-p^{k}(t)\big){\mathrm{diag}}(\lambda)X_t\,{{\mathrm d} }t +p^{k}(t)(QX_t\,{{\mathrm d} }t + {{\mathrm d} }M_t),$$ and with $\pi^{k}(t)={\mathbb{E}}(p^{k}(t)X_t)$ we find $$\begin{aligned}
\dot{\pi}^{k}(t) & ={\mathrm{diag}}(\lambda)\pi^{k-1}(t)+(Q-{\mathrm{diag}}(\lambda))\pi^{k}(t).\end{aligned}$$ For $k=0$, one immediately finds the solution $\pi^0(t)=\exp(Q_\lambda t)x(0)$. For $k>0$ there seems to be no simply expression in terms of exponential of $Q$ and $Q_{k\lambda}$ as in Proposition \[proposition:pink\], not even for $k=1$, although one has $$\pi^1(t)=\int_0^t\exp(-Q_\lambda(t-s)){\mathrm{diag}}(\lambda)\exp(Q_\lambda s)\,{{\mathrm d} }s\,x(0).$$ However, it is possible to get a formula for the vector $$\Pi^n(t) = \begin{pmatrix}
\pi^0(t) \\
\vdots \\
\pi^n(t)
\end{pmatrix},$$ since it satisfies the ODE $$\dot{\Pi}^n(t)={\mathbf{Q}}_n\Pi^n(t),$$ where ${\mathbf{Q}}_n\in{\mathbb{R}}^{(n+1)d\times (n+1)d}$ is given by $${\mathbf{Q}}_n=
\begin{pmatrix}
Q-{\mathrm{diag}}(\lambda) & 0 & \cdots & \cdots & 0 \\
{\mathrm{diag}}(\lambda) & Q-{\mathrm{diag}}(\lambda) & 0 & & 0 \\
0 & {\mathrm{diag}}(\lambda) & \ddots & \ddots & \vdots \\
\vdots & & \ddots & Q-{\mathrm{diag}}(\lambda) & 0 \\
0 & \cdots & 0 & {\mathrm{diag}}(\lambda) & Q-{\mathrm{diag}}(\lambda)
\end{pmatrix}.$$ Together with the initial conditions $\pi^k(0)=\delta_{k0}x(0)$, one obtains $$\Pi^n(t)=\exp({\mathbf{Q}}_nt)(e^n_0\otimes x(0)),$$ where $e^n_0$ is the first basis vector of ${\mathbb{R}}^{n+1}$. An elementary expression for $\exp({\mathbf{Q}}_nt)$ is not available due to the fact that $Q-{\mathrm{diag}}(\lambda)$ and ${\mathrm{diag}}(\lambda)$ do not commute. Besides, ${\mathbf{Q}}_n$ is block lower triangular with identical blocks on the main diagonal and therefore cannot be block diagonalized.\
However, in the present case there is a nice expression for the characteristic function $\phi(t,u)={\mathbb{E}}\exp({\mathrm{i}}uN_t)X_t$, unlike the situation of Corollary \[cor:multiphi\]. To determine $\phi(t,u)$, we apply the Itô formula (note that $[N,X]=0$) and obtain $$\label{eq:phinx}
{{\mathrm d} }\exp({\mathrm{i}}uN_t)X_t = (e^{{\mathrm{i}}u}-1)e^{{\mathrm{i}}u N_{t-}}X_{t-}{{\mathrm d} }N_t + e^{{\mathrm{i}}u N_{t-}}{{\mathrm d} }X_{t},$$ which yields after taking expectations and using the dynamics of $X$ and $N$ $$\dot{\phi}(t,u)=((e^{{\mathrm{i}}u}-1){\mathrm{diag}}(\lambda)+Q)\phi(t,u).$$ Hence $$\phi(t,u)=\exp\big(((e^{{\mathrm{i}}u}-1){\mathrm{diag}}(\lambda)+Q)t\big)x(0).$$ Contrary to the $\pi^k(t)$ of Proposition \[proposition:pink\] we thus found a *simple* formula for $\phi(t,u)$. This formula is in line with [@ASM Proposition 1.6] for Markovian arrival processes.
It is possible to obtain the above results as limits from results in Section \[section:mmmo\], by replacing there $\lambda$ by $\lambda/n$ and letting $n\to\infty$.
If we look at the moment generating functions $\psi(t,v)={\mathbb{E}}\exp(-vN_t)X_t$, we have $\psi(t,v)=\exp\big((Q-(1-e^{-v}){\mathrm{diag}}(\lambda))t\big)x(0)$. Replace in Remark \[remark:bin\] the parameter $\lambda$ with $\lambda/n$ and let $n\to\infty$ and write $B_n$ instead of $B$. Then we have $\psi_n(t,v)={\mathbb{E}}\exp\big((Q-{\mathrm{diag}}(\lambda)B_n/n)t\big)x(0)$. As $B_n/n\to 1-e^{-v}$ a.s., we obtain $\exp\big((Q-{\mathrm{diag}}(\lambda)B_n/n)t\big)\to\exp\big((Q-{\mathrm{diag}}(\lambda)(1-e^{-v}))t\big)$ a.s. Since the exponentials are bounded, we also have convergence of the expectations by dominated convergence. Replacing $-v$ with ${\mathrm{i}}u$ gives the characteristic function.
Conditional probabilities {#conditional-probabilities}
-------------------------
Mimicking the approach of Section \[section:mmmo\], we consider again the $\nu^k_t={{\bf 1}}_{\{N_t=k\}}$. Let $${\bar{\nu}}^n_t=\begin{pmatrix}
\nu^0_t \\
\vdots \\
\nu^n_t
\end{pmatrix}.$$ Then ${\bar{\nu}}^n$ still satisfies Equation . Combining this with the dynamics of $N$, we obtain the semimartingale decomposition $${{\mathrm d} }{\bar{\nu}}^n_t=\lambda^\top X_t(J-I){\bar{\nu}}^n_t\,{{\mathrm d} }t + {{\mathrm d} }M_t.$$ Letting ${\bar{\zeta}}^n_t={\bar{\nu}}^n_t\otimes X_t$, then we can derive, similar to the approach of Section \[section:mmmo\], $${{\mathrm d} }{\bar{\zeta}}^n_t= {\mathbf{Q}}_n{\bar{\zeta}}^n_t\,{{\mathrm d} }t + {{\mathrm d} }M_t.$$ This is for each $n$ a finite dimensional system, which can be extended to an infinite dimensional system for $\zeta_t$. The resulting infinite coefficient matrix will be lower triangular again, $${{\mathrm d} }\zeta_t={\mathbf{Q}}_\infty\zeta_t\,{{\mathrm d} }t +{{\mathrm d} }M_t,$$ where ${\mathbf{Q}}_\infty=I_\infty\otimes Q_\lambda-J_\infty\otimes {\mathrm{diag}}(\lambda)$ with $I_\infty$ the infinite dimensional identity matrix and $J_\infty$ the infinite dimensional counterpart of the earlier encountered matrix $J$. It follows that for the vector of conditional probabilities we have $${\mathbb{E}}[\zeta_t|{\mathcal{F}}_s]=\exp({\mathbf{Q}}_\infty(t-s)){\bar{\zeta}}_s.$$ This looks like an infinite dimensional expression, but ${\mathbb{E}}[{{\bf 1}}_{\{N_t=n\}}X_t|{\mathcal{F}}_s]$ can be computed from ${\mathbb{E}}[\bar\zeta^n_t|{\mathcal{F}}_s]=\exp({\mathbf{Q}}_n(t-s))\bar\zeta^n_s$, which effectively reduces the infinite dimensional system to a finite dimensional one. One can now also compute, with $\ell_n^\top=\begin{pmatrix} 0 & \cdots & 0 & 1 \end{pmatrix}\in{\mathbb{R}}^{1\times (n+1)}$, $${\mathbb{P}}(N_t=n,X_t=e_j|{\mathcal{F}}_s)=(\ell_n^\top\otimes e_j^\top)\exp({\mathbf{Q}}_n(t-s)){\bar{\zeta}}^n_s.$$
Conditional characteristic function
-----------------------------------
Our aim is to find an expression for $\phi_{t|s}:={\mathbb{E}}[\exp({\mathrm{i}}uN_t)X_t|{\mathcal{F}}_s]$. Since we deal in the present section with the MM Poisson process $N$, the bivariate process $(X,N)$, unlike its counterpart in Section \[section:mm\], is an instance of a Markov additive process [@ASM], and ${\mathbb{E}}[\exp({\mathrm{i}}u(N_t-N_s))X_t|{\mathcal{F}}_s]$ will only depend on $X_s$. We first follow the forward approach.
It holds that $$\label{eq:phits}
\phi_{t|s}=\exp\left(((e^{{\mathrm{i}}u}-1){\mathrm{diag}}(\lambda) + Q)(t-s)\right)e^{{\mathrm{i}}u N_{s}}X_s.$$
Starting point is Equation . We use the dynamics of $N$ and $X$ to get the semimartingale decomposition $$\begin{aligned}
{{\mathrm d} }\exp({\mathrm{i}}uN_t)X_t & = (e^{{\mathrm{i}}u}-1)e^{{\mathrm{i}}u N_{t}}{\mathrm{diag}}(\lambda)X_t\,{{\mathrm d} }t + e^{{\mathrm{i}}u N_{t}}QX_t\,{{\mathrm d} }t +{{\mathrm d} }M_{t}\\
& = ((e^{{\mathrm{i}}u}-1){\mathrm{diag}}(\lambda) + Q)e^{{\mathrm{i}}u N_{t}}X_t\,{{\mathrm d} }t +{{\mathrm d} }M_{t}.\end{aligned}$$ Let $t\geq s$. We obtain (differentials w.r.t. $t$) $${{\mathrm d} }\phi_{t|s}=((e^{{\mathrm{i}}u}-1){\mathrm{diag}}(\lambda) + Q)\phi_{t|s}\,{{\mathrm d} }t,$$ which has the desired solution.
Next we outline the backward approach. Observe first that $\phi_{t|s}$ is a martingale in the $s$-parameter and that due to the fact that $(N,X)$ is Markov, we can write for some function $\Phi$, $\phi_{t|s}=\Phi(t-s,N_s)X_s$. We identify $\Phi$ as follows, using the Itô formula w.r.t. $s$. We obtain $$\begin{aligned}
{{\mathrm d} }\phi_{t|s} & = \left(-\dot{\Phi}(t-s,N_s)\,{{\mathrm d} }s+ (\Phi(t-s,N_{s-}+1)-\Phi(t-s,N_{s-})){{\mathrm d} }N_s\right)X_{s-} \\
& \quad\mbox{}\quad +\Phi(t-s,N_{s-})\,{{\mathrm d} }X_s \\
& = \left(-\dot{\Phi}(t-s,N_s)+ (\Phi(t-s,N_{s}+1)-\Phi(t-s,N_{s})){\mathrm{diag}}(\lambda)\right)X_{s}\,{{\mathrm d} }s \\
& \quad\mbox{}\quad +\Phi(t-s,N_{s})Q X_s\,{{\mathrm d} }s+{{\mathrm d} }M_s.\end{aligned}$$ The above mentioned martingale property leads to the system of ODEs ($n\geq 0$) $$\label{eq:phin}
\dot{\Phi}(t,n)=\Phi(t,n+1){\mathrm{diag}}(\lambda)+\Phi(t,n)\left(Q-{\mathrm{diag}}(\lambda)\right).$$ We have the initial conditions $\Phi(0,n)=\exp({\mathrm{i}}u n)$. To know $\Phi(t,n)$ it seems necessary to know $\Phi(t,n+1)$, which suggest that the ODEs are difficult to solve constructively. Instead, we pose a solution, we will verify that $$\Phi(t,n)=\exp\left(((e^{{\mathrm{i}}u}-1){\mathrm{diag}}(\lambda) + Q)t\right)e^{{\mathrm{i}}u n}.$$ Differentiation of the given expression for $\Phi(t,n)$ gives $$\dot{\Phi}(t,n)= \Phi(t,n)((e^{{\mathrm{i}}u}-1){\mathrm{diag}}(\lambda) + Q).$$ Note that $\Phi(t,n+1)=\Phi(t,n)e^{{\mathrm{i}}u}$. Insertion of this into the ODE gives $$\dot{\Phi}(t,n)=\Phi(t,n)(e^{{\mathrm{i}}u}{\mathrm{diag}}(\lambda)+\left(Q-{\mathrm{diag}}(\lambda)\right)),$$ which coincides with .
Filtering {#section:filtering}
=========
Let $N$ be a counting process with predictable intensity process $\lambda$. In many cases it is conceivable that $\lambda$ is an unobserved process and expressions in terms of $\lambda$ are not always useful. Let $\hat\lambda_t={\mathbb{E}}[\lambda_t|{\mathcal{F}}^N_t]$. Then the semimartingale decomposition of $N$ w.r.t. the filtration $\mathbb{F}^N$ is given by $${{\mathrm d} }N_t=\hat\lambda_t\,{{\mathrm d} }t + {{\mathrm d} }\hat m_t,$$ where $\hat m$ is a (local) martingale w.r.t. $\mathbb{F}^N$. The general filter of the Markov chain $X$, $\hat X_t={\mathbb{E}}[X_t|{\mathcal{F}}^N_t]$ satisfies the following well known formula (see [@bremaud], originating from [@schuppen]) with $Q$ as in Section \[section:mm1\] $${{\mathrm d} }\hat{X}_t=Q\hat{X}_t\,{{\mathrm d} }t +\hat{\lambda}_{t-}^+(\widehat{X\lambda}_{t-}-\hat{X}_{t-}\hat{\lambda}_{t-})({{\mathrm d} }N_t-\hat\lambda_t\,{{\mathrm d} }t),$$ where $\widehat{X\lambda}_{t} ={\mathbb{E}}[X_t\lambda_t|{\mathcal{F}}^N_t]$ and where we use the notation $x^+={{\bf 1}}_{x\neq 0}/x$ for a real number $x$. For any of the previously met models for the counting process $N$ we have a predictable intensity process of the form $\lambda_t=\lambda^\top X_{t-} f(N_{t-})$, where $f$ depends on the specific model at hand. It follows that $\hat\lambda_t=\lambda^\top\hat{X}_{t-}f(N_{t-})$. In all cases we consider it happens that $f(N_t)$ remains zero after it has reached zero, and hence $N$ stops jumping as soon as $f(N_t)=0$. Since $\lambda^\top X_t>0$, with the convention $\frac{0}{0}=0$ the above filter equation reduces to $$\label{eq:filter}
{{\mathrm d} }\hat{X}_t=Q\hat{X}_t\,{{\mathrm d} }t +\frac{1}{\lambda^\top\hat{X}_{t-}}({\mathrm{diag}}(\lambda)\hat{X}_{t-}-\hat{X}_{t-}\lambda^\top\hat{X}_{t-})({{\mathrm d} }N_t-\hat\lambda_t\,{{\mathrm d} }t).$$ For the specific models we have encountered we give in the next sections more results on $\hat{X}$.
Filtering for the MM multiple point process {#section:filter2}
-------------------------------------------
The notation of this section is as in Section \[section:mmmo\] and subsequent sections. Let $\hat{\zeta}_t={\mathbb{E}}[\zeta_t|{\mathcal{F}}^N_t]$. Then $\hat{\zeta}_t=\nu_t\otimes \hat{X}_t$, where $\hat{X}_t={\mathbb{E}}[X_t|{\mathcal{F}}^N_t]$. For $\hat{X}_t$ we have from , $${{\mathrm d} }\hat{X}_t=Q\hat{X}_t\,{{\mathrm d} }t+\frac{1}{\lambda^\top \hat{X}_{t-}}\left({\mathrm{diag}}(\lambda)\hat{X}_{t-}- \hat{X}_{t-}\hat{X}_{t-}^\top\lambda\right)\,({{\mathrm d} }N_t-(n-N_t)\lambda^\top \hat{X}_{t}\,{{\mathrm d} }t).$$ At the jump times $\tau_k$ ($k=1,\ldots,n$) (these are the order statistics of the original default times $\tau^i$) of $N$ we thus have $$X_{\tau_k}=\frac{1}{\lambda^\top \hat{X}_{\tau_k-}}{\mathrm{diag}}(\lambda)\hat{X}_{\tau_k-}$$ Between the jump times, $\hat{X}$ evolves according to the ODE $$\frac{{{\mathrm d} }\hat{X}_t}{{{\mathrm d} }t}=Q\hat{X}_t-(n-N_t)({\mathrm{diag}}(\lambda)\hat{X}_{t-}- \hat{X}_{t-}\hat{X}_{t-}^\top\lambda),$$ which is also valid after the last jump of $N$. It follows that for $t\geq \tau_n$ we have $\hat{X}_t=\exp(Q(t-\tau_n))\hat{X}_{\tau_n}$.\
Below we need $[\nu,\hat{X}]^\otimes_t=\sum_{s\leq t}\Delta \nu_s\otimes \Delta\hat{X}_s$. Using the equations for $\nu$ and $\hat{X}$, we find $${{\mathrm d} }[\nu,\hat{X}]^\otimes_t=\frac{1}{\lambda^\top\hat{X}_{t-}}((J-I)\otimes ({\mathrm{diag}}(\lambda)-\lambda^\top\hat{X}_{t-}I))\hat{\zeta}_{t-}{{\mathrm d} }N_t.$$ For $\hat{\zeta}_t$ we have, using the product formula for tensors, $${{\mathrm d} }\hat{\zeta}_t = {{\mathrm d} }\nu_t\otimes \hat{X}_{t-}+\nu_{t-}\otimes {{\mathrm d} }\hat{X}_t+{{\mathrm d} }[\nu,\hat{X}]^\otimes_t.$$ This yields after some tedious computations the following semimartingale decomposition for $\hat\zeta$ $$\begin{aligned}
{{\mathrm d} }\hat{\zeta}_t & =\big(I\otimes Q+(n-N_t)(J-I)\otimes{\mathrm{diag}}(\lambda)\big)\hat{\zeta}_{t}\,{{\mathrm d} }t \\
& \quad \mbox{ } +\frac{1}{\lambda^\top\hat{X}_{t-}}\big(J\otimes{\mathrm{diag}}(\lambda)-\lambda^\top\hat{X}_{t-}I\otimes I\big)\hat{\zeta}_{t-}\,{{\mathrm d} }\hat{m}_t \\
& = {\mathbf{Q}}\hat\zeta_t\,{{\mathrm d} }t + \frac{1}{\lambda^\top\hat{X}_{t-}}\big(J\otimes{\mathrm{diag}}(\lambda)-\lambda^\top\hat{X}_{t-}I\otimes I\big)\hat{\zeta}_{t-}\,{{\mathrm d} }\hat{m}_t,\end{aligned}$$ where ${{\mathrm d} }\hat{m}_t={{\mathrm d} }N_t-(n-N_t)\lambda^\top \hat{X}_{t}\,{{\mathrm d} }t$ and ${\mathbf{Q}}$ as in Section \[section:mmmo\].\
Here are two applications. One can now compute $${\mathbb{P}}(N_t=k|{\mathcal{F}}^N_s)={{\bf 1}}^\top{\mathbb{E}}[\zeta^k_{t|s}|{\mathcal{F}}^N_s]={{\bf 1}}^\top\hat\zeta^k_{t|s},$$ for which we can use $\hat\zeta_{t|s}=\exp({\mathbf{Q}}(t-s))\hat\zeta_s$. Formula yields for the conditional characteristic function of $N_t$ given its own past until time $s<t$ the explicit expression $${\mathbb{E}}[e^{{\mathrm{i}}uN_t}|{\mathcal{F}}^N_s]=\sum_{k=0}^n\sum_{j=k}^n {n-k \choose j-k}(1-e^{{\mathrm{i}}u})^{n-j}e^{{\mathrm{i}}uj}{{\bf 1}}^\top\exp(Q_{(n-j)\lambda} (t-s))\hat{X}_s\nu^k_s.$$ In case $n=1$ the above formulas simplify considerably. Here are a few examples, where we use the notation of Section \[section:mm1\]. Suppose that only $Y$ is observed. Let ${\mathcal{F}}^Y_t=\sigma(Y_s,0\leq s\leq t)$. With $Z_t:=Y_tX_t$ we want to compute $\hat{Z}_{t|s} :={\mathbb{E}}[Z_t|{\mathcal{F}}^Y_s]$ for $t\geq s$. Let $\hat{X}_t={\mathbb{E}}[X_t|{\mathcal{F}}^Y_t]$, then obviously, $\hat{Z}_{t|s}=\hat{X}_{t|s}Y_s$. Moreover, one has from $$\begin{aligned}
\hat{Z}_{t|s}
& = \exp(Q(t-s))\hat{X}_s-\exp(Q_\lambda (t-s))\hat{X}_{s}(1-Y_s).\end{aligned}$$ As a consequence we have for $\hat{Y}_{t|s}={{\bf 1}}^\top \hat{Z}_{t|s}$ $$\hat{Y}_{t|s}= 1-{{\bf 1}}^\top \exp(Q_\lambda (t-s))\hat{X}_{s}(1-Y_s).$$
Filtering for the MM Poisson process
------------------------------------
The filter equations now take the familiar form $${{\mathrm d} }\hat{X}_t=Q\hat{X}_t\,{{\mathrm d} }t+\frac{1}{\lambda^\top \hat{X}_{t-}}\left({\mathrm{diag}}(\lambda)\hat{X}_{t-}- \hat{X}_{t-}\hat{X}_{t-}^\top\lambda\right)\,({{\mathrm d} }N_t-\lambda^\top \hat{X}_{t}\,{{\mathrm d} }t).$$ For $\bar\nu_t$ we have the infinite dimensional analogue of . This leads for $\hat\zeta_t=\bar\nu_t\otimes \hat{X}_t$ as in a Section \[section:filter2\] to $${{\mathrm d} }\hat{\zeta}_t = {\mathbf{Q}}_\infty\hat\zeta_t\,{{\mathrm d} }t + \frac{1}{\lambda^\top\hat{X}_{t-}}\big(J_\infty\otimes{\mathrm{diag}}(\lambda)-\lambda^\top\hat{X}_{t-}I_\infty\otimes I_\infty\big)\hat{\zeta}_{t-}\,({{\mathrm d} }N_t-\lambda^\top \hat{X}_{t}\,{{\mathrm d} }t).$$ Note that this system is infinite dimensional, but for each $n$ we also have for $\hat{\bar{\zeta}}^n_t={\mathbb{E}}[\hat\zeta^n_t|{\mathcal{F}}^N_t]$ the truncated finite dimensional system $${{\mathrm d} }\hat{\bar{\zeta}}^n_t = {\mathbf{Q}}_n\hat{\bar\zeta}^n_t\,{{\mathrm d} }t + \frac{1}{\lambda^\top\hat{X}_{t-}}\big(J\otimes{\mathrm{diag}}(\lambda)-\lambda^\top\hat{X}_{t-}I\otimes I\big)\hat{\bar{\zeta}}^n_{t-}\,({{\mathrm d} }N_t-\lambda^\top \hat{X}_{t}\,{{\mathrm d} }t).$$ For the conditional characteristic function ${\mathbb{E}}[\exp({\mathrm{i}}u N_t)X_t|{\mathcal{F}}^N_s]$ we have $${\mathbb{E}}[\exp({\mathrm{i}}u N_t)X_t|{\mathcal{F}}^N_s]=\exp\left(((e^{{\mathrm{i}}u}-1){\mathrm{diag}}(\lambda) + Q)(t-s)\right)e^{{\mathrm{i}}u N_{s}}\hat{X}_s,$$ whereas $\psi_t=e^{{\mathrm{i}}u N_{t}}\hat{X}_t$ satisfies the equation (${{\mathrm d} }\hat m_t={{\mathrm d} }N_t-\lambda^\top\hat X_t\,{{\mathrm d} }t$) $${{\mathrm d} }\psi_t=(\frac{e^{{\mathrm{i}}u}}{\lambda^\top\hat X_{t-}}{\mathrm{diag}}(\lambda)-I)\psi_{t-}{{\mathrm d} }\hat m_t + \left(Q+(e^{{\mathrm{i}}u}-1){\mathrm{diag}}(\lambda)\right)\psi_t\,{{\mathrm d} }t.$$
Rapid switching {#section:rapid}
===============
In this section we present some auxiliary results that we shall use in obtaining limits for the various default processes when the Markov chain evolves under a rapid switching regime, i.e. the transition matrix $Q$ will be replaced with $\alpha Q$, where $\alpha >0$ tends to infinity. In the first two results and their proofs we use the notation $C(M)$ for the matrix of cofactors of a square matrix $M$. Throughout this section we write $\lambda_\infty$ for $\lambda^\top\pi$.
\[lemma:C\] Let $Q$ have a unique invariant vector $\pi$. Then $$C(Q)=q\,\pi{{\bf 1}}^\top,$$ where the constant $q$ can be computed as $\det(\hat{Q})$, where $\hat{Q}$ is obtained from $Q$ by replacing its last row with ${{\bf 1}}^\top$.
Note first that $\pi$ can be obtained as the solution to $\hat{Q}\pi=e_d$, where $e_d$ is the last basis vector of ${\mathbb{R}}^d$. By Cramer’s rule $\pi$ can be expressed using the cofactors of $\hat{Q}$. In particular, $\pi_d=\hat{C}_{dd}/\det(\hat{Q})$, where $\hat{C}$ is the cofactor matrix of $\hat{Q}$. But $\hat{C}_{dd}=C_{dd}$, so $\pi_d=C_{dd}/\det(\hat{Q})$.
Write $C=C(Q)$ and recall that $CQ=\det(Q)$ and hence zero. It follows that every row of $C$ is a left eigenvector of $Q$. Since $Q$ has rank $d-1$ by its assumed irreducibility, every row of $C$ is a multiple of ${{\bf 1}}^\top$. Hence $C=\alpha{{\bf 1}}^\top$, for some $\alpha\in{\mathbb{R}}^{d\times 1}$. By similar reasoning, $C=\pi\beta$ for some $\beta\in{\mathbb{R}}^{1\times d}$. We conclude that $C=q\pi{{\bf 1}}^\top$ for some real constant $q$. Use now $C_{dd}=q\pi_d$ and the above expression for $\pi_d$ to arrive at $q=\det(\hat{Q})$.
\[prop:inv\] Let $Q$ have a unique invariant vector $\pi$ and let all $\lambda_i$ be positive. Then $(\alpha Q-{\mathrm{diag}}(\lambda))^{-1}\to -\frac{\pi{{\bf 1}}^\top}{\lambda_\infty}$ for $\alpha\to\infty$.
We have seen in Section \[section:mm1\] that $Q-{\mathrm{diag}}(\lambda)$ is invertible if all $\lambda_i>0$ and so the same is true for $\alpha Q-{\mathrm{diag}}(\lambda)$. Both $\det(\alpha Q-{\mathrm{diag}}(\lambda))$ and the cofactor matrix of $\alpha Q-{\mathrm{diag}}(\lambda)$ are polynomials in $\alpha$ and we compute the leading term. The determinant is computed by summing products of elements of $\alpha Q-{\mathrm{diag}}(\lambda)$, from each row and each column one. The $\alpha^d$ term in this determinant has coefficient $\det(Q)$, which is zero. Consider the term with $\alpha^{d-1}$. It is seen to be equal to $-\sum_{i=1}^d\lambda_iC(\alpha Q-{\mathrm{diag}}(\lambda))_{ii}=-\alpha^{d-1}\sum_{i=1}^d\lambda_iC(Q-{\mathrm{diag}}(\lambda/\alpha))_{ii}$. For the cofactor matrix itself a similar procedure applies. We get $C(\alpha Q-{\mathrm{diag}}(\lambda))=\alpha^{d-1}C(Q-{\mathrm{diag}}(\lambda)/\alpha)$ and it results from Lemma \[lemma:C\] that for $\alpha\to\infty$ $$\frac{C(\alpha Q-{\mathrm{diag}}(\lambda))}{\det(\alpha Q-{\mathrm{diag}}(\lambda))}\to\frac{C(Q)}{-\sum_{i=1}^d\lambda_iC(Q)_{ii}}=-\frac{q\pi{{\bf 1}}^\top}{q\sum_{i=1}^n\lambda_i\pi_i}=-\frac{\pi{{\bf 1}}^\top}{\lambda_\infty}.$$
\[proposition:exp\] For $\alpha\to\infty$ it holds that $$\exp\big((\alpha Q-{\mathrm{diag}}(\lambda))t\big)\to\exp(-\lambda_\infty t)\pi{{\bf 1}}^\top.$$
For any analytic function $f:{\mathbb{C}}\to{\mathbb{C}}$, $f(z)=\sum_{k=0}^\infty a_kz^k$, one defines $f(M):=\sum_{k=0}^\infty a_kM^k$ for $M\in{\mathbb{C}}^{d\times d}$ (assuming that the power series converges on the spectrum of $M$). It then holds (see also Higham [@higham Definition 1.11], where this is taken as a definition of $f(M)$) that $$f(M)=\frac{1}{2\pi{\mathrm{i}}}\oint_\Gamma (zI-M)^{-1}f(z)\,{{\mathrm d} }z,$$ where $\Gamma$ is a closed contour such that all eigenvalues of $M$ are inside it. Take $M=\alpha Q-{\mathrm{diag}}(\lambda)$. It follows from Proposition \[prop:inv\], note that also $\lambda_\infty$ lies inside $\Gamma$ as it is a convex combination of the $\lambda_i$, that $(zI-\alpha Q+{\mathrm{diag}}(\lambda))^{-1}\to\frac{1}{z+\lambda_\infty}\pi{{\bf 1}}^\top$. Hence $$f(\alpha Q-{\mathrm{diag}}(\lambda))\to\pi{{\bf 1}}^\top f(-\lambda_\infty).$$ Apply this to $f(z)=\exp(tz)$.
Rapid switching for the MM multiple point process
-------------------------------------------------
Suppose we scale the $Q$ matrix with $\alpha\geq 0$, and we let $X^\alpha$ be Markov with transition matrix $\alpha Q$. Many (random) variables below will be indexed by $\alpha$ as well. Here is a way to get accelerated dynamics for $N^\alpha_t$ (previously denoted $N_t$).
Suppose that one takes the original Markov chain $X$ and replaces the dynamics of $N$ with one in which $X$ is accelerated, $$\label{eq:yalpha}
N^\alpha_t=\int_0^t(n-N^\alpha_s)\lambda^\top X_{\alpha s}\,{{\mathrm d} }s +m_t.$$ Indeed the process $X^\alpha$ defined by $X^\alpha_t=X_{\alpha t}$ has intensity matrix $\alpha Q$, and its invariant measure is $\pi$ again. Recall that, conditionally on ${\mathcal{F}}^X$, $N^\alpha_t$ has a Bin$(n,1-\exp(-\int_0^t\lambda^\top X_{\alpha s}\,{{\mathrm d} }s))$ distribution and that its unconditional distribution is Bin$(n,1-{\mathbb{E}}\exp(-\int_0^t\lambda^\top X_{\alpha s}\,{{\mathrm d} }s))$.
The ergodic property of $X$ gives $\int_0^tX_{\alpha s}\,{{\mathrm d} }s = \frac{1}{\alpha}\int_0^{\alpha t}X_s\,{{\mathrm d} }s\to \pi t$ a.s. and hence by dominated convergence for the expectations we have that the limit distribution of $N^\alpha_t$ for $\alpha\to\infty$ is Bin$(n,1-\exp(-\lambda_\infty t))$. One immediately sees that the default times $\tau^{\alpha,k}$ convergence in distribution to $\tau^k$ that are independent and have an exponential distribution with parameter $\lambda_\infty$. Keeping this in mind, the other results in this section are easily understandable.\
We recall the content of Proposition \[proposition:exp\]. Replacing $\lambda$ with $k\lambda$ for $k\geq 0$ (zero included) yields $$\label{eq:qn}
\exp\big((\alpha Q-k{\mathrm{diag}}(\lambda))t\big)\to\exp(-k\lambda_\infty t)\pi{{\bf 1}}^\top.$$ To express the dependence of the matrix ${\mathbf{Q}}$ given by on $\alpha $ in the present section, we write ${\mathbf{Q}}^\alpha $ (so ${\mathbf{Q}}^\alpha=A\otimes {\mathrm{diag}}(\lambda) + I\otimes \alpha Q$) and $F^\alpha (t)$ instead of $F(t)$ as given in Lemma \[lemma:diag\].
\[lemma:f\] The solution $F^\alpha $ to the equation $\dot{F}={\mathbf{Q}}^\alpha F$, has for $\alpha \to\infty$ limit $F^\infty$ given by its blocks $$F^\infty_{ij}(t)=f^\infty_{ij}(t)\pi{{\bf 1}}^\top,$$ where the $f^\infty_{ij}(t)$ are the binomial probabilities on $n-i$ ‘successes’ of a Bin$(n-j,\exp(-\lambda_\infty t))$ distribution, $$f^\infty_{ij}(t)={n-j \choose n-i} \exp(-(n-i)\lambda_\infty t)(1-\exp(-\lambda_\infty t))^{i-j}.$$
We depart from Lemma \[lemma:diag\] and the expression for $F^\alpha_{ij}(t)$ given there when we replace $Q$ with $\alpha Q$. Taking limits for $\alpha \to\infty$ yields $$\begin{aligned}
F^\infty_{ij}(t) & = {n-j \choose n-i} \sum_{k=j}^i(-1)^{i-k} {i-j\choose i-k}\exp(-(n-k)\lambda_\infty t)\pi{{\bf 1}}^\top \\
& = {n-j \choose n-i} (-1)^{i-j}\exp(-(n-j)\lambda_\infty t)\sum_{l=0}^{i-j} {i-j\choose l}(-\exp(\lambda_\infty t))^l\pi{{\bf 1}}^\top \\
& = {n-j \choose n-i} \exp(-(n-i)\lambda_\infty t)(1-\exp(-\lambda_\infty t))^{i-j}\pi{{\bf 1}}^\top,\end{aligned}$$ from which the assertion follows.
One can also use this proposition to show that $N^\alpha_t$ in the limit has the Bin$(n,1-\exp(-\lambda_\infty t))$ distribution. Indeed, since $\nu^i_0=\delta_{i0}$, we get ${\mathbb{P}}(N^\alpha_t=i,X_t=e_j)\to F^\infty_{i0}(t)=f^\infty_{i0}(t)\pi$ and hence ${\mathbb{P}}(N^\alpha_t=i)\to f^\infty_{i0}(t)$.
For conditional probabilities one has the following result.
Let $N$ be a process like in Equation , with $\lambda$ replaced with $\lambda_\infty$. For $\alpha \to\infty$ one has in the limit $\zeta^i_{t|s}=0$ for $i<N_s$ and for $i\geq N_s$ $$\zeta^i_{t|s}={n-N_s \choose n-i} \exp(-(n-i)\lambda_\infty\,(t-s))(1-\exp(-\lambda_\infty\,(t-s)))^{i-N_s} \pi.$$ It follows that, conditional on ${\mathcal{F}}_s$, $N_t-N_s$ has a Bin$(n-N_s,1-\exp(-\lambda_\infty\,(t-s)))$ distribution. In fact, one has weak convergence of the $N^\alpha$ to $N$.
We compute in the limit $\zeta^i_{t|s}={\mathbb{E}}[\nu^i_tX_t|{\mathcal{F}}_s]$ and obtain from Lemma \[lemma:f\] $$\begin{aligned}
\zeta^i_{t|s} & = \sum_{j=0}^nF^\infty_{ij}(t-s)\zeta^j_s\\
& = \sum_{j=0}^nf^\infty_{ij}(t-s)\nu^j_s\pi\\
& = \sum_{j=0}^i {n-j \choose n-i} \exp(-(n-i)\lambda_\infty\,(t-s))(1-\exp(-\lambda_\infty\,(t-s)))^{i-j} \nu^j_s\pi\\
& = {n-N_s \choose n-i} \exp(-(n-i)\lambda_\infty\,(t-s))(1-\exp(-\lambda_\infty\,(t-s)))^{i-N_s} \pi,\end{aligned}$$ from which the first assertion follows.
Weak convergence can be proved in many ways. Let us first look at the case of one obligor, $n=1$. The integral in Equation is, with $\tau^{\alpha}=\tau^{1,\alpha}$ equal to $$\frac{1}{\alpha}\int_0^{\alpha(\tau^\alpha\wedge t)}\lambda^\top X_u\,{{\mathrm d} }u.$$ Replacing the upper limit of the integral by $t$, this almost surely converges to $\lambda_\infty t$ for $\alpha\to\infty$. In fact this convergence is a.s. uniform. Having already established the convergence in distribution of the $\tau^\alpha$, and by switching to an auxiliary space on which the $\tau^\alpha$ a.s. converge to $\tau^\infty$, we get $$\frac{1}{\alpha}\int_0^{\alpha(\tau^\alpha\wedge t)}\lambda^\top X_u\,{{\mathrm d} }u\to \int_0^{\alpha(\tau^\infty\wedge t)}\lambda^\top X_u\,{{\mathrm d} }u.$$ This is sufficient, see [@kls] or [@js Section VIII.3d] to conclude the weak convergence result for the case $n=1$.
For the general case, one first notices that the process $N^\alpha$ is a sum of MM one point processes that are conditionally independent given ${\mathcal{F}}^X$ and become independent in the limit. Combine this with the result for $n=1$. Alternatively, one could apply the results in [@js Section VII.3d] again, although the computations will now be more involved.
Rapid switching for the MM Poisson process
------------------------------------------
As before we replace $Q$ with $\alpha Q$ and let $\alpha\to\infty$ and denote $N^\alpha$ the corresponding counting process. We apply Proposition \[proposition:exp\] to the matrix exponential $\exp\left(((e^{{\mathrm{i}}u}-1){\mathrm{diag}}(\lambda) + \alpha Q)(t-s)\right)$, and we find that the limit for $\alpha\to\infty$ equals $\exp((e^{{\mathrm{i}}u}-1)\lambda_\infty(t-s))\pi{{\bf 1}}^\top$. Hence, by virtue of , we obtain ${\mathbb{E}}[\exp({\mathrm{i}}uN^\alpha_t)X_t|{\mathcal{F}}_s]$ $\to \exp((e^{{\mathrm{i}}u}-1)\lambda_\infty(t-s))\pi$ for the limit of the conditional characteristic function. This is just one of the many ways that eventually lead to the conclusion that for $\alpha\to\infty$ the process $N^\alpha$ converge weakly to an ordinary Poisson process with constant intensity $\lambda_\infty$. In [@kls] one can find the stronger result that the variational distance between the MM law of $N^\alpha_t, t\in [0,T]$ and the limit law is of order $\alpha^{-1}$.
[99]{}
A. Ang and G. Bekaert (2002). Regime switches in interest rates. [*Journal of Business and Economic Statistics*]{} [**20**]{}, 163–182.
S. Asmussen (2003). *Applied Probability and Queues*, 2nd edition. Springer, New York.
S. Asmussen and H. Albrecher (2010). *Ruin probabilities*, 2nd edition. World Scientific, New Jersey.
T.R. Bielecki and M. Rutkowski (2002). *Credit risk: modelling, valuation and hedging*. Springer-Verlag, Berlin.
J. Blom, K. de Turck, and M. Mandjes (2015). Analysis of Markov-modulated infinite-server queues in the central-limit regime. *Probability in the Engineering and Informational Sciences* [**29**]{}, 433–459.
J. Blom, O. Kella, M. Mandjes, and H. Thorsdottir (2014). Markov-modulated infinite-server queues with general service times. *Queueing Systems* [**76**]{}, 403–424.
P. Brémaud (1980). *Point processes and Queues*. Springer-Verlag, New York-Berlin.
A. Chen and L. Delong (2015). Optimal Investment for a Defined-Contribution Pension Scheme under a Regime Switching Model, *ASTIN Bulletin*, March 2015, 1–23.
D. Duffie, D. Filipovi[ć]{}, and W. Schachermayer (2003). Affine processes and applications in finance. *Ann. Appl. Probab.* [**13**]{}, 984–1053.
R.J. Elliott, L. Aggoun and J.B. Moore (1995). *Hidden Markov Models*. Springer-Verlag, New York.
R. Elliott and R. Mamon (2002). An interest rate model with a Markovian mean-reverting level. [*Quantitative Finance*]{} [**2**]{}, 454–458.
R. Elliott and T.K. Siu (2009). On Markov-modulated exponential-affine bond price formulae. [*Applied Mathematical Finance*]{} [**16**]{}, 1–15.
D. Filipovi[ć]{} (2009). *Term-structure models*. Springer-Verlag, Berlin.
A. L. Goel and K. Okumoto (1978). Bayesian software prediction models, vol. 1: An imperfect debugging model for reliability and other quantitative measures of software systems. Rome Air Develop. Center, NY, *RADC-TR-78-155*.
G. Di Graziano (2014). [ Optimal Trading Stops and Algorithmic Trading]{}. SSRN [http://ssrn.com/abstract=2381830]{}.
J. Hamilton (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. *Econometrica* [**57**]{}, 357–384.
N.J. Higham (2008). *Functions of Matrices: Theory and Computation*. SIAM, Philadelphia.
G. Huang, M. Mandjes and P. Spreij (2014). Weak convergence of [M]{}arkov-modulated diffusion processes with rapid switching. *Statist. Probab. Lett.* [**86**]{}, [74–79]{}.
Gang Huang, Marijn Jansen, Michel Mandjes, Peter Spreij, Koen De Turck (2015). [ Markov-modulated Ornstein-Uhlenbeck processes]{}. *Advances in Applied Probability* [**48(1)**]{}, 235-254.
J. Jacod and A.N. Shiryaev (2003). *Limit theorems for stochastic processes, 2nd edition*. Grundlehren der Mathematischen Wissenschaften 288, Springer-Verlag, Berlin.
Z. Jelinski and P. B. Moranda (1972). Software reliability research. In *Statistical Computer Performance Evaluation*, W. Freiberger, Ed. NewYork: Academic, 465– 484.
Z. Jiang, M.R. Pistorius (2008). On perpetual American put valuation and first passage in a regime- switching model with jumps. *Finance and Stochastics* [**12**]{}, 331–355.
Z. Jiang, M. Pistorius (2012). Optimal dividend distribution under Markov regime switching. *Finance and Stochastics* [**16**]{}, 449–476.
Y.M. Kabanov, R.S. Liptser, and A.N. Shiryaev (1983). Weak and strong convergence of distributions of counting processes. *Teor. Veroyatnost. i Primenen.*, [**28**]{}, 288–319.
G. Koch, P.J.C. Spreij (1983). Software Reliability as an Application of Martingale & Filtering Theory. *IEEE Transactions on Reliability* [**R-32**]{}, 342–345.
B. Littlewood (1980). Theories of Software Reliability: How Good Are They and How Can They Be Improved? *IEEE Transactions on Software Engineering* [**SE-6**]{}, 489–500.
M. Neuts (1981). *Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach.* Courier Dover Publications, New York.
J.H. van Schuppen (1977). Filtering, prediction and smoothing for counting process observations, a martingale approach. *SIAM J. Appl. Math.* [**32**]{}, 552–570.
G. Yin and C. Zhu (2010). [ Hybrid Switching Diffusions: Properties and Applications]{}. Springer, New York.
G. Yin (2009). Asymptotic expansions of option price under regime-switching diffusions with a fast-varying switching process, *Asymptotic Anal.* [**65**]{}, 203–222.
X.Y. Zhou and G. Yin (2003). Markowitz mean-variance portfolio selection with regime switching: A continuous-time model, *SIAM J. Control Optim.* [**42**]{}, 1466–1482.
[^1]: This argument has been provided by Koen de Turck, University of Ghent.
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abstract: 'Starting from the NRQCD Lagrangian the heavy quark-antiquark potential is written in terms of field strength insertions on a static Wilson loop. The relevant matching coefficients are given at the present status of knowledge. The short-range, perturbatively dominated, behaviour of the spin-dependent terms is discussed.'
address:
- 'Institut für Theoretische Physik, Univ. Wien, Boltzmanngasse 5, A-1090 Vienna, Austria'
- 'Institut für Hochenergiephysik, Öster. Akad. der Wiss., Nikolsdorfergasse 18, A-1050 Vienna, Austria'
author:
- 'Nora Brambilla [^1] and Antonio Vairo'
title: 'Some aspects of the quark-antiquark Wilson loop formalism in the NRQCD framework'
---
NRQCD AND THE WILSON LOOP FORMALISM
===================================
Heavy quark bound states provide an extremely difficult but at the same time appealing system to test QCD. The difficulties are obvious. One is the mixing of different energy scales. This is a typical feature of any bound state problem in quantum field theory and makes tricky even a purely perturbative solution of it. An other conceptual difficulty is connected with the nonperturbative nature of low-energy QCD. This suggests that nonpertubative contributions have to be taken into account in almost all QCD bound states. The reason why heavy mesons are appealing is that the existence of an expansion parameter (the inverse of the mass $m$ in the Lagrangian and the velocity $v$ of the quark as a dynamical defined power counting parameter) makes possible to handle the first difficulty and to keep under control the second one. The tool is provided by NRQCD [@lepage]. This is an effective theory equivalent to QCD and obtained from QCD by integrating out the hard energy scale $m$. The Lagrangian comes from the original QCD Lagrangian via a Foldy–Wouthuysen transformation. The ultraviolet regime of QCD (at energy scale $m$) is perturbatively encoded order by order in the coupling constant $\alpha_{\rm s}$ in the matching coefficients which appear in front of the new operators of the effective theory. This ensures the equivalence between the effective theory and the original one at a given order in $1/m$ and $\alpha_{\rm s}$. At order $1/m^2$ the NRQCD Lagrangian describing a bound state between a quark of mass $m_1$ and an antiquark of mass $m_2$ is [@manohar; @pineda] $$\begin{aligned}
& & \!\!\!\!\!\!\!\!\!\!\!\! L = Q_1^\dagger\!\left(\!iD_0 + c^{(1)}_2 {{\bf D}^2\over 2 m_1} +
c^{(1)}_4 {{\bf D}^4\over 8 m_1^3} + c^{(1)}_F g { {\bf \sigma}\cdot {\bf B} \over 2 m_1} \right.
\nonumber\\
& & \!\!\!\!\!\!\!\!\!\!\!\! + c^{(1)}_D g { {\bf D}\!\cdot\!{\bf E} - {\bf E}\!\cdot\!{\bf D} \over 8 m_1^2}
\left. \! + i c^{(1)}_S g {{\bf \sigma} \!\!\cdot \!\!({\bf D}\!\times\!{\bf E} - {\bf E}\!\times\!{\bf D})
\over 8 m_1^2} \!\right)\!Q_1
\nonumber\\
& & \!\!\!\!\!\!\!\!\!\!\!\!
+ \hbox{\, antiquark terms}\, (1 \leftrightarrow 2)
\nonumber\\
& & \!\!\!\!\!\!\!\!\!\!\!\!+ {d_1\over m_1 m_2} Q_1^\dagger Q_2 Q_2^\dagger Q_1
+ {d_2\over m_1 m_2} Q_1^\dagger {\bf \sigma} Q_2 Q_2^\dagger {\bf \sigma} Q_1
\nonumber\\
& & \!\!\!\!\!\!\!\!\!\!\!\!+ {d_3\over m_1 m_2} Q_1^\dagger T^a Q_2 Q_2^\dagger T^a Q_1
\nonumber\\
& & \!\!\!\!\!\!\!\!\!\!\!\!
+ {d_4\over m_1 m_2} Q_1^\dagger T^a {\bf \sigma} Q_2
Q_2^\dagger T^a {\bf \sigma} Q_1.
\label{nrqcd}\end{aligned}$$ This is the relevant Lagrangian in order to calculate the bound state observables up to order $O(v^4)$. A discussion of the operators appearing in (\[nrqcd\]) in terms of powers of the quark velocity can be found in [@lepage; @vairo]. The coefficients $c^{(j)}_2$, $c^{(j)}_4$, ... are evaluated at a matching scale $\mu$ for a particle of mass $m_j$.
Nonperturbative contributions to the heavy meson observables can be evaluated directly from the Lagrangian (\[nrqcd\]) via lattice simulations [@lepage]. Typically, since the hard degrees of freedom have been integrated out explicitly, the needed lattice cut-off $\mu$ is expected to be larger (smaller in terms of energy) than the usual one with a clear reduction in the computation time. Despite the advantages, there are also some drawbacks in this method. In particular in this way we do not learn very much on our “analytic” knowledge on the QCD vacuum structure. Moreover computations on coarse lattices are not always under control. Therefore it is worthwhile to use the Lagrangian of NRQCD as a starting point and to work out the quark-antiquark interaction in the so-called Wilson loop formalism [@brown]. The advantage in doing so is that all the nonperturbative dynamics will be contained in gauge field averages of field strength insertions on a static Wilson loop. These can be very easily evaluated by means of some QCD vacuum model [@bv], or via traditional lattice simulations [@bali] providing in this way a powerful method in order to discriminate between different models. The derivation of the quark-antiquark potential in the Wilson loop formalism from the NRQCD Lagrangian was first suggested in this context in [@chen] and is discussed with details in [@vairo]. Here we present only some results. The heavy quark-antiquark potential (assumed that it exists) is given by $$\begin{aligned}
& & \!\!\!\!\!\!\!\!\!\!\!\!
V(r) = \lim_{T \to \infty} { i \log W \over T}
\nonumber\\
& & \!\!\!\!\!\!\!\!\!\!\!\!
+ \left( {{\bf S}^{(1)}\cdot{\bf L}^{(1)}\over m_1^2} +
{{\bf S}^{(2)}\cdot{\bf L}^{(2)}\over m_2^2} \right)\!
{2 c^+_F V_1^\prime(r) + c^+_S V_0^\prime(r) \over 2r}
\nonumber\\
& & \!\!\!\!\!\!\!\!\!\!\!\!
+ { {\bf S}^{(1)}\cdot{\bf L}^{(2)} +
{\bf S}^{(2)}\cdot{\bf L}^{(1)} \over m_1 m_2} {c^+_F V_2^\prime(r) \over r}
\nonumber\\
& & \!\!\!\!\!\!\!\!\!\!\!\!
+ \left( {{\bf S}^{(1)}\cdot{\bf L}^{(1)}\over m_1^2} -
{{\bf S}^{(2)}\cdot{\bf L}^{(2)}\over m_2^2} \right) \!
{2 c^-_F V_1^\prime(r) + c^-_S V_0^\prime(r) \over 2r}
\nonumber\\
& & \!\!\!\!\!\!\!\!\!\!\!\!
+ { {\bf S}^{(1)}\cdot{\bf L}^{(2)} -
{\bf S}^{(2)}\cdot{\bf L}^{(1)} \over m_1 m_2} {c^-_F V_2^\prime(r) \over r}
\nonumber\\
& & \!\!\!\!\!\!\!\!\!\!\!\!
+{1\over 8}\left( {c_D^{(1)} \over m_1^2}
+ {c_D^{(2)} \over m_2^2} \right) (\Delta V_0(r) + \Delta V_a^E(r))
\nonumber\\
& & \!\!\!\!\!\!\!\!\!\!\!\!
+{1\over 8}\left( {c_F^{(1)} \over m_1^2}
+ {c_F^{(2)} \over m_2^2} \right) \Delta V_a^B(r)
\nonumber\\
& & \!\!\!\!\!\!\!\!\!\!\!\!
+{c_F^{(1)}c_F^{(2)}\over m_1 m_2} \left(
{{\bf S}^{(1)}\cdot{\bf r} {\bf S}^{(2)}\cdot{\bf r} \over r^2} -
{{\bf S}^{(1)}\cdot {\bf S}^{(2)} \over 3} \right) \! V_3(r)
\nonumber\\
& & \!\!\!\!\!\!\!\!\!\!\!\!
+ {{\bf S}^{(1)}\cdot {\bf S}^{(2)} \over 3 m_1 m_2}
\left( c_F^{(1)} c_F^{(2)} V_4(r) -48 \pi \alpha_{\rm s} C_F \, d \, \delta^3(r)\right)
\label{pot}\end{aligned}$$ The “potentials" $V_1$, $V_2$, ... are scale dependent gauge field averages of electric and magnetic field strength insertions on the static Wilson loop and are explicitly given in [@vairo; @bali]. $W$ is the gauge average of the non-static Wilson loop. The expansion of it around the static Wilson loop gives the static potential $V_0$ plus velocity (non-spin) dependent terms. ${\bf S}^{(j)}$ and ${\bf L}^{(j)}$ are the spin and orbital angular momentum operators of the particle $j$. The matching coefficients are defined as $2 c^{\pm}_{F,S} \equiv c^{(1)}_{F,S} \pm c^{(2)}_{F,S}$ and $d$ is the relevant contribution to the mixing coming from the four quark operators in Eq. (\[nrqcd\]) and will be given in the next section. Apart from the matching Eq. (\[pot\]) is equivalent to the potential derived in [@brown]. In the next section we will give explicitly the matching coefficients and discuss briefly the relevance of the matching in order to have a short range consistent potential.
MATCHING COEFFICIENTS
=====================
Since for reparameterization invariance $c_S = 2 c_F -1$ [@manohar], all the spin dependent potentials given in Eq. (\[pot\]) turn out to depend only on $c_F$ (if the mass of the particle is irrelevant we will omit to indicate it). This coefficient is known up to two loop in the anomalous dimension [@amoros]: $$\begin{aligned}
c_F &=& \left( {\alpha_{\rm s}(m)\over \alpha_{\rm s}(\mu) } \right)^{\gamma_0/2\beta_0}
\left[ 1 + {\alpha_{\rm s}(m)\over 4\pi} c_1 \right.
\nonumber\\
&+& \left. {\alpha_{\rm s}(m) - \alpha_{\rm s}(\mu) \over 4\pi}
{\gamma_1\beta_0 - \gamma_0\beta_1 \over 2 \beta_0^2} \right]
\nonumber\end{aligned}$$ where $\beta_j$ are the usual $\beta$-function coefficients, $\gamma_0 = 2 C_A$, $\gamma_1 = 68 C_A^2/9 - 26 C_A N_f/9$, $c_1 = 2 (C_A + C_F)$, $N_f$ is the number of flavors, $C_F$ is the Casimir of the fundamental representation and $C_A$ is the Casimir of the adjoint representation. At the lattice scale used in [@bali] the numerical values of this coefficient at the bottom and charm mass are $c_F(m_b) \simeq 1.06 \times (1 + 0.15) = 1.22$ and $c_F(m_c) \simeq 1.27 \times (1 + 0.25) = 1.59$ respectively. The corrections due to the one loop matching are relevant (15 % in the bottom case and 25 % in the charm case) and therefore of the same order of the next power in the velocity in the Lagrangian (\[nrqcd\]) (usually accepted values are $\langle v^2_b \rangle \sim 0.07$ and $\langle v^2_c \rangle \sim 0.24$).
An evaluation of the coefficient $c_D$ associated with the Darwin term in the NRQCD Lagrangian is given in [@balzereit]: $$\begin{aligned}
c_D &=& \left({7\over4} - 8 {C_F\over C_A} \right) \left( {\alpha_{\rm s}(m)\over \alpha_{\rm s}(\mu) }
\right)^{2 C_A / 3\beta_0}
\nonumber\\
&-& {5 \over 4} \left( {\alpha_{\rm s}(m)\over \alpha_{\rm s}(\mu) }
\right)^{2 C_A / \beta_0} + {1\over 2} + 8 {C_F\over C_A}.
\nonumber\end{aligned}$$ This corrects a previous wrong evaluation given in [@chen]. At the lattice scale used in [@bali] the numerical values of this coefficient at the bottom and charm mass are $c_D(m_b) \simeq 0.76$ and $c_D(m_c) \simeq -0.08$ respectively. As pointed out in [@bali], since the potential $\Delta V_a^E$ manifests a $1/r$ behaviour, this term gives a flavor-dependent contribution to the central potential. However this contribution is suppressed in the bottom case by the bottom mass (see Eq. (\[pot\])) and in the charm case by the smallness of the corresponding matching coefficient.
Finally, the contributions coming from the four-fermion operators are usually suppressed either in $\alpha_{\rm s}$ or in powers of the quark velocity $v$ [@lepage; @pineda; @vairo]. Nevertheless under RG transformation the contribution to the spin-spin potential coming from the chromomagnetic operator in the NRQCD Lagrangian mixes with some of the local four quark operators. In order to take into account this mixing the delta contribution to the spin-spin potential has been added in Eq. (\[pot\]) though it would be suppressed in $\alpha_{\rm s}$. The coefficient $d$ has been evaluated in [@chen]: $$d = {1\over 8} \!\left( {\alpha_{\rm s}(m_1)\over \alpha_{\rm s}(m_2) }
\right)^{C_A / \beta_0} \!\! \left[1 - \left( {\alpha_{\rm s}(m_2)\over \alpha_{\rm s}(\mu) }
\right)^{2 C_A / \beta_0} \right].$$
As noticed in [@chen] the presence of the matching coefficients in the expression for the potential makes possible the agreement in the short range region between the potential derived here with the traditional QCD one loop perturbative calculation, e.g. in [@ng]. Let us focus on the spin-orbit terms $V_1$ and $V_2$. Comparing properly with [@ng] we get for $V_2$ the perturbative contribution $$\begin{aligned}
V_{2,pert}^\prime(r) &=& {C_F\alpha_{\rm s}(\mu)\over r^2}
\left\{ 1 + {\alpha_{\rm s} \over \pi} \left[ - {\beta_0 \over 12} - {2C_A\over 3}
\right.\right.
\nonumber\\
&+& \left.\left. {\beta_0 - C_A \over 2} (\log (\mu r) + \gamma_E -1) \right]\right\},
\nonumber\end{aligned}$$ where $\gamma_E$ is the Euler constant. This expression agrees very well with the lattice measurement of the same quantity shown in Fig. \[figv2lat\].
=7.5truecm -1 truecm
In the same way we get for $V_1$ the perturbative contribution $$V_{1,pert}^\prime(r) = - {\alpha_{\rm s}^2 \over \pi} {1\over r^2} {C_A C_F\over 2}
(\log (\mu r) + \gamma_E).$$ It is extremely interesting to compare the above expression with the short-range behaviour of the $V_1$ potential as given by the lattice measurement shown in Fig. \[figv1lat\]. Apart an overall shift proportional to the string tension and therefore of nonperturbative origin the agreement is very good. This is quite significant since the perturbative part of $V_1$ is entirely due to loop corrections. As a consequence $V_1$ is more sensitive than $V_2$ to the matching scale $\mu$. Notice that at very short distances the function $-V_1^\prime$, just because the $\log (\mu r)$ term, is expected to become negative, but up to now no lattice data are available in this region.
=7.5truecm -1 truecm
As a last comment we notice that due to the so-called Gromes relation $V_2^\prime - V_1^\prime = V_0^\prime$ [@gromes] a $V_5$ potential (in the notation of [@ng]) emerges also in Eq. (\[pot\]) by collecting the contributions coming from the fourth and fifth line. The perturbative expression we get agrees with that one given in [@ng]: $$V_{5,pert}(r) = {c_F^- V_2^\prime(r) \over r} = {\alpha_{\rm s}^2 \over \pi}
{1 \over r^3} {C_A C_F \over 4} \log {m_2 \over m_1} .$$
CONCLUSIONS
===========
In the framework of NRQCD and at the present status of the matching we have given the expression for the heavy quark potential in terms of field strength insertions on a static Wilson loop. This has the advantage that traditional lattice calculations can be used in order to evaluate nonperturbative contributions. Moreover in this way a comparison between different QCD vacuum models can be performed directly in terms of Wilson loop expectation values. This approach has been developed with some extent in [@vairo]. Here we have emphasized the role played by the matching coefficients in order to make consistent the short range behaviour of the potential that we obtain with the usual scattering matrix derived potential. We noticed that present lattice data are sensitive to one loop corrections and to the matching scale.
As a conclusion, let us mention two open problems. In order to have a 10 % accuracy in the quarkonium spin splitting it is necessary to add to the Lagrangian (\[nrqcd\]) higher order operators [@lepage]. The inclusion (if possible) of such operators in an expression like Eq. (\[pot\]) is still to do. Moreover in order to obtain Eq. (\[pot\]) we have implicitly assumed the existence of a potential. Non-potential terms surely exist in perturbative QCD. How to treat it in a system affected by nonperturbative physics is still unclear. Interesting developments could come from a promising approach recently proposed in [@pnrqcd].
[**Acknowledgments**]{} We thank Antonio Pineda for valuable discussions.
[999]{} G. P. Lepage, L. Magnea, C. Nakhleh, U. Magnea and K. Hornbostel, Phys. Rev. [**D 46**]{} (1992) 4052. A. V. Manohar, Phys. Rev. [**D 56**]{} (1997) 230. A. Pineda and J. Soto, hep-ph/9802365 (1998). N. Brambilla, J. Soto and A. Vairo in preparation; A. Vairo in Proceedings of “Quark Confinement and the Hadron Spectrum III”, World Scientific, Singapore, 1998. L. S. Brown and W. I. Weisberger, Phys. Rev. [**D 20**]{} (1979) 3239; E. Eichten and F. Feinberg, Phys. Rev. [**D 23**]{} (1981) 2724; M. E. Peskin in Proceeding of the 11th SLAC Institute, SLAC Report No. 207, ed. P. Mc Donough (1983) 151; N. Brambilla, P. Consoli, G. Prosperi, Phys. Rev. [**D 50**]{} (1994) 5878. N. Brambilla and A. Vairo, Phys. Rev. [**D 55**]{} (1997) 3974. G. S. Bali, K. Schilling and A. Wachter, Phys. Rev. [**D 56**]{} (1997) 2566. Y. Chen, Y. Kuang and R. J. Oakes, Phys. Rev. [**D 52**]{} (1995) 264. G. Amoros, M. Beneke and M. Neubert, Phys. Lett. [**B 401**]{} (1997) 81. C. Balzereit and T. Ohl, Phys. Lett. [**B 386**]{} (1996) 335. Y. J. Ng, J. Pantaleone and S. H. Tye, Phys. Rev. Lett. [**55**]{} (1985) 916. D. Gromes, Zeit. Phys. [**C 22**]{} (1984) 265. A. Pineda and J. Soto, Nucl. Phys. Proc. Suppl. [**B 64**]{} (1998) 428.
[^1]: Marie Curie fellow, TMR contract n. ERBFMBICT961714
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abstract: 'We extend Culler and Shalen’s construction of detecting essential surfaces in 3-manifolds to 3-orbifolds. We do so in the setting of the ${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ character variety, and following Boyer and Zhang in the ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ character variety as well. We show that any slope detected on a canonical component of the $\mathrm{(P)SL}_2({\ensuremath{\mathbb{C}}})$ character variety of a one cusped hyperbolic 3-manifold with symmetries must be the slope of a symmetric surface. As an application, we show that for each symmetric double twist knot there are slopes which are strongly detected on the character variety but not on the canonical component.'
author:
- 'J. Leach and K. L. Petersen'
bibliography:
- 'myrefs.bib'
title: Symmetries and Detection of Surfaces by the Character Variety
---
Introduction
============
Culler and Shalen [@MR683804] developed a beautiful theory showing how to associate surfaces in 3-manifolds to representations of their fundamental groups. To certain points and ideal points in the ${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ character variety of a 3-manifold there are valuations that value negatively. Such a valuation induces a non-trivial action on a Bass-Serre tree. One can associate essential surfaces dual to such an action. We say such a surface is [*detected*]{} by the action. This theory was extended to the ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ setting by Boyer and Zhang [@MR1670053]. We study how the existence of a symmetry of the manifold effects the detection of surfaces.
The ${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ character variety of a manifold $M$, $X(M)$, is the set of representations of $\pi_1(M)$ to ${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ up to trace equivalence, and the ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ character variety $\bar{X}(M)$ can be defined similarly. These character varieties are ${\ensuremath{\mathbb{C}}}$ algebraic sets. An irreducible component of the ${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ character variety of a hyperbolic 3-manifold $M$ is called a [*canonical component*]{}, and is written $X_0(M),$ if it contains the character of a discrete and faithful representation. Canonical components are often called geometric components because they encode a wealth of geometric information about the underlying manifold. Thurston [@thurston] showed that all but finitely many representations coming from Dehn filling of a single cusp are on a canonical component. We will call a surface in $M$ [*symmetric*]{} if it is fixed set-wise by the group of orientation preserving isometries of $M$, and [*non-symmetric*]{} otherwise. In Section \[mainsection\] we prove the main goal of this paper.
[thm]{}[maintheorem]{} \[thm:maintheorem\] Let $M$ be a finite volume, orientable, hyperbolic 3-manifold with a single cusp, and $G$ a subgroup of the group of orientation preserving isometries of $M$ with the property that the orbifold quotient $M/G$ has a flexible cusp. Let $v$ be a discrete valuation on a field $F$ such that there is a representation $\phi:\pi_1(M) \rightarrow {\text{SL}}_2(F)$ with $v({\text{tr}}(\phi(\gamma)))<0$ for some $\gamma \in \pi_1(M)$. Then $v$ detects an essential surface in $M$. If this representation $\phi$ is associated to a point (or ideal point) on a canonical component $X_0(M)$ then $v$ detects a symmetric essential surface in $M$.
Essential surfaces are detected in a highly non-canonical way. A single point can detect multiple surfaces. This theorem shows that if $M$ has a symmetry, then any point (or ideal point) on a canonical component which detects an essential surface detects a symmetric essential surface.
For affine points on the character variety, we consider any valuation. One example is the $\pi$-adic valuation for a prime ideal $\pi$ associated to an algebraic non-integral (ANI) point. For ideal points we consider the order of vanishing valuation. An ideal point is a point in a smooth projective completion that is not in the character variety, a point at infinity. (See Section \[section:PSLOrbs\].) We will work with the ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ character variety for this proof. There are obstructions to lifting a ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ representation to an ${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ representation which make the ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ character variety is more widely applicable, especially when studying the quotient orbifold $M/G$.
A key step in the proof of Theorem \[thm:maintheorem\] is our extension of Culler and Shalen’s construction of essential surfaces in 3-manifolds from actions on trees [@MR683804; @MR1886685] to the orbifold setting. We show that this theory can be extended to orbifolds, in both the ${\text{SL}}$ and ${\text{PSL}}$ setting. In Section \[section:PSLOrbs\] we discuss the valuations for orbifold groups in both the ${\text{SL}}$ and ${\text{PSL}}$ setting, recalling how these valuations give rise to actions on trees. And we prove the following.
[thm]{}[orbifoldmaintheorem]{} \[orbifoldTreeAction\] Let $Q$ be a compact, orientable, irreducible 3-orbifold. If $\pi_1^{orb}(Q)$ acts non-trivially and without inversions on a tree $T$, then there exists an essential $2$-suborbifold $F$ in $Q$ dual to this action.
A similar result was proven independently by Yokoyama [@MR3501267]. However, we require some specific properties for our applications to surface detection. For example, we assume that the orbifolds are orbifold irreducible and show that the constructed $2$-suborbifolds can always be made transverse to the singular set of the orbifold. This is necessary to show that these $2$-suborbifolds lift to essential surfaces in covering manifolds, which we show in Lemma \[suborbliftinglemma\] in Section \[section:lifting\].
A slope $r\in {\ensuremath{\mathbb{Q}}}\cup \{1/0\}$ is called a [*boundary slope*]{} if there is an essential surface $F$ in $M$ such that $\partial F \cap \partial M$ is a non-empty set of parallel simple closed curves on $\partial M$ of slope $r$. A boundary slope $r$ is called [*strict*]{} if there is an essential surface $F$ in $M$ which is not a fiber or semi-fiber such that $\partial F \cap \partial M$ has slope $r$. It is known [@MR881270 Proposition 1.2.7] that fibered and semi-fibered surfaces are never detected by ideal points of components of the character variety which contain irreducible representations. Let $x$ be a point in a smooth projective closure of $X(M)$. The boundary slope $r$ is said to be [*detected*]{} at $x$ if a surface with boundary slope $r$ is detected at $x$. We call a slope $r$ [*strongly detected*]{} by $x$ if $x$ detects no closed essential surface, and otherwise we say $r$ is [*weakly detected*]{} by $x$. (Some authors use the term detected exclusively for detection at an ideal point.) We will call a boundary slope which is the slope of a symmetric essential surface [*symmetric*]{}, and we will call all other boundary slopes [*non-symmetric*]{}. In [@ccgls] it was shown that the detected boundary slopes at ideal points are the slopes of the Newton polygon of the $A$-polynomial.
For knot complements in $S^3$ with the standard framing the boundary slope $0/1$ is detected by characters of abelian representations. This corresponds to the longitude slope, which is the slope of a Seifert surface. Motegi [@MR953952] showed that there are closed graph manifolds that contain essential tori not detected by ideal points of the character variety. Boyer and Zhang [@MR1670053 Theorem 1.8] showed that there are infinitely many closed hyperbolic 3-manifolds whose character varieties do not detect closed essential surfaces at ideal points. Schanuel and Zhang gave an example of a closed hyperbolic 3-manifold with a closed essential surface that is not detected by an ideal point but is ANI-detected. They also constructed a family of graph manifolds with boundary slopes that are not strongly detected at ideal points, but are weakly detected. Chesebro and Tillmann [@MR2395254] used mutants to construct examples of (cylindrical) knot complements containing strict boundary slopes which are weakly detected at ideal points, but not strongly detected at ideal points. Chesebro [@MR3073906] also demonstrated a connection between module structures on the coordinate ring of an irreducible component of the character variety and the detection of closed essential surfaces. Casella, Katerba, and Tillmann [@CKT] used this to show that there are closed essential surfaces in hyperbolic knot complements which are not detected by ideal points of the character variety. These surfaces are also not ANI-detected.
The following corollary about boundary slopes follows immediately from Theorem \[thm:maintheorem\].
Let $M$ be a finite volume, orientable, hyperbolic 3-manifold with a single cusp, such that the orbifold quotient $M/G$ has a flexible cusp. Any boundary slope detected on $X_0(M)$ is a symmetric slope.
We will prove the following in Section \[mainsection\].
\[maincor\] Let $M$ be a finite volume, orientable, hyperbolic 3-manifold with a single cusp, such that the orbifold quotient $M/G$ has a flexible cusp. There are at least two distinct symmetric essential surfaces in $M$ with different slopes.
In Section \[examplesection\], we use these results to prove a general statement for hyperbolic two-bridge knots.
\[cor:twobridge\] Every hyperbolic two-bridge knot complement contains at least two symmetric essential surfaces with different boundary slopes.
In Section \[examplesection\] we study the symmetric double twist knots, an infinite family of two-bridge knots. Let $K_n$ be the double twist knot with two twist regions with $n$ full twists, as defined in Section \[examplesection\], and $X(K_n)$ the ${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ character variety of its complement. We show that every boundary slope is detected by some ideal point on the character variety, and all symmetric slopes are detected by ideal points on a canonical component. However, for each $n$, there is a boundary slope not detected on the canonical component.
[thm]{}[symmetricdoubletwistthm]{} \[mainresultTheoremdetectedslopesstrict\] Let $K_n$ be a hyperbolic symmetric double twist knot. The ideal points on $X(K_n)$ detect the slopes $0$, $-8n+2$ and $-4n$; these are all of the boundary slopes in $S^3-K_n$. The ideal points on the canonical component $X_0(K_n)$ detect slopes $0$ and $-8n+2$; these are all of the symmetric boundary slopes of $S^3-K_n$.
Organization
------------
In Section \[section:orbifolds\] we recall the definition and construction of an orbifold and the orbifold fundamental group. We collect facts about 3-orbifolds and suborbifolds. In Section \[actionontreesection\] we prove Theorem \[orbifoldTreeAction\] and then prove Lemma \[suborbliftinglemma\], which shows that we can lift a detected $2$-suborbifold in $M/G$ to an essential surface in $M$. We review the construction of the ${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ and ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ character varieties in Section \[section:charactervarieties\]. We also prove Theorem \[thm:symmetiesandcanonicalcomponent\] which says that a canonical component is fixed by the induced action of a symmetry. In Section \[section:inducedmaps\] we discuss induced maps on the character variety. We also discuss ${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ and ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ Culler Shalen theory for manifolds and orbifolds in Section \[section:PSLOrbs\]. In Section \[mainsection\] we use this and Theorem \[orbifoldTreeAction\] to prove Theorem \[thm:maintheorem\]. Section \[examplesection\] is devoted to the double twist knots.
Orbifolds {#section:orbifolds}
==========
We refer the reader to [@MR1778789] for a comprehensive treatment of orbifolds. Associated to the orbifold $Q$, we let $X_Q$ denote the [*underlying space*]{}, and $\Sigma(Q)$ be the [*singular locus*]{} of $Q$. The singular locus consists of the points $x\in X_Q$ with non-trivial [*local group*]{} $G_x$.
An *orbifold covering map* $p:P\to Q$ is a continuous map of underlying spaces from $X_P$ to $X_Q$ with the following property. Each point $x\in X_Q$ has a neighborhood $U$ which necessarily is diffeomorphic to $\tilde{U}/D$ for some affine patch $\tilde{U}$ and finite group $D$ of diffeomorphisms of $\tilde{U}$. We require that each component of $p^{-1}(U)$ is isomorphic to $\tilde{U}/D'$ for some $D'<D$. Every orbifold $Q$ has a universal covering $p:\tilde{Q}\to Q$ such that for every orbifold covering map $f:X\to Q$ there is an orbifold covering map $p':\tilde{Q}\to P$ such that $f\circ p':\tilde{Q}\to Q$ is also an orbifold covering map. The *orbifold fundamental group*, $\pi_1^{orb}(Q)$, is the group of deck transformations of the orbifold universal cover $\tilde{Q}$. An orbifold is *(very) good* if it is (finitely) covered by a manifold. Otherwise, it is *bad*.
If $M$ is a manifold and a group $G$ acts on $M$ properly discontinuously, then $M/G$ has an induced orbifold structure. In the context of Theorem \[thm:maintheorem\] we will consider orbifolds which are the quotient of compact, orientable, irreducible 3-manifolds by a finite symmetry group. All such orbifolds are themselves compact, orientable, and orbifold irreducible. Therefore, we will restrict our attention to orbifolds that are compact, orientable, and orbifold irreducible. For our application to Theorem \[thm:maintheorem\] we only need to consider good orbifolds, but the proof of Theorem \[orbifoldTreeAction\] does not require the orbifold to be good.
The assumption that the orbifold $Q$ is orientable gives a useful description of the singular set.
[@MR1778789 Theorem 2.5]\[thm:singularset\] Let $Q$ be a orientable $3$-orbifold. Then the underlying space $X_Q$ is an orientable 3-manifold, and the singular $\Sigma(Q)$ set consists of a trivalent graph where the three edges at any vertex have orders $(2,2,k)$, $(2,3,3)$, $(2,3,4)$, or $(2,3,5)$.
The 2-dimensional orbifolds which are 2-spheres with 3 cone points and cone angles $(\pi, \pi/2, \pi/2)$, $(\pi, 2\pi /3, \pi/3)$ and $(2\pi/3, 2\pi /3, 2\pi /3)$ are the Euclidean turnovers $S^2(2,4,4)$, $S^2(2,3,6)$, and $S^2(3,3,3)$, respectively. The 2-dimensional orbifold which is 2-sphere with 4 cone points all of cone angle $\pi$ is the pillowcase, $S^2(2, 2, 2, 2)$. If $X$ is an orientable, non-compact finite volume hyperbolic 3-orbifold, then a cusp of $X$ has the form $Q \times [0, \infty)$, where $Q$ is a Euclidean orbifold. In fact (see [@MR1291531]) $Q$ is either a torus, a pillowcase or a turnover. The cusp is said to be [*rigid*]{} if $Q$ is a Euclidean turnover, otherwise it is called [*flexible*]{}.
Suborbifolds
------------
We begin with the manifold setting. Let $M$ denote an orientable compact, irreducible, hyperbolic 3-manifold. Let $F$ be a (not necessarily connected) surface in $M$. A [*bicollaring*]{} of $F$ is a homeomorphism $h$ of $F\times [ -1,1 ]$ onto a neighborhood of $F$ in $M$ such that $h(x,0)=x$ for every $x\in F$ and $h(F\times [ -1,1 ])\cap \partial M=h(\partial F\times [ -1,1 ])$. A surface $F$ is *bicollared* if $F$ admits a bicollaring. A surface $F$ in $M$ is [*boundary parallel*]{} if there exist a homotopy of $F$ that takes $F$ into the boundary of $M$.
A surface $F$ is *essential* if it has the following properties: $F$ is bicollared; every component of $F$ is $\pi_1$-injective in $M$; no component of $F$ is a $2$-sphere; no component of $F$ is boundary parallel; and $F$ is nonempty.
In the orbifold setting we have the following. A $2$-suborbifold $F$ in a $3$-orbifold $Q$ is *bicollared* if $X_F$ is a bicollared as an embedded surface in the manifold $X_Q$. An *orbifold disc* (or *ball*) is the quotient of a $2$-disc (or $3$-ball) by a finite group. A [*spherical*]{} 2-orbifold is a quotient of a 2-sphere by a finite group. A $2$-suborbifold $F$ in a $3$-orbifold $Q$ is called *orbifold incompressible* if for any component $P$ of $F$, $P$ has the following properties, where $\chi$ is the orbifold Euler characteristic (see [@MR1778789]):
- $\chi(P)>0$ and $P$ doesn’t bound an orbifold ball in $Q$.\
and
- $ \chi(P)\leq 0$ and any $1$-suborbifold in $P$ that bounds a orbifold disc in $Q-P$ also bounds an orbifold disc in $P$.
We will call incompressible $2$-suborbifold *essential* if it is bicollared and has no boundary parallel or spherical components. We call a $3$-orbifold *orbifold irreducible* if it contains no bad $2$-suborbifold or essential spherical $2$-suborbifold.
Dual $2$-suborbifolds to Actions on Trees {#actionontreesection}
==========================================
The action of a group $\Gamma$ on a tree $T$ is said to be [*trivial*]{} if there is a vertex of $T$ fixed by the entire group $\Gamma$, otherwise the action is called non-trivial. The group $\Gamma$ acts [*without inversions*]{} if $\Gamma$ does not reverse the orientation of any invariant edge. We will now assume that $Q$ is a compact, orientable, and orbifold-irreducible $3$-orbifold and $\pi_1^{orb}(Q)$ acts non-trivially and without inversions on a tree $T$. Our proof of Theorem \[orbifoldTreeAction\] is modeled the work in [@MR942518], [@MR881270] and [@MR1886685] for surfaces in 3-manifolds, and uses ideas from [@MR1065604].
We now outline the proof. First, we make the action simplicial. In Lemma \[treeproof1\] we obtain a triangulation of the orbifold $Q$ that lifts to a triangulation of the universal orbifold cover $\tilde{Q}$. In Lemma \[treeproof2\] we use this triangulation to construct from $\tilde{Q}$ a $\pi_1^{orb}(Q)$-equivariant, simplicial map $\tilde{f}$ from $\tilde{Q}$ into $T$. For $E$ the set of midpoints of edges in $T$, we look at the sets $\tilde{F}=\tilde{f}^{-1}(E)\subset \tilde{Q}$ and $F=p(\tilde{F}) \subset Q$. We show that they are bicollared, and after altering $\tilde{f}$ by homotopy show that we can make $F$ an essential $2$-suborbifold. We will also show that we can take the suborbifold to be transverse to the singular set, which will allow us to lift the suborbifold to an essential surface in a covering space.
Let $\tilde{Q}$ be the universal covering space of $Q$ with covering map $p:\tilde{Q}\rightarrow Q$.
\[treeproof1\] There is a triangulation $\Delta$ of $X_Q$ such that the singular set of $Q$ is contained in the $1$-skeleton of $\Delta$. This triangulation lifts to a triangulation $\tilde{\Delta}$ of $\tilde{Q}$.
Let $\Delta$ be any triangulation of $X_Q$, the underlying space of $Q$. We will show that we can perform homotopy and barycentric subdivision to obtain a triangulation $\Delta'$ of $X_Q$ with the property that the singular set of $Q$ is contained in the $1$-skeleton of $\Delta'$. As a result, this triangulation $\Delta'$ will lift to a triangulation of $\tilde{Q}$.
By Theorem \[thm:singularset\] due to [@MR1778789] the singular set of $Q$ is a finite trivalent graph. Therefore, we can perform a homotopy on $\Delta$ to make the singular set transverse to every $2$-cell. We then perform barycentric subdivision on the triangulation until the interior of every $3$-cell contains at most $1$ vertex of the singular set of $Q$ and every $2$-cell intersects the singular set at most once (while maintaining the transverse property). Therefore, any singular set that intersects the interior of a $3$-cell either intersects as a line segment with endpoints on different faces, or as three line segments meeting at a vertex (again with endpoints on different faces). We now can perform barycentric subdivision so that the singular set is contained in the new $1$ and $2$ cells of the triangulation. After homotopy, it is contained in the $1$-skeleton of the barycentric subdivision.
Given a triangulation $\Delta$ as in Lemma \[treeproof1\], let $\tilde{\Delta}^i$ represent the $i$-skeleton of $\tilde{\Delta}$. Let $S^i$ be a complete system of orbit representatives for the action of $\pi_1^{orb}(Q)$ on $\tilde{\Delta}^i$. For each $s\in S^0$, define the subgroup stabilizing $s$ as $$\Gamma_s=\{\gamma\in\pi_1^{orb}(Q) \mid \gamma \cdot s = s \}.$$ Then $\Gamma_s$ is a subgroup of the local group $G_{p(s)}$, and is a finite group. If $p(s)$ is not in the singular locus of $Q$, then $\Gamma_s$ is trivial. Let $E$ denote the set of midpoints of edges of $T$.
We begin with a map $h_0:S^0\rightarrow T^0$ with the property that $h_0(s)$ is stabilized by $\Gamma_s$. This local group is finite and so there is at least one vertex in $T$ stabilized by the whole group. Therefore the condition that $h_0(s)$ is stabilized by $\Gamma_s$ can be easily achieved. For example, when $ Q$ is hyperbolic then the space $\tilde{Q}={\ensuremath{\mathbb{H}}}$ is contractible, so we can embed $T$ in $\tilde{Q}$ and take $h:\tilde{Q}\rightarrow T$ to be a deformation retraction. Since $T$ is necessarily an infinite tree, in practice, we can alter $h$ by homotopy so that $h_0$ sends $S^0$ to $T^0$.
\[treeproof2\] Given a map $h_0:S^0\rightarrow T^0$ such that for each point $s\in S^0,$ $h_0(s)$ is stabilized by $\Gamma_s$, there is a $\pi_1^{orb}(Q)$-equivariant map $\tilde{f}:\tilde{Q}\rightarrow T$ that restricts to $h_0$ and is transverse to $E$.
The proof will proceed as follows. We will define a continuous and $\pi_1^{orb}$-equivariant map $\tilde{f_i}$ from the $i$-skeleton of $\tilde{Q}$ to $T$ and then extend it to the $(i+1)$-skeleton until we have a map $\tilde{f}=\tilde{f}_3$ defined $f$ on $\tilde{Q}$. The final equivariant map $\tilde{f}$ is not unique as many choices are involved in the construction of this map.
We begin by constructing $\tilde{f}_0$. The set $S^0\subset \tilde{\Delta}^0\subset \tilde{Q}$ and every orbit for the action of $\pi_1^{orb}$ on $\tilde{\Delta}^0$ intersects $S^0$ in precisely one point. Because the action of any finite group on a tree is trivial, $\Gamma_s$ acts trivially on $T$ and fixes at least one vertex of $T$.
Given $s'\in \tilde{\Delta}^0-S^0$ then $s'=\gamma\cdot s$ for some $\gamma \in \pi_1^{orb}$ and we extend $h_0$ by defining $\tilde{f}_0(s')=\gamma\cdot h^0(s)$. Given such an $s'$, if $s'=\gamma \cdot s = \gamma' \cdot s$ then $\gamma^{-1} \gamma' \cdot s = s$ and so $ \gamma^{-1} \gamma' \in \Gamma_s <G_{p(s)}$. By construction, $h_0(s)$ is fixed by $\gamma^{-1} \gamma' $. From this it follows that $\gamma\cdot h_0(s)=\gamma' \cdot h_0(s)$ and indeed $\tilde{f}_0$ is well-defined. The map $\tilde{f}_0$ is unique because the $\pi_1^{orb}$ equivariant condition necessitates that $\tilde{f}_0(\gamma \cdot s)= \gamma \cdot h_0(s)$. We conclude that $h_0$ can be uniquely extended to a $\pi_1$-equivariant map $\tilde{f}_0$ from $\tilde{\Delta}^0$ to $T^0$.
Next we extend the map $\tilde{f}_0$ to $\tilde{f}_1: \tilde{\Delta}^1 \rightarrow T$. We start by choosing $S^1$. For each $1$-simplex $\sigma\in S^1$ let $h_{\sigma}:\sigma\to T$ be a continuous map that agrees with $\tilde{f}_0$ on $\partial\sigma$ and maps into the unique line segment in $T$ connecting the two points of $\tilde{f}_0(\partial\sigma)$. If both endpoints of $\sigma$ are mapped to the same vertex of $T$ by $\tilde{f}_0$, then $h_{\sigma}$ also maps every point in $\sigma$ to that vertex. Let $\tilde{f}_1$ be the unique $\pi_1^{orb}$-equivariant map from $S^1$ into $T$ given by $$\tilde{f}_1(\gamma \cdot s) = \gamma \cdot h_{\sigma}(s)$$ for all $\gamma\in\pi^{orb}_1(Q)$, $\sigma\in S^1$, and $s\in\sigma$. Note that this results in a simplicial map.
Again we need to check that this function is well-defined. It agrees with $\tilde{f}_0$ on the boundary of every simplex, so it is enough to consider $s' \in \tilde{\Delta}^1-S^1$. Assume that $s' = \gamma \cdot s = \gamma'' \cdot s''$ for some $s\in \sigma,$ a $1$-simplex in $\tilde{\Delta}^1$ and $s'' \in \sigma'',$ a $1$-simplex in $\tilde{\Delta}^1$. Then $\sigma = \sigma''$ because $S^1$ is a complete system of orbit representatives, and we have $ \gamma \cdot s = \gamma '' \cdot s''$ so that $s = \gamma^{-1} \gamma'' \cdot s''$. It follows that $\gamma^{-1} \gamma''$ sends $\sigma$ to itself. First, consider the case when $s\neq s''$. Then the action of $\gamma^{-1} \gamma''$ on $\tilde{\Delta}^1$ interchanges the endpoints of $\sigma$, so the action of $\gamma^{-1} \gamma''$ must do so as well under $\tilde{f}_0$ since it is a $\pi_1^{orb}$-equivariant map. Because $\pi_1^{orb}(Q)$ acts on $T$ without inversions, both endpoints of $\sigma$ must be mapped to the same point, $x$, in $T$, and so all of $\sigma$ is mapped to $x$ and necessarily $\gamma^{-1} \gamma''$ fixes $x$. Therefore, considering $s'=\gamma \cdot s$ we have $$\tilde{f}_1(s') = \tilde{f}_1( \gamma \cdot s) = \gamma \cdot h_{\sigma}(s) = \gamma \cdot x.$$ Similarly, with $s'= \gamma'' \cdot s''$ we have $\tilde{f}_1(s') = \gamma'' \cdot x$, which equals $\gamma \cdot x$ since $\gamma^{-1} \gamma ''$ fixes $x$. It now suffices to consider the case when $s=s''$, so that $s' = \gamma \cdot s = \gamma'' \cdot s$ and $\gamma^{-1} \gamma '' $ fixes $s$. Since the action is without inversions, it cannot be the case that $\gamma^{-1} \gamma ''$ fixes $s$ only and interchanges the endpoints of the $1$-simplex $\sigma$, so $\gamma^{-1} \gamma ''$ fixes $\sigma$ point-wise. Therefore, $h_{\sigma}$ is the identity. It follows that $\gamma \cdot h_{\sigma}(s)= \gamma'' \cdot h_{\sigma}(s)$, and so $\tilde{f}_1(s')$ is well-defined.
Now we extend this map to $\tilde{f}_2:\tilde{\Delta}^2\to T$. For any simplex $\sigma\in S^2$ we have a map $\tilde{f}_2|_{\partial\sigma}:\partial\sigma\to T$ by restricting to $\tilde{f}_1$. Because $T$ is contractible, this map can be continuously extended to a map $h_{\sigma}:\sigma\to T$. Now let the map $\tilde{f}_2:\tilde{\Delta}^2\to T$ be defined as $$\tilde{f}_2(\gamma \cdot s)= \gamma \cdot h_{\sigma}(s)\text{ for all $\gamma\in\pi_1^{orb}(Q)$, $\sigma\in S^2$, and $s\in\sigma$.}$$ We have already shown that this is well-defined on the boundaries of every simplex. Because every point in the interior of a simplex is not a lift of a singular point (those are all in the $1$-skeleton by construction) and $\pi_1^{orb}(Q)$ acts freely on all points of $\tilde{Q}$ that are not lifts of singular points, $\tilde{f}_2$ is well-defined. By the simplicial approximation theorem [@MR1325242] $\tilde{f}_2$ can be made simplicial.
The construction of $\tilde{f}=\tilde{f}_3$ is identical to that of $\tilde{f}_2$. Note that $\tilde{f}$ is $\pi_1$-equivariant by construction. We can perform a homotopy to ensure the transverse condition.
The next step is to use the map $\tilde{f}$ to construct a bicollared $2$-suborbifold in $Q$. We will assume that we have a triangulation of $\tilde{Q}$ that is $\pi_1^{orb}(Q)$ invariant so that $\tilde{f}$ is simplicial. Let $\tilde{f}$ be as in Lemma \[treeproof2\], and let $\tilde{F}=\tilde{f}^{-1}(E)\subset \tilde{Q}$, and $F=p(\tilde{F})\subset Q$.
\[lemma:bicollared\] Consider an action of $\pi_1^{orb}(Q)$ on a tree $T$ that is non-trivial and without inversions. The set $\tilde{F}$ is a two sided and bicollared surface in $\tilde{Q}$, and the set $F$ is a two sided and bicollared $2$-suborbifold in $Q$.
Let $s\in T-T^0$ contained in edge $e$, and consider $P=\tilde{f}^{-1}(s)\subset \tilde{Q}$. Let $\sigma$ be an $i$-simplex of $\tilde{Q}$ and consider $P \cap \sigma$. If $\tilde{f}$ does not map $\sigma$ onto $e$ then $P\cap \sigma = \emptyset$. Otherwise, if $\tilde{f}$ does map $\sigma$ onto $e$, no vertex of $\sigma$ is mapped to $s$ since $s$ is not a vertex and the action is simplicial. Therefore, the set $P \cap \sigma$ is an $(i-1)$-cell which is properly embedded in $\sigma$. We see that $P$ intersects every simplex of $\tilde{Q}$ either trivially or in a properly embedded codimension 1 cell, and so $P$ is a 2-manifold. Since for each $s\in T-T^0$ the set $\tilde{f}^{-1}(s)$ is a properly embedded 2-manifold in $\tilde{Q}$, it follows that $\tilde{F}$ is a properly embedded 2-manifold in $\tilde{Q}$. Each point of $E$ is two sided in $T$ and therefore $\tilde{F}=\tilde{f}^{-1}(E)$ is two sided in $\tilde{Q}$ since $\tilde{f}$ is transverse to $E$. The surface $\tilde{F}$ is bicollared in $\tilde{Q}$ because $\tilde{F}$ is bicollared in each cell of $\tilde{\Delta}$ and the action of $T$ is without inversions.
The set $E$ is $\pi_1^{orb}$ invariant, and $\tilde{f}$ is $\pi_1^{orb}$ equivariant, so $\tilde{F}$ is invariant under the action of $\pi_1^{orb}$ on $\tilde{Q}$ and therefore $F$ is a $2$-suborbifold. Let $\bar{p}:T\rightarrow T/\pi_1^{orb}(Q)$ be the natural map. The map $\tilde{f}$ induces a unique map $f:Q\rightarrow T/\pi_1^{orb}(Q)$ with the property that $f\circ p = \bar{p} \circ \tilde{f}$. Since $\tilde{f}$ is transverse to $E$ it follows that $f$ is transverse to $\bar{p}(E)$. Each point of $\bar{p}(E)$ is two sided in $T/\pi_1^{orb}(Q)$, and it follows that $F$ is two sided in $X_Q$. Since the singular set $\Sigma(Q)$ is contained in the 1-skeleton of $\Delta$ and $\tilde{f}$ acts simplicially, it follows that $\tilde{F}$ is two sided in $Q$.
The singular set $\Sigma(Q)$ consists of edges and vertices where 3 edges meet. We have constructed a triangulation so that the singular set is contained in the 1-skeleton. The set $\tilde{F}$ is transverse to the 1-skeleton of $\tilde{Q}$, and since $p^{-1}(\Sigma(Q))$ is contained in the 1-skeleton of $\tilde{Q}$ we conclude that $\tilde{F}$ is transverse to $p^{-1}(\Sigma (Q))$. The image $F=p(\tilde{F})$ in $Q$ is a $2$-suborbifold. Because of the transversality, $F$ only contains a finite number of singular points, and the bicollaring can be extended to all of $F$. Thus $F$ is bicollared in $Q$.
Following Culler and Shalen, we say that a bicollared 2-suborbifold $F$ in $Q$ is [*dual*]{} to an action of $\pi_1^{orb}(Q)$ on $T$ if is arises from this construction for some $\pi_1^{orb}(Q)$ equivariant map that is transverse to $E$.
If $Q$ is a 3-orbifold and $F$ is a sub-orbifold then there is a natural map $i_*:\pi_1^{orb}(F) \rightarrow \pi_1^{orb}(Q)$ induced by inclusion $i:F\rightarrow Q$. (And similarly for a component $C_i$ of $Q-F$.) See [@MR1065604 Page 167] for details. Many standard theorems about submanifolds carry through to the orbifold case, for example [@MR1065604 Lemma 3.10] if $F$ is compact, two sided and incompressible then the induced map is injective. These $i_*$ maps are well-defined up to conjugation if no base point is specified.
\[lemma:stabilizers\] Let $F$ be a dual 2-suborbifold to an action of $\pi_1^{orb}(Q)$ on a tree $T$. Then
1. \[lemma:stabilizers:vertex\] For each component $C_i$ of $Q-F$ the subgroup $i_*(\pi_1^{orb}(C_i))$ of $\pi_1^{orb}(Q)$ is contained in the stabilizer of some vertex of $T$.
2. \[lemma:stabilizers:edge\] For each component $F_j$ of $F$, the subgroup $i^*(\pi-1^{orb}(F_j))$ of $\pi_1^{orb}(Q)$ is contained in the stabilizer of some edge of $T$.
Let $\tilde{f}:\tilde{Q} \rightarrow T$ be a $\pi_1^{orb}(Q)$-equivariant map transverse to the set $E$ of midpoints of edges of $T$ so that $F=f^{-1}(E)$. Let $\Gamma_i$ be $i_*(\pi_1^{orb}(C_i) ) < \pi_1^{orb}(Q)$.
By construction of $F$, $\tilde{f}$ maps any connected lift of $C_i$ into a component of $T-E$. The subgroup $\Gamma_i$ stabilizes a component $\tilde{C}$ of $\tilde{Q}-\tilde{F}$. Since $\tilde{f}$ is $\pi_1^{orb}(Q)$ equivariant, $\Gamma_i$ stabilizes $\tilde{f}( \tilde{C} )$. Since this is contained in a component of $T-E$, $\tilde{f}$ stabilizes the lone vertex in this component.
For the second item, we argue as above and conclude that $i_*(\pi_1^{orb}(F_j))$ stabilizes a component of $T-T^0$, where $T^0$ is the vertex set. Since the action is simplicial, and without inversions this group must stabilize a whole edge.
We want this construction to give an incompressible 2-suborbifold in $Q$, so the next step is to make $F$ orbifold incompressible. This will complete the proof of Theorem \[orbifoldTreeAction\].
As before, we have a $\pi_1^{orb}(Q)$ equivariant map $\tilde{f}:\tilde{Q} \rightarrow T$ that is transverse to $E$ where $F=f^{-1}(E)$ and $\tilde{f}^{-1}(E)=p^{-1}(F)$ in $\tilde{Q}$. If $F$ were empty the $Q-F$ consists of all of $Q$ and by Lemma \[lemma:stabilizers\] (\[lemma:stabilizers:edge\]) we conclude that $\pi_1^{orb}(Q)$ fixes a vertex of $T$, which by definition means that this action is trivial. Therefore, $F$ is non-empty.
From the construction thus far, and Lemma \[lemma:bicollared\] in particular, we may assume that $F$ is a bicollared 2-suborbifold. We may assume that $F$ is not essential, and so either $F$ is not incompressible or $F$ has boundary parallel or spherical components. We will first consider the case when $F$ is not incompressible. We will determine another $\pi_1^{orb}(Q)$-equivariant map $\tilde{f}':\tilde{Q}\rightarrow T$ which is also transverse to $E$ such that $(\tilde{f}')^{-1}(E) = p^{-1}(F')$ for a 2-suborbifold $F'$ dual to the action.
If $F$ is not incompressible, then there is a component $F_0$ of $F$ that contains a $1$-suborbifold that bounds an orbifold disc $D$ in $Q-F_0$ which does not bound an orbifold disc in $F_0$. If $D$ intersects more than one component of $F$ then we can chose an innermost intersection of $D$ with a component of $F$ to get a new compressing orbifold disc $D'$ for the component $F_0'$. Thus we will assume that $D\cap F=\partial D\subset F_0$.
We may assume that $\partial D$ does not intersect $\Sigma(Q)$ by pushing $\partial D$ off the singular points. Let $A$ be an annular neighborhood of $\partial D$ in $F_0$ (we choose $A$ such that it contains no singular points). Let $D_1$ and $D_2$ be two parallel copies of $D$, whose boundaries are the two components of $\partial A$. Note that the union of $D_1$ and $D_2$ and $A$ is a $2$-sphere in the underly space $X_Q$. Then because $Q$ is orbifold irreducible the union of $D_1$ and $D_2$ and $A$ is a spherical suborbifold that bounds an orbifold ball in $Q$.
![Compressing orbifold disk[]{data-label="compressingSurgery"}](compressingdisc){width="3in"}
Let $B$ be a nice neighborhood of $D$ in $Q$ such that $B$ is an orbifold ball and meets $F$ along the boundary of the annulus $A$, so $B\cap F_0 = \partial A$. Because the singular locus is a trivalent graph in $Q$, we can perform a homotopy so that $D$ does not contain a vertex of this graph and therefore choose $B$ to intersect only one ‘edge’ of the singular locus. Therefore, $B$ is locally the quotient of a ball by a finite cyclic group.
Then $B$ is the union of a solid torus $X^+$ and an orbifold ball $X^-$, where $X^+\cap X^- = A$. We may take $D_1$ and $D_2$ to be properly embedded in $X^+$. Now let $\tilde{B}$ be a component of $p^{-1}(B)$, and let $\tilde{A}$, $\tilde{D_1}$, $\tilde{D_2}$, $\tilde{X}^+$, and $\tilde{X}^-$ denote the inverse images of $A$, $D_1$, $D_2$, $X^+$, and $X^-$ in $\tilde{B}$.
Because $\tilde{f}$ is transverse to $E$ and $\tilde{A}=\tilde{B}\cap\tilde{F}^{-1}(E)$ and since $T$ is a tree, $\tilde{f}$ maps $\tilde{X}^+$ and $\tilde{X}^-$ to the closure of two different components of $T-E$. Lets call these components $Y^+$ and $Y^-$ respectfully.
We are now going to construct a new $\pi_1$-equivariant map $\tilde{f}'$ by removing the compressing orbifold disk. Let $\tilde{f}'$ be equal to $\tilde{f}$ for all of $\tilde{Q}-p^{-1}(B)$. First define $\tilde{f}'$ to be constant on both $\tilde{D_1}$ and $\tilde{D_2}$ and equal to the unique point of $E$ that is in the intersection of $Y^+$ and $Y^-$. Extend this map to the rest of $p^{-1}(D_1\cup D_2)$ by having $\tilde{f}'(\gamma \cdot x)=\gamma\cdot \tilde{f}'(x)$ for all $x\in p^{-1}(D_1\cup D_2)$ and $\gamma\in\pi_1(Q)$. To see that this is well-defined notice that if $\gamma \cdot x =\gamma' \cdot x'$, then $x=\gamma^{-1}\gamma' \cdot x'$, meaning $\gamma^{-1}\gamma'$ stabilizes $\tilde{D_1}$ (or $\tilde{D_2}$) in $\tilde{Q}$, and $\gamma \cdot \tilde{f}'(x)=\gamma' \cdot \tilde{f}'(x')$. If $\gamma \cdot \tilde{f}'(x') \neq \gamma' \cdot \tilde{f}'(x')$, then $\gamma^{-1}\gamma'$ does not stabilize $Y^+ \cap Y^-$. However, $\gamma^{-1}\gamma'$ stabilizes $\tilde{D_1}$ ($\tilde{D_2}$) and thus $\partial\tilde{D_1}$ ($\partial\tilde{D_2}$). Thus $\gamma^{-1}\gamma'$ does stabilize $Y^+ \cap Y^-$. Since it stabilizes $\partial\tilde{D_1}$ ($\partial\tilde{D_2}$) and $\tilde{f}$ is a $\pi_1$-equivariant map that sends $\partial\tilde{D_1}$ ($\partial\tilde{D_2}$) onto $Y^+ \cap Y^-$, we conclude that $\tilde{f}'$ is so far well-defined.
Now $\tilde{f}'$ needs to be extended over the three orbifold balls that $\tilde{D_1}$ and $\tilde{D_2}$ divides $\tilde{B}$ into, Call these $B_1$,$B_2$ and $B_3$. To extend the map over $B_i$ start with any triangulation of $p(B_i)$ that lifts to a triangulation of $B_i$ (which is fine enough such that no $3$-cell has all of its vertices on the boundary of $p(B_i)$ and every $1$-cell and $2$-cell that has all of its vertices on the boundary are completely contained in the boundary). Now follow the construction of a $\pi_1^{orb}$-equivariant map we used before sending all $0$-cells that are not in $\partial B_i$ to the vertex of $Y^+$ or $Y^-$ (note that for all $B_i$ the map $\tilde{f}$ sends $\partial B_i$ to the closure of either $Y^+$ or $Y^-$). It should be noted that no points in the interior of $B_i$ are mapped to a point of $E$ by this construction. This gives us a $\pi_1$-equivariant map $\tilde{f}':\tilde{Q}\to T$ that agrees with $\tilde{f}$ outside of $p^{-1}(B)$, and for which the compressing disk has been removed from $p(\tilde{f}'^{-1}(E))$.
Repeat the above steps until there are no more compressing orbifold disks for $F$. Each time the process is performed the component that was cut either remains connected or is split into two new components of $F$. When $F_0$ remains connected the genus of $F_0$ in the underlying space $X_Q$ is reduced. This process cannot increase the genus of a component. If a component is split into two, the sum of the genera of the two components is at most the genus of the original component. Therefore, it is only possible it is only possible to perform this process a finite number of times.
When we perform this process and split a component $F_0$ into two components $F_0'$ and $F_1'$ by the compressing orbifold disc $D$, we do so because the curve $\partial D$ in $F_0$ does not bound an orbifold disc in $F_0$. As such, the new components must have a positive genus, a non-empty boundary, or contain at least two singular points (at least three singular points if the disk $D$ has a singular point). Otherwise, one of the components of $F_0-\partial D$ is an open orbifold disk in $Q$ whose closure is a closed orbifold disk in $F$ bounded by $\partial D$. Suppose that $F$ has a combined genus $g$, boundary components $b$ and singular points $s$ across all its components. Then from the genus and boundary components the number of compressing disks it is possible to remove is bounded above by $2g-1+b$, each of which potentially adds a new singular point to $F$. The maximum number of singular points $F$ can have is bounded above by $4g+2b$ when not yet counting those potentially added when creating a new component without boundary that has genus $0$. When compressing along a disk that splits a component $F_0$ into two components $F_0'$ and $F_1'$ with at least one, say $F_0'$, having genus $0$ and no boundary, $F_1'$ will have less singular points than $F_0$ and $F_0'$ will have at most one more singular point than $F_0$. However, both $F_0'$ and $F_1'$ will have less singular points than $F_0$ unless $F_1'$ has positive genus or a non-empty boundary. The maximum number of compressing disks possible to remove without separating $F_0$ into two genus $0$ components with empty boundary is bounded above by $6g+3b+s$. The maximum number of compressing disks removable from $F$ is bound above by $12g+3b+2s$.
We conclude that after a finite number of steps $F$ will no longer have any compressing orbifold disks and we can move on to removing boundary parallel and spherical suborbifold components from $F$. As a result, we now handle the case when $F$ has boundary parallel or spherical suborbifold components.
Suppose $F_0$ is an innermost boundary parallel component of $F$. Then there exists a deformation retract $\rho$ of $Q$ onto a component of $Q-F_0$ that is constant and injective on neighborhoods of every other component of $F$. Then there is a $\pi_1(Q)$-equivariant map $\tilde{\rho}:\tilde{Q}\to \tilde{Q}$ such that $p^{-1}(F_0)\cap\tilde{\rho}(\tilde{Q})=\emptyset$. Let $\tilde{f}'=\tilde{f}(\tilde{\rho})$. Then $F_0\not\subset p(\tilde{f}'^{-1}(E))\subset F$.
Suppose $F_0$ is some spherical $2$-orbifold component of $F$. Then $F_0$ is contained in the interior of some $3$-orbifold ball $B$ such that $\tilde{f}$ maps each component of $p^{-1}(\partial B)$ to a single point of $T$. Define $\tilde{f}'$ to agree with $\tilde{f}$ for all $\tilde{Q}-p^{-1}(B)$ and let $\tilde{f}'$ map each component $\tilde{B}$ of $p^{-1}(B)$ to the point $\tilde{f}'(\partial\tilde{B})$.
The $2$-suborbifold $F$ in $Q$ defined by $\tilde{f}$ is incompressible, non-boundary parallel, and contains no essential orbifold $2$-spheres. It remains to prove that $F$ is non-empty. From the construction of $F$ from the map $\tilde{f}$, $F$ can only be empty if $\tilde{f}$ maps all of $\tilde{Q}$ into a component $V$ of $T-E$. However, because $\pi_1^{orb}(Q)$ acts non-trivially on $T$ no component of $T-E$ is stabilized by $\pi_1^{orb}(Q)$. Thus there exists some $\gamma\in\pi_1^{orb}(Q)$ such that $\gamma \cdot V=V'\neq V$. Since $\tilde{f}$ is $\pi_1^{orb}$-equivariant, the image of $\tilde{f}$ must also contain a point in $V'$. Because the image of $\tilde{f}$ is connected it contains a point in $E$. Therefore, $F$ is a non-empty essential $2$-suborbifold.
Lifting essential $2$-suborbifolds {#section:lifting}
-----------------------------------
We will now show that when $Q$ is good the suborbifolds obtained from Theorem \[orbifoldTreeAction\] lift to essential surfaces in the covering manifold.
\[suborbliftinglemma\] Let $F$ be an essential $2$-suborbifold in a good, orientable, orbifold-irreducible $3$-orbifold $Q$ such that $F$ contains only a finite number of singular points. Let $p:M\to Q$ be an orbifold covering map by an irreducible manifold $M$. Then $p^{-1}(F)$ is a symmetric essential surface in $M$.
We will begin by proving that $F$ lifts to a bicollared surface in $M$. Since $Q$ is covered by a manifold $M$, the universal cover of $Q$ is also a manifold. Away from the singular points on $F$, $F$ is bicollared surface. So, outside a neighborhood of each singular point on $F$, $F$ lifts to a bicollared surface in $M$. Neighborhoods of the singular points can be made sufficiently small so that the orbifold structure in the neighborhood is that of a $2$-sphere modulo an action by a finite cyclic group.
Suppose some component $S$ of $p^{-1}(F)$ is a $2$-sphere. Then $p(S)$ would be an essential spherical $2$-suborbifold in $Q$ because it is a $2$-sphere modulo the action of a finite group. This contradicts the fact that $Q$ is orbifold-irreducible. Thus $p^{-1}(F)$ contains no $2$-sphere components.
It follows from [@MR1926138 Proposition $2.8$] that every component of $p^{-1}(F_0)$ is incompressible for every component $F_0$ of $F$. Thus, every component of $p^{-1}(F)$ is $\pi_1$-injective.
Suppose a component $S$ of $p^{-1}(F)$ is boundary parallel. Then there exists a homotopy $h$ of $S$ that takes $S$ into the boundary of $M$. Then by the composition of continuous functions $p(h)$ is a homotopy that takes $S$ into the boundary of $Q$, contradicting the fact that $F$ has no boundary parallel components. Thus we conclude that $p^{-1}(F)$ contains no boundary parallel components, which completes the proof.
The surface is symmetric by construction since it is the pullback of a surface in $Q=M/G$.
Character Varieties {#section:charactervarieties}
====================
${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ Character Varieties {#section:SL}
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We will use the standard terminology in the field, where a variety refers to an algebraic set which is not necessarily irreducible or smooth. The [*${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ representation variety*]{} of the group $\Gamma$ is the set $$R(\Gamma)=\{\rho:\Gamma\to {\text{SL}}_2(\mathbb{C})\}.$$ Roughly speaking, these representations correspond to geometric structures on $M$, and conjugate representations correspond to isometric hyperbolic structures on $M$. As such it is natural to work with the character variety, which is $R(\Gamma)$ up to trace equivalence. Formally, the [*${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ character variety*]{} of $\Gamma$ is $$X(\Gamma) = \{ \chi_{\rho}: \rho\in R(\Gamma)\}$$ where the character function $\chi_{\rho}:\Gamma \rightarrow {\ensuremath{\mathbb{C}}}$ is defined as $\chi_{\rho}(\gamma) = {\text{tr}}(\rho(\gamma))$ for all $\gamma \in \Gamma$. The natural surjection $t:R(\Gamma) \rightarrow X(\Gamma)$ which sends a representation to its trace is a regular map. Both $R(\Gamma)$ and $X(\Gamma)$ are complex algebraic sets defined over ${\ensuremath{\mathbb{Q}}}$. Isomorphic group yield isomorphic varieties, so we will often write $R(M)$ and $X(M)$ to denote $R(\pi_1(M))$ and $X(\pi_1(M))$ up to isomorphism. A representation is called [*reducible*]{} if it is conjugate to an upper triangular representation, and [*irreducible*]{} otherwise. Two irreducible representations into ${\text{SL}}_2(\mathbb{C})$ have the same trace function if and only if they are conjugate representations (see [@MR683804]). A point $x\in X(\Gamma)$ equals $t(\rho)$ for some representation $\rho$ and we will often write it as $\chi_{\rho}$ with the understanding that $\rho$ is not uniquely determined. We will make use of the regular functions $$I_{\gamma}:X(\Gamma) \rightarrow {\ensuremath{\mathbb{C}}}$$ defined by $I_{\gamma}(\chi_{\rho})= \chi_{\rho}(\gamma)={\text{tr}}(\rho(\gamma))$ for a fixed $\gamma\in \Gamma$. The $I_{\gamma}$ functions extend to rational functions on a smooth projective completion of $X(M)$.
In general, the algebraic sets $R(M)$ and $X(M)$ are not irreducible (as algebraic sets) and have multiple components. For example, the set of characters of reducible representations is easily seen to be an algebraic set in its own right. We call a component a [*canonical component*]{} and write $R_0(M)$ (or $X_0(M)$) if it contains a (character of a) discrete and faithful representation. We write $X(M)=X_{red}(M)\cup X_{irr}(M)$ where $X_{red}(M)$ contains the characters of irreducible representations, and $X_{irr}$ is the (affine) Zariski closure of $X(M)-X_{red}(M)$. That is, $X_{irr}(M)$ is the Zariski closure of the union of components which each contain the character of irreducible representation.
We will call the character variety defined as above a [*natural model*]{} to distinguish it from a smooth projective completion. A point in the Zariski closure of $X(M)$ is an [*ideal point*]{} if it does not correspond to a trace function. That is, the set of ideal points consists of those points in the projective closure of $X(M)$ that are not in $X(M)$.
We will call a $\chi_{\rho} \in X(\Gamma)$ an [*algebraic non-integral*]{} (or ANI) point if $\rho(\Gamma)\subset {\text{SL}}_2(\bar{{\ensuremath{\mathbb{Q}}}})$, and there is a $\gamma \in \Gamma $ such that ${\text{tr}}(\rho(\gamma)) $ is not contained in $\mathcal{O}_K$, where $\mathcal{O}_K$ is the ring of integers of a number field $K$. Culler and Shalen [@MR683804; @MR881270] showed the following for an ideal or ANI point $x$ and Shanuel and Zhang [@MR1835066 Corollary 3] generalized this to other valuations. Either
1. There is a unique primitive element $\gamma$ such that $I_{\gamma}(x)\in {\ensuremath{\mathbb{C}}}$,\
or
2. $I_{\gamma}(x)\in {\ensuremath{\mathbb{C}}}$ for all $\gamma \in \pi_1(\partial M)$.
Strongly detected slopes correspond to the $\gamma$ to case $(1)$ and weakly detected slopes to case $(2)$. In $(2)$, $\chi_{\rho}$ detects a closed essential surface.
${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ Character Varieties {#section:PSL}
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There are several different constructions for the ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$-representation and character varieties of $\Gamma$; we refer the reader to [@MR1670053 Section 3], [@MR1739217 Section 2.1], and [@MR1248117].
Let $\bar{R}(\Gamma)$ be the set of representations of $\Gamma$ into ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$. This also has the structure of a complex algebraic set. The natural quotient map $$\Phi:{\text{SL}}_2({\ensuremath{\mathbb{C}}}) \rightarrow {\text{PSL}}_2({\ensuremath{\mathbb{C}}})$$ induces a regular map $\Phi_*:R(\Gamma) \rightarrow \bar{R}(\Gamma)$. Each fiber is either empty or an orbit of the free ${\ensuremath{{\mathrm{Hom}}}}(\Gamma, {\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}})$ action. This is the sign change action on $R(\Gamma)$ defined by $(\epsilon \cdot \rho)(\gamma) = \epsilon(\gamma) \rho(\gamma)$ for $\epsilon \in {\ensuremath{{\mathrm{Hom}}}}(\Gamma, {\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}})$. A ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ representation $\bar{\rho}$ lifts to an ${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ representation $\rho$ exactly when the second Stiefel-Whitney class $\omega_2(\overline{\rho}) \in H^2(\Gamma; {\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}})$ vanishes. The isomorphism class of $\omega_2(\bar{\rho})$ depends only on the component of $\bar{\rho}\in \bar{R}(\Gamma)$ and $\Phi_*$ defines a regular cover from $R(\Gamma)$ to its image.
As in the ${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ case, the action of ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ on $\bar{R}(\Gamma)$ has a quotient $\bar{X}(\Gamma)$ with quotient map $\bar{t}:\bar{R}(\Gamma)\rightarrow \bar{X}(\Gamma)$ which is constant on conjugacy classes of representations. The set $\bar{X}(\Gamma)$ is an affine algebraic set and is determined by the stipulation that its coordinate ring is isomorphic to the ring of invariants of the natural action of ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ on ${\ensuremath{\mathbb{C}}}[\bar{R}(\Gamma)]$. For each $\gamma \in \Gamma$ the map $\bar{t}:\bar{X}(\Gamma) \rightarrow {\ensuremath{\mathbb{C}}}$ given by $\bar{t}(\bar{\rho}) = ({\text{tr}}(\bar{\rho}(\gamma)))^2$ is regular, and we write $\bar{t}(\bar{\rho})=\chi_{\bar{\rho}}$.
We call $\bar{X}(\Gamma)$ the [*${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ character variety of $\Gamma$*]{}, and if $\Gamma$ is the fundamental group of a 3-manifold $M$ or orbifold $Q$ we use $\bar{X}(M)$ or $\bar{X}(Q)$ to denote $\bar{X}(\pi_1(M))$ and $\bar{X}(\pi_1^{orb}(Q))$ up to isomorphism. We call an irreducible component a [*canonical component*]{} if it contains $\bar{t}(\bar{\rho})$ where $\bar{\rho}$ is a discrete and faithful representation and write $\bar{X}_0$.
The function $\Phi$ induces a regular map $\Phi_{\#}:X(\Gamma) \rightarrow \bar{X}(\Gamma)$ with fibers that are either empty or the orbits of the ${\ensuremath{{\mathrm{Hom}}}}(\Gamma, {\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}})$ action as $\epsilon \cdot \chi_{\rho}=\chi_{\epsilon\cdot \rho}$. Indeed (see [@MR1670053] Lemma 3.1) two characters $\chi_{\bar{\rho}} $ and $\chi_{\bar{\rho'}}$ are equal exactly when the trace of $\rho(\gamma)$ equals the trace of $\rho'(\gamma)$ up to sign for all $\gamma \in \Gamma$.
Similar to the $I_{\gamma}$ functions defined above, for any $\gamma\in \Gamma$ we define $f_{\gamma}:\bar{X}(\Gamma) \rightarrow {\ensuremath{\mathbb{C}}}$ by $$f_{\gamma}(\chi_{\bar{\rho}}) = {\text{tr}}(\Phi^{-1}(\bar{\rho}(\gamma)))^2-4$$ where the squaring is necessary because the sign is not well-defined.
We call a point $\bar{\rho}\in \bar{R}(\Gamma)$ an [*algebraic non-integral*]{} (or ANI) point if $\rho(\Gamma)\subset {\text{PSL}}_2(\bar{{\ensuremath{\mathbb{Q}}}})$ and there is a $\gamma \in \Gamma $ such that ${\text{tr}}(\Phi^{-1}(\bar{\rho}(\gamma)))^2$ is not contained in $ \mathcal{O}_K$, where $\mathcal{O}_K$ is the ring of integers of a number field $K$.
Induced Maps {#section:inducedmaps}
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A homomorphism $\phi:A\rightarrow B$ between two groups $A$ and $B$ induces a map $$\begin{aligned}\phi^*: R(B) & \rightarrow R(A) \\ \rho & \mapsto \rho \circ \phi. \end{aligned}$$ This descends to the ${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ character variety as $$\begin{aligned}\hat{\phi}: X(B) & \rightarrow X(A) \\ \chi_{\rho} & \mapsto \chi_{\phi^*(\rho)} \end{aligned}$$ where $\chi_{\phi^*(\rho)} =\chi_{ \rho \circ \phi}.$ This is defined for all (affine) points on $R(B)$ or $X(B)$. We obtain analogous maps in the ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ setting. Re-using notation, we have the induced map $$\begin{aligned}\phi^*: \bar{R}(B) & \rightarrow \bar{R}(A) \\ \bar{\rho} & \mapsto \bar{\rho} \circ \phi. \end{aligned}$$ and for the ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ character variety we have $$\begin{aligned}\hat{\phi}: \bar{X}(B) & \rightarrow \bar{X}(A) \\ \chi_{\bar{\rho}} & \mapsto \chi_{\bar{\rho}\circ \phi} \end{aligned}$$ where $\chi_{\bar{\rho}}=\bar{t}(\bar{\rho})$, and $\chi_{\bar{\rho}\circ \phi}=\bar{t}(\bar{\rho}\circ \phi)$.
If $\hat{\phi}$ is dominant (its image is Zariski dense) then $\hat{\phi}$ induces a ${\ensuremath{\mathbb{C}}}$-algebra homomorphism (see [@MR0463157 page 25] ) between function fields ${\ensuremath{\mathbb{C}}}(X(A))\rightarrow {\ensuremath{\mathbb{C}}}(X(B))$. For a regular function $f\in {\ensuremath{\mathbb{C}}}(X(A))$ its image is $f\circ \phi$ where it’s understood that this is defined on an open subset of $X(A)$. If the mapping $\hat{\phi}$ is birational then it induces an isomorphism between ${\ensuremath{\mathbb{C}}}(X(A))$ and ${\ensuremath{\mathbb{C}}}(X(B))$. Considering the case when $X(A)$ and $X(B)$ (or $\bar{X}(A))$ and $\bar{X}(B)$) are curves, it follows that $\hat{\phi}(X)$ is either Zariski dense in $X(A)$ (or $\bar{X}(A)$) or it is a point.
Let $Q=M/G$ and consider the branched, or orbifold covering $p:M\rightarrow Q$ induced by the natural quotient. This $p$ is an orbi-map (as defined in [@MR1065604] on pages 161-162) and therefore by [@MR1065604 Proposition 2.4] the induced homomorphism $p_*:\pi_1(M) \rightarrow \pi_1^{orb}(Q)$ is injective. If $\tilde{\alpha}$ is a path in the underlying space of the universal cover of $M$, then $p_*([\tilde{\alpha}])= [\tilde{\alpha}] $ where this second $[\tilde{\alpha}]$ is in the universal cover of $Q$.
Symmetries and the Canonical Component {#section:symmandcanonical}
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A symmetry $\sigma$ of a knot $K$ is an orientation preserving homeomorphism of $S^3$ that send $K$ to itself. As a consequence of Mostow rigidity the full group of symmetries of $K$ is finite and acts on $M=S^3-K$ by isometries (see [@thurston] Corollary 5.7.4). The symmetry group of a knot in $S^3$ is either cyclic or dihedral [@MR1646740]. By the Gordon-Luecke theorem [@MR965210] any homeomorphism of knot complements must take meridian to meridian, and therefore take an (un-oriented) longitude to itself as well. In general, if $M$ is a one cusped hyperbolic 3-manifold we define a [*symmetry*]{} of $M$ to be any orientation-preserving isometry of $M$. We will only consider symmetries that preserve the framing of the the cusp.
If $\sigma:M\rightarrow M$ is a symmetry of $M$ then $\sigma$ induces an automorphism of $\sigma_*$ of $\pi_1(M)$. This induces an automorphism $\sigma^*$ of the representation variety $R(M)$ by $\sigma^*(\rho)=\rho\circ \sigma_*$, and an automorphism $\hat{\sigma}$ of the ${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ character variety $X(M)$ by $\hat{\sigma}(\chi_{\rho})=\chi_{\sigma^*(\rho)}$. An analogous statement can be made in the ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ setting. These are all regular maps.
In this section, we prove the following theorem.
\[thm:symmetiesandcanonicalcomponent\] Let $\sigma$ be a symmetry of the one cusped hyperbolic manifold $M$ with induced symmetry $\hat{\sigma}$ on $\bar{X}(M)$. A canonical component $\bar{X}_0(M)$ of the ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ character variety is a subset of the fixed point set of $\hat{\sigma}$. For $g\in \pi_1(M)$, and a point $\chi_{\rho}$ on a canonical component of the ${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ character variety, $X_0(M)$, we have $\hat{\sigma}(\chi_{\rho}(g)) = \pm \chi_{\rho}(g)$. If for all $g\in \pi_1(M)$, $\sigma_*(g)$ and $g$ have the same image in the surjection $\pi_1(M) \rightarrow {\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}}$ then $X_0(M)$ is a subset of the fixed point set of $\hat{\sigma}$.
Since $M$ has only one cusp, $\sigma$ induces a symmetry on $\partial M$ which retains the framing of $\partial M$. Let $M(p/q) = M\cup_{\partial M} T$ be $p/q$ Dehn filling of $M$. It follows that $\sigma $ extends to a symmetry of $M(p/q)$.
By Thurston’s hyperbolic Dehn surgery theorem, all but finitely many Dehn surgeries on $M$ are hyperbolic. For such a $M(p/q)$, by Mostow rigidity there is a unique complete hyperbolic structure on $M(p/q)$ therefore representations of $\pi_1(M(p/q))$ correspond to a holonomy representation to ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$. Any two such representations differ only by conjugation and perhaps complex conjugation. Complex conjugation corresponds to an orientation reversal.
By the Seifert-Van Kampen theorem, $\pi_1(M(p/q))$ is isomorphic a quotient of $\pi_1(M)$, so there is a surjection $p:\pi_1(M) \rightarrow \pi_1(M(p/q))$. Let $\rho:\pi_1(M)\rightarrow \mathrm{(P)SL}_2({\ensuremath{\mathbb{C}}})$ be a representation. Then we obtain a representation $\rho \circ p: \pi_1(M)\rightarrow \mathrm{(P)SL}_2({\ensuremath{\mathbb{C}}})$. It follows that $R(M(p/q))$ naturally embeds in $R(M)$, and similarly $X(M(p/q))\subset X(M)$ and $\bar{X}(M(p/q))\subset \bar{X}(M)$. Thurston’s theorem shows that there are faithful representations of $\pi_1(M(p/q))$ for all but finitely many $p/q$ in any Riemannian neighborhood of any discrete and faithful representation of $\pi_1(M)$. Therefore, there are infinitely many of these points on any $X_0(M)$ or $\bar{X}_0(M)$ and they form a Zariski dense set. We conclude that it is enough to show that the Dehn surgery points on $X_0(M)$ (or $\bar{X}_0(M)$) are fixed by $\hat{\sigma}$.
Assume that $M(p/q)$ is hyperbolic and consider a loop $\gamma\in M(p/q)$. As $\sigma$ extends to $M(p/q)$, the length of a geodesic representative for $\gamma$, and the length of a geodesic representative for $\sigma(\gamma)$ must be equal. Since $M(p/q)$ is a closed hyperbolic 3-manifold all loops correspond to loxodromic elements in the holonomy representation where $\gamma$ corresponds to $\pm A \in {\text{PSL}}_2({\ensuremath{\mathbb{C}}})$. Let $\ell_0(\gamma)$ denote the (complex) translation length of a geodesic representative for $\gamma$. Then (see [@MR1937957] p 372) $$\pm {\text{tr}}\gamma/2 = \cosh(\ell_0(\gamma)/2).$$ If a holonomy representation takes $\sigma(\gamma)$ to a matrix $\pm B$ we see that $ \pm {\text{tr}}A = {\text{tr}}B.$ Therefore, for all $\rho:\pi_1(M)\rightarrow {\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ that correspond to holomony representations of hyperbolic Dehn fillings (since $\sigma$ is orientation preserving) and all $[\gamma] \in \pi_1(M(p/q))$ we have $\chi_{\rho}([\gamma]) = \chi_{\rho}(\sigma_*([\gamma])$. In other words, $\hat{\sigma}(\chi_{\rho})=\chi_{\rho}$. Therefore, $\hat{\sigma}$ is fixed in $\bar{X}_0(M)$.
It follows that $\hat{\sigma}:X_0(M) \rightarrow X_0(M)$ satisfies $\hat{\sigma}(\chi_{\rho}(g)) = \chi_{\rho}(\sigma_*(g)) = \pm \chi_{\rho}(g)$ for all $g\in \pi_1(M)$. By the discussion in Section \[section:PSL\] we conclude that the signs will be equal if $g$ and $\sigma(g)$ are in the same coset. By definition, this occurs when $g$ and $\sigma_*(g)$ have the same image in the map to ${\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}}$.
If $\sigma$ is a symmetry that fixes unoriented free homotopy classes of loops, then the induced action $\hat{\sigma}$ on $\bar{X}(M)$ is trivial; the induced action on the representation variety sends an element of $\mathrm{(P)SL}_2({\ensuremath{\mathbb{C}}})$ to a conjugate or the inverse of a conjugate. Indeed there are examples [@MR3078072] where the induced action $\hat{\sigma}$ is trivial even when $\sigma$ does not fix unoriented free homotopy classes of loops.
When $M$ is a knot complement in $S^3$ then the condition that for all $g\in \pi_1(M)$, $\sigma_*(g)$ and $g$ have the same image in the surjection $\pi_1(M) \rightarrow {\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}}$ holds, so we have the following immediate corollary.
\[cor:symmetiesandcanonicalcomponent\] Let $K$ be a hyperbolic knot in $S^3$ with exterior $M$. A canonical component of the ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ character variety, $\bar{X}_0(M)$ is a subset of the fixed point set of $\hat{\sigma}$. A canonical component of the ${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ character variety, $X_0(M)$, is a subset of the fixed point set of $\hat{\sigma}$.
$\mathrm{(P)SL}_2({\ensuremath{\mathbb{C}}})$ Culler-Shalen Theory for Orbifolds {#section:PSLOrbs}
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We have proven Theorem \[orbifoldTreeAction\], which says that given an action of $\pi_1^{orb}(Q)$ on a tree that is non-trivial and without inversions, we can detect an essential 2-suborbifold that is dual to the action. Now we show how to use valuations from ${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ and ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ character varieties to obtain such actions. Specifically, we show that the valuation theory aspect of Culler and Shalen’s work holds for orbifolds in both the ${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ and ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ settings.
Let $F$ be a field with discrete valuation $v:F^*\rightarrow {\ensuremath{\mathbb{Z}}}$. The valuation ring is $${\ensuremath{\mathcal{O}}}_v=\{0\} \cup \{ a\in F^*: v(a)\geq 0\}.$$We construct a simplicial tree $T=T_{F,v}$ on which ${\text{SL}}_2(F)$ and ${\text{PSL}}_2(F)$ act simplicially and without inversions.
The tree $T$ is defined as follows. The vertices of $T$ are the homothety classes of lattices in $F^2$ and for lattices $\Lambda_1$ and $\Lambda_2$ the distance between the vertices $[\Lambda_1]$ and $[\Lambda_2]$ is $v(\det A)$ where $A:F^2\rightarrow F^2$ is a linear transformation taking $\Lambda_1$ to $\Lambda_2$. The group ${\text{SL}}_2(F)$ acts on $T$ by linear automorphisms. By Theorem \[orbifoldTreeAction\], an action that is non-trivial and without inversions (non-canonically) identifies essential $2$-suborbifolds dual to it. Note that the action of $-I\in {\text{SL}}_2(F)$ fixes every vertex of $T$ as it takes any lattice $\Lambda$ to itself. Let $\Phi:{\text{SL}}_2(F) \rightarrow {\text{PSL}}_2(F)$ be the natural map. It follows that there is a well-defined action of ${\text{PSL}}_2(F)$ on $T$ as follows. Let $\bar{A} \in {\text{PSL}}_2(F)$; we define $\bar{A}\cdot T= A\cdot T$ where $A\in \Phi^{-1}(A)$. (Given $\psi:\Gamma \rightarrow {\text{PSL}}_2(F)$, we are not assuming that $\psi(\Gamma)\subset {\text{PSL}}_2(F)$ lifts to ${\text{SL}}_2(F)$; we are simply using the pull-back.)
The action of an element $A \in {\text{SL}}_2(F)$ on $T$ fixes a vertex if and only if $A$ is conjugate in $\mathrm{GL}_2(F)$ to an element in ${\text{SL}}_2({\ensuremath{\mathcal{O}}}_v)$. This is equivalent to ${\text{tr}}(A) \in {\ensuremath{\mathcal{O}}}_v$. Therefore if there is an $A \in {\text{SL}}_2(F)$ such that $v(A)<0$ this $A$ cannot stabilize any vertex and the action is non-trivial.
Let ${\rm GL}_2(F)^+$ be the kernel of the map ${\rm{GL}}_2(F)\rightarrow {\ensuremath{\mathbb{Z}}}\rightarrow {\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}}$. By [@MR0476875 Section II 1.2-1.3], the group ${\rm GL}_2(F)$ acts with inversions, but ${\rm GL}_2(F)^+$ acts on $T$ without inversions. Since, ${\text{SL}}_2(F)\subset {\rm GL}_2(F)^+,$ both ${\text{SL}}_2(F)$ and ${\text{PSL}}_2(F)$ act without inversions.
With Theorem \[orbifoldTreeAction\] we have the following, as in [@MR1835066 Lemma 1].
\[prop:valuationgivesessential\] Let $v$ be a discrete valuation on a field $F$ and let $T=T_{F,v}$ be the associated tree. If a finite volume 3-orbifold $Q$ has a representation $\psi:\pi_1^{orb}(Q)\rightarrow \mathrm{(P)SL}_2(F)$ with $v({\text{tr}}(\psi(\gamma)))<0$ for some $\gamma \in \pi_1^{orb}(Q)$ then this induces a non-trivial action of $\pi_1^{orb}(Q)$ on $T$ and dual to this action is an essential $2$-suborbifold of $Q$.
Similarly, the following is essentially [@MR1835066 Corollary 3] using [@MR1835066 Lemma 2] and follows from Lemma \[lemma:stabilizers\].
\[prop:generaltype12\] Let $Q$ be finite volume 3-orbifold. Let $v$ be a discrete valuation on a field $F$, let $T=T_{F,v}$ be the associated tree and let $\psi:\pi_1^{orb}(Q)\rightarrow \mathrm{(P)SL}_2(F)$ such that there is a $\gamma \in \pi_1^{orb}(Q)$ with $v({\text{tr}}(\psi(\gamma)))<0$. Then
1. There is a unique primitive element $\gamma \in \pi_1^{orb}(\partial Q)$ such that $v({\text{tr}}(\psi(\gamma))) \geq 0$. Then $Q$ contains an essential $2$-suborbifold dual to the action with peripheral element $\gamma$.
2. For all $\gamma \in \pi_1^{orb}(\partial Q)$, we have $v({\text{tr}}(\psi(\gamma)))\geq 0$. Then $Q$ contains a closed essential $2$-suborbifold dual to the action.
Note that $(1)$ corresponds to having a unique primitive element $\gamma$ with $\psi(\gamma) \in {\ensuremath{\mathcal{O}}}_F$, and $(2)$ corresponds to all $\psi(\gamma) \in {\ensuremath{\mathcal{O}}}_F$. Now we discuss valuations from ANI points and points at infinity in a bit more detail.
ANI-Point
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See [@MR1835066 page 51] for a discussion of this in the ${\text{SL}}_2({\ensuremath{\mathbb{C}}})$ case for manifolds.
Let $\Gamma$ be a finitely generated group, and assume that there is a representation $\rho:\Gamma \rightarrow {\text{SL}}_2(F)$, or $\bar{\rho}:\Gamma \rightarrow {\text{PSL}}_2(F)$ where $F$ is a number field. An [*algebraic non-integral or ANI point*]{} is a $\chi_{\rho}$ or $\chi_{\bar{\rho}}$ where there is $\gamma \in \Gamma$ such that ${\text{tr}}(\rho(\gamma))\not \in {\ensuremath{\mathcal{O}}}_F$, or $\pm {\text{tr}}(\bar{\rho}(\gamma)) \not \in {\ensuremath{\mathcal{O}}}_F$. This is equivalent to the definition given earlier in Sections \[section:SL\] and \[section:PSL\]. Therefore, there is some prime ideal $\pi$ of ${\ensuremath{\mathcal{O}}}_F$ such that the $\pi$-adic valuation $v$ has $v( {\text{tr}}(\rho(\gamma)))<0$, or similarly $v(\pm {\text{tr}}(\bar{\rho}(\gamma)))<0$. Since $v(-1)=1$ this is well-defined. It immediately follows that this action is non-trivial and without inversions.
In this case, we can rephrase Proposition \[prop:generaltype12\] as the following. Given an ANI point $x$ in $X(Q)$ (or $\bar{X}(Q)$) then either
1. There is a unique primitive element $\gamma\in \pi_1^{orb}(\partial Q)$ such that $I_{\gamma}(x)$ (or $f_{\gamma}(x)$) is in ${\ensuremath{\mathbb{C}}}$. Then $Q$ contains an essential 2-suborbifold dual to the action with peripheral element $\gamma$.\
or
2. For all $\gamma \in \pi_1^{orb}(\partial Q)$ we have $I_{\gamma}(x)$ (or $f_{\gamma}(x)$) in ${\ensuremath{\mathbb{C}}}$. Then $Q$ contains a closed essential 2-suborbifold dual to the action.
This is analogous to the statement for manifolds.
Ideal Point
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Let $\Gamma$ be a finitely generated group. Let $X$ be a curve component of $X(\Gamma)$ and identify $X$ with a smooth projective model. Then $I_{\gamma}$ is an element of the function field of $X$, and we can consider $I_{\gamma}$ to be a meromorphic function. For a fixed $x\in X$ there is a valuation $v_x$ on the function field ${\ensuremath{\mathbb{C}}}(X)$ which assigns to each rational function its order at $x$. This valuation extends to a valuation $w$ on $K$, the function field of the subvariety $R$ of $R(M)$ that maps to $X$ under the trace map. Using $w$, ${\text{SL}}_2(K)$ acts on $T=T_w$ by restricting the action of ${\text{GL}}_2(K)$.
Using the tautological representation $\mathcal{P}:\Gamma \rightarrow {\text{SL}}_2(K)$, the action of ${\text{SL}}_2(K)$ on $T$ can be pulled back via $\mathcal{P}$ to an action of $\Gamma$ on $T$. This action is without inversions, and is non-trivial if $x$ is an ideal point. This works as follows. The tautological representation is $\mathcal{P}:\Gamma \rightarrow {\text{SL}}_2({\ensuremath{\mathbb{C}}}[X])$ where ${\ensuremath{\mathbb{C}}}[X]$ is the coordinate ring of $X$, is defined by $$\mathcal{P}(\gamma) = {\left(\begin{array}{cc}a_{\gamma} & b_{\gamma} \\ c_{\gamma} & d_{\gamma} \end{array}\right)}$$ where the coordinates are functions of the representations. For example, $a_{\gamma}$ is the element of the coordinate ring ${\ensuremath{\mathbb{C}}}[X]$ that for an $x \in X$ produces $a_{\gamma}(x)\in {\ensuremath{\mathbb{C}}}$. This gives an action of $\Gamma$ on $T_w$ by $$\gamma \cdot [\Lambda] = [\mathcal{P}(\gamma) \cdot \Lambda ]$$ specifically, if $\{e,f\}$ is a basis for $\Lambda$ we have $ \gamma \cdot [\Lambda] $ given by $${\left(\begin{array}{cc}a_{\gamma} & b_{\gamma} \\ c_{\gamma} & d_{\gamma} \end{array}\right)} \left( \begin{array}{c} e \\ f \end{array} \right) = \left( \begin{array}{c} a_{\gamma} e +b_{\gamma}f \\ c_{\gamma}e + d_{\gamma} f\end{array} \right)$$ which determines a basis for a lattice. (The result is the homothety class of this lattice.) Note that $a_{\gamma}, b_{\gamma}, c_{\gamma}, d_{\gamma}, e,$ and $f$ are all in $K$.
Boyer and Zhang [@MR1670053 Section 4] extended this theory to ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ character varieties. Using a central ${\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}}$ extension, they define a compatible tautological representation $\bar{\mathcal{P}}: \Gamma \rightarrow {\text{PSL}}_2({\ensuremath{\mathbb{C}}})$. If there is a curve component $X$ in $X(\Gamma)$ that under the natural map, maps to $\bar{X}$ in $\bar{X}(\Gamma)$, then we see that for an order of vanishing valuation (at an ideal point), $I_{\gamma}$ blows up at an ideal point $x\in X$ if and only if $f_{\gamma}$ blows up at the image of $x$ in $\bar{X}$.
The construction is as follows. They show (Section 3) that for a finitely generated group $\Gamma$ and representation $\bar{\rho}:\Gamma \rightarrow {\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ there is a central extension $\phi:\hat{\Gamma}\rightarrow \Gamma$ of $\Gamma$ by ${\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}}$ and a representation $\hat{\rho}:\hat{\Gamma}\rightarrow {\text{SL}}_2({\ensuremath{\mathbb{C}}})$ such that the commutative diagram $$\begin{tikzcd}
1 \ar[r] & {\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}}\ar[d,equal] \ar[r] & \hat{\Gamma} \ar[d, "\hat{\rho}"] \ar[r,"\phi"]& \Gamma \ar[d,"\bar{\rho}"] \ar[r] & 1 \\
1 \ar[r] & {\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}}\ar[r] & {\text{SL}}_2({\ensuremath{\mathbb{C}}}) \ar[r, "\Phi"] & {\text{PSL}}_2({\ensuremath{\mathbb{C}}}) \ar[r] & 1
\end{tikzcd}$$ is exact. We call $(\hat{\Gamma}, \phi, \hat{\rho})$ the central ${\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}}$ extension of $\Gamma$ lifting $\bar{\rho}$. As mentioned in [@MR1670053] the central extensions of $\Gamma$ by ${\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}}$ are classified by $H^2(\Gamma, {\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}})$ and $\bar{\rho}$ determines an element $w_2(\bar{\rho}) \in H^2(\Gamma, {\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}})$. This is zero if and only if $\hat{\Gamma} \cong \Gamma \times {\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}}$. (That is, if $\hat{\rho}$ lifts to an element of $R(\Gamma)$.) When $H^2(\Gamma, {\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}})$ is trivial, $\hat{\Gamma}\cong \Gamma \times {\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}}$ for every $\bar{\rho}$ and each $\bar{\rho}$ lifts to $R(\Gamma)$.
Let $\Phi:{\text{SL}}_2({\ensuremath{\mathbb{C}}})\rightarrow {\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ be the natural quotient. Let $\bar{X} \subset \bar{X}(\Gamma)$ be a curve. Then there is a subvariety $\bar{R}\subset \bar{t}^{-1}(\bar{X})$ which is 4-dimensional. Let $\bar{\rho}$ be a smooth point in $\bar{R}$ and $(\hat{\Gamma}, \phi, \hat{\rho})$ the central extension. Then there is a variety $S\subset R(\hat{\Gamma})$ containing $\hat{\rho}$ and a dominating regular map $\phi_*:S\rightarrow \bar{R}$. The function field ${\ensuremath{\mathbb{C}}}(S)$ is a finitely generated extension of ${\ensuremath{\mathbb{C}}}(\bar{X})$ by identifying $f\in {\ensuremath{\mathbb{C}}}(\bar{X})$ with $f\circ \bar{t} \circ \phi_* \in {\ensuremath{\mathbb{C}}}(S)$.
There is a tautological representation $\hat{\mathcal{P}}:\hat{\Gamma}\rightarrow {\text{SL}}_2( {\ensuremath{\mathbb{C}}}(S))$ as discussed above. The tautological representation $\bar{\mathcal{P}}:\Gamma \rightarrow {\text{PSL}}_2({\ensuremath{\mathbb{C}}}(S))$ is given by $$\bar{\mathcal{P}}(\gamma) = \Phi( \hat{\mathcal{P}}(\phi^{-1}(\gamma))).$$ So that $$\bar{\mathcal{P}}(\gamma)(\hat{{\rho}}) = \bar{\rho}(\gamma)$$ for every $\gamma\in \Gamma$ and $\bar{\rho}\in \bar{R}$ and $\hat{\rho}\in S$ such that $\phi_*( \hat{\rho})=\bar{\rho}$.
We can rephrase Proposition \[prop:generaltype12\] as the following. Given an ideal point $x$ in $X(Q)$ (or $\bar{X}(Q)$) then either
1. There is a unique primitive element $\gamma\in \pi_1^{orb}(\partial Q)$ such that $I_{\gamma}(x)$ (or $f_{\gamma}(x)$) is in ${\ensuremath{\mathbb{C}}}$. Then $Q$ contains an essential 2-suborbifold dual to the action with peripheral element $\gamma$.\
or
2. For all $\gamma \in \pi_1^{orb}(\partial Q)$ we have $I_{\gamma}(x)$ (or $f_{\gamma}(x)$) in ${\ensuremath{\mathbb{C}}}$. Then $Q$ contains a closed essential 2-suborbifold dual to the action.
This is analogous to [@MR1670053 Proposition 4.7]
The proof of Theorem \[thm:maintheorem\] {#mainsection}
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If $F$ is detected by an action of $\pi_1^{orb}(Q)$ on a tree $T$ we now show that there is an action of $\pi_1(M)$ on $T$ which detects a symmetric essential surface in $M$. Let $p_Q:\tilde{Q}\rightarrow Q$, $p_M:\tilde{Q}\rightarrow M$, and $p:M\rightarrow Q$ be the covering maps so that $p\circ p_M=p_Q$. The action of $\pi_1^{orb}(Q)$ on $T$ gives a map $\tilde{f}:\tilde{Q}\rightarrow T$, which induces the maps $f_Q= \tilde{f} \circ p_Q^{-1} $ and $f_M= \tilde{f} \circ p_M^{-1}$ where $f_Q:Q\rightarrow T$ and $f_M:M\rightarrow T$. $$\begin{tikzcd}
\tilde{Q} \ar[dr,"\tilde{f}"] \ar[d,"p_M"] \arrow[dd, bend right=60, swap, "p_Q"] & {} \\
M \ar[r,"f_M"] \ar[d,"p"] & T \\
Q \ar[ur, swap,"f_Q"] & {}
\end{tikzcd}$$ The map $f_Q$ is well-defined because of the $\pi_1^{orb}(Q)$ equivariance of $\tilde{f}$. The same is true of $f_M$ because $\pi_1(M)$ injects into $\pi_1^{orb}(Q)$.
\[lemma:liftedsurfaceisdetected\]
If $F_Q$ is detected by an action of $\pi_1^{orb}(Q)$ on a tree $T$ then $p^{-1}(F_Q)$ is a symmetric essential surface in $M$ detected by the induced action of $\pi_1(M)$ on $T$.
By Section \[actionontreesection\], the essential 2-suborbifold $F_Q$ of $Q$ is associated to a $\pi_1^{orb}(Q)$ equivariant map $\tilde{f}:\tilde{Q}\rightarrow T$ which is transverse of $E$. The map $\tilde{f}$ induces the maps $f_Q$ and $f_M$ as above.
By Theorem \[orbifoldTreeAction\], $\tilde{F}=\tilde{f}^{-1}(E)$ is an essential surface in $\tilde{Q}$ and $F_Q=p_Q(\tilde{F})=f_Q^{-1}(E).$ Now, $\tilde{f}$ is $\pi_1^{orb}(Q)$ equivariant, and since $\pi_1(M)$ injects into $\pi_1^{orb}(Q)$ it is also $\pi_1(M)$ equivariant. By Lemma \[suborbliftinglemma\], $F_M=p_M(\tilde{F})=f_M^{-1}(E)$ is a symmetric essential surface in $M$ because $F_M=p^{-1}(F_Q)$. Because $F_M=f_M^{-1}(E)$ it is the surface detected by the induced action of $\pi_1(M)$ on $T$.
\[lemma:SL=PSL\] Assume that $\bar{X}(M)$ lifts to $X(M)$, and let $x\in X(M)$ or an ideal point of $X(M)$. Let $\bar{x}$ be the image of $x$ under the covering map $X(M)\rightarrow \bar{X}(M)$. Assume that associated to $x$ there is a discrete valuation $v$ on a field $F$ where $T$ is the associated ${\text{SL}}_2$-tree, and there is a representation $\phi:\pi_1(M) \rightarrow {\text{SL}}_2(F)$ with $v({\text{tr}}(\phi(\gamma)))<0$ for some $\gamma \in \pi_1(M)$. Then associated to $\bar{x}$ there is an action on $T$ and if $x$ detects a surface $F$ then $\bar{X}$ also detects $F$.
Similarly, assume that associated to $\bar{x}$ there is a discrete valuation $v$ on a field $F$ where $T$ is the associated tree, and there is a representation $\phi:\pi_1(M) \rightarrow {\text{PSL}}_2(F)$ with $v({\text{tr}}(\phi(\gamma)))<0$ for some $\gamma \in \pi_1(M)$. Then $x$ has an action on $T$ and if $\bar{x}$ detects $F$ then $x$ also detects $F$.
The points $x$ and $\bar{x}$ are either both affine points or both ideal points. If they are affine points, the representations associated to $x$ and $\bar{x}$ are different only by signs, so any valuation associated to one point induces an identical valuation on the other. If they are ideal points, the order of vanishing valuations are exactly the same. Since $-I$ acts on the tree by fixing every vertex, the lemma follows.
Now we are ready to prove Theorem \[thm:maintheorem\], which we restate:
By Lemma \[lemma:SL=PSL\] it is enough to consider the ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ character variety. By Section \[actionontreesection\] there is a $\pi_1(M)$ equivariant map $\tilde{f}:\tilde{Q} \rightarrow T$ that is transverse to $E$ such that the surfaces $\tilde{F}=\tilde{f}^{-1}(E)$ and $F_M= p_M(\tilde{F})=f_M^{-1}(E)$ are two sided and bicollared and $F_M$ is essential.
Recall that $p:M\rightarrow Q$ is the covering map which induces $p_*:\pi_1(M) \rightarrow \pi_1^{orb}(Q)$. The map $\hat{p}:\bar{X}(Q) \rightarrow \bar{X}(M)$ is given by the following: if $\chi_{\bar{\rho}} \in \bar{X}(Q)$ then $\hat{p}(\chi_{\bar{\rho}}) = \chi_{\bar{\rho}'}$ where for $\gamma\in \pi_1(M)$ we have $$\chi_{\bar{\rho}'}(\gamma) = \chi_{\bar{\rho}\circ p_*}(\gamma) = \chi_{\bar{\rho}}(p_*(\gamma)).$$ It follows that $\hat{p}$ is defined for all affine points of $\bar{X}(Q)$, and if $\chi_{\bar{\rho}}$ is an affine point, then so is its image, $\chi_{\bar{\rho}'}$.
Long and Reid [@MR1739217 Theorem 3.4] proved that given a covering $p:M\to Q$ as above where $Q$ has a flexible cusp, then the induced map $\hat{p}$ is a birational equivalence. Let $X$ be a smooth projective completion of $\bar{X}(Q)$. Then $X$ is also a smooth projective completion of $\bar{X}(M)$, and $\hat{p}$ induces an automorphism of $X$. If $\bar{x}$ is an ideal point of $\bar{X}(Q)$ then there is a $\gamma \in \pi_1^{orb}(Q)$ such that $f_{\gamma}(\bar{x})$ blows up. Assume that $\hat{p}(\bar{x})=\bar{x}'$. Then the function $f_{p_*(\gamma)}(\bar{x}')$ is the image of $f_{\gamma}(\bar{x})$ under the induced mapping. The map $p_*$ is an injection, and since the cover is finite, $p_*(\pi_1(M))$ is a finite index subgroup of $\pi_1^{orb}(Q)$. But since $f_{\gamma}(\bar{x})$ blows up, so does $f_{\gamma^n}(\bar{x})$ for all $n$. In particular, $\gamma$ cannot be torsion. Moreover, there are affine points $\chi_{\bar{\rho}_i}$ converging to $\bar{x}$ where $f_{\gamma}(\chi_{\bar{\rho}_i})$ is unbounded. The images, under the map induced by $p$, of infinitely many $\chi_{\bar{\rho}_i}$ are in any neighborhood of the image of $\bar{x}$. Taking $n$ large enough so that $ \gamma^n\in p_*(\pi_1(M))$ we let $\delta\in \pi_1(M)$ such that $p_*(\delta)=\gamma^n$. The corresponding evaluation functions are given by $$f_{\delta}(\chi_{\bar{\rho}_i\circ p_*}) = {\text{tr}}(\bar{\rho}_i(p_*(\delta)))^2-4$$ and we see that these evaluation functions are unbounded. We conclude that if $\bar{x}$ is an ideal point of $\bar{X}(Q)$ then its image is an ideal point of $\bar{X}(M)$. We have shown that $\hat{p}$ is defined on all of $\bar{X}(Q)$, and affine points map to affine points and (in the natural extension of $\hat{p}$ to smooth projective models) ideal points map to ideal points.
First, we consider the case when $\phi$ is associated to an ideal point $\bar{x}$ and the corresponding valuation is the order of vanishing valuation. Below we will consider representations as extended to $\bar{X}$, a smooth projective completion of $\bar{X}_0(M)$. Let $\Gamma=\pi_1(M)$ and let $\Gamma'=\pi_1^{orb}(Q)$. We refer the reader to [@MR1670053 Section 4] for details about the following construction. The ideal point $\bar{x}$ detects a surface from the action of $\pi_1(M)$ on the tree $T$. Let $\bar{R}\subset \bar{t}^{-1}(\bar{X})$ so that $\bar{R}\subset \bar{R}(\Gamma)$. For $\bar{\rho} \in \bar{R}$ such that $\bar{t}(\rho)=\bar{x}$, let $(\hat{\Gamma}, \phi, \hat{\rho})$ be the ${\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}}$ central extension of $\Gamma$ lifting $\bar{\rho}$. Then there is a $S\subset R(\hat{\Gamma})$ containing $\hat{\rho}$ and a dominating regular map $\phi_*:S\rightarrow \bar{R}$. The function field ${\ensuremath{\mathbb{C}}}(S)$ is a finitely generated extension of the function field ${\ensuremath{\mathbb{C}}}(\bar{X})$. An analogous construction follows for $\Gamma'$ where we will embellish the relevant spaces with a $'$ symbol. Here the relevant central extension will be associated to the representation $p_*(\bar{\rho}):\Gamma'\rightarrow {\text{PSL}}_2({\ensuremath{\mathbb{C}}})$.
The action corresponding to $\bar{x}$ is induced by the tautological representation $\bar{\mathcal{P}}:\pi_1(M)\rightarrow {\text{PSL}}_2({\ensuremath{\mathbb{C}}}(S))$. Let $\bar{x}'\in \bar{X}(Q)$ such that $\hat{p}(\bar{x}')=\bar{x}$. Such an $\bar{x}'$ exists because the map extends to an automorphism $\bar{X}\rightarrow \bar{X}$ since $\bar{X}$ is a smooth projective completion of both $\bar{X}(Q)$ and $\bar{X}(M)$. The function field of $\bar{x}'$ and the function field of $\bar{x}$ are therefore isomorphic and are contained in ${\ensuremath{\mathbb{C}}}(S)$.
Both $\bar{x}$ and $\bar{x}'$ with the order of vanishing valuation define actions on the tree $T$ for ${\text{SL}}_2({\ensuremath{\mathbb{C}}}(S))$. The fact that $\bar{x}$ has a negative valuation means that in the unique extension of the valuation of ${\ensuremath{\mathbb{C}}}(S)$ it also values negatively. As stated above, since $\bar{x}$ is an ideal point, so is $\bar{x}'$, and so it has a negative valuation that extends to ${\ensuremath{\mathbb{C}}}(S)$ as well. By Theorem \[orbifoldTreeAction\] there is a surface $F_Q$ in $Q$ dual to the action associated to $\bar{x}'$. By Lemma \[lemma:liftedsurfaceisdetected\] the surface, $F_Q$ lifts to a symmetric surface $F_M$ in $M$ which is detected by the induced action of $\Gamma=\pi_1(M)$ on $T$. It remains to show that this induced action is an action at $\bar{x}$. That is, the induced action from Lemma \[lemma:liftedsurfaceisdetected\] is the restriction $\mathcal{P}'\hspace{-0.1cm}\mid_{\Gamma}$ of the tautological representation $\mathcal{P}':\Gamma'\rightarrow {\text{PSL}}_2({\ensuremath{\mathbb{C}}}(S))$. It is left to show that this is the tautological representation $\mathcal{P}:\Gamma\rightarrow {\text{PSL}}_2({\ensuremath{\mathbb{C}}}(S))$ associated to $\bar{x}$. If $\bar{x}$ is associated to the representation $\bar{\rho}:\Gamma\rightarrow {\text{PSL}}_2({\ensuremath{\mathbb{C}}})$, then the tautological representation satisfies $\mathcal{P}(\gamma)(\hat{\rho})=\bar{\rho}(\gamma)$ for all $\gamma\in \Gamma$. The restriction of $\mathcal{P}'$ to $\Gamma$ satisfies $\mathcal{P}'(p^*(\gamma))(\hat{\rho}')=\bar{\rho}'(p^*(\gamma))$, so it is enough to see that $$\bar{\rho}'(p^*(\gamma)) = \bar{\rho}(\gamma)$$ which is the case since $\hat{p}(\bar{x}')=\bar{x}$ so that $p_*(\bar{\rho}')=\bar{\rho}$ and therefore $p_*(\bar{\rho}')(\gamma) = \bar{\rho}(p^*(\gamma))$ as needed.
Finally, we consider the case when $\phi$ is associated to an affine point $\chi_{\bar{\rho}} \in \bar{X}(M)$. Then $\bar{\rho}:\pi_1(M) \rightarrow {\text{PSL}}_2(F)$ for some field $F\subset {\ensuremath{\mathbb{C}}}$ and there is a valuation $v$ on $F$ such that $ v(\pm {\text{tr}}(\bar{\rho}(\gamma)))<0$. Assume that $F$ is the smallest field containing the image. By the above discussion there is an affine point $\chi_{\bar{\rho}'}\in \bar{X}(Q)$ such that $\hat{p}(\chi_{\bar{\rho}'})=\chi_{\bar{\rho}}$. Therefore, $\bar{\rho}':\pi_1^{orb}(Q) \rightarrow {\text{PSL}}_2(F')$ for some $F'\subset {\ensuremath{\mathbb{C}}}$. Again, assume that $F'$ is the smallest such field. Recall that $\hat{p}(\chi_{\bar{\rho}'}) = \chi_{\bar{\rho}' \circ p_*}$. If $\gamma \in \pi_1(M)$ then $$\chi_{\bar{\rho}}(\gamma) = \chi_{\bar{\rho}' \circ p_*}(\gamma) = \chi_{\bar{\rho}'}(p_*(\gamma))\in F'.$$ We conclude that $F\subset F'$.
Let $\gamma\in \pi_1(M)$ be such that $v(\pm {\text{tr}}(\bar{\rho}(\gamma)))<0$. Since $\bar{\rho}'(p_*(\gamma)))=\bar{\rho}(\gamma)$ we have $v(\pm {\text{tr}}(\bar{\rho}'(p_*(\gamma))))<0$ as well. The valuation $v$ on $F$ extends to a valuation $v'$ on $F'$. Let $T=T_{v',F'}$. By Theorem \[orbifoldTreeAction\] there is a surface $F_Q$ in $Q$ dual to the action associated to $\chi_{\bar{\rho}'}$. By Lemma \[lemma:liftedsurfaceisdetected\] the surface, $F_Q$ dual to the action from $\chi_{\bar{\rho}'}$ lifts to a symmetric surface $F_M$ in $M$ which is detected by the induced action of $\pi_1(M)$ on $T$. The proof that this induced action of $\pi_1(M)$ on $T$ is the same as the action from the valuation corresponding to $\chi_{\bar{\rho}}$ follows similar to the above.
The condition that the quotient orbifold has a flexible cusp should be thought of as a generic condition. The only known knot complements in $S^3$ which cover orbifolds with rigid cusps are the dodecahedral knots [@MR1184399] and the figure-eight knot.
The following corollary follows immediately because ideal points on a canonical component of the character variety of a one cusped manifold only detect surfaces with a single boundary slope.
Let $M$ be a compact, orientable, one cusped, hyperbolic 3-manifold. If $G$ is a finite group of orientation preserving symmetries of $M$ such that the orbifold $M/G$ has a flexible cusp, then for every slope detected at an ideal point of $\bar{X}_0(M)$ there exists an essential surface with that slope that is fixed set wise by every element of $G$.
We now prove Corollary \[maincor\].
By [@MR2303551], the ideal points of $X_0(M)$ detects at least two distinct strict boundary slopes. Then the ideal points of $\bar{X}_0(M)$ detect at least two distinct boundary slopes of $M$. Because $M$ is one cusped, each ideal point of $\bar{X}_0(M)$ only essential surfaces of a single slope. Then $\bar{X}_0(M)$ has at least two ideal points that strongly detect distinct slopes. Then by Theorem \[thm:maintheorem\] these two ideal points detect essential surfaces which have distinct boundary slopes and are preserved by every orientation preserving symmetry of $M$.
Two-Bridge Knots {#examplesection}
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Two-bridge knots are those knots admitting a projection with only two maxima and two minima as critical points. To each rational number $p/q$ with $q$ odd we associate a two-bridge knot; when $q$ is even this procedure produces a two component link. There are various ways to see this association, we will demonstrate it through continued fractions (see [@MR1959408]). Continued fractions for $p/q$ can be used to determine surfaces in the knot complement by [@MR778125]. There are different conventions for the continued fractions. We follow the treatment in [@MR778125] with exception of signs. (Instead of the treatment below, they have a $-$ after each $a_i$ instead of a $+$. In the notation below, this changes the sign on the even $a_i$ terms.)
![Two-bridge knot []{data-label="figure:twobridge"}](fourplatgen.pdf)
For each $p/q$ we consider continued fractions that satisfy $$\frac{p}q = r+ \frac{1}{a_1+ \cfrac{1}{a_2+\cfrac{1}{ \ddots+\cfrac{1}{a_s}}}}$$ for any integer $r$. We write $p/q=r+[a_1,a_2,\dots, a_s]$. The knot $ K(p,q)$ is the boundary of the surface obtained by plumbing together $s$ bands in a row, the $i^{\mathrm{th}}$ band having $(-1)^{i-1}a_i$ half-twists, right-handed for a positive integer and left handed for a negative integer. (See Figure 2 of [@MR778125 page 227].) Let $\Sigma[a_1,\dots, a_s]$ be the corresponding branched surface. Therefore, every two-bridge knot has a diagram as in Figure \[figure:twobridge\] where the numbers in each box indicate the number of half twists. The knot $K(p,q)$ is the same as the knot $K(p',q')$ exactly when $q=q'$ and $p'\equiv p^{\pm1} \pmod q.$
As mentioned, many continued fractions give the same $p/q$. Hatcher and Thurston showed that all $\pi_1$-injective surfaces in a two-bridge knot can be found by computing all the continued fraction expansions for $p/q$. Every continued fraction satisfying $|a_i|\geq 2$ for every $i$ corresponds to a branched surface in the knot complement from which some incompressible surface can be seen using plumbing. Surfaces obtained from different continued fractions are never isotopic. The boundary slopes of the surfaces correspond to the continued fraction in the following way ([@MR778125] Proposition 2). For a given continued fraction $[a_1,a_2,\cdots,a_s]$ (with $|a_i| \geq 2$) the surfaces obtained from the corresponding branched surface all have slope $$2((n^+-n^-)-(n_0^+-n_0^-)),$$ where $n^+$ and $n^-$ are the number of positive and negative terms respectfully in $$[a_1,-a_2,a_3,-a_4,\cdots,\pm a_s],$$ and where $n_0^+$ and $n_0^-$ are the number of positive and negative terms respectfully in the unique continued fraction with all even terms after swapping the signs of the even numbered terms (i.e. swap the sign of the 2nd term, 4th term, etc.).
Each two-bridge knot has a symmetry group of order 4 or order 8. The order 4 symmetry group is isomorphic to the Klein four group. Every symmetry in this group acts trivially on the free homotopy classes of unoriented loops in $S^3-K(p,q)$ and therefore the induced action on the character variety is trivial. (See Section \[section:symmandcanonical\].) When the continued fraction $[a_1,a_2, \dots, a_s]$ representing the knot $K(p,q)$ is palindromic there is a symmetry $\sigma$ of the knot that turns the 4-plat upside down, and these knots have order 8 symmetry group. One can see this from the plumbing description of $K(p,q)$ above. The following lemma allows us to identify which surfaces in a two-bridge knot with this type of symmetry are preserved by $\sigma$.
\[lemma:twistsymmetries\] Let $K(p,q)$ be a two-bridge knot with palindromic twist region pattern, and let $\sigma$ be as above. The action of $\sigma$ takes the branched surface $\Sigma[b_1, b_2 \dots, b_{s-1}, b_s]$ to $(-1)^{s+1} \Sigma[ b_s, b_{s-1}, \dots,b_2, b_1]$.
The action of $\sigma$ turns the 4-plat upside down. For a branched surface given by $[b_1,b_2,\dots, b_s]$ this flips the twist regions with the conventions above, $b_i$ corresponds to $(-1)^{k+1}b_i$ twists which corresponds to the sign differences above.
Now we prove Corollary \[cor:twobridge\], that every hyperbolic two-bridge knot contains at least two symmetric essential surfaces.
First, suppose that $K$ is the figure-eight knot. Then $K$ has exactly one essential surface, $F_4$, with boundary slope $4$, and $K$ has exactly one essential surface, $F_{-4}$, with boundary slope $-4$. Because an orientation preserving homeomorphism of $K$ takes essential surfaces in $K$ to essential surfaces with the same boundary slope, every symmetry of $K$ takes $F_4$ to itself and $F_{-4}$ to itself.
Now, assume that $K$ is a hyperbolic two-bridge knot other than the figure-eight knot. By Reid [@MR1099096], $K$ is not arithmetic. As a result, $K$ admits no hidden symmetries by Reid and Walsh [@MR2443107 Theorem 3.1]. By Margulis [@MR1090825], since $K$ is not arithmetic, there is a unique minimal element in the commensurability class of $S^3-K$. The minimal orientable element is called the commenturator quotient. Neumann and Reid [@MR1184416 Prop. 9.1] show that a non-arithmetic knot has hidden symmetries if and only if this commensurator quotient has a rigid cusp. Therefore, the commensurator quotient of $S^3-K$ has a flexible cusp. It follows that any orbifold that covers $S^3-K$ has a flexible cusp since the torus and pillowcase cannot be covered by a turnover. We conclude that the orbifold obtained from the quotient of the group of orientation preserving symmetries on $K$ has a flexible cusp. The statement now follows from Corollary \[maincor\].
Double Twist Knots
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![A $J(k,l)$ link []{data-label="doubleTwistKnot"}](jkl2.pdf)
A double twist link, often denoted $J(k,l)$, is a link with $k$ and $l$ half twist regions as seen in Figure \[doubleTwistKnot\]. For $J(k,l)$ to be a knot $kl$ must be even; otherwise it is a two component link. It was shown in [@MR2827003] that $\text{(P)SL}_2({\ensuremath{\mathbb{C}}})$ character variety of $J(k,l)$ consists of one component (of characters of irreducible representations) if $k\neq l$ and if $k=l$ then it has two components. We will now concentrate on the case when $k=l$ as these are the only double twist knots which have a component of the character variety that does not consist of characters of abelian representations and is non-canonical. Let $K_n=J(2n,2n)$ be a symmetric double twist knot.
We will explicitly calculate the slopes that correspond to essential surfaces in $S^3-K_n$ and then prove that the symmetric slopes [*are*]{} detected on the canonical component, whereas the non-symmetric slopes [*are not*]{} detected on the canonical component.
The double twist knot $J(l,m)$ is equivalent to the two-bridge knot $K(p,q)$ with $\frac{p}{q}=\frac{l}{1-lm}$ in $\mathbb{Q}/\mathbb{Z}$ (see [@MR2827003]). Therefore, $K_n = K(2n, 1-4n^2)$ in two-bridge notation. We will calculate the slopes of the essential surfaces in the $K_n$ knots by using the continued fraction expansions of their two-bridge knot notation.
Continued Fraction Lemmas
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If a sequence of numbers $a_i,a_{i+1},\cdots a_{i+j}$ repeats $k$ times in a continued fraction we will write this sequence as $(a_i,a_{i+1},\cdots, a_{i+j})_k$. For example $$[2,-2,2,-2]=[(2,-2)_2]$$ and $$[5,3,2,3,2,7,2,3,2,3,2,3,2,3,2]=[5,(3,2)_2,7,(2,3)_4,2]=[5,(3,2)_2,7,2,(3,2)_4].$$
\[lemmaNegativeContinuedFraction\] The following relationship holds between negations of continued fractions $$[a_1,a_2,\cdots,a_s] = -[-a_1,-a_2,\cdots,-a_s].$$
Let $[a_1,a_2,\cdots,a_s]=\frac{p}{q}$. Then $$\frac{-p}{q} = \frac{-1}{a_1+ \frac{1}{a_2+ \cdots +\frac{1}{a_s} } } =\frac{1}{-a_1+ \frac{-1}{a_2+ \cdots +\frac{1}{a_s} } }= \frac{1}{-a_1+ \frac{1}{-a_2+ \frac{-1}{a_3 + \cdots +\frac{1}{a_s}} } }.$$ Continuing in this fashion, $$\frac{-p}{q} = \frac{1}{-a_1+ \frac{1}{-a_2+ \cdots +\frac{1}{-a_s} } }=[-a_1,-a_2,\cdots,-a_s].$$
[\[repeatfractionlemma1\]]{} For any positive integer $s$, $$[(-2,2)_s] = \frac{-2s}{2s+1} \ \text{ and } \ [(2,-2)_s] = \frac{2s}{2s+1}.$$
This follows by induction on $s$. Notice that $[-2,2]=\frac{-2}{3}$. Assuming that $[(-2,2)_s]= \frac{-2s}{2s+1}$, we see that $${[(-2,2)_{s+1}]=\frac{1}{-2+ \frac{1}{2+ \frac{-2s}{2s+1}}} }=\frac{1}{-2+\frac{2s+1}{2s+2}}= \frac{2s+2}{-2s-3}$$ as needed. The second assertion follows from Lemma \[lemmaNegativeContinuedFraction\].
In what follows, we extend the definition of our continued fraction notation to include real number entries. We treat any real number as if it were an integer with respect to its placement in the continued fraction.
[\[repeatfractionlemma2\]]{} For any non-negative integer $k$ and real number $x$, $$[(2,-2)_k, 2,x] = \frac{(2k+1)x+2k}{(2k+2)x+2k+1}.$$
Direct computation shows that $[2,x] = \frac{1}{2+\frac{1}{x}}= \frac{x}{2x+1}$. We proceed by induction on $k$. Assume that $[(2,-2)_k, 2,x] = \frac{(2k+1)x+2k}{(2k+2)x+2k+1}$. An elementary computation shows that $$[(2,-2)_{k+1}, 2,x]=\frac{1}{2+\frac{1}{-2+\frac{(2k+1)x+2k}{(2k+2)x+2k+1}}}= \frac{(2k+3)x+2k+2}{(2k+4)x+2k+3}$$ as needed.
\[repeatfractioncompositionlemma\] Given any continued fraction of length $s$ and any integer $i$ such that $1<i<s$, $$[a_1,a_2,\cdots,a_i,a_{i+1},\cdots,a_s] =[a_1,a_2,\cdots,a_{i-1},\frac{1}{[a_i,a_{i+1},\cdots,a_s]} ].$$
Let $[a_1,a_2,\cdots,a_s]$ be a continued fraction. Then $$[a_1,a_2,\cdots,a_s]=\frac{1}{a_1+\frac{1}{a_2+\cdots+\frac{1}{a_{i-1}+\frac{1}{a_i+\cdots +\frac{1}{a_s}}} }}.$$ Note that $$\frac{1}{a_i+\cdots +\frac{1}{a_s}}= [a_i,\cdots,a_s].$$ Therefore, $$[a_1,a_2,\cdots,a_s]=\frac{1}{a_1+\frac{1}{a_2+\cdots+\frac{1}{a_{i-1}+[a_i,\cdots,a_s]} }}= [a_1,a_2,\cdots,a_{i-1},\frac{1}{[a_i,a_{i+1},\cdots,a_s]} ].$$
Boundary Slopes of Symmetric Double Twist Knots
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The main result of this section, which we will use in Section \[examplesections\] to prove Theorem \[mainresultTheoremdetectedslopesstrict\], is the following.
\[slopeOfEssentialSurfaces\] For $|n|>1$ the slopes of essential surfaces in the knot $K_n$ are exactly $0$, $-4n$, and $-8n+2$. Surfaces of slope $-4n$ are not preserved by the symmetry that turns the 4-plat upside down.
Proposition \[slopeOfEssentialSurfaces\] will follow directly from Proposition \[proposition:continuedfractions\] which establishes that $0$, $-4n$, and $-8n+2$ are the only boundary slopes of $K_n$, and Lemma \[lemma:-4nnotsymmetric\] which shows that $-4n$ is not a symmetric slope.
\[proposition:continuedfractions\] The continued fractions for the knot $K_n$ when $n>1$ are $$[2n,-2n],[2n-1,2,(-2,2)_{n-1}],[(-2,2)_{n-1},-2,-2n+1],[(-2,2)_{n-1},-3,(2,-2)_{n-1}]$$ and these correspond to branched surfaces that give slopes of $0$,$-4n$, $-4n$, and $-8n+2$ respectfully. \[propcontinuedfrac\]
The rational number associated to the two-bridge knot $K_n$ is $\frac{2n}{4n^2-1}$. We will first show that these four continued fractions are expansions of this rational number (modulo ${\ensuremath{\mathbb{Z}}}$) and then show that there are no other continued fraction expansions which are associated to branched surfaces.
The fact that $[2n,-2n]= \frac{2n}{4n^2-1}$ is an elementary computation. This is the unique continued fraction with all even terms. We conclude that $n_0^+=2$ and $n_0^-=0$ since these are the number of positive and negative (respectfully) terms in $[2n,-(-2n)]$. For this expansion $n^+=n_0^+$ and $n^-=n_0^-$ so the associated slope is $2((n^+-n^-)-(n_0^+-n_0^-))=0$.
The second continued fraction is $[2n-1,2,(-2,2)_{n-1}]$. From Lemma \[lemmaNegativeContinuedFraction\] we have $$[2n-1,2,(-2,2)_{n-1}]=-[-2n+1,-2,(2,-2)_{n-1}].$$ By Lemma \[repeatfractionlemma1\] this is $$\frac{-1}{(-2n+1)+\frac{1}{-2+[(2,-2)_{n-1}]}}= \frac{-1}{(-2n+1)+\frac{1}{-2+\frac{2n-2}{2n-1}}}=\frac{2n}{4n^2-1}$$ as needed. We immediately see that $n^+=1$ and $n^-=1+2(n-1)=2n-1$. This continued fraction corresponds to surfaces with slope $2((n^+-n^-)-(n_0^+-n_0^-))=-4n.$
The third continued fraction is $[(-2,2)_{n-1},-2,-2n+1]$. By Lemma \[lemmaNegativeContinuedFraction\] $$[(-2,2)_{n-1},-2,-2n+1]= -[(2,-2)_{n-1},2,2n-1].$$ Therefore we will consider $[(2,-2)_{n-1},2,2n-1]$. By Lemma \[repeatfractionlemma2\] $$[(2,-2)_{n-1}, 2,2n-1] = \frac{(2n-1)^2+2n-2}{(2n)(2n-1)+2n-1}=1-\frac{2n}{4n^2-1}.$$ Therefore, the original continued fraction is $-1+(2n /(4n^2-1))$, which gives the correct fraction. Here $n^+=1$ and $n^-=2n-1$ so it corresponds to surfaces with slope $-4n$ as well.
The fourth continued fraction is $[(-2,2)_{n-1},-3,(2,-2)_{n-1}]$. Call this $\tfrac{p}q$. From Lemma \[repeatfractioncompositionlemma\], $\tfrac{p}q=[(-2,2)_{n-2},-2,\frac{1}{[2,-3,(2,-2)_{n-1}]}].$ By Lemma \[repeatfractionlemma1\], $[(2,-2)_{n-1}]=(2n-2)/(2n-1)$ and so $$[2,-3,(2,-2)_{n-1}]=\frac{1}{2+\frac{1}{-3+[(2,-2)_{n-1}]}} = \frac{4n-1}{6n-1}.$$ Therefore, $\tfrac{p}q=[(-2,2)_{n-2},-2,\frac{6n-1}{4n-1}]$. By Lemma \[lemmaNegativeContinuedFraction\], we can negate the entries and have that $\tfrac{p}{q}= -[(2,-2)_{n-2},2,-\frac{6n-1}{4n-1}].$ Now we can apply Lemma \[repeatfractionlemma2\], obtaining $$\tfrac{p}{q} = -\frac{(2n-3)\frac{-6n+1}{4n-1}+2n-2}{(2n-2)\frac{-6n+1}{4n-1}+2n-1} =-1+\frac{2n}{4n^2-1}.$$ For this continued fraction $n^+=0$ and $n^-=4(n-1)+1=4n-3$. Thus it corresponds to surfaces with slope $-8n+2$.
It now suffices to show that these are the only continued fractions associated to branched surfaces for these knots. Every rational number has a unique continued fraction that contains all positive terms and does not end in a $1$. For $n>1$ this continued fraction is $\frac{2n}{4n^2-1}=[1,2n-2,1,2n-1]$. However, this does not correspond to a branched surface in the knot complement because it has terms with 1’s. Since this continued fraction is of the form $[1,a,1,a-1]$, it follows from [@MR2441951 Lemma 7] that there are a total of four continued fractions that correspond to surfaces in the knot complement, and two of those correspond to surfaces with the same boundary slopes (note that [@MR2441951] uses slightly different notation for continued fractions).
\[lemma:-4nnotsymmetric\] The symmetry $\sigma$ does not fix any surface of slope $-4n$.
By Proposition \[proposition:continuedfractions\] it suffices to analyze the action of $\sigma$ on the branched surfaces. By Lemma \[lemma:twistsymmetries\], $\sigma$ fixes the continued fractions $$[2n,-2n] \ \text{ and} \ [(-2,2)_{n-1},-3, (2,-2)_{n-1}]$$ and swaps $$[2n-1,2,(-1,2)_{n-1}] \ \text{ and } \ [(-2,2)_{n-1},-2,-2n+1].$$ It follows that no surface of slope $-4n$ is fixed by $\sigma$, because in Hatcher and Thurston’s classification ([@MR778125] Theorem 1 part d)) surfaces correspond to unique continued fractions.
Detected Slopes of the $K_n$ Double Twist Knots {#examplesections}
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The character varieties for the knots $K_n$ ($|n|>1$) all have exactly two components which contain characters of irreducible representations, by [@MR2827003]. Let $X_0(n)$ denote the canonical component, and $X_1(n)$ the other component with $X(n)=X_0(n)\cup X_1(n)$; the algebraic set $X(n)$ is the Zariski closure of the irreducible characters. In this section we will determine which slopes are detected by the ideal points of the character variety of the $K_n$ knots.
We will use the following theorem of Ohtsuki from [@MR1248091]. The ideal points in the statement are ideal points of components of the ${\text{PSL}}_2({\ensuremath{\mathbb{C}}})$ character variety which contain characters of irreducible representations (see [@MR1248091 Section 1]).
\[OhtsukiTheorem1\] There is a 1 to 1 correspondence between the ideal points of $X(K(p,q))$ and the elements of the set:
$$\bigcup_{[n_j]}(\{ (k_1,\cdots,k_N) |k_i\in\mathbb{Z}/(n_i),k_i\neq 0, \exists i \text{ such that } k_i\neq \frac{n_i}{2} \}/ \sim ).$$ Here the union is taken over all continued fraction expressions for $\frac{p}{q}$ and the equivalence relation is generated by $(k_j)\sim(-k_j)$ and $(k_j)\sim((-1)^j k_j)$.
We now prove Theorem \[mainresultTheoremdetectedslopesstrict\] which we restate for convenience.
By Proposition \[slopeOfEssentialSurfaces\] we know that the strict boundary slopes for the knot $K_n$ are $0$, $-4n$, and $-8n+2$. As a consequence of Theorem \[OhtsukiTheorem1\] all strict boundary slopes in two-bridge knots are detected by ideal points of the character variety. Specifically, by Ohtuski’s theorem, the continued fractions $$[2n-1,2,(-2,2)_{n-1}] \text{ and } [(-2,2)_{n-1},-2,-2n+1]$$ both have exactly $n-1$ ideal points corresponding to them.
For completeness, we now show that a canonical component $X_0(M)$ always detects at least two boundary slopes, although this is well known. The evaluation function $I_{\gamma}:X_0(M) \rightarrow {\ensuremath{\mathbb{C}}}$ defined by $I_{\gamma}(\rho)=\chi_{\rho}(\gamma)$ is non-constant on $X_0(M)$ for all $\gamma \in \pi_1(\partial M)$ [@MR735339 Proposition 2]. As a result, the image $A_0(M)$ of $X_0(M)$ under the eigenvalue map to the $A$-polynomial curve cannot correspond to a factor of the type $f(L^aM^b)$ where $f(x)$ is a polynomial. Therefore the Newton polygon associated to $A_0(M)$ is two-dimensional. By [@ccgls] the boundary slopes of the Newton polygon are the boundary slopes detected by the character variety, so any canonical component $X_0(M)$ detects at least two slopes.
It remains to show that $-4n$ is not detected on the canonical component. We first show that $S^3-K_n$ has a flexible cusp. By [@MR3575575], if $M$ is a non-arithmetic hyperbolic two-bridge link complement, then $M$ admits no hidden symmetries. By [@MR1099096] the figure-eight is the only arithmetic knot. Therefore, for $|n|>1$ the complement of $K_n$ admits no hidden symmetries. It follows that the corresponding quotient orbifold has a flexible cusp by [@MR1184416]. The fact that $S^3-K_n$ has a flexible cusp follows. By Lemma \[lemma:-4nnotsymmetric\] no surface of slope $-4n$ is preserved under the symmetry $\sigma$ which turns the 4-plat upside down. Therefore, by Theorem \[thm:maintheorem\], a canonical component cannot detect any surface of slope $-4n$. From the above, it must detect at least two boundary slopes, so it must detect slopes 0 and $-8n+2$. Therefore, there must be surfaces of these slopes which are preserved by the full symmetry group of the manifold.
We believe that the slopes detected on the non-canonical component $X_1(M)$ of the character variety are exactly $0$ and $-4n$. By direct computation, using [@MR3425625], we found the corners of the $A$-polynomial for the $K_n$ knot for $n$ up to $10$ to be $(0,12n-1),(2,12n-1),(1,4n),(3,8n-2),(2,0),(4,0)$. This combined with a calculation of the Newton polygon of the canonical component of the character variety also based on [@MR3425625] shows that the Newton polygon for the non-canonical component of the $A$-polynomial for these knots has corners at $(0,8n-2),(1,8n-2),(1,0),(2,0)$.
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abstract: 'Many-body entanglement is at the heart of the Kondo effect, which has its hallmark in quantum dots as a zero-bias conductance peak at low temperatures. It signals the emergence of a conducting singlet state formed by a localized dot degree of freedom and conduction electrons. Carbon nanotubes offer the possibility to study the emergence of the Kondo entanglement by tuning many-body correlations with a gate voltage. Here we show an undiscovered side of Kondo correlations, which counterintuitively tend to block conduction channels: inelastic cotunneling lines in the magnetospectrum of a carbon nanotube strikingly disappear when tuning the gate voltage. Considering the global [$SU(2)$]{} $\otimes $ [$SU(2)$]{} symmetry of a nanotube coupled to leads, we find that only resonances involving flips of the Kramers pseudospins, associated to this symmetry, are observed at temperatures and voltages below the corresponding Kondo scale. Our results demonstrate the robust formation of entangled many-body states with no net pseudospin.'
author:
- Michael Niklas
- Sergey Smirnov
- Davide Mantelli
- Magdalena Margańska
- 'Ngoc-Viet Nguyen'
- '[Wolfgang Wernsdorfer]{}'
- 'Jean-Pierre Cleuziou'
- 'Milena Grifoni\*'
title: 'Blocking transport resonances via Kondo many-body entanglement in quantum dots'
---
The ubiquity of Kondo resonances in quantum dots relies on the fact that their occurrence requires only the presence of degenerate dot states, whose degeneracy is associated to degrees of freedom which are conserved during the tunneling onto and out of the dot [@Hewson1997]. Finite magnetic fields can be used to break time-reversal symmetry related degeneracies and unravel the deep nature of the Kondo state by tracking the magnetic field evolution of split Kondo peaks [@G-Gordon1998; @Nygaard2000; @Sasaki2004; @nature-jarillo:484; @Jarillo-Herrero2005; @Quay2007; @Makarovski2007; @Grap2011; @Lan2012; @Tettamanzi2012]. In a recent work [@Schmid2015], the striking report was made that specific transport resonances were not observable in nonlinear magnetoconductance measurements of split Kondo peaks in carbon nanotubes (CNTs), despite being expected from theoretical predictions [@Choi2005; @Fang2008; @Fang_Erratum]. Even more intriguing is that those resonances were recorded in inelastic cotunneling measurements in the weak coupling regime [@Jespersen2011]. Because in [@Schmid2015] no comparative measurement for the weak-coupling regime was reported, the missing of resonances could not be unambigously interpreted as a signature of the Kondo effect. From a closer inspection of other experimental reports for the Kondo regime [@nature-jarillo:484; @Quay2007; @Lan2012; @Cleuziou2013], we notice that the absence of some resonances seems systematic.
In the following we study the low-temperature nonlinear electron transport in a very clean CNT quantum dot [@Laird2015]. By simply sweeping a gate voltage [@nmat-cao:745; @Makarovski2007], we could tune the same CNT device from a weak coupling regime, where Coulomb diamonds and inelastic cotunneling are observed, to a Kondo regime with strong many-body correlations to the leads. Then, using nonlinear magnetospectroscopy, transport resonances have been measured. The two regimes have been described using accurate transport calculations based on perturbative and nonperturbative approaches in the coupling, respectively. The missing resonances in the Kondo regime have been clearly identified, and their suppression fully taken into account by the transport theory. Accounting for both spin and orbital degrees of freedom, we discuss a global [$SU(2)$]{} $\otimes $ [$SU(2)$]{} symmetry related to the presence of two Kramers pairs in realistic carbon nanotube devices with spin-orbit coupling (SOC) [@Ando2000; @Kuemmeth2008; @prb-delvalle:165427; @ncomms-steele:1573] and valley mixing [@Kuemmeth2008; @Jespersen2011; @GroveRasmussen2012; @Izumida2015; @Marganska2015]. In virtue of an effective exchange interaction, virtual transtions which flip the Kramers pseudospins yield low-energy many-body singlet states with net zero Kramers pseudospin. This result in turn reveals that the transport resonances suppressed in the deep Kondo regime are associated to virtual processes which do not flip the Kramers pseudospin.
R {#r .unnumbered}
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[**[Measurement and modelling of transport regimes.]{}**]{} The device under study consists of a semiconducting CNT, grown *in-situ* on top of two platinum contacts, used as normal metal source and drain leads. Details of the device fabrication were reported previously [@Cleuziou2013] (see also the Methods). The CNT junction is suspended over an electrostatic gate and can be modelled as a single semiconducting quantum dot of size imposed by the contact separation ($\approx$ 200 nm). All the measurements were performed at a mixing chamber temperature of about $T_{\rm exp}$= 30 mK, which sets a lower bound to the actual electronic temperature. The set-up includes the possibility to fully rotate an in-plane magnetic field up to 1.5 T.
The CNT level spectrum is depicted in Figs. 1a and 1b. Transverse bands, represented by the coloured hyperbolae in Fig. 1a, emerge from the graphene Dirac cones as a consequence of the quantization of the transverse momentum $k_\perp$. Bound states (bullets) are due to the quantization of the longitudinal momentum $k_\parallel$. Four-fold spin-valley degeneracy yields the exotic spin plus orbital [$SU(4)$]{} Kondo effect [@Choi2005; @nature-jarillo:484; @Jarillo-Herrero2005; @Makarovski2007; @Anders2008; @Cleuziou2013; @Ferrier2015]. The SOC removes the spin degeneracy of the transverse bands in the same valley (red and blue hyperbolae), and hence the [$SU(4)$]{} symmetry [@Jarillo-Herrero2005; @Fang2008; @Fang_Erratum; @Galpin2010; @Lan2012; @Cleuziou2013; @Schmid2015; @Ferrier2015]. Due to time reversal symmetry, for each $k_\parallel$ a quartet of states consisting of two Kramers pairs splitted by the energy $\Delta=$ [[$\Delta_{\text{SO}}$]{}]{} arises. When also valley mixing is present, with the energy scale [[$\Delta_{\text{KK'}}$]{}]{}, orbital states are formed which are superpositions of valley states. A quartet now consists of two Kramers doublets at energies $\varepsilon_{\rm d}=\pm\Delta/2$, with $\Delta=\sqrt{{{\ensuremath{\Delta_{\text{SO}}}}}^2+{{\ensuremath{\Delta_{\text{KK'}}}}}^2}$, see Fig. 1b.
By sweeping the gate voltage, the chemical potential is moved from above (electron sector) to below (hole sector) the charge neutrality point and quadruplets of states are thus successively emptied. This pattern is visible in a typical measurement of the differential conductance $dI/dV$ versus the bias voltage $V_{\rm sd}$ and the gate voltage $V_{\rm g}$, Figs. 1c, 1d, which exhibits a characteristic four-fold periodicity. Figure 1c displays such a stability diagram for the electron sector, where Coulomb diamonds and inelastic cotunneling excitation lines are visible. Owing to significantly different ratios $\Gamma/U$ of the tunnel coupling to the charging energy in the valence and conduction regimes, Kondo physics dominates for odd hole number in the hole sector shown in Fig. 1d. In order to investigate the dominant transport mechanisms, we have performed transport calculations for both regimes, using a standard minimal model for a longitudinal mode of a CNT quantum dot with SOC and valley mixing terms [@Jespersen2011; @Laird2015]. The explicit form of the model Hamiltonian $\hat H_{\rm CNT}$ and the parameters used for the transport calculations are provided in the Methods. The transport calculations in the electron regime implement a perturbation theory (PT) which retains all tunneling contributions to the dynamics of the CNT reduced density matrix up to second order in the tunnel coupling $\Gamma$. This approximation thus accounts for Coulomb blockade (first order in $\Gamma) $ and leading order cotunneling processes (second order in $\Gamma$), and it is expected to give accurate results for small ratios $\Gamma/k_{\rm B}T$ and $\Gamma/U$ [@Koller2010]. The results of the calculations for the differential conductance are shown in Fig. 1g; a gate trace in Fig. 1e. The perturbative theory reproduces the position of the inelastic cotunneling thresholds (panels 1c and 1g). In the gate trace of Fig. 1e the experimental peaks are wider than the theoretical ones. Because in the latter the broadening is solely given by the temperature, this indicates that higher order terms are responsible for a broadening of the order $\Gamma$ and for a Lamb shift of the experimental peaks [@Koenig1996; @Pedersen2005; @Dirnaichner2015]. In this work we are interested only in the evolution of the cotunneling resonances in magnetic field, which is well captured by the perturbative approach as long as Kondo ridges have not yet formed. This situation radically changes in the hole sector where the gate trace reveals Kondo ridges for odd hole numbers. The theoretical trace in Fig. 1f is the outcome of a nonperturbative numerical DM-NRG calculation [@Bulla2008] which uses the same model Hamiltonian but with slightly different parameters. The strong suppression of the conductance in the valley with even hole occupancy is an indication of the breaking of the [$SU(4)$]{} symmetry in the presence of SOC and valley mixing to an ${\ensuremath{SU(2)}}\otimes{\ensuremath{SU(2)}}$ one [@Galpin2010; @Mantelli2015]. In the DM-NRG calculations the two-particles exchange $J$ was not included due to high computational costs. The latter further reduces the symmetry in the 2h valley (see e.g the spectrum in Fig. 2b), and hence the experimental conductance is more rapidly suppressed in that valley than as predicted by our simulations. On the other hand, $J$ is not relevant for describing the spectrum in the 3h and 1h cases (Figs. 2a, 2c), which is the focus of the present work.
In the DM-NRG calculations the fit to the experiment was done assuming a temperature of $T=30$ mK. From the so extracted parameters we evaluate the temperature dependence of the conductance at $-\varepsilon_{\rm d} =U/2 -\Delta/2$, and $-\varepsilon_{\rm d}=5U/2 +\Delta/2$, corresponding to gate voltage values located roughly in the middle of the 1h and 3h valleys, respectively, and extract the Kondo temperatures, see Fig. 1h. At such values of $\varepsilon_{\rm d}$ the Kondo temperature takes its minimal value in a given valley, which sets a lower bound for $T_{\rm K}$ [@Mantelli2015]. We find $T_{\rm K}=84$ mK and $T_{\rm K}=160$ mK for the 1h and 3h valleys, respectively. Correspondingly, $0.1 <T_{\rm exp}/T_{\rm K}<1$, suggesting that the experiment is in the so-called Kondo crossover regime [@Hewson1997] also for the actual electronic and Kondo temperatures.\
[**Virtual transitions revealed by magnetospectroscopy.**]{} Having set the relevant energy scales for both the electron and hole sectors, we proceed now with the investigation of magnetotransport measurements at finite source-drain bias, which have been performed for different fillings. A magnetic field ${\bf B}$ breaks time reversal symmetry and thus the Kramers degeneracies. By performing inelastic cotunneling spectroscopy, we can get information on the lowest lying resonances of our interacting system. The magnetospectrum corresponding to electron filling $n_{\rm e}=1,2,3$ of a longitudinal quadruplet, as expected for the perturbative regime, is shown in Figs. 2a - 2c. For the case of odd occupancies, we call ${\cal T}$ transitions processes within a Kramers pair; $\cal{C}$ and $ \cal{P}$ operations are associated to inter-Kramers transitions, as shown in Figs. 2a and 2c. Panels 2d-2f and 2g-2i show magnetotransport measurements and theoretical predictions for the electron and hole regimes, respectively. In these panels the current second derivative d$^2I/{\rm d}V^2$ is reported. We have preferred this quantity over the more conventional d$I/{\rm d}V$ (shown in the Supplementary Figures 4 and 5 and discussed in the Supplementry Note 4) to enhance eye visibility of the excitation spectra. In panels 2d-2f as well as 2h we have used our perturbative approach [@Koller2010]. The calculations in Figs. 2g, 2i, in contrast, are based on the Keldysh effective action (KEA) method [@Smirnov_2013; @Smirnov_2013a] and are non-perturbative. The nature of the dominant inelastic transitions is clearly identified by simply looking at the excitation spectrum (dashed lines in Figs. 2d-2i). All inelastic transitions from the ground state are resolved in the cotunneling spectroscopy performed in the low coupling electron regime, similar to previous reports [@Jespersen2011]. When inspecting the hole regime, though, it is clear that only for the 2h case, panel 2h, the experimental data can be interpreted by means of a simple cotunneling excitation spectrum; moreover, the 2e and 2h cotunneling spectra are very similar. In the 1h and 3h cases shown in panels 2g, 2i Kondo correlations dominate the low energy transport, and differences with respect to the electron sector are seen. The zero-bias Kondo peak does not immediately split as the field is applied; rather the splitting occurs at a critical field such that the energy associated to the inelastic $\cal{T}$ transition is of the order of the Kondo temperature [@Hewson1997]. In the 1h valley the lowest pair of levels merges again for values of the field of about 1.2 Tesla, yielding a Kondo revival [@nature-jarillo:484; @Galpin2010]. Bias traces of the differential conductance highlighting the revival are shown in the Supplementary Figure 3 and analyzed in the Supplementary Note 3. Striking here is the observation that, in contrast to the 1e and 3e cases, [*only one*]{} of the two inter-Kramers transitions is resolved in the experimental data for the 3h and 1h valley. However, in particular for the 1h case, the $\cal{P}$ and ${\cal C}$ excitation lines as expected from the excitation spectrum should be separated enough to be experimentally distinguishable, similar to the 3e case. By comparing with the excitation spectrum (dashed lines in panels 2g, 2i), we conclude that it is the ${\cal P}$ transition which is not resolved. Our KEA transport theory qualitatively reproduces these experimental features.\
Magnetotransport measurements performed for other quadruplets both in the conduction and valence regimes exhibit qualitatively similar features (see Supplementary Figures 6-8, Supplementary Table 1 and Supplementary Note 5), and hence confirm the robustness of the suppression of $\cal{ P}$ transitions in the Kondo regime. Our results naturally reconcile the apparently contradictory observations in Refs. [@Schmid2015] and [@Jespersen2011]. Furthermore, they suggest that the inhibition of selected resonances in the Kondo regime is of fundamental nature. [**Fundamental symmetries of correlated CNTs.**]{} To understand the experimental observations microscopically, we have analyzed those symmetries of an isolated CNT which also hold in the presence of on-site Coulomb repulsion typical of Anderson models.\
In the absence of a magnetic field, one finds a $U(1)\otimes U(1)\otimes SU (2) \otimes SU (2)$ symmetry related to the existence of two pairs of time-reversal degenerate doublets, see Fig. 1b, called in the following upper $({\rm u})$ and lower $({\rm d})$ Kramers channels. The $U(1)$ symmetries reflect charge conservation in each Kramers pair with generators $\hat Q_\kappa=\frac{1}{2}\sum_{j \in \kappa}(\hat n_j-\frac{1}{2})$ which measure the charge of the pair with respect to the half-filling. Here is $j= (1,2)$ or $ (3,4)$ for $\kappa={\rm u}$ or ${\rm d}$. The [$SU(2)$]{} symmetries are generated by the spin-like operators ${ \hat{\bf J}_\kappa}=\frac{1}{2}\sum_{j,j'\in \kappa}\hat d^\dagger_j {\boldsymbol\sigma}_{j,j'}\hat d_{j'}$. Here ${\boldsymbol\sigma}$ is the vector of Pauli matrices. Physically, $\hat J^z_{\rm u}=(\hat n_1-\hat n_2)/2$ and $ \hat J^z_{\rm d}=(\hat n_4-\hat n_3)/2$ account for the charge unbalance within the Kramers pair. Thus, an isolated CNT with one electron or a hole only in the quadruplet has a net Kramers pseudospin (and charge). Fig. 3a shows the [ two]{} degenerate groundstate configurations $\vert\Downarrow;-\rangle_{}$, $\vert\Uparrow;- \rangle_{}$ of the isolated CNT with an unpaired effective spin ($\Downarrow$ or $\Uparrow$) in the lowest Kramers pair and no occupation (symbol “-”) of the upper Kramers pair. In the weak coupling regime, a perturbative approach to linear transport accounts for elastic cotunneling processes involving the doubly degenerate groundstate pair [@Grabert]. These virtual transitions are denoted ${\cal I}$ or ${\cal T}$ when they involve the same state or its Kramers partner, respectively, see Fig. 3a. A finite magnetic field breaks the [$SU(2)$]{} symmetries. However, former degenerate CNT states can still be characterized according to the eigenvalues of the $\hat Q_\kappa$ and $\hat J^z_\kappa$ operators, since they commute with the single-particle CNT Hamiltonian which has in the Kramers basis the form (see Methods): $$\hat H_{0}=\sum_{\kappa =\pm}\left(\bar\varepsilon ({\bf B})+\kappa \frac{\bar\Delta ({\bf B})}{2}\right) \hat N_\kappa + (2\delta \varepsilon ({\bf B})+\kappa\delta \Delta ({\bf B})) \hat J^z_\kappa ,$$ where u/d$=+/-$, $\hat N_\kappa=2\hat Q_\kappa +1$, and at zero field is $\bar\Delta({ B}=0)=\Delta$, $\bar\varepsilon(B=0)=\varepsilon_{\rm d}$, $\delta\varepsilon=\delta\Delta=0$. Hence our finite-bias and finite magnetic field spectroscopy allows us to clearly identify the relevant elastic and inelastic virtual processes according to the involved Kramers charge and spin. As illustrated in Fig. 3b, in the weak tunneling regime only energy differences matter, and hence both intra-Kramers (${\cal I}$, ${\cal T}$) and inter-Kramers (${\cal P}$, $\cal{C}$) transitions are expected in transport. In the Kondo regime this picture changes. As we shall demonstrate, emerging Kondo correlations lead to the progressive screening of the Kramers pseudospin of the dot by the conduction electrons. To this aim we observe that, when a sizeable tunnel coupling to the leads is included, the CNT charge and pseudospin operators $\hat Q_\kappa$ and $\hat {\bf J}_\kappa$ are no longer symmetries of the coupled system, since the tunneling does not conserve the dot particle number. The occurrence of the Kondo effect, however, suggests that the CNT quantum numbers $j=1,2,3,4$ are carried also by the conduction electrons and conserved during tunneling [@Choi2005]. This is the case when the dot is only a segment of the CNT (see Supplementary Figure 1). Following [@Mantelli2015], we hence introduce charge, $\hat{\cal { Q}}_\kappa=\hat Q_\kappa +\hat Q_{{\rm L},\kappa}$, and pseudospin, $ \hat {\boldsymbol{\cal J}}_{\kappa}$=$\hat {{\bf J}}_{\kappa}+\hat {{\bf J}}_{{\rm L},\kappa}$, operators of the coupled CNT plus leads (L) system. Under the assumption that the tunneling couplings are the same within each Kramers channel $\kappa = \rm{u,d}$, the total Hamiltonian (see Supplementary Methods) commutes with the charge and pseudospin operators $\hat{\cal { Q}}_\kappa$ and $ \hat {\boldsymbol{\cal J}}_{\kappa}$, which hence generate a $U(1)\otimes U(1)\otimes SU(2) \otimes SU(2)$ symmetry of the coupled system. As a consequence, many-body states can be characterized by the quadruplet of eigenvalues $({\cal Q}_{\rm d},{\cal Q}_{\rm u}; {\cal J}_{\rm d},{\cal J}_{\rm u})$, where the highest eigenvalue ${\cal J}_\kappa$ of $\hat {\cal J}^z_\kappa$ is indicated in the quadruplet. This notation gives direct access to the eigenvalues $ {\cal J}_{\kappa} ( {\cal J}_{\kappa}+1) $ of $ \hat {\boldsymbol{\cal J}}_{\kappa}^2$. Such quadruplets can be numerically calculated within our scheme for the Budapest DM-NRG code [@Thot2008], and yield (for the valleys with one electron or one hole) a *singlet* ground state characterized by the quadruplet $(0,0;0,0)$. Thus “0” is also eigenvalue of ${\hat {\boldsymbol{\cal J}}}_{\rm u}^2$ and ${\hat{\boldsymbol{\cal J}}}_{\rm d}^2$. [*I.e., we find a unique ground state with no net pseudospin.*]{} This situation is illustrated in Fig. 3c: due to ${\cal Q}_\kappa=0$, the Kramers channels are half-filled (two charges per channel), whereby one charge arises from the electron trapped in the CNT itself. For $\Delta=0$ this CNT charge is equally distributed among the two channels, while for large values of $\Delta/T_{\rm K}(\Delta)$, as in our calculation (see Fig. 1h), it is mainly in the lowest Kramers channel. Thus at zero temperature the localized CNT pseudospin is fully screened by an opposite net pseudospin in the leads. In the orthonormal basis $\{ \vert m\rangle_{}\otimes\vert n\rangle_{\rm L}\}$ spanned by the pseudospin eigenstates of CNT and leads this ground state is characterized by the entangled configuration $\frac{1}{\sqrt{2}}[\vert\Uparrow;-\rangle_{}\otimes\vert\Downarrow;\Downarrow,\Uparrow\rangle _{\rm L}-\vert\Downarrow;-\rangle_{}\otimes\vert \Uparrow;\Downarrow,\Uparrow \rangle_{\rm L}]$ of dot and leads pseudospins.
In the standard spin-1/2 Kondo effect the appearance of a unique singlet ground state with no net spin is the result of the screening of the quantum impurity spin by the conduction electrons spins, due to the antiferromagnetic character of the coupling constant between such degrees of freedom [@Hewson1997]. Triplets are (highly) excited states of the system. To interpret the spin 1/2 Kondo effect in quantum dots, it is possible to derive from an Anderson model an effective Kondo Hamiltonian [@Schrieffer1966] given by the product of the quantum dot spin and the conduction electrons spin. The coupling constant for this product is positive and thus antiferromagnetic. Also for the more complex case of a CNT effective Kondo Hamiltonians have been derived, with positive coupling constants for Kramers channels identified by orbital and spin degrees of freedom [@Choi2005; @Lim2006]. The antiferromagnetic character of the coupling constants remains also when, as in our case, the more abstract Kramers pseudospin is used. A natural consequence of the antiferromagnetic nature of the correlations is that at low temperatures and zero bias elastic virtual transitions which flip the pseudospin, i.e., ${\cal T}$ transitions, are favoured, as depicted in Fig. 3c. Similarly, ${\cal C}$ transitions are inelastic processes which flip the pseudospin and become accessible at finite bias, as shown in Fig. 3d. They connect the singlet ground state to an excited state where the CNT charge is located in the upper Kramers channel. Our results suggest that $ {\cal P}$ transitions are inhibited because they involve virtual transitions which conserve the pseudospin.\
[**Entanglement of Kramers pseudospins.**]{} To further confirm that it is the Kramers pseudospins and not distinct spin or orbital degrees of freedom which should be considered in the most general situations, we report results for the differential conductance as a function of the angle $\theta$ formed by the magnetic field and the CNT’s axis. The combined action of SOC, valley mixing and non collinear magnetic field mixes spin and valley degrees of freedom which, in general, are no longer good quantum numbers to classify CNTs states. Nevertheless, the three discrete ${\cal T}$, $\cal{ P}$ and $\cal{C}$ operations still enable us to identify the inelastic transitions in the 1h and 3h case, [*independent*]{} of the direction of the magnetic field. The angular dependence of both energy and excitation spectrum for a fixed magnetic field amplitude is shown in Figs. 4a, 4b for the 3h and 1h fillings, respectively. The corresponding transport spectra are shown in Figs. 4c, 4d, respectively. A perpendicular magnetic field almost restores (for our parameter set) Kramers degeneracy, thus revitalizing the Kondo resonance for this angle. As the field is more and more aligned to the CNT’s axis, the degeneracy is removed, which also enables us to distinguish between ${\cal P}$ and ${\cal C}$ transitions. As in the axial case of Fig. 2, only the inelastic resonance associated to the ${\cal C}$ transition is clearly resolved in both the experiment and theory.\
[**Entropy and specific heat.**]{} Usually, quantum entanglement suffers from decoherence effects [@Buchleitner2009; @Akulin2005]. The Kondo-Kramers singlets, however, are associated to a global symmetry of the quantum dot-plus lead complex, and are robust against thermal fluctuations or finite bias effects as long as the impurity is in the Fermi liquid regime [@Hewson1997] ($T< 0.01$ $ T_{\rm K}$ for our experiment). For larger energy scales, $0.01< T/T_{\rm K} <1$ the impurity is not fully screened, but Kondo correlations persist yielding universal behavior of relevant observables, as seen e.g. in Fig. 1h at the level of the linear conductance. In order to further investigate the impact of thermal fluctuations on Kondo correlations, we have calculated the temperature dependence of the impurity entropy $S_{\rm CNT}=S_{\rm tot}-S_{\rm L}$, where the $S_i$ are thermodynamic entropies, and of the impurity specific heat [@Merker2012] (see Supplementary Note 1 and Supplementary Figure 2). The conditional entropy $S_{\rm CNT}(T)$ remains close to zero up to temperatures $T\approx 0.01$ $T_{\rm K}$, indicating that the system is to a good approximation in the singlet ground state. At higher temperatures the impurity entropy grows, but universality is preserved up to temperatures close to $ T_{\rm K}$, at which the entropy approaches the value $k_{\rm B} \log 2$.
D {#d .unnumbered}
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Our results show that specific low-energy inelastic processes, observed in the perturbative cotunneling regime, tend to be blocked in the Kondo regime due to antiferromagnetic-like correlations, which at zero temperature yield a many-body ground state with net zero Kramers pseudospin. This signature of the Kondo effect is universal, in the sense that it does not depend on the degree of the spin-orbit coupling or valley mixing specific to a given CNT. As such, it is also expected for [$SU(4)$]{} correlated CNTs, which explains the missing inelastic resonance in the seminal work [@nature-jarillo:484]. Furthermore, we believe that such pseudospin selective suppression should be detectable also in a variety of other tunable quantum dot systems with emergent [$SU(4)$]{} and [$SU(2)$]{} $\otimes$ [$SU(2)$]{} Kondo effects [@Borda2003; @Sasaki2004; @Tettamanzi2012; @Minamitami2012; @Keller2013; @Crippa2015].
Because the screening is progressively suppressed by increasing the temperature or the bias voltage, it should be possible to recover such inelastic transitions by continuosly tuning those parameters. Indeed, signatures of the re-emergence of the ${\cal P}$ transition are seen in the KEA calculations and experimental traces at fields around 0.9 T in the form of an emerging shoulder, see Supplementary Figure 3. Experiments at larger magnetic fields, not accessible to our experiment, are required to record the evolution of this shoulder, and thus the suppression of (non-equilibrium) Kondo correlations by an applied bias voltage.
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[**Experimental fabrication.**]{} Devices were fabricated from degenerately doped silicon ${\rm Si/SiO_{2}/Si_{3}N_{4}}$ wafers with a 500 nm thick thermally grown ${\rm SiO_{2}}$ layer and 50 nm ${\rm Si_{3}N_{4}}$ on top. Metal leads separated by 200 nm were first defined by electron-beam lithography and deposited using electron-gun evaporation. A thickness of 2 nm Cr followed by 50 nm Pt was used. A 200 nm deep trench was created using both dry-etching and wet-etching. A second step of electron-beam lithography was used to design a 50 nm thin metallic local gate at the bottom of the trench. Catalyst was then deposited locally on top of the metal leads. Carbon nanotubes were then grown by the CVD technique to produce as clean as possible devices. Only devices with room temperature resistances below 100 k$\Omega$ were selected for further studies at very low temperature. A scanning electron microscopy of a device similar to the one measured in this work is shown in the Supplementary Figure 1.\
[**Transport methods.**]{} For the transport calculations, three different approaches have been used: the density-matrix numerical renormalization group (DM-NRG) method, a real time diagrammatic perturbation theory (PT) for the dynamics of the reduced density, and the analytical Keldysh effective action (KEA) approach. Further details are discussed in the Supplementary Note 2.\
[**Model CNT Hamiltonian.**]{} In our calculations we have used the standard model Hamiltonian for the longitudinal mode of a CNT accounting for spin-orbit coupling (SOC), valley mixing, onsite and exchange Coulomb interactions, and an external magnetic field [@Laird2015]. Regarding both SOC and the valley mixing as perturbations breaking the [$SU(4)$]{} symmetry of the single particle CNT Hamiltonian, it has the general form $${\hat{H}}_{{\text{CNT}}}={\hat{H}}_{{\text{d}}}+{\hat{H}}_{{\text{SO}}}+{\hat{H}}_{\rm KK'}+{\hat{H}}_{\rm U}+{\hat{H}}_{\rm J}+{\hat{H}}_{{\text{B}}},
\label{eq:H_CNT}$$ where ${\hat{H}}_{{\text{d}}}+{\hat{H}}_{\rm U}$ is the [$SU(4)$]{} invariant component. In the basis set $\{K' \uparrow,K' \downarrow, K \uparrow, K \downarrow\}$ indexed by the valley and spin degrees of freedom $\tau =K',K=\pm$ and $\sigma=\uparrow,\downarrow=\pm$, respectively, it reads $${\hat{H}}_{{\text{d}}}+{\hat{H}}_{\rm U}=\varepsilon_{{\text{d}}}\sum_{\tau,\sigma=\pm}{\hat{d}}^{\dagger}_{\tau,\sigma}{\hat{d}}_{\tau,\sigma}+\frac{U}{2}\sum_{(\tau,\sigma)\neq (\tau',\sigma')}\hat n_{\tau,\sigma}\hat n_{\tau',\sigma'},$$ with $\varepsilon_{{\text{d}}}$ the energy of the quantized longitudinal mode, which can be tuned through the applied gate voltage, and $U$ accounting for charging effects. Valley mixing and SOC break the [$SU(4)$]{} symmetry with characteristic energies [[$\Delta_{\text{KK'}}$]{}]{} and [[$\Delta_{\text{SO}}$]{}]{}, respectively. The corresponding contributions read: $$\label{eq:dot_hamiltonian}
{\hat{H}}_{\rm KK'}+ {\hat{H}}_{{\text{SO}}}=\frac{\Delta_{{\text{KK'}}}}{2}\sum_{\tau,\sigma=\pm}{\hat{d}}^{\dagger}_{\tau,\sigma}{\hat{d}}_{-\tau,\sigma} +
\frac{\Delta_{{\text{SO}}}}{2}\sum_{\tau,\sigma=\pm}\sigma\tau\hat n_{\tau,\sigma}.$$ The SOC term is a result of the atomic spin-orbit interaction in carbon, and thus exists also for ideally infinitely long CNTs [@Ando2000]. The valley mixing, in contrast, is absent in long and defect free CNTs. It only arises due to scattering off the boundaries in finite length CNTs or due to disorder [@Kuemmeth2008; @Izumida2015; @Marganska2015]. It is expected to be zero in disorder-free CNTs of the zig-zag class, due to angular momentum conservation rules, and finite in CNTs of the armchair class [@Marganska2015]. In our experiments, according to Table \[tab:parameters\], the valley mixing is very small, which suggests a tube of the zig-zag class.
Similar to the SOC and valley mixing, the exchange interaction preserves time reversal symmetry. Its microscopic form is not known for abritrary chiral angles. It has been evaluated so far for the case of pure armchair tubes [@Mayrhofer2008], and for the zig-zag class [@Secchi2009; @Laird2015] CNTs. Because the experiments suggest that our tube is of the zig-zag class, we choose in the following a form suitable to describe this case. It reads $${\hat{H}}_{\rm J}=-\frac{ J}{2}\sum_{\sigma=\pm}\{\hat n_{K,\sigma}\hat n_{K',\sigma}+ {\hat{d}}^{\dagger}_{K,\sigma} {\hat{d}}^{\dagger}_{K',-\sigma}{\hat{d}}_{K,-\sigma}{\hat{d}}_{K',\sigma}\},$$ with $J<0$ the exchange coupling. Finally, contributions arising from a magnetic field [**B**]{} contain both Zeeman and orbital parts. Decomposing [**B**]{} into components parallel and perpendicular to the tube axis, $B_\parallel=B\cos\theta$ and $B_\perp=B\sin\theta$, respectively, one finds: $$\begin{aligned}
{\hat{H}}_{{\text{B}}}&=&{\hat{H}}_{{\text{B}}}^{\rm Z}+{\hat{H}}_{{\text{B}}}^{\rm orb}\nonumber\\
&=&B_\parallel\sum_{\tau,\sigma=\pm}\big(\frac{g_{\rm s}}{2}\mu_{\rm {\text{B}}}\sigma
+\mu_{\rm orb}\tau \big){\hat{d}}^{\dagger}_{\tau,\sigma}{\hat{d}}_{\tau,\sigma}\nonumber\\
&+&\frac{g_{\rm s}}{2}\mu_{\rm B}B_{\perp}\sum_{\tau,\sigma=\pm}{\hat{d}}^{\dagger}_{\tau,\sigma}{\hat{d}}_{{\tau},-\sigma}.\end{aligned}$$ Notice that the spin and valley remain good quantum numbers in the presence of an axial field ($\theta=0,\pi$), while a perpendicular component flips the spin degrees of freedom. The parameters of the CNT Hamiltonian used to fit the experimental data shown in Figs. 1, 2 and 4 are listed in Table \[tab:parameters\].\
[**Kramers charge and pseudospin representation.** ]{} We call Kramers basis the quadruplet $\{\vert i\rangle\}$, $ i=1,2,3,4$ (shown in Fig. 1b ) which diagonalizes the single particle part $\hat H_{0}={\hat{H}}_{\rm d}+{\hat{H}}_{{\text{KK'}}} +{\hat{H}}_{{\text{SO}}} + {\hat{H}}_{ {\text{B}}}$ of the CNT Hamiltonian. For magnetic fields parallel or perpendicular to the CNT axis, this Hamiltonian is easily diagonalized, see e.g. [@Schmid2015]. For other orientations of the field, because of the combined action of SOC and valley mixing, such states are a linear superposition of all the basis states $\{\vert \tau,\sigma \rangle \}$, such that neither the spin nor the valley are in general good quantum numbers any more. One has to resort to numerical tools to find both the eigenvectors $\{ \vert i \rangle \}$ and the eigenvalues $\varepsilon_i$, $i=1,2,3,4$. The angular dependence of these eigenenergies is sketched in Fig. 4.
Despite the complexity inherent in the Hamiltonian $\hat H_{0}$, a closer inspection reveals the existence of conjugation relations among the quadruplet of states $i=1,2,3,4$ generated by the time-reversal operator $\hat{\cal T}$, as well as by the particle-hole like and chirality operators $\hat{\cal P}$ and $\hat{\cal C}=\hat{\cal P}\hat{\cal T}^{-1}$, respectively [@Schmid2015]. Specifically, the states are ordered such that $(1,2)$ and $(3,4)$ are time-reversal partners, while $(1,4)$ and $(2,3)$ are particle-hole partners. In the $\{ \vert \tau,\sigma \rangle \}$ basis the operators read $$\begin{aligned}
\label{eq:CPT}
\hat{\cal T}&=&\hat\kappa\sum_{\tau,\sigma}\sigma{\hat{d}}^{\dagger}_{-\tau,-\sigma}{\hat{d}}_{\tau,\sigma},\\ \hat{\cal P}&=&\hat\kappa\sum_{\tau,\sigma}\sigma\tau{\hat{d}}^{\dagger}_{-\tau,\sigma}{\hat{d}}_{\tau,\sigma},\\ \hat{\cal C}&=&\sum_{\tau,\sigma} (-\tau){\hat{d}}^{\dagger}_{\tau,-\sigma}{\hat{d}}_{\tau,\sigma},\end{aligned}$$ where $\hat\kappa$ stands for the complex conjugation operator. In the absence of a magnetic field $\hat{\cal T}$ commutes with the total CNT Hamiltonian, yielding a single-particle spectrum with two degenerate Kramers doublets (1,2) and (3,4) separated by the inter-Kramers splitting $\Delta=\sqrt{\Delta_{\rm SO}^2+\Delta_{\rm KK'}^2}$ (see Fig. 1b). As far as the $\hat{\cal P}$ and $\hat{\cal C}$ operators are concerned, at zero magnetic field they are symmetries only in the absence of SOC and valley mixing. Since both anticommute with $\hat H_{\rm SO}+\hat H_{\rm K K'}$, it holds for ${\cal P}$-conjugated pairs, $\varepsilon_{1,2}(\Delta)=\varepsilon_{4,3} (-\Delta)$. A magnetic field breaks the time-reversal symmetry; however, because $\hat H_{ {\text{B}}}$ anticommutes with $\hat {\cal T}$, formerly degenerate Kramers states are still related to each other by Kramers conjugation. For an arbitrary magnetic field [**B**]{} time-reversal conjugation and particel-hole conjugation imply [@Schmid2015]: $$\begin{aligned}
\varepsilon_{1,4}({\bf B})&=&\varepsilon({\bf B}) \pm \frac{1}{2} \Delta ({\bf B}),\\
\varepsilon_{2,3}({-\bf B})&=&\varepsilon_{1,4}(\bf{B}),\end{aligned}$$ where $\varepsilon ({\bf B})$ and $\Delta ({\bf B})$ reduce to the longitudinal energy and Kramers splitting $\varepsilon_{\rm d}$ and $\Delta $, respectively, at zero field.
These relations clearly suggest the introduction of auxiliary charge $\hat N_{ij}:=\hat n_i+\hat n_j$ and pseudospin $\hat J^z_{ij}=(\hat n_i-\hat n_j)/2$ operators, in terms of which we can write $$\begin{aligned}
\hat H_{0} &=&\varepsilon({\bf B})\hat N_{14}+\Delta({\bf B})\hat J_{14}^z +\varepsilon(-{\bf B}) \hat N_{23} +\Delta (-{\bf B})\hat J^z_{23} . \end{aligned}$$ Introducing the average quantities $\bar\Delta ({\bf B}) :=(\Delta ({\bf B}) + \Delta (-{\bf B}) )/2$, $\bar\varepsilon ({\bf B}):= (\varepsilon ({\bf B}) + \varepsilon (-{\bf B}) )/2$, as well as the differences $\delta\Delta ({\bf B}) :=(\Delta ({\bf B}) - \Delta (-{\bf B}) )/2$, $\delta\varepsilon ({\bf B}):= (\varepsilon ({\bf B}) - \varepsilon (-{\bf B}) )/2$, the CNT Hamiltonian can be easily recast in terms of total charge and pseudospin of a Kramers pair. It reads: $$\begin{aligned}
\hat H_{0}&=& \left(\bar\varepsilon({\bf B}) +\frac{\bar\Delta ({\bf B})}{2}\right) \hat N_{12}+[2\delta\varepsilon ({\bf B}) +\delta\Delta({\bf B})]\hat J_{12}^z\\
&+& \left(\bar\varepsilon({\bf B}) -\frac{\bar\Delta ({\bf B})}{2}\right) \hat N_{43}+[2\delta\varepsilon ({\bf B}) -\delta\Delta({\bf B})]\hat J_{43}^z .\end{aligned}$$ Such equation is Eq. (1) in the main part of the manuscript upon calling $\hat J^z_{43}=\hat J^z_{\rm d}$, $\hat J^z_{12}=\hat J^z_{\rm u}$, and similarly $\hat N_{43}=\hat N_{\rm d}$, $\hat N_{12}=\hat N_{\rm u}$.\
[**Data availability**]{} The data that support the main findings of this study are available from the corresponding author upon request.\
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A {#a .unnumbered}
=
The authors acknowledge fruitful discussions with C. Strunk, A. H" uttel, G. Zaránd and C. P. Moca as well as financial support by the Deutsche Forschungsgemeinschaft via SFB 689 and GRK 1570, and by the ERC Advanced Grant MolNanoSpin No. 226558.
A {#a-1 .unnumbered}
=
M.N. performed the perturbative non-equilibrium calculations, S.S. evaluted the differential conductance in the Kondo regime using the non-equilibrium Keldysh effective action approach, while D.M. did the equilibrium DM-NRG simulations. M.M. evaluated the magnetospectrum of the isolated nanotube and devised all the figures. N.-V.N. helped to fabricate and characterize the devices, J.-P.C. carried out and analyzed the experiments while W.W. supervised them. M.G. performed the theoretical analysis and wrote the manuscript with critical comments provided by all authors.
A {#a-2 .unnumbered}
=
[**Supplementary information**]{} is available in the online version of this paper.\
[**Competing financial interests**]{} The authors declare no competing financial interests.\
{width="16cm"}
{width="16cm"}
{width="\textwidth"}
{width="16cm"}
------------------------------------ -- -- --------------------------------- -----------------
holes electrons
(shell $N_h$=6) (shell $N_e$=6)
[[$\Delta_{\text{SO}}$]{}]{}(meV) -0.21 -0.4
[[$\Delta_{\text{KK'}}$]{}]{}(meV) 0.08 0.04
[$\mu_{\mathrm{orb}}$]{}(meV/T) 0.51 (3h), 0.51 (2h), 0.55 (1h) 0.43
$U$ (meV) PT 26,5
$U$ (meV) NRG 4,7
$U$ (meV) KEA $\infty$ (3h, 1h)
$J$ (meV) PT -1.35 -1.4
$\Delta_\mu B_\parallel$ (meV/T) -0.05 -0.06
$e\Delta V_{\rm sd}$ (meV) 0.12 0.28
------------------------------------ -- -- --------------------------------- -----------------
: \[tab:parameters\] [**Parameter set**]{}. The table shows the parameters used to fit the electronic transport spectra of the CNT in the gate voltage region shown in the main text. It corresponds to the valence quadruplet $N_h$=6 (hole transport), and the conduction quadruplet $N_e$ =6 (electron transport), counting the Coulomb diamonds from the band gap. The abbreviations PT, NRG and KEA refer to the three theoretical methods used in our calculations (see text). The experimental data for each Coulomb valley are offset by $\Delta V_{\rm sd}$, and tilted in the magnetic field by $\Delta_\mu B_\parallel$, resulting in an asymmetry between the measurement in fields parallel and antiparallel to the CNT axis. In all the plots presented in the work both the offset and the tilt have been removed.
|
[**[Discovery and spectroscopic study of the massive Galactic cluster Mercer 81. ]{}\
Diego de la Fuente$^{1}$, Francisco Najarro$^{1}$, Ben Davies$^{2}$ and Donald F. Figer$^{3}$** ]{}\
$^{1}$ Centro de Astrobiología (CSIC/INTA), ctra. de Ajalvir km. 4, 28850 Torrejón de Ardoz, Madrid, Spain\
$^{2}$ Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK\
$^{3}$ Center for Detectors, Rochester Institute of Technology, 54 Lomb Memorial Drive, Rochester, NY 14623, USA
Abstract {#abstract .unnumbered}
========
During the last decade, hundreds of young massive cluster candidates have been detected in the disk of the Milky Way. We investigate one of these candidates, Mercer 81, which was discovered through a systematic search for stellar overdensities, with follow-up NICMOS/HST infrared narrow-band photometry to find emission-line stars and confirm it as a massive cluster. Surprisingly, the brightest stars turned out to be a chance alignment of foreground stars, while a real massive cluster was found among some fainter stars in the field. From a first spectroscopic study of four emission-line stars (ISAAC/VLT), it follows that Mercer 81 is a very massive young cluster, placed at the far end of the Galactic bar. Additionally, in this work we present some unpublished spectra from a follow-up observation program which confirm that the cluster hosts several Nitrogen-rich Wolf-Rayet stars (WN) and blue supergiants.
Introduction
============
Since the 1990s, infrared astronomy has experienced a considerable development, allowing to observe in detail the most extincted regions of the Milky Way. As a consequence, many obscured young massive clusters (YMCs) have been found in the Galactic disk, revealing a great amount of previously unknown star formation. YMCs are notable for hosting a population of massive stars, which typically consists of OB, Wolf-Rayet and hypergiant stars. YMCs are also ideal test beds to measure the high-mass region of the IMF ([@imf]) and the evolution of massive stars ([@evolmassive]), as well as useful tools to map the Galactic metallicity ([@diskmetals]).
Only a few Galactic YMCs have been investigated extensively, e.g. Westerlund 1 ([@wd1]), the Arches Cluster ([@arches]) or the Quintuplet Cluster ([@quintuplet]). However, several systematic searches for IR clusters have been carried out ([@survey1], [@survey3], [@survey4]), yielding hundreds of new cluster candidates that mostly remain unstudied. The usual way to confirm such candidates as real YMCs consists of finding their young massive stars, whose high mass-loss rates allow to track them as highly reddened emission-line stars, especially by means of their Paschen-$\alpha$ emission excess ([@paschena]). A subsequent spectroscopic analysis of these stars eventually lead to a complete characterization of each cluster.
Discovery and rediscovery \[discovery\]
=======================================
![\[fig1\] RGB image of Mercer 81, composed of NICMOS/HST photometric data as follows: F222M (red), F160W (green) and \[F187N$-$F190N\] (blue). From [@mc81]. ](delafuentedF1.pdf){width="13.5cm"}
The cluster candidate Mercer 81 was found in 2005 by [@survey2] in an algorithmic search for stellar overdensities in the GLIMPSE point-source catalog. In 2008, this candidate was observed with the instrument NICMOS onboard the Hubble Space Telescope as part of the observing program \#11545, whose main goal was to find highly reddened emission-line stars in candidate clusters. The strategy consisted of obtaining images through the filters F160W and F222M in order to measure the reddening; as well as narrow-band images at the wavelengths of Paschen-$\alpha$ (F187N) and the continuum region near $P_\alpha$ (F190N), in such a way that the subtraction image F187N$-$F190N would pinpoint the massive stars.
![\[fig2\] The two only spectra that could be completed in the first observation program at ISAAC/VLT. From [@mc81]. ](delafuentedF2.pdf){width="15.5cm"}
Results of the aforementioned photometry were published by [@mc81], who presented a composite false-color image of Mercer 81 (Fig. \[fig1\]; see caption for the RGB-color explanation) where stars can be easily identified by means of their color. The most reddened stars must present red/orange colors, while the $P_\alpha$ excess corresponds to a blue color; therefore, the massive members of the cluster are expected to have a combination of these colors, appearing pink/magenta sources. On the other hand, the unreddened foreground stars look yellow/green. Fig. \[fig1\] shows that the brightest stars in the field constitute a chance alignment of foreground stars, which probably were crucial for the candidate detection. However, a fainter but numerous group of highly reddened stars, including nine emission-line stars, can be seen in the north half of the field
Spectroscopy
============
In 2009, H- and K-band spectroscopy of the emission-line stars at ISAAC/VLT was proposed (program ID: 083.D-0765); unfortunately, due to a temporary failure of the instrument, only the spectra of two cluster members could be completed. These spectra (Fig. \[fig2\]), which consisted of a late-B/early-A supergiant and a Nitrogen-rich Wolf-Rayet star (spectral type: WN7-8), have been published by [@mc81], where a model for the WN achieved with the CMFGEN code ([@cmfgen1], [@cmfgen2]) was included.
![\[fig3\] New H- and K-band spectra of Mercer 81 cluster members. Objects follow the naming scheme of [@mc81], with the addition of new cluster member Mc81-11. ](delafuentedF3.pdf){width="15.5cm"}
In 2011, we could complete the observations with additional spectra of 8 emission-line stars at ISAAC/VLT (program ID:087.D-0957), which are presented in Fig. \[fig3\]. All these objects show H and He broad emission lines, indicating extended winds and high mass-loss rates.
Three of the new spectra (Mc81-2, Mc81-5 and Mc81-7) are almost identical to Mc81-3, therefore having the same spectral type (WN7-8). Three others (Mc81-6, Mc81-9 and Mc81-10) are intermediate-O supergiants, as clearly showed by He<span style="font-variant:small-caps;">ii</span> features along with narrow C<span style="font-variant:small-caps;">iv</span> emission lines. Spectrum Mc81-11 might be as well classified as a O supergiant, although the low signal-to-noise ratio does not allow to confirm it. Finally, spectrum Mc81-8 shows intermediate features between a O supergiant and WN, perhaps entailing this object is in transition between these evolutionary stages ([@transition]).
Discussion and future work
==========================
Based on photometry (section \[discovery\]) and spectroscopic analysis of the 2 firstly observed spectra, [@mc81] presented a first characterization of Mercer 81 that now can be improved by means of the more complete spectroscopic data presented here. Particularly, the total mass of Mercer 81 (a few $\times 10^4$) given by [@mc81] was estimated assuming that all the unobserved emission-line stars were WN and that the cluster has the same evolutionary stage than Westerlund 1 ([@wd1evol]). However, our new data clearly suggest that Mercer 81 is younger, as we have detected earlier spectral types (O5-6 I) with respect to Westerlund 1 (O9 I, maximum), as well as the presence of emission-line stars other than WNs.
Since we are currently in the process of modeling the observed spectra, we cannot present here definitive results yet. Ongoing NLTE spherical atmosphere models will yield accurate spectral types and measure stellar and wind properties, including chemical abundances. This will result in a complete characterization of Mercer 81 which could be crucial to understand the chemodynamics of the inner disk due to its privileged location, at the far end of the Galactic Bar ([@mc81]). On the other hand, the extreme and uncommon objects belonging to this cluster apparently form an evolutionary sequence that may turn Mercer 81 into an ideal laboratory to study the final stages of massive stars.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was partially supported by the Ministerio de Ciencia e Innovación through grants AYA2008-06166-C03-02 and AYA2010-21697-C05-01. D.F. also acknowledges financial support from the FPI-MICINN predoctoral fellowship BES-2009-027786.
|
**Convergence properties of fixed-point search**
**with general but equal phase shifts for any number of iterations**
[^1]
Dafa Li$^{a}$[^2], Xiangrong Li$^{b}$, Hongtao Huang$^{c}$, Xinxin Li$^{d}$
$^a$ Department of mathematical sciences, Tsinghua University, Beijing 100084 CHINA
$^b$ Department of Mathematics, University of California, Irvine, CA 92697-3875, USA
$^c$ Electrical Engineering and Computer Science Department
University of Michigan, Ann Arbor, MI 48109, USA
$^d$ Department of computer science, Wayne State University, Detroit, MI 48202, USA
The correspondence Author Dafa Li,
Phone Number is (8610)62773561
Fax No. is (8610) 62785847
Abstract
Grover presented the fixed-point search by replacing the selective inversions by selective phase shifts of $\pi /3$. In this paper, we investigate the convergence behavior of the fixed-point search algorithm with general but equal phase shifts for any number of iterations.
PACS number: 03.67.Lx
Keywords: Amplitude amplification, the fixed-point search, quantum computing.
Introduction
============
Grover’s quantum search algorithm is used to find a target state in an unsorted database of size $N$[@Grover97][@Grover98]. The Grover’s quantum search algorithm can be considered as a rotation of the state vectors in two-dimensional Hilbert space generated by the start ($s$) and target ($t$) vectors[@Grover98]. The amplitude of the target state increases monotonically towards its maximum and decreases monotonicallyafter reaching the maximum [@LDF05]. This search algorithm is called the amplitude amplification algorithm. For the size $N=2^{n}$ of the database, quantum search algorithm requires $O(\sqrt{N})$ steps to find the target state. As mentioned in [@Grover05] [@Brassard97], unless we stop when it is right at the target state, it will drift away. A fixed-point search algorithm was presented in [@Grover05] to avoid drifting away from the target state. The fixed-point search algorithm obtained by replacing the selective inversions by selective phase shifts of $\pi /3$, converges to the target state irrespective of the number of iterations. The main advantage of the fixed-point search with equal phase shifts of $\pi /3$ is that it performs well for small but unknown initial error probability and the fixed-point behavior leads to robust quantum search algorithms [Grover05]{}. However, the target state is the limit state when the number of iterations tends to the infinite.
For readability, we introduce the fixed-point search algorithm as follows. In [@Grover05] the transformation $UR_{s}^{\pi /3}U^{+}R_{t}^{\pi /3}U$, where $U$ is any unitary operator, was applied to the start state $|s\rangle
$,
$$\begin{aligned}
R_{s}^{\pi /3} &=&I-[1-e^{i\frac{\pi }{3}}]|s\rangle \langle s|, \notag \\
R_{t}^{\pi /3} &=&I-[1-e^{i\frac{\pi }{3}}]|t\rangle \langle t|,
\label{grover1}\end{aligned}$$
where $|t\rangle $ stands for the target state. The transformation $UR_{s}^{\pi /3}U^{+}R_{t}^{\pi /3}U$ is denoted as Grover’s the Phase-$\pi
/3$ search algorithm in [@Tulsi].
Let us consider the fixed-point search algorithm with general but equal phase shifts as follows.
$$\begin{aligned}
R_{s}^{\theta } &=&I-[1-e^{i\theta }]|s\rangle \langle s|, \notag \\
R_{t}^{\theta } &=&I-[1-e^{i\theta }]|t\rangle \langle t|. \label{grover2}\end{aligned}$$
The transformation $UR_{s}^{\theta }U^{+}R_{t}^{\theta }U$ was called as the Phase-$\theta $ search algorithm and studied in [@LDF07a]. It is enough to let $\theta $ be in $[0,\pi ]$.
Note that if we apply $U$ to the start state $|s\rangle $, then the amplitude of reaching the target state $|t\rangle $ is $U_{ts}$[Grover98]{}, where $\left\vert \left\vert U_{ts}\right\vert \right\vert
^{2}=1-\epsilon $. As indicated in [@Grover98], in the case of database search, $|U_{ts}|$ is almost $1/\sqrt{N}$, where $N$ is the size of the database. Thus, $\epsilon $ is almost $1-1/N$ and $\epsilon $ is close to $1$ for the large size of database.
Apply the operations $U$, $R_{s}^{\theta }$, $U^{+}$, $R_{t}^{\theta }$, and $U$ to the start $|s\rangle $ and let $D(\theta )$ be the deviation of the state $UR_{s}^{\theta }U^{+}R_{t}^{\theta }U|s\rangle $ from the $t$ state for any phase shifts of $\theta $. The deviation $D(\theta )$ was reduced in [@LDF07a] and is rewritten as follows. $$D(\theta )=4(1-\cos \theta )^{2}\epsilon (\epsilon -d)^{2}, \label{dev1}$$where $d=\frac{1-2\cos \theta }{2(1-\cos \theta )}$. It was shown that $D(\theta )$ is between $0$ and $1$ in [@LDF07a]. For the Phase-$\pi /3$ search algorithm, $D(\pi /3)=\epsilon ^{3}$[@Grover05].
$\allowbreak $In [@LDF06], we explored the performance of the fixed-point search with general but different phase shifts for one iteration. In [@LDF07a], we discussed the performance of the fixed-point search with general but equal phase shifts for one iteration.
In this paper, we investigate convergence behavior of the fixed-point search with general but equal phase shifts for any number of iterations. It is useful for designing fixed-point search algorithms for different choices of the phase shift parameter $\theta $. The following results are established in Section 2.
(1). The fixed-point search with equal phase shifts of $\theta \leq \pi /2$ converges to the target state.
(2). The fixed-point search with equal phase shifts of $\theta $, where $\pi
/2<\theta \leq \arccos (-1/4)$, converges the target state with the probability of at least $80\%.$
(3). The fixed-point search with equal phase shifts of $\theta $, where $\arccos (-1/4)<\theta \leq 2\pi /3$, converges the target state with the probability of among $66.6\%$ and $80\%.$
(4). The fixed-point search with equal phase shifts of $\theta $, where $2\pi /3<\theta \leq \pi $, does not converge.
In section 3, we analyze the convergence rate for different values of $\theta $. It is demonstrated that the Phase-$\pi /3$ is not always optimal and the convergence rate can be improved by choosing $\theta >\pi /3$. In section 4, we show that for the size $N=2^{n}$ of the database, $O(n)$ iterations of the Phase-$\theta $ search can find the target state. However, as indicated in [@Grover05], $O(n)$ iterations of the Phase-$\theta $ search involve the exponential queries.
Convergence performance of the Phase-$\protect\theta $ search for any number of iterations
==========================================================================================
Let $\epsilon _{0}=\epsilon $ and $0<\epsilon <1$. Then, from Eq. ([dev1]{}) one can obtain the following iteration equation
$$\epsilon _{m+1}=4(1-\cos \theta )^{2}\epsilon _{m}(\epsilon _{m}-d)^{2}.
\label{g-iteration}$$
In this section, we discuss the convergence behavior of the Phase-$\theta $ search for any number of iterations. For the Phase-$\pi /3$ search, after recursive application of the basic iteration for $m$ times, the failure probability $\epsilon _{m}=$ $\epsilon ^{3^{m}}$ and the success probability $\left\vert U_{m,ts}\right\vert =1-\epsilon ^{3^{m}}$[@Grover05]. The $\epsilon _{m}$ in Eq. (\[g-iteration\]) is the failure probability of the Phase-$\theta $ search algorithm after $m$ iterations.
From inference \[11\] in [@LDF07a], Eq. (\[g-iteration\]) has the following fixed-points: $0$, $1$ ($\theta \neq 0$), $a$, where $a=\cos
\theta /(\cos \theta -1)$ ($\theta \neq 0$). In other words, if the sequence $\{\epsilon _{m}\}$ in Eq. (\[g-iteration\]) has a limit then the limit must be $0$, $1$ or $a$. Clearly $a<d$. To study the convergence performance for any number of iterations, we need the following results which are listed in the following paragraphs (A), (B), and (C).
(A). From Eq. (\[g-iteration\]), we obtain the following,
$$\epsilon _{m}-\epsilon _{m-1}=4\epsilon _{m-1}\left( \cos \theta -1\right)
^{2}\left( 1-\epsilon _{m-1}\right) (a-\epsilon _{m-1}). \label{iteration-2}$$
Eqs. (\[g-iteration\]) and (\[iteration-2\]) imply the following convergence property. Property 1.
(1.1) If $\epsilon _{m}=d$, then $\epsilon _{m+l}=0$, for any $l>0$.
(1.2). if $\epsilon _{m-1}>a$ and $\epsilon _{m-1}\neq 0$, $\epsilon
_{m}<\epsilon _{m-1}$;
(1.3). If $\epsilon _{m-1}<a$ and $\epsilon _{m-1}\neq 0$, $\epsilon
_{m}>\epsilon _{m-1}$.
(B). When $\pi /2\leq \theta \leq \pi $, we have the following equation.
$$\epsilon _{m}-a=4(\cos \theta -1)^{2}(\epsilon _{m-1}-a)(\epsilon
_{m-1}-b)(\epsilon _{m-1}-c), \label{fp-eq1}$$
where $b=\frac{1}{2}-\frac{\sqrt{-\cos \theta (2-\cos \theta )}}{2(1-\cos
\theta )}$, and $c=\frac{1}{2}+\frac{\sqrt{-\cos \theta (2-\cos \theta )}}{2(1-\cos \theta )}$. When $\epsilon _{i}=a$, $b$ or $c$, $\epsilon _{i+l}=a$ for any $l>0$. Note that $b<d$, $a<d$, and $d<c$.
$\allowbreak $(C). Let
$$f(x)=4(1-\cos \theta )^{2}x(x-d)^{2}. \label{iterat}$$
Then, the derivative of $f(x)$ is
$$f^{\prime }(x)=12(1-\cos \theta )^{2}(x-d)(x-d/3). \label{deriv}$$
From Eq. (\[deriv\]), (1). $f^{\prime }(x)=0$ at $r$ and $d$, where $r=d/3$; (2).$\ f^{\prime }(x)<0$ when $r<x<d$; (3). $f^{\prime }(x)>0$ when $x<r$ or $x>d$; (4). When $\theta >\pi /3$, $f(x)$ has a relative maximum $g=\frac{2(1-2\cos \theta )^{3}}{27(1-\cos \theta )}$ at $r$ and a relative minimum $0 $ at $d$.
When $0<\protect\theta \leq \protect\pi /2$, for any $\protect\epsilon _{0}\in (0,1)$, the Phase-$\protect\theta $ search converges to the target state.
--------------------------------------------------------------------------------------------------------------------------------------------------------
Note that $0$ is an attractive fixed-point when $0<\theta <\pi /2$ and $0$ is also a semi-attractive fixed-point when $\theta =\pi /2$. See inference \[11\] in [@LDF07a].
(1). $0<\theta \leq \pi /3$
For this case, $d\leq 0$ and $a<0$. In Eq. (\[g-iteration\]), $d=0$ means $\epsilon _{m+1}=\epsilon _{m}^{3}$, which is Grover’s Phase-$\pi /3$ search. From $d<0$ and Eq. (\[g-iteration\]), $\epsilon _{m}>0$. By property (1.2), always $\epsilon _{m}<\epsilon _{m-1}$ when $0<\theta \leq \pi /3$. That is, the sequence $\{\epsilon _{m}\}$ in Eq. (\[g-iteration\]) decreases monotonically$.$Therefore, for any $\epsilon _{0}$ in $(0$, $1)$ $\lim_{m\rightarrow \infty }\epsilon _{m}=0$.
(2). $\pi /3<\theta \leq \pi /2$
For this case, $a\leq 0$, $0<d\leq 1/2$. Hence, from Eq. (\[g-iteration\]) $0\leq \epsilon _{i}<1$. By property (1.2), always $\epsilon _{m}\leq
\epsilon _{m-1}$ when $\pi /3<\theta \leq \pi /2$. That is, the sequence $\{\epsilon _{m}\}$ in Eq. (\[g-iteration\]) decreases. Factually, the sequence $\{\epsilon _{m}\}$ in Eq. (\[g-iteration\]) decreases monotonically and $\epsilon _{m}>0$, or is of the form $\epsilon
_{0}>\epsilon _{1}>...>\epsilon _{k}=0$ and $\epsilon _{l}=0$ for any $l>k$. Therefore, for any $\epsilon _{0}$ in $(0$, $1)$ $\lim_{m\rightarrow \infty
}\epsilon _{m}=0$.
Example 1. For the Phase-$\pi /2$ search, $\epsilon _{m}=\epsilon
_{m-1}(2\epsilon _{m-1}-1)^{2}$. Let $\epsilon _{0}=0.99999$. See Fig. 1.
$\epsilon _{1}=\allowbreak 0.999\,95$, $\ \ \epsilon _{2}=\allowbreak
0.999\,75$, $\epsilon _{3}=\allowbreak 0.998\,75$, $\ \ \epsilon
_{4}=\allowbreak 0.993\,76$,
$\epsilon _{5}=\allowbreak 0.969\,11$, $\ \ \epsilon _{6}=\allowbreak
0.853\,07$, $\epsilon _{7}=\allowbreak 0.425\,37$, $\ \ \epsilon
_{8}=\allowbreak 9.\,\allowbreak 476\,6\times 10^{-3}$.
When $\protect\pi /2<\protect\theta <\arccos (-1/4)$, the Phase-$\protect\theta $ search converges the target state with the success probability of $(1-a)>80\%$.
------------------------------------------------------------------------------------------------------------------------------------------------------------------
For the Phase-$\theta $ search, $a<g<r<b<d<c$. Note that $a$ is an attractive fixed-point. See inference \[11\] in [@LDF07a]. From Eq. ([fp-eq1]{}), we have the following property.
Property 2.
(2.1). $a<\epsilon _{m}\leq g$ whenever $a<\epsilon _{m-1}<b$;
(2.2). $0\leq \epsilon _{m}<a$ whenever $b<\epsilon _{m-1}<c$ or $\epsilon
_{m-1}<a$.
**The convergence region of the Phase-**$\theta $** search**
(A). When $\epsilon _{0}\in (0,c]$ and $\epsilon _{0}\neq d$, the deviation from the target state converges to the fixed-point $a$.
There are four cases. The argument is the following.
Case 1. When $\epsilon _{0}=a$ or $b$ or $c$, it is trivial by Eq. ([fp-eq1]{}).
Case 2. $\epsilon _{0}<a$. By property (2.2), $0<\epsilon _{m}<a$ for any $m$. By property 1, the sequence $\{\epsilon _{m}\}$ increases monotonically. Hence, the sequence $\{\epsilon _{m}\}$ converges to $a$ from below.
Case 3. $a<\epsilon _{0}<b$. By property (2.1), always $a<\epsilon _{m}\leq
g $ for any $m>0$, and by property 1, the sequence $\{\epsilon _{m}\}$ decreases monotonically. Hence, the sequence $\{\epsilon _{m}\}$ converges to $a$ from above.
Case 4. $b<\epsilon _{0}<c$ and $\epsilon _{0}\neq d$. By property (2.2), $0<\epsilon _{1}<a$. Then, it turns to case 2.
Conclusively, when $\epsilon _{0}\in (0,c]$ and $\epsilon _{0}\neq d$, from the above four cases, $\epsilon _{m}\neq d$, hence $\lim_{m\rightarrow
\infty }\epsilon _{m}=a$.
(B). When $\epsilon _{0}\in (c,1)$, the deviation from the target state converges to the fixed-points $a$ or $0$.
By property (1.2), $\epsilon _{0}>...>\epsilon _{j^{\ast }-1}(>c)>\epsilon
_{j^{\ast }\text{ }}(\leq c)$. If $\epsilon _{j^{\ast }\text{ }}=d$, then $\epsilon _{m}=0$ for any $m>j^{\ast }$. Otherwise, $\lim_{m\rightarrow
\infty }\epsilon _{m}=a$ by the above (A).
Phase-$\arccos (-1/4)$ search converges the target state with the success probability of $80\%$.
-------------------------------------------------------------------------------------------------
For the Phase-$\arccos (-1/4)$ search, $a=1/5$ is an attractive fixed-point, see inference \[11\] in [@LDF07a]. $b=a=1/5$, $d=3/5$, and $c=4/5$. The iteration equation is $\epsilon _{m}=\epsilon _{m-1}(\allowbreak 5\epsilon
_{m-1}-3)^{2}/4$. Eq. (\[fp-eq1\]) becomes the following.
$\allowbreak $$$\epsilon _{m}-1/5=\allowbreak \frac{25}{4}\left( \epsilon _{m-1}-4/5\right)
\left( \epsilon _{m-1}-1/5\right) ^{2}. \label{eq2}$$
From Eq. (\[eq2\]) we have the following property.
Property 3.
(3.1). $\epsilon _{m}<1/5$ when $\epsilon _{m-1}<4/5$ and $\epsilon
_{m-1}\neq 1/5$.
(3.2). $\epsilon _{m}>1/5$ when $\epsilon _{m-1}>4/5$.
**The convergence region of the Phase-**$\arccos (-1/4)$** search**
(A). When $\epsilon _{0}\in (0,4/5]$ and $\epsilon _{0}\neq 3/5$, the deviation from the target state converges to the fixed-point $1/5$.
When $\epsilon _{0}=1/5$ or $4/5$, it is trivial by Eq. (\[fp-eq1\]). When $\epsilon _{0}\in (0,4/5)$ and $\epsilon _{0}\neq 1/5$, always $\epsilon
_{m}<1/5$ for $m>0$ by property (3.1) and the sequence $\{\epsilon _{m}\}$ increases monotonically from $m>0$ by property (1.3). Therefore, the sequence $\{\epsilon _{m}\}$ converges to $1/5$ from below. (B) When $\epsilon _{0}\in (4/5,1)$, the deviation from the target state converges to the fixed-points $1/5$ or $0$.
By property (1.2), $\epsilon _{0}>\epsilon _{1}>....>\epsilon _{m}(\leq 4/5)$. Case 1. If $\epsilon _{m}=3/5$, then $\epsilon _{i}=0$ for any $i>m$. Case 2. Otherwise, by the above (A), $\lim_{m\rightarrow \infty }\epsilon
_{m}=1/5 $.
$\allowbreak $
Example 2. Let $\epsilon _{0}=0.9999;$
$\epsilon _{1}=\allowbreak 0.999\,4$, $\ \ \epsilon _{2}=\allowbreak
0.996\,4 $, $\ \ \ \ \ \ \ \ \epsilon _{3}=\allowbreak 0.978\,55$, $\
\epsilon _{4}=\allowbreak 0.876\,41$,
$\epsilon _{5}=\allowbreak 0.418\,50$, $\ \epsilon _{6}=\allowbreak
8.\,\allowbreak 616\,5\times 10^{-2}$, $\ \epsilon _{7}=\allowbreak
0.142\,19 $, $\ \epsilon _{8}=\allowbreak 0.186\,26$,
$\epsilon _{9}=\allowbreak 0.199\,28$, $\ \epsilon _{10}=\allowbreak 0.2$.
$\allowbreak $When $\arccos (-1/4)<\protect\theta \leq 2\protect\pi /3$, the Phase-$\protect\theta $ search converges the target state with the success probability of $(1-a)$, where $66\%\leq (1-a)<80\%$.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
For the Phase-$\theta $ search, $b<r<a<g<d<c$. Note that $a$ is an attractive fixed-point when $\arccos (-1/4)<\theta <2\pi /3$ and $1/3$ is a semi-attractive fixed-point when $\theta =2\pi /3$. See inference \[11\] in [@LDF07a]. From Eq. (\[fp-eq1\]), we have the following property.
Property 4.
(4.1). $a<\epsilon _{m}\leq g$ whenever $b<\epsilon _{m-1}<a$;
(4.2). $0\leq \epsilon _{m}<a$ whenever $a<\epsilon _{m-1}<c$ or $\epsilon
_{m-1}<b$.
**The convergence region of the Phase-**$\theta $** search**
(A). When $\epsilon _{0}\in (0,c]$ and $\epsilon _{0}\neq d$, the deviation from the target state converges to the fixed-point $a$.
There are seven cases. We argue them as follows.
Case 1. If $\epsilon _{0}=a$ or $b$ or $c$, then it is trivial.
Case 2. $a<\epsilon _{0}\leq g$. The proof is put in Appendix A.
Case 3. $\epsilon _{0}<b$. By property (1.3), $\epsilon _{j}$ increases monotonically from $\epsilon _{0}$ until $\epsilon _{j^{\ast }-1}<b$ and $b\leq \epsilon _{j^{\ast }}<f(b)=a$ since $f^{\prime }(x)>0$ when $x<b$. If $\epsilon _{j^{\ast }}=b$, it is trivial. Otherwise, by property (4.1), $\epsilon _{j^{\ast }+1}$ is in $(a,g]$. Now it turns to case 2.
Case 4. $b<\epsilon _{0}\leq r$. When $\epsilon _{0}=r$, $\epsilon
_{m}=f^{(m-1)}(g)$. From the proof of case 2, $\lim_{m\rightarrow \infty
}f^{(m-1)}(g)=a$. Next consider that $b<\epsilon _{0}<r$. Since $f^{\prime
}(x)>0$ when $b<x<r$, $f(b)<f(\epsilon _{0})<f(r)$. That is, $a<\epsilon
_{1}<g$. It turns to case 2.
Case 5. $r<\epsilon _{0}<a$. Since $f^{\prime }(x)<0$ when $r<x<a$, $a<\epsilon _{1}<g$. It turns to case 2.
Case 6. $g<\epsilon _{0}<d$. Since $f^{\prime }(x)<0$ when $g<x<d$ and $a<g$, $0<\epsilon _{1}<f(g)<a$. Then, it turns to cases 1, 3, 4, 5.
Case 7. $d<\epsilon _{0}<c$. Since $f^{\prime }(x)>0$ when $d<x<c$, $0<\epsilon _{1}<a$. Then, it turns to cases 1, 3, 4, 5.
(B). When $\epsilon _{0}\in (c,1)$, the deviation from the target state converges to the fixed-points $a$ or $0$.
When $\epsilon _{0}>c$, by property (1.2) the sequence $\{\epsilon _{i}\}$ decreases monotonically from $\epsilon _{0}$ to $\epsilon _{i^{\ast }\text{ }}\leq c$. Case 1, if $\epsilon _{i^{\ast }\text{ }}=d$, then $\epsilon
_{i}=0 $ for any $i>i^{\ast }$. Case 2. Otherwise, by the above (A) $\lim_{m\rightarrow \infty }\epsilon _{m}=a$.
Example 3. For the Phase-$2\pi /3$ search, $a=1/3$. The iteration equation becomes $\epsilon _{m}=\epsilon _{m-1}(3\epsilon _{m-1}-2)^{2}$. Let $\epsilon _{0}=0.99999$. We have the following iterations. See Fig. 1.
$\epsilon _{1}=0.999\,93$, $\ \epsilon _{2}=0.999\,51$, $\epsilon
_{3}=0.996\,57$, $\ \epsilon _{4}=0.976\,17$,
$\epsilon _{5}=0.841\,59$, $\ \epsilon _{6}=0.231\,76$, $\ \ \epsilon
_{7}=\allowbreak 0.394\,52$, $\ \epsilon _{8}=\allowbreak 0.263\,50$,
$\epsilon _{9}=\allowbreak 0.385\,47$, $\ \epsilon _{10}=\allowbreak
0.274\,32$, $\epsilon _{11}=\allowbreak 0.380\,05$, $\ \epsilon
_{12}=\allowbreak 0.280\,99$,
$\epsilon _{13}=\allowbreak 0.376\,17$.
$2\protect\pi /3<\protect\theta \leq \protect\pi $, the Phase-$\protect\theta $ search does not converge.
---------------------------------------------------------------------------------------------------------
For the Phase-$\theta $ search, $b<r<a<d<c$. From Eq. (\[fp-eq1\]), we have the following property.
Property 5
(5.1). $a<\epsilon _{m}\leq g$ whenever $b<\epsilon _{m-1}<a$;
(5.2). $0\leq \epsilon _{m}<a$ whenever $a<\epsilon _{m-1}<c$ or $\epsilon
_{m-1}<b$.
For large $\epsilon $, by property (1.2), the sequence $\{\epsilon _{i}\}$ decreases monotonically from $\epsilon _{0}$ to $\epsilon _{i^{\ast }}(\leq
c)$. If $\epsilon _{i^{\ast }}=d$, then $\epsilon _{i}=0$ for any $i>i^{\ast
}$. If $\epsilon _{i^{\ast }}=a$, $b$, or $c$, then $\epsilon _{i}=a$ when $i>i^{\ast }$. Otherwise, when $i>i^{\ast }$, $\epsilon _{i}$ oscillate around the fixed point $a$ by property 1. However, the sequence $\{\epsilon
_{i}\}$ does not converges because $a$, $0$ and $1$ are repulsive fixed-points.
Example 4. For the Phase-$\pi $ search, the iteration equation becomes $\epsilon _{m}=\epsilon _{m-1}(4\epsilon _{m-1}-3)^{2}$, $a=1/2$. Let $\epsilon =0.99999$. We have the following iterations. See Fig. 1.
$\epsilon _{1}==0.999\,91$, $\ \ \epsilon _{2}=0.999\,19,$ $\ \epsilon
_{3}=0.992\,73$, $\ \epsilon _{4}=0.935\,83,$
$\epsilon _{5}=0.517\,07$, $\ \ \ \epsilon _{6}=0.448\,87$, $\epsilon
_{7}=0.651\,25$, $\ \epsilon _{8}=0.101\,61,$
$\epsilon _{9}=0.683\,49$, $\ \ \ \epsilon _{10}=4.\,837\,6\times 10^{-2}$.
Clearly, the sequence $\{\epsilon _{i}\}$ monotonically decreases from $\epsilon _{0}$ to $\epsilon _{6}$. Note that after the sixth iteration, $\epsilon _{m}$ oscillate around the fixed point $1/2$.
A comparison of rates of convergence after any number of iterations
===================================================================
For the Phase-$\pi /3$ search, let the iteration equation be $\epsilon
_{m}(\pi /3)=(\epsilon _{m-1}(\pi /3))^{3}$, where $\epsilon _{m}(\pi /3)$ is the failure probability of the Phase-$\pi /3$ search algorithm after $m$ iterations. For the Phase-$\theta $ search, we can rewrite Eq. ([g-iteration]{}) as $\epsilon _{m}(\theta )=4(1-\cos \theta )^{2}\epsilon
_{m-1}(\theta )(\epsilon _{m-1}(\theta )-d)^{2}$, where the $\epsilon
_{m}(\theta )$ is the failure probability of the Phase-$\theta $ search algorithm after $m$ iterations. We want to compare the failure probability of the Phase-$\theta $ ($\neq \pi /3$) search algorithm with the one of the Phase-$\pi /3$ search after $m$ iterations. It is known that the less the failure probability is, the faster the algorithm converges. By factoring,
$$\begin{aligned}
\epsilon _{m}(\theta )-\epsilon _{m}(\pi /3) &=& \notag \\
&&\epsilon _{m-1}(\theta )(2\cos \theta -1)(1-\epsilon _{m-1}(\theta
))(3-2\cos \theta )\ast \notag \\
&&(\epsilon _{m-1}(\theta )-\frac{1-2\cos \theta }{3-2\cos \theta })+\epsilon _{m-1}^{3}(\theta )-\epsilon _{m-1}^{3}(\pi /3). \label{compare}\end{aligned}$$
We have the following results.
(1). $\pi /3<\theta \leq \pi $
Case 1. For large $\epsilon $, the Phase-$\theta $ search converges faster than the Phase-$\pi /3$ search for $m$ iterations until $\epsilon
_{m-1}(\theta )<$ $\frac{1-2\cos \theta }{3-2\cos \theta }$.
In [@LDF07a], we show if $\epsilon _{0}(\theta )=\epsilon _{0}(\pi
/3)=\epsilon >\frac{1-2\cos \theta }{3-2\cos \theta }$ then $\epsilon
_{1}(\theta )<\epsilon _{1}(\pi /3)=\epsilon ^{3}$. If $\epsilon
_{m-1}(\theta )>\frac{1-2\cos \theta }{3-2\cos \theta }$ and $\epsilon
_{m-1}(\theta )<\epsilon _{m-1}(\pi /3)$, then by Eq. (\[compare\])$\
\epsilon _{m}(\theta )<\epsilon _{m}(\pi /3)$. Thus, $\epsilon _{i}(\theta
)<\epsilon _{i}(\pi /3)$, where $i=1$, $2$, ..., $m-1$, until $\epsilon
_{m-1}(\theta )<$ $\frac{1-2\cos \theta }{3-2\cos \theta }$. It says that after $m$ iterations, the failure probability of the Phase-$\theta $ search is less than the one of the Phase-$\pi /3$ search until $\epsilon
_{m-1}(\theta )<$ $\frac{1-2\cos \theta }{3-2\cos \theta }$. It suggests us first to use the fixed-point search with large phase shifts for the large size of database.
Case 2. For small $\epsilon $, the Phase-$\pi /3$ search converges faster than the Phase-$\theta $ search for $m$ iterations until $\epsilon
_{m-1}(\theta )>\frac{1-2\cos \theta }{3-2\cos \theta }$.
In [@LDF07a], we show if $\epsilon _{0}(\theta )=\epsilon _{0}(\pi
/3)=\epsilon <\frac{1-2\cos \theta }{3-2\cos \theta }$ then $\epsilon
_{1}(\theta )>\epsilon _{1}(\pi /3)=\epsilon ^{3}$. If $\epsilon
_{m-1}(\theta )<\frac{1-2\cos \theta }{3-2\cos \theta }$ and $\epsilon
_{m-1}(\theta )>\epsilon _{m-1}(\pi /3)$, then by Eq. (\[compare\])$\
\epsilon _{m}(\theta )>\epsilon _{m}(\pi /3)$.
(2). When $0<\theta <\pi /3$, the Phase-$\pi /3$ search converges faster than the Phase-$\theta $ search for any $\epsilon $ for any number of iterations.
When $\epsilon _{0}(\theta )=\epsilon _{0}(\pi /3)=\epsilon $, in [LDF07a]{} we show $\epsilon _{1}(\theta )>\epsilon _{1}(\pi /3)$. Assume that $\epsilon _{m-1}(\theta )>\epsilon _{m-1}(\pi /3)$. From Eq. (\[compare\]), it is easy to see that $\epsilon _{m}(\theta )>\epsilon _{m}(\pi /3)$. Therefore, $\epsilon _{m}(\theta )>\epsilon _{m}(\pi /3)$ for any $m$. Hence, when $0<\theta <\pi /3$, the Phase-$\pi /3$ search converges faster than the Phase-$\theta $ search for any $\epsilon $ for any number of iterations.
For any known $\protect\epsilon $, $O(n)$ iterations can find the target state.
===============================================================================
Assume that a database has $N=2^{n}$ states (items). Then a state (an item) is found with the probability of $1/N$[@Grover98]. In other words, the failure probability $\epsilon =1-1/N$. It is known that the Phase-$\pi /3$ search converges the target state. In this section, we investigate how to use the fixed-point search to find the target state in a database when $\epsilon $ is known. As discussed in [@Grover05], the fixed-point search is a recursive algorithm, therefore the number of queries grows exponentially with the number of recursion levels. For example, the Phase-$\pi /3$ search at $i$-level recursion involves $q_{i}=(3^{i}-1)/2$ queries [@Tulsi]. This implies that $O(n)$ iterations of the Phase-$\theta $ search involve the exponential queries.
When $\protect\epsilon \leq 3/4$, only one iteration is needed to find the target state.
----------------------------------------------------------------------------------------
When $0\leq \epsilon \leq \frac{3}{4}$, $\left\vert 1-\frac{1}{2(1-\epsilon )}\right\vert \leq 1$. Let $\cos \theta =1-\frac{1}{2(1-\epsilon )}$. Then $D(\theta )=0$. Therefore, if $\epsilon $ is fixed and $0\leq \epsilon \leq
\frac{3}{4}$, then we choose $\theta =\arccos [1-\frac{1}{2(1-\epsilon )}],$ which is in $(\pi /3$ ,$\pi ]$, as phase shifts. The Phase-$\arccos [1-\frac{1}{2(1-\epsilon )}]$ search will obviously make the deviation vanish. It means that one iteration will reach $t$ state if the $\theta $ is chosen as phase shifts. Ref. [@LDF07a].
When $\protect\epsilon >3/4$, $O(n)$ iterations can find the target state.
--------------------------------------------------------------------------
### First use the Phase-$\protect\pi /3$ search
For the Phase-$\pi /3$ search, $\epsilon _{n}=\epsilon ^{3^{n}}$. There exists the least natural number $n^{\ast }$ such that $\epsilon ^{3^{n^{\ast
}}}\leq 3/4$. By calculating, $n^{\ast }=\lceil (\ln \ln \frac{4}{3}-\ln \ln
\frac{1}{\epsilon })/\ln 3\rceil $.
Lemma 1. For the Phase-$\pi /3$ search, $n^{\ast }=$ $O(n)$.
Proof. In the case of database search, Let $N=2^{n}$. Then $\epsilon
=1-2^{-n}$, and $\lim_{n\rightarrow +\infty }\frac{n^{\ast }}{n}=\frac{\ln 2}{\ln 3}$. Almost $\frac{n\ln 2}{\ln 3}\approx \allowbreak \lceil
0.63n\rceil $. Let $N=10^{n}$. Then $\epsilon =1-10^{-n}$, and $\lim_{n\rightarrow +\infty }\frac{n^{\ast }}{n}=\frac{\ln 10}{\ln 3}=\frac{1}{\lg 3}$. Almost $\frac{n}{\lg 3}\approx 2n$. Thus, $n^{\ast }=O(n)$. Hence, when $\epsilon >3/4$, after $n^{\ast }$ iterations of the Phase-$\pi
/3$ search the failure probability $\epsilon _{n^{\ast }}\leq $ $3/4$. Then, after one iteration of the Phase-$\arccos [1-\frac{1}{2(1-\epsilon _{n^{\ast
}})}]$ search by using the result in section 4.1, it will reach $t$ state.
Example 5. Let $N=10^{4}$. Then $\epsilon =1-10^{-4}$, $n^{\ast }=8$,$\
\epsilon _{7}=\allowbreak 0.803\,32$, $\epsilon _{8}=\allowbreak 0.518\,4$. See Fig.1. However, for this purpose, it only needs 4 iterations for the Phase-$\pi $ search. See example 7.
Example 6. Let $N=2^{10}$. Then $\epsilon =1-2^{-10}$, $n^{\ast }=6$, $\epsilon _{5}=\allowbreak 0.788\,56$, $\epsilon _{6}=\allowbreak 0.490\,35$.
### First use the Phase-$\protect\theta $ ($\neq \protect\pi /3$) search
Let $\epsilon >3/4$. Then, by property (1.2), for the Phase-$\theta $ search, there exists the least natural number $m^{\ast }(\theta )$ such that $\ \epsilon _{0}>\epsilon _{1}>...>\epsilon _{m^{\ast }(\theta
)-1}(>3/4)>\epsilon _{m^{\ast }(\theta )}(\leq 3/4)$.... Thus, after $m^{\ast }(\theta )$ iterations of the Phase-$\theta $ search, the failure probability $\epsilon _{m^{\ast }(\theta )}\leq 3/4$. Then, after one iteration for the Phase-$\arccos [1-\frac{1}{2(1-\epsilon _{m(\theta )^{\ast
}})}]$ search by using the result in section 4.1, it will reach $t$ state.
Next let us calculate $m^{\ast }(\theta )$. Let $\delta =1-\epsilon $, where $\delta $ is the success probability. When $\epsilon $ is close to $1$, $\delta $ is close to $0$. Then, for large $\epsilon $, by induction $\epsilon _{l}=1-[1+4(1-\cos \theta )]^{l}\delta +O(\delta ^{2})$. Thus, $\epsilon _{l}\approx 1-[1+4(1-\cos \theta )]^{l}\delta $. By this approximate formula of $\epsilon _{l}$, $m^{\ast }(\theta )$ $\approx
M^{\ast }(\theta )=\lceil \frac{-2\lg 2-\lg \delta }{\lg (1+4(1-\cos \theta
))}\rceil $.
In the case of database search, let $N=2^{n}$. Then $\epsilon =1-2^{-n}$, $\delta =2^{-n}$, and $m^{\ast }(\theta )$ $\approx $ $M^{\ast }(\theta
)=\lceil \frac{(n-2)\lg 2}{\lg (1+4(1-\cos \theta ))}\rceil $. For the Phase-$\pi /3$ search, $M^{\ast }(\pi /3)=\lceil \frac{n\ln 2}{\ln 3}-\frac{2\ln 2}{\ln 3}\rceil $. Note that $\frac{2\ln 2}{\ln 3}=\allowbreak
1.\,\allowbreak 261\,9$. Therefore, when $n$ is large enough $M^{\ast }(\pi
/3)\approx $ $m^{\ast }(\pi /3)=n^{\ast }$. For the Phase-$\pi $ search, $m^{\ast }(\pi )\approx M^{\ast }(\pi )=\lceil (n-2)\lg 2/(2\lg 3)\rceil
\approx (\lg 2)n$. See Table (I).
Let $N=10^{n}$. Then $\epsilon =1-10^{-n}$, $\delta =10^{-n}$, and $m^{\ast
}(\theta )\approx M^{\ast }(\theta )=\lceil \frac{n-2\lg 2}{\lg (1+4(1-\cos
\theta ))}\rceil $. For the Phase-$\pi /3$ search, $M^{\ast }(\pi /3)=\lceil
\frac{n}{\lg 3}-\frac{2\lg 2}{\lg 3}\rceil $. Note that $\frac{2\lg 2}{\lg 3}=\frac{2\ln 2}{\ln 3}$. Therefore, when $n$ is large enough $M^{\ast }(\pi
/3)\approx $ $m^{\ast }(\pi /3)=n^{\ast }$. For the Phase-$\pi $ search, $m^{\ast }(\pi )\approx M^{\ast }(\pi )=\lceil (n-2\lg 2)/(2\lg 3)\rceil
\approx n$. See
Example 7. Let $N=10^{4}$. Then $\epsilon =1-10^{-4}$, $M^{\ast }(\pi )=4$, $\epsilon _{4}=0.475\,32$. See Table (II).
Lemma 2. For the Phase-$\theta $ $(\neq \pi /3)$ search, $m^{\ast }(\theta
)= $ $O(n)$.
Proof. When $\pi /3<\theta \leq \pi $, as discussed in case 1 of (1) in Sec. 3, $m^{\ast }(\theta )<n^{\ast }$. By lemma 1, this lemma holds. When $0<\theta <\pi /3$, from the approximate formula of $m^{\ast }(\theta )$, $m^{\ast }(\theta )=$ $O(n)$.
Remark. $M^{\ast }(\theta )$ monotonically decreases as $\theta $ increases from $0$ to $\pi $, especially $\frac{M^{\ast }(\pi )}{n^{\ast }}\approx 1/2$. Therefore, we suggest first to use Phase-$\pi $ search for $m^{\ast }(\pi
) $ times to get the failure probability $\epsilon _{m^{\ast }}\leq 3/4$.
Summary
=======
In this paper, we investigate convergence performance of the Phase-$\theta $ search for any number of iterations. We discuss the convergence region and rate of the Phase-$\theta $ search and study the convergence behavior of the Phase-$\theta $ search for different initial $\epsilon _{0}$.
**Acknowledgement**
We want to thank the reviewer of [@LDF07a] for suggesting us to study the convergence behavior of the fixed-point search with general but equal phase shifts for any number of iterations.
Appendix A
==========
Proof. Since $f^{\prime }(x)<0$ when $r<x<d$, $f(g)\leq \epsilon _{1}<a$. Note that $r<f(g)$. Thus, $r<f(g)\leq \epsilon _{1}<a$. Let $f^{(k)}(x)=f(f^{(k-1)}(x))$. Since $f^{\prime }(x)<0$, $a<\epsilon _{2}\leq
f^{(2)}(g)<f(r)=g$ and $r<f(g)<f^{(3)}(g)\leq \epsilon _{3}<a$. By induction, generally $a<\epsilon _{2k}\leq
f^{(2k)}(g)<f^{(2k-2)}(g)<...f^{(2)}(g)<g$ and $r<f(g)<...<f^{(2k-1)}(g)<f^{(2k+1)}(g)\leq \epsilon _{2k+1}<a$. That is, $\epsilon _{i}$ oscillate around the fixed point $a$ by property 1 and between $f^{(2k)}(g)$ and $f^{(2k+1)}(g)$. It is plain that the sequence $\{f^{(2k)}(g)\}$ decreases monotonically as $k$ increases while the sequence $\{f^{(2k+1)}(g)\}$ increases monotonically as $k$ does. Hence, the sequences $\{f^{(2k)}(g)\}$ and $\{f^{(2k+1)}(g)\}$ have limits. Let $\lim_{k\rightarrow \infty }f^{(2k)}(g)=\alpha $ and $\lim_{k\rightarrow
\infty }f^{(2k+1)}(g)=\beta $. Clearly, $\alpha $, $\beta <d$. From Eq. ([g-iteration]{}), $f^{(2k)}(g)=4(1-\cos \theta
)^{2})f^{(2k-1)}(g)(f^{(2k-1)}(g)-d)^{2}$ and $f^{(2k+1)}(g)=4(1-\cos \theta
)^{2})f^{(2k)}(g)(f^{(2k)}(g)-d)^{2}$. By taking the limits, we obtain $\alpha =4(1-\cos \theta )^{2})\beta (\beta -d)^{2}$ and $\beta =4(1-\cos
\theta )^{2})\alpha (\alpha -d)^{2}$. By substituting, $\beta =[4(1-\cos
\theta )^{2})]^{2}\beta (\beta -d)^{2}(\alpha -d)^{2}$. By cancelling, $[4(1-\cos \theta )^{2})]^{2}(\beta -d)^{2}(\alpha -d)^{2}=1$. Then, there are two cases. Case 1. $4(1-\cos \theta )^{2})(d-\beta )(d-\alpha )=1$. By solving this equation, $\alpha =\beta =1$ or $\alpha =\beta =a$. Since $\alpha $, $\beta <d<1$, then $\alpha =\beta =a$. Case 2. $4(1-\cos \theta
)^{2})(d-\beta )(d-\alpha )=-1$. There is no solution because $\alpha $, $\beta <d$. Therefore, $\lim_{k\rightarrow \infty
}f^{(2k)}(g)=\lim_{k\rightarrow \infty }f^{(2k+1)}(g)=a$. Then, $\lim_{k\rightarrow \infty }f^{(k)}(g)=a$, and also $\lim_{m\rightarrow
\infty }\epsilon _{m}=a$. We finish the proof.
[9]{} L.K.Grover, Phys. Rev. Lett. 79 (1997) 325.
L.K.Grover, Phys. Rev. Lett. 80 (1998) 4329.
D. Li et al., Theor. Math. Phys. 144(3) (2005) 1279-1287.
L.K.Grover, Phys. Rev. Lett. 95 (2005) 150501.
G. Brassard, Science 275 (1997) 627.
T.Tulsi, L. Grover, and A. Patel, quant-ph/0505007. Also, Quant. Inform. and Comput. 6(6) (2006) 483–494.
D. Li et al., Eur. Phys. J. D 45 (2007) 335-340.
D. Li et al., Phys. Lett. A 362 (2007) 260-264. Also see quant-ph/0604062.
[^1]: The paper was supported by NSFC(Grants No. 60433050 and 60673034), the basic research fund of Tsinghua university NO: JC2003043.
[^2]: email address:dli@math.tsinghua.edu.cn
|
---
abstract: 'Experimental studies of hypernuclear dynamics, besides being essential for the understanding of strong interactions in the strange sector, have important astrophysical implications. The observation of neutron stars with masses exceeding two solar masses poses a serious challenge to the models of hyperon dynamics in dense nuclear matter, many of which predict a maximum mass incompatible with the data. In this article, it is argued that valuable new insight may be gained extending the experimental studies of kaon electro production from nuclei to include the $\isotope[208][]{\rm Pb}(e,e^\prime K^+) \isotope[208][\Lambda]{\rm Tl}$ process. The connection with proton knockout reactions and the availability of accurate $\isotope[208][]{\rm Pb}(e,e^\prime p) \isotope[207][]{\rm Tl}$ data can be exploited to achieve a largely model-independent analysis of the measured cross section. A framework for the description of kaon electro production based on the formalism of nuclear many-body theory is outlined.'
author:
- Omar Benhar
title: 'Extracting Hypernuclear Properties from the $(e, e^\prime K^+)$ Cross Section'
---
Introduction
============
Experimental studies of the $(e,e^\prime K^+)$ reaction on nuclei have long been recognised as a valuable source of information on hypernuclear spectroscopy. The extensive program of measurements performed or approved at Jefferson Lab [@E94-107; @E12-15-008]encompassing a variety of nuclear targets ranging from $\isotope[6][]{\rm Li}$ to $\isotope[40][]{\rm Ca}$ and $\isotope[48][]{\rm Ca}$ has the potential to shed new light on the dynamics of strong interactions in the strange sector, addressing outstanding issues such as the isospin-dependence of hyperon-nucleon interactions and the role of three-body forces involving nucleons and hyperons. In addition, because the appearance of hyperons is expected to become energetically favoured in dense nuclear matter, these measurements have important implications for neutron star physics.
The recent observation of two-solar-mass neutron stars [@demorest; @antonio] the existence of which is ruled out by many models predicting the presence of hyperons in the neutron star core [@isaac_etal]suggests that the present understanding of nuclear interactions involving hyperons is far from being complete. In the literature, the issue of reconciling the calculated properties of hyperon matter with the existence of massive stars is referred to as [*hyperon puzzle*]{} [@puzzle].
Owing to the severe difficulties involved in the determination of the potential describing hyperon-nucleon (YN) interactions from scattering data, the study of hypernuclear spectroscopy has long been regarded as a very effective alternative approach to obtain much needed complementary information.
In this context, the $(e,e^\prime K^+)$ process offers clear advantages. The high resolution achievable by $\gamma$-ray spectroscopy can only be exploited to study energy levels below nucleon emission threshold, while $(K^-,\pi^-)$ and $(\pi^+, K^+)$ reactions mainly provide information on non-spin-flip interactions. Moreover, compared to hadron induced reactions, kaon electro production allows for a better energy resolution, which may in turn result in a more accurate identification of the hyperon binding energies [@E94-107]. However, the results of several decades of study of of the $(e,e^\prime p)$ reaction [@Benhar:NPN] show that to achieve this goal the analysis of the measured cross sections must be based on a theoretical model taking into account the full complexity of electron-nucleus interactions. Addressing this issue will be critical for the extension of the Jefferson Lab program to the case of a heavy target with large neutron excess, such as $\isotope[208][]{\rm Pb}$, best suited to study hyperon dynamics in an environment providing the best available proxy of the neutron star interior.
This article is meant to be a first step towards the development of a comprehensive framework for the description of the $(e,e^\prime K^+)$ cross section within the formalism of nuclear many-body theory, which has been extensively and successfully employed to study the proton knockout reaction [@Benhar:NPN]. In fact, the clear connection between $(e,e^\prime p)$ and $(e,e^\prime K^+)$ processes, that naturally emerges in the context of the proposed analysis, shows that the missing energy spectra measured in $(e,e^\prime p)$ experiments provide the baseline needed for a model-independent determination of the hyperon binding energies.
The text is structured as follows. In Sect.\[Axsec\] the description of kaon electro-production from nuclei in the kinematical regime in which factorisation of the nuclear cross section is expected to be applicable is reviewed, and the relation to the proton knockout process is highlighted. The main issues associated with the treatment of the elementary electron-proton vertex and the calculation of the nuclear amplitudes comprising the structure of the $\isotope[208][]{\rm Pb}(e,e^\prime K^+) \isotope[208][\Lambda]{\rm Tl}$ cross section are discussed in Sect. \[Pbxsec\]. Finally, the summary and an outlook to future work can be found in in Sect. \[summary\].
The ${\rm A}(e, e^\prime K^+){_\Lambda}{\rm A}$ cross section {#Axsec}
=============================================================
Let us consider the kaon electro-production process $$\begin{aligned}
\label{eek:A}
e(k) + {\rm A}(p_{\rm A}) \to e^\prime(k^\prime) + K^+(p_K) + {_\Lambda}{\rm A}(p_R) \ , \end{aligned}$$ in which an electron scatters off a nucleus of mass number ${\rm A}$, and the hadronic final state $$\begin{aligned}
\label{def:F}
| F \rangle = | K^+ {_\Lambda}{{\rm A}} \rangle \ , \end{aligned}$$ comprises a $K^+$ meson and the recoiling hypernucleus, resulting from the replacement of a proton with a $\Lambda$ in the target nucleus. The incoming and scattered electrons have four-momenta $k \equiv (E,{\bf k})$ and $k^\prime \equiv(E^\prime,{\bf k}^\prime)$, respectively, while the corresponding quantities associated with the kaon and the recoiling hypernucleus are denoted $p_K \equiv (E_K,{\bf p}_k)$ and $p_R \equiv(E_R,{\bf p}_R)$. Finally, in the lab reference framein which the lepton kinematical variables are measured $p_A \equiv(M_A,0)$.
The differential cross section of reaction can be written in the form $$\begin{aligned}
\label{A:xsec}
d \sigma_A \propto L_{\mu\nu} W^{\mu\nu} \ \delta^{(4)}( p_0 + q - p_F) \ , \end{aligned}$$ with $\lambda, \mu = 1,2,3$, where $q = k - k^\prime$ and $p_F~=~p_K + p_R$ are the four-momentum transfer and the total four-momentum carried by the hadronic final state, respectively. The tensor $L_{\mu\nu}$, fully specified by the electron kinematical variables, can be written in the form [@AFF]
$$\begin{aligned}
L = \left(
\begin{array}{ccc}
\eta_+ & 0 & -\sqrt{\epsilon_L \eta_+} \\
0 & \eta_- & 0 \\
-\sqrt{\epsilon_L \eta_+} & 0 & \epsilon_L \\
\end{array}
\right) \ ,\end{aligned}$$
with $\eta_\pm = \left( 1 \pm \epsilon \right)/2$ and $$\begin{aligned}
\epsilon = \left( 1 + 2 \frac{|{\bf q}|^2}{Q^2}\ \tan^2 \frac{\theta_e}{2} \right)^{-1} \ \ ,\end{aligned}$$ where $\theta_e$ is the electron scattering angle, $q \equiv ( \omega, {\bf q} )$, $Q^2 = - q^2$, and $\epsilon_L = \epsilon Q^2 / \omega^2$.
All the information on hadronic, nuclear and hypernuclear dynamics in contained in the nuclear response tensor, defined as $$\begin{aligned}
\label{A:tensor}
W^{\mu\nu} = \langle 0 | {J_{\rm A}^\mu}^\dagger(q) | F \rangle \langle F | J_{\rm A}^\nu(q) | 0 \rangle \ ,\end{aligned}$$ where $|0 \rangle$ denotes the target ground state and the final state $|F\rangle$ is given by Eq..
Equation shows that the theoretical calculation of the cross section requires a consistent description of the nuclear and hypernuclear wave functions, as well as of the nuclear current operator appearing in the transition matrix element, $ J_{\rm A}^\mu$. This problem, which in general involves non trivial difficulties, greatly simplifies in the kinematical region in which the impulse approximation can be exploited.
Impulse Approximation and Factorisation {#IA}
---------------------------------------
Figure \[graph\] provides a diagrammatic representation of the $(e,e^\prime K^+)$ process based on the factorisation [*ansatz*]{}. This scheme is expected to be applicable in the impulse approximation regime, corresponding to momentum transfer such that the wavelenght of the virtual photon, $\lambda~\sim~1/|{\bf q}|$, is short compared to the average distance between nucleons in the target nucleus, $d_{NN}~\sim~1.5 \ {\rm fm}$.
![Schematic representation of the scattering amplitude associated with the process of Eq. in the impulse approximation regime.[]{data-label="graph"}](Fig1.pdf)
Under these condition, which can be easily met at Jefferson Lab, hereafter JLab, the beam particles primarily interacts with individual protons, the remaining ${\rm A}-1$ nucleons acting as spectators. As a consequence, the nuclear current operator reduces to the sum of one-body operators describing the electron-proton interaction $$\begin{aligned}
J_{\rm A}^\mu(q) = \sum_{i=1}^A j^\mu_i(q)\ , \end{aligned}$$ and the hadronic final state takes the product form $$\begin{aligned}
| F \rangle = | K^+ \rangle \otimes \vert {_\Lambda}{{\rm A}} \rangle \ , \end{aligned}$$ with the outgoing $K^+$ being described by a plane wave, or by a distorted wave obtained from a kaon-nucleus optical potential [@E94-107].
From the above equations, it follows that the nuclear transition amplitude $$\begin{aligned}
{\mathcal M}^\mu = \langle K^+ {_\Lambda}{{\rm A}} | J_{\rm A}^\mu(q) | 0 \rangle \ , \end{aligned}$$ can be written in factorised form through insertion of the completeness relations $$\begin{aligned}
\int \frac{d^3p}{(2\pi)^3} | {\bf p} \rangle \langle {\bf p} | = \int \frac{d^3p_\Lambda}{(2\pi)^3} | {\bf p}_\Lambda \rangle \langle {\bf p}_\Lambda |
= \openone \ ,\end{aligned}$$ where the integrations over the momenta carried by the proton and the $\Lambda$ also include spin summations, and $$\begin{aligned}
\sum_n | ({\rm A}-1)_n \rangle \langle ({\rm A}-1)_n | = \openone , \end{aligned}$$ the sum being extended to all eigenstates of the $({\rm A}-1)$-nucleon spectator system.
The resulting expression turns out to be
$$\begin{aligned}
\label{factorisation}
\mathcal{M}^\mu & = \langle K^+ {_\Lambda}{\rm A} | J_{\rm A}^\mu | 0 \rangle
= \sum_{i=1}^{\rm A} \sum_n \int \frac{d^3p}{(2\pi)^3} \frac{d^3p_\Lambda}{(2 \pi)^3} \
{\mathcal M}^\star_{ _\Lambda {\rm A} \to ({\rm A}-1)_n + \Lambda} \ \langle {\bf p}_K {\bf k}_\Lambda | j_i^\mu | {\bf p} \rangle
\ {\mathcal M}_{{\rm A} \to ({\rm A}-1)_n + p} \ \ , \end{aligned}$$
where the current matrix element describes the elementary electromagnetic process $\gamma^* + p \to K^+ + \Lambda$.
The nuclear and hypernuclear amplitudes in the right-hand side of Eq., labelled ${\mathcal M}_N$ and ${\mathcal M}_\Lambda$ in Fig. \[graph\], are given by $$\begin{aligned}
\label{ampl:N}
{\mathcal M}_{{\rm A} \to ({\rm A}-1)_n + p} = \{ \langle {\bf p} | \otimes \langle ({\rm A}-1)_n | \} | 0 \rangle \ , \end{aligned}$$ and $$\begin{aligned}
\label{ampl:Y}
{\mathcal M}_{_\Lambda {\rm A} \to ({\rm A}-1)_n + \Lambda} = \{ \langle {\bf p}_\Lambda | \otimes \langle ({\rm A}-1)_n | \} | _\Lambda{{\rm A}} \rangle \ .\end{aligned}$$ In the above equations, the states $ \vert ({\rm A}-1)_n \rangle$ and $ \vert _\Lambda{\rm A} \rangle$ describe the $({\rm A}-1)$-nucleon spectator system, appearing as an intermediate state, and the final-state $\Lambda$-hypernucleus, respectively.
The amplitudes of Eq. determine the Green’s function describing the propagation of a proton in the target nucleus, $G({\bf k},E)$, and the associated spectral function, defined as $$\begin{aligned}
\label{SF:N}
P({\bf k},E) & = - \frac{1}{\pi} {\rm Im} \ G({\bf k},E) \\
\nonumber
& = \sum_n \vert {\mathcal M}_{{\rm A }\to ({\rm A}-1)_n + p} \vert^2
\ \delta(E + M_A-m-E_n) \ , \end{aligned}$$ where $m$ is the nucleon mass and $E_n$ denotes the energy of the $({\rm A}-1)$-nucleon system in the state $n$. The spectral function describes the [*joint*]{} probability to remove a nucleon of momentum ${\bf k}$ from the nuclear ground state leaving the residual system with excitation energy $E>0$.
Within the mean-field approximation underlying the nuclear shell model, Eq. reduces to the simple form $$\begin{aligned}
\label{SF:N:MF}
P({\bf k},E) = \sum_{\alpha \in \{F\}} |\varphi({\bf k})|^2 \delta(E - |\epsilon_\alpha|) \ ,\end{aligned}$$ where $\alpha \equiv \{ nj\ell \}$ is the set of quantum numbers specifying the single-nucleon orbits. The sum is extended to all states belonging to the Fermi sea, the momentum-space wave functions and energies of which are denoted $\varphi_\alpha({\bf k})$ and $\epsilon_\alpha$, respectively, with $\epsilon_\alpha<0$.
Equation shows that within the independent particle model the spectral function reduces to a set of $\delta$-function peaks, representing the energy spectrum of single-nucleon states. Dynamical effects beyond the mean field shift the position of the peaks, that also acquire a finite width. In addition, the occurrence of virtual scattering processes leading to the excitation of nucleon pairs to states above the Fermi surface leads to the appearance of a sizeable continuum contribution yo the Green’s funcion, accounting for $\sim 20 \%$ of the total strength. As a consequence, the normalisation of a shell model state $\varphi_\alpha$, referred to as spectroscopic factor, is reduced from unity to a value $Z_\alpha <1$.
The nuclear spectral functions have been extensively studied measuring the cross section of the $(e,e^\prime p)$ reaction, in which the scattered electron and the knocked out nucleon are detected in coincidence. The results of these experiments, carried out using a variety of nuclear targets, have unambiguous identified the states predicted by the shell model, highlighting at the same time the limitations of the mean-field approximation and the effects of nucleon-nucleon correlations [@FruMug; @Benhar:NPN].
In analogy with Eqs. and , the amplitudes of Eq. comprise the spectral function $$\begin{aligned}
\label{SF:L}
P_\Lambda({\bf k}_\Lambda,E_\Lambda) & = \sum_n \vert {\mathcal M}_{_\Lambda A \to (A-1)_n + \Lambda} \vert^2 \\
\nonumber
& \times \delta(E_\Lambda+ M_{_\Lambda {\rm A}} - M_\Lambda - E_n) \ ,\end{aligned}$$ describing the joint probability to remove a $\Lambda$ from the hypernucleus $_\Lambda{\rm A}$ leaving the residual system with energy $E_\Lambda$. Here $M_\Lambda$ and $M_{_\Lambda {\rm A}}$ denote the mass of the $\Lambda$ and the hypernucleus, respectively.
The observed $(e,e^\prime K^+)$ cross section, plotted as a function of the missing energy $$\begin{aligned}
\label{def:emiss}
E^\Lambda_{\rm miss} = \omega - E_{K^+} \ . \end{aligned}$$ exhibits a collection of peaks, providing the sought-after information on the energy spectrum of the $\Lambda$ in the final state hypernucleus[^1] .
Note that both the electron energy loss, $\omega$, and the energy of the outgoing kaon,$E_{K^+}$, are [*measured*]{} kinematical quantities.
Kinematics {#Kin}
----------
The expression of $E^\Lambda_{\rm miss}$, Eq., can be conveniently rewritten considering that the $\delta$-function of Eq. implies the condition $$\begin{aligned}
\label{full:encons}
\omega + M_A = E_{K^+} + E_{_\Lambda{\rm A}} \ .\end{aligned}$$ Combining the above relation with the requirement of conservation of energy at the nuclear and hypernuclear vertices, dictating that $$\begin{aligned}
\label{cons:ampl}
M_A = E_p + E_n \ \ \ , \ \ \ E_\Lambda + E_n = E_{_\Lambda{\rm A}} \ , \end{aligned}$$ we find $$\begin{aligned}
\label{cons:vert}
\omega + E_p = E_{K^+} + E_\Lambda \ .\end{aligned}$$ Finally, substitution into Eq. yields $$\begin{aligned}
\label{lambda:emiss}
E^\Lambda_{\rm miss} = E_\Lambda - E_p \ .\end{aligned}$$
The above equation, while providing a relation between the [*measured*]{} missing energy and the binding energy of the $\Lambda$ in the final state hypernucleus, defined as $B_\Lambda~=~-E_\Lambda$, [*does not*]{} allow for a model independent identification of $E_\Lambda$. The position of a peak observed in the missing energy spectrum turns out to be determined by the difference between the energies needed to remove a $\Lambda$ from the final state hypernucleus, $E_\Lambda$, or a proton from the target nucleus, $E_p$, leaving the residual $(A-1)$-nucleon system in the same bound state, specified by the quantum numbers collectively denoted $n$.
The proton removal energies, however, can be independently obtained from the missing energy [*measured*]{} in proton knockout experiments, in which the scattered electron and the ejected proton are detected in coincidence, defined as $$\begin{aligned}
\label{def:emissp}
E^p_{\rm miss} = \omega - E_{p^\prime} = - E_p \ .\end{aligned}$$ where $E_{p^\prime}$ is the energy of the outgoing proton. Note that, consistently with Eq., in the right-hand side of the above equation the kinetic energy of the recoiling nucleus has been omitted.
From Eqs. and it follows that the $\Lambda$ binding energy can be determined in a fully model independent fashion from $$\begin{aligned}
B_\Lambda = - E_\Lambda = - ( E^\Lambda_{\rm miss} - E^p_{\rm miss} ) \ ,\end{aligned}$$ combining the information provided by the missing energy spectra measured in $(e,e^\prime K^+)$ and $(e,e^\prime p)$ experiments.
The $\isotope[208][]{\rm Pb}(e, e^\prime K^+)\isotope[208][\Lambda]{\rm Tl}$ Cross Section {#Pbxsec}
==========================================================================================
In view of astrophysical applications, it will be of outmost importance to extend the ongoing experimental studies of kaon electro-production, to include heavy nuclear targets with large neutron excess, such as $\isotope[208][]{Pb}$, that provide the best available proxy for neutron star matter. In this section, I will briefly discuss the main elements needed to carry out the calculation of the $\isotope[208][]{\rm Pb}(e, e^\prime K^+)\isotope[208][\Lambda]{\rm Tl}$ cross section within the factorisation scheme illustrated in Section \[IA\].
The $e+p \to e^\prime + K^+ + \Lambda$ process
----------------------------------------------
The description of the elementary $e+p \to e^\prime + K^+ + \Lambda$ process involving an isolated proton at rest has been obtained from the isobar model [@Adam; @isobar_model], in which the hadron current is derived from an effective Lagrangian comprising baryon and meson fields. Different implementations of this model are characterised by the intermediate states appearing in processes featuring the excitation of resonances [@Sotona; @petr1; @petr2]. The resulting expressionsinvolving a set of free parameters determined by fitting the available experimental datahave been employed to obtain nuclear cross sections within the approach based on the nuclear shell model and the frozen-nucleon approximation [@Sotona; @E94-107]
In principle, the calculation of the nuclear cross section within the scheme outlined in Sect. \[IA\] should be performed taking into account that the elementary process involves a bound, moving nucleon, with four-momentum $p \equiv (E_p,{\bf p})$ and energy $$\begin{aligned}
\label{offshell:momentum}
E_p = m - E \ , \end{aligned}$$ as prescribed by Eq. . However, the generalisation to off-shell kinematics of phenomenological approaches constrained by free proton data, such as the isobar model of Refs. [@Sotona; @petr1; @petr2], entails non trivial difficulties.
A simple procedure to overcome this problem is based on the observation that in the scattering process on a bound nucleon, a fraction $\delta \omega$ of the energy transfer goes to the spectator system. The amount of energy given to the struck proton, the expression of which naturally emerges from the impulse approximation formalism, turns out to be [@benhar_RMP] $$\begin{aligned}
\label{omegatilde}
{\widetilde \omega} & = \omega - \delta \omega \\
\nonumber
& = \omega + m - E - \sqrt{ m^2 + {\bf p}^2 } \ .\end{aligned}$$ Note that from the above equations it follows that $$\begin{aligned}
\label{omegatilde2}
E_p + \omega = \sqrt{ m^2 + {\bf p}^2 } + {\widetilde \omega} \ , \end{aligned}$$ implying in turn $$\begin{aligned}
\label{omegatilde2}
(p + q )^2 = ( {\widetilde p} + {\widetilde q} )^2 = W^2\ , \end{aligned}$$ with ${\widetilde q} \equiv ( {\widetilde \omega} , {\bf q})$ and ${\widetilde p} \equiv ( \sqrt{ m^2 + {\bf p}^2 }, {\bf p})$.
The above equations show that the replacement $q \to {\widetilde q}$ allows to establish a correspondence between scattering on an off-shell moving proton leading to the appearance of a final state of invariant mass $W$, and the corresponding process involving a proton in free space.
It has to be mentioned that, although quite reasonable on physics grounds, the use of ${\widetilde q}$ in the hadron current leads to a violation of current conservation. This problem is inherent in the impulse approximation scheme, which does not allow to simultaneously conserve energy and current in correlated systems. A very popular and effective workaround for this issue, widely employed in the analysis of $(e,e^\prime p)$ data, has been first proposed by de Forest in the 1980s [@forest].
In view of the fact that the extension of the work of Refs.[@petr1; @petr2] to the case of a moving proton does not involve severe conceptual difficulties, the consistent application of the formalism developed for proton knock-out processes the case of kaon electro production appears to be feasible. In this context, it should also be pointed out that the factorisation scheme discussed in Sect. \[Axsec\] allows for a fully relativistic treatment of the electron-proton vertex, which is definitely required in the kinematical region accessible at JLab [@benhar_RMP].
Nuclear and Hypernuclear Dynamics
---------------------------------
Vauable information needed to obtain $\Lambda$ removal energies from the $\isotope[208][]{Pb}(e, e^\prime K^+)\isotope[208][\Lambda]{Tl}$ cross section, using the procedure described in Sect. \[Kin\], has been gained by the high-resolution studies of the $\isotope[208][]{Pb}(e,e^\prime p)\isotope[207][]{Tl}$ reaction performed at NIKHEF-K in the late 1980s and 1990s [@Quint1; @Quint2; @Irene1; @Irene2]. The available missing energy spectrameasured with a resolution of better than 100 KeV and extending up to $\sim 30$ MeVprovide both position and width of the peaks corresponding to the bound states of the recoiling $\isotope[207][]{Tl}$ nucleus.
It is very important to realise that a meaningful interpretation of NIKHEF-K data requires the use of a theoretical framework taking into account effects of nuclear dynamics beyond the mean-field approximation. This issue is clearly illustrated in Figs. \[deviations\] and \[spectrum\].
Figure \[deviations\] displays the difference between the energies corresponding to the peaks in the measured missing energy spectrum, $\langle E^p_\alpha \rangle$, and the predictions of the mean-field model reported in Ref. [@meanfield], $E_\alpha^{HF}$. It is apparent that the discrepancy, measured by the quantity $$\begin{aligned}
\label{def:delta}
\Delta_\alpha = | E_\alpha^{HF} - \langle E^p_\alpha \rangle | \ , \end{aligned}$$ where the index $\alpha \equiv \{ nj\ell \}$ specifies the state of the recoiling system, is sizeable, and as large as $\sim 3$ MeV for deeply bound states.
![Difference between the energies corresponding to the peaks of the missing energy spectrum of the $\isotope[208][]{Pb}(e,e^\prime p)\isotope[207][]{Tl}$ reaction reported in Ref. [@Quint1] and the results of the mean-field calculations of Ref. [@meanfield], displayed as a function of the proton binding energy $E_p = -E_{\rm miss}$. The states are labeled according to the standard spectroscopic notation[]{data-label="deviations"}](Fig2.pdf)
In Fig. \[spectrum\], the spectroscopic factors extracted from NIKHEF-K data are compared to the results of the theoretical analysis of Ref. [@BFF0]. The solid line, exhibiting a remarkable agreement with the experiment, has been obtained combining theoretical nuclear matter results, displayed by the dashed line, and a phenomenological correction to the nucleon self-energy, accounting for finite size and shell effects. The energy dependence of the spectroscopic factors of nuclear matter at equilibrium density has been derived from a calculation of the pole contribution to the spectral function of Eq. , carried out using Correlated Basis Function (CBF) perturbation theory and a microscopic nuclear Hamiltonian including two- and three-nucleon potentials [@GF1].
The results of Fig. \[spectrum\] show that the spectroscopic factors of the deeply bound proton states of $\isotope[208][]{\rm Pb}$ are largely unaffected by surface and shell effect, and can be accurately estimated using the results of nuclear matter calculations. Finite size effects, mainly driven by long-range nuclear dynamics, are more significant in the vicinity of the Fermi surface, where they account for up to $\sim$ 35% of the deviation from the mean-field prediction, represented by the solid horizontal line.
![Spectroscopic factors of the shell model states of $\isotope[208][]{\rm Pb}$, obtained from the analysis of the $\isotope[208][]{{\rm Pb}}(e,e^\prime K^+) \isotope[208][\Lambda]{Tl}$ cross section measured at NIKHEF-K [@Quint1]. The dashed line represent the results of theoretical calculations of the spectroscopic factors of nuclear matter, while the solid line has been obtained including corrections taking into account finite size and shell effects in $\isotope[208][]{\rm Pb}$ [@BFF0]. For comparison, the horizontal line shows the prediction of the independent particle model. The deviations arising form short- and long-range correlations are highligted and labelled SCR and LRC, respectively.[]{data-label="spectrum"}](Fig3.pdf)
In addition to the nucleon spectral function, the analysis of the $\isotope[208][]{\rm Pb}(e, e^\prime K^+)\isotope[208][\Lambda]{\rm Tl}$ cross section requires a consistent description of the $\Lambda$ spectral function, defined by Eq. . Following the pioneering nuclear matter study of Ref. [@wim], microscopic calculations of $P_\Lambda({\bf k}_\Lambda,E_\Lambda)$ in a variety of hypernucleiranging from $\isotope[5][\Lambda]{\rm He}$ to $\isotope[208][\Lambda]{\rm Pb}$have been recently carried out by the author of Ref. [@Isaac]. In this work, the self-energy of the $\Lambda$ was obtained from $G$-matrix perturbation theory in the Brueckner-Hartree-Fock approximation, using the Jülich [@julich1; @julich2] and Nijmegen [@nijmegen1; @nijmegen2; @nijmegen3] models of the YN potential .
The generalisation of the approach of Ref. [@Isaac]needed to treat $\isotope[207][]{\rm Tl}$ using Hamiltonians including both YN and YNN potentialsdoes not appear to involve severe difficulties, of either conceptual or technical nature. Therefore, a consistent description of the $\isotope[208][]{\rm Pb}(e, e^\prime K^+)\isotope[208][\Lambda]{\rm Tl}$ process within the factorisation scheme described in the previous section is expected to be achievable within the time frame relevant to the JLab experimental program.
Summary and outlook {#summary}
===================
The results discussed in this article suggest that precious new information on hypernuclear dynamics can be obtained from a largely model independent analysis of the measured $\isotope[208][]{{\rm Pb}}(e,e^\prime K^+) \isotope[208][\Lambda]{Tl}$ cross section, and that a consistent theoretical framework, allowing to exploit the data to constrain YN and YNN potential models, can be developed within the well established approach based on nuclear many-body theory and the Green’s function formalism.
More recent computational approaches, mostly based on the Monte Carlo method [@Carlson:2014vla], have been very successful in obtaining ground-state expectation values of Hamiltonians involving nucleons and hyperons, needed to model the equation of state of strange baryon matter, see, e.g. Ref. [@puzzle]. However, the present development of these techniques does not allow the calculation of either $(e,e^\prime p)$ or $(e,e^\prime K^+)$ cross sections, most notably in the kinematical regime in which the underlying non-relativistic approximation is no longer applicable. On the other hand, the approach based on factorisation, allowing for a fully relativistic treatment of the electron-proton interaction, has proved very effective for the interpretation of the available $(e,e^\prime p)$ data.
Owing to the extended region of constant density, $\isotope[208][]{{\rm Pb}}$ is the best available proxy for uniform nuclear matter. This feature, which also emerges from the results displayed in Fig. \[spectrum\], will be critical to acquire new information on three-body forces, complementary to that obtainable using a Calcium target.
The results of accurate many-body calculations of the ground-state energies of finite nuclei [@CVMC] and isospin-symmetric nuclear matter [@APR]performerd with the [*same*]{} nuclear Hamiltonian including the Argonne $v_{18}$ [@AV18] and Urbana IX [@UIX] NN and NNN interaction models, respectivelyshow that the potential energy per nucleon arising from three-nucleon interactions is a monotonically increasing function of A whose value changes sign, varying from -0.23 MeV in $\isotope[40][]{{\rm Ca}}$ to 2.78 MeV in nuclear matter at equilibrium density. In view of astrophysical applications, constraining three-body forces in the mass region in which they change from attractive to repulsive in the non-strange sector appears to be needed.
The solution of the “hyperon puzzle” is likely to require a great deal of theoretical and experimental work for many years to come. The results discussed in this article strongly suggest that the extension of the JLab kaon electro production program to $\isotope[208][]{{\rm Pb}}$ will allow to collect data useful to broaden the present understanding of hypernuclear dynamics in nuclear matter.
This work was supported by the Italian National Institute for Nuclear Research (INFN) under grant TEONGRAV. The author is deeply indebted to Petr Byd[ž]{}ovsk[ý]{}, Franco Garibaldi and Isaac Vidaña for many illuminating discussions on issues related to the subject of this article.
[^1]: In principle, the right-hand side of Eq. should also include a term accounting for the kinetic energy of the recoiling hypernucleus. However, for heavy targets this contribution turns out to be negligibly small, and will be omited.
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abstract: 'Coexistence of phases, characterized by different electronic degrees of freedom, commonly occurs in layered superconductors. Among them, alkaline intercalated chalcogenides are model systems showing microscale coexistence of paramagnetic (PAR) and antiferromagnetic (AFM) phases, however, temporal behavior of different phases is still unknown. Here, we report the first visualization of the atomic motion in the granular phase of K$_{x}$Fe$_{2-y}$Se$_2$ using X-ray photon correlation spectroscopy. Unlike the PAR phase, the AFM texture reveals an intermittent dynamics with avalanches as in martensites. When cooled down across the superconducting transition temperature T$_c$, the AFM phase goes through an anomalous slowing behavior suggesting a direct relationship between the atomic motions in the AFM phase and the superconductivity. In addition of providing a compelling evidence of avalanche-like dynamics in a layered superconductor, the results provide a basis for new theoretical models to describe quantum states in inhomogeneous solids.'
author:
- 'A. Ricci'
- 'G. Campi'
- 'B. Joseph'
- 'N. Poccia'
- 'D. Innocenti'
- 'C. Gutt'
- 'M. Tanaka'
- 'H. Takeya'
- 'Y. Takano'
- 'T. Mizokawa'
- 'M. Sprung'
- 'N.L. Saini'
title: 'Intermittent dynamics of antiferromagnetic phase in inhomogeneous iron-based chalcogenide superconductor'
---
Introduction
============
The observation of superconductivity in iron-based chalcogenides [@PNAS] has opened new frontiers in the field of layered materials with interesting interplay of atomic defects, magnetism, and superconductivity [@Dai]. Such an interplay has been widely discussed for other layered systems [@1; @Campi_Nat15; @Bianconi_SUST15; @Poccia_APL; @Poccia_SUST]. Among iron-based chalcogenides, A$_{x}$Fe$_{2-y}$Se$_2$ (A = K, Rb, Cs) system [@Guo; @Mizuguchi; @Ying; @Ming-Hu] is a good example showing an intrinsic phase separation [@Ricci; @LiWei; @Yuan; @Wang2; @Chen; @Bendele14; @MTanaka] and a delicate balance between a magnetic phase due to iron vacancy order and the coexisting metallic phase. A$_{x}$Fe$_{2-y}$Se$_2$ shows superconductivity below a transition temperature T$_c$ of $\sim$32 K and manifests a peculiar microstructure with coexisting antiferromagnetic (AFM) phase having stoichiometry of A$_{0.8}$Fe$_{1.6}$Se$_2$ (245) and paramagnetic (PAR) metallic phase of A$_{x}$Fe$_{2}$Se$_2$ (122). A variety of experiments have studied the phase separation properties [@Guo; @Mizuguchi; @Ying; @Ming-Hu; @Ricci; @LiWei; @Yuan; @Wang2; @Chen; @Bendele14; @MTanaka], revealing a wealth of information on the microstructure of the system. For example, space resolved micro X-ray diffraction ($\mu$XRD) on K$_{x}$Fe$_{2-y}$Se$_{2}$ [@Ricci] has identified a $\sqrt{5}\times\sqrt{5}$ superstructure due to iron-vacancy order in the average tetragonal lattice to occur below $\sim$ 580 K and a phase separation to appear below $\sim$ 520 K. The earlier is a second order transition while the latter transition has primarily of first order character [@Ricci_sust]. Depending on the growth conditions, the system contains about $\sim$70-90% of insulating AFM phase with $\sqrt{5}\times\sqrt{5}$ superstructure while the remaining minority phase is metallic and is characterized a compressed in-plane lattice. This peculiar phase separation puts these chalcogenides in the class of granular systems in which dynamics in the microscopic granules has large effect on their macroscopic properties. Here, we have used X-ray photon correlation spectroscopy (XPCS), a diffraction based technique [@xpcs1; @xpcs2; @xpcs3], to probe the atomic dynamics in the coexisting phases of superconducting K$_{x}$Fe$_{2-y}$Se$_{2}$. XPCS exploits the temporal evolution of X-ray speckle pattern generated by coherent radiation. The speckle patterns represent a direct fingerprint of the nano scale phase disorder in the material. If the material fluctuates in time, the speckle pattern does the same, and a measurement of the speckle intensity fluctuation reveals the dynamics of the system.
Experimental details.
=====================
The single crystal samples of K$_x$Fe$_{2-y}$Se$_2$ were prepared using the Bridgman method [@Mizuguchi]. After the growth, the single crystals were sealed into a quartz tube and annealed for 12 hours at 600$^\circ$C. Well characterized sample of size 3$\times$3 mm$^2$, having composition K$_{0.65}$Fe$_{1.65}$Se$_2$ was used for the present measurements. The electric and magnetic characterizations were performed by temperature dependent measurements of resistivity using a physical property measurement system (PPMS - Quantum Design) and magnetization using a superconducting quantum interference device (SQUID) magnetometer (Quantum Design). The sample exhibits a sharp superconducting transition at T$_{c}$ of $\sim$32 K.
The XPCS experiments were carried out in the $\theta$/2$\theta$ reflection geometry with beam falling parallel to the b-axis of the single crystal sample having tetragonal symmetry (see, e.g. Fig. 1(a) showing the experimental geometry). The measurements were carried out at the Coherence Beamline P10 of PETRA III synchrotron radiation source in Hamburg where the X-ray beam, produced by a 5m long undulator (U29), is monochromatized using a Si(111) double crystal monochromator. X-ray photon beam of energy 8 keV with a bandwidth dE/E$\sim$1.4$\times$10$^{-4}$ was used. At this energy the tranverse coherence lengths is 277 $\mu$m in vertical direction and 46 $\mu$m in horizontal direction. The collimated coherent X-ray beam was focused using a beryllium compound refractive lens (CRL) transfocator [@crl] to a size of about 2$\times$2 $\mu$m$^2$ on the sample positioned at 1.6 m down stream of the transfocator center. The incident flux on the sample was $\sim$ 10$^{11}$ photons/s. The exit window of the heating chamber and He-cryostat (see, e.g., supplemental material [@SuppMat] and reference [@Sup_ref1] therein) as well as the entrance window of the detector flight path was covered by a 25 $\mu$m thick Kapton sheet. The scattered signal was detected at a distance of $\sim$5 m using a large horizontal scattering set-up. A PILATUS 300 detector (7 ms readout time) was used for the alignment and MAXIPIX 2$\times$2 detector (0.3 ms readout time) was used to record the X-rays scattered by the sample with an angular resolution of 6.228 $\times 10^{-4}$ degree.
Results and discussions
=======================
![**X-ray Photon Correlation Spectroscopy (XPCS) measurements on K$_{x}$Fe$_{2-y}$Se$_2$.** a) Schematic diagram of the experimental setup with the sample mounted on a copper block inside an evacuated heat chamber. b) A typical speckle pattern of the (004) reflection at 517 K, showing two different phases characterized by their c-axis, i.e. expanded and compressed c-axis for AFM and PAR phases respectively (see the cartoon picture). Line profiles of the intensity distributions of the speckles corresponding to AFM (left) and PAR (right) phases alongwith the averaged profiles are also shown. The upper profile shows the two phases characterized by different c-axis. c) Temperature evolution of the AFM (red squares) and PAR (blue dots) phases across the phase separation temperature $\sim$ 520 K.[]{data-label="Fig1"}](Fig1.jpg){width="8"}
Figure 1(a) shows a schematic picture of the XPCS setup. The single crystal sample of K$_x$Fe$_{2-y}$Se$_2$ sample is mounted on a copper block inside an evacuated chamber. More details on the sample environment and the experimental setup are shown in supplemental material [@SuppMat] (see, also, reference [@Sup_ref1] therein). The sample shows phase separation while it is cooled across a temperature of $\sim$ 520 K. A speckled (004) Bragg reflection, measured on K$_x$Fe$_{2-y}$Se$_2$ crystal at a constant temperature of 517 K, is displayed in Fig. 1(b). The reflection is a direct indicator of the phase separation [@Ricci; @Ricci_sust] in the block antiferromagnetic tetragonal phase due to iron vacancy order (space group I4/mmm with a=b=4.01 Å, c=13.84 Å, hereafter called AFM phase) and c-axis expanded tetragonal paramagnetic phase (hereafter called PAR phase). The profiles (004) peak are shown in Fig. 1(b) displaying the typical speckles due to coherent X-rays. Temperature dependence of the normalized intensity for the two phases is shown in Fig. 1(c). The majority AFM phase contributes $\sim$80-90% while the remaining $\sim$10-20% is the PAR phase.
After 100 seconds of measurements at 517 K, the sample temperature was raised quickly by 1 K to bring the system in a non-equilibrium state. Figure 2 displays time evolution of the two phases before and after a temperature step of 1 K. The relaxation can be seen in Fig. 2(a) displaying the time evolution of the integrated intensities corresponding to the two phases (normalized with respect to the total intensity in the equilibrium state). The time evolution of (004) reflections mean profiles for the two phases is shown in Fig. 2 of supplimental material [@SuppMat]. The sample temperature was kept constant (at 518$\pm$ 0.2 K) during whole time series measurements (see, e.g. the temporal fluctuations of temperature plotted in Fig. 2(a)). The speckled pattern evolves with time after the temperature stimulation (i.e., a quick change of temperature by 1 K after $\sim$ 100 sec), shown for different instants in Fig. 2(b). There are some apparent changes as a function of time in the two regions of the speckled pattern while the system is relaxing from the non-equilibrium state.
![**Time evolution of different phases before and after the temperature pulsed step** of 1 K from the equilibrium state at 517 K. a) Time evolution of the AFM (red squares) and PAR (blue dots) phases. The time series collection started at constant temperature (517 K) in the equilibrium state. After 100 seconds the temperature was changed rapidly by 1 K (black curve shown as inset) and 1000 additional diffraction patterns were collected to study the complex non-equilibrium dynamic. b) Speckle patterns for some time delays during the time series for the AFM (upper panels) and PAR (lower panels) phases. c-b) Two time correlation functions. The PAR phase (panel c) shows a normal (quasi-static) dynamics while the situation for the AFM phase (panel d) is different, revealing avalanches in the domains transformations.[]{data-label="Fig2"}](Fig2.jpg){width="8"}
The complex non-equilibrium dynamics in highly heterogeneous systems can be visualized in the best way through two-time correlation functions ($ttcf$) [@twotime; @Sutton; @Ruta1; @Ruta2; @Fluerasu; @growMod]. The two-time correlation functions are calculated correlating all possible pairs of diffraction patterns collected during the time series described above. Following equation is used to calculate the $ttcf = C (I(t_1), I(t_2))$ [@Gutt]: $$\frac{\sum_m (I_m (t_1) - \langle I(t_1)\rangle) (I_m (t_2) - \langle I(t_2) \rangle)}{\sqrt{\sum_m (I_m (t_1) - \langle I(t_1) \rangle)^2 (I_m (t_2) - \langle I(t_2) \rangle)^2}}$$ Here, $C (I(t_1), I(t_2))$ is the two-time correlation function ($ttcf$), $I_m (t_1)$ and $I_m (t_2)$ are intensities measured in the detector pixel $m$ at time $t_1$ and $t_2$, $\langle I(t_1)\rangle$ and $\langle I(t_2) \rangle$ are respectively the mean intensities measured over all pixels of images recorded at time $t_1$ and $t_2$. To minimize the overlap, we tried different regions of interest around (004) reflections corresponding to the two phases in K$_x$Fe$_{2-y}$Se$_2$ before judging for the regions of interest shown as squares in Fig. 2(b) for the calculations of $ttcf$. The two-time correlation images for the two phases are displayed in Fig. 2(c) and 2(d). The PAR phase displays a normal (quasi-static) dynamics revealing the system to evolve from a locked-static state to the next one defined by a close minimum in the energy landscape [@ELS; @ELS1; @mg26e32]. In fact, the intensity distribution in the two-time correlation image is spreading out with time. The diagonal width of the two-time correlation image provides information on the correlation time, i.e., the time scale in which given atomic configuration (characterized by a well-defined wave vector) is no longer corresponds to the one measured at a later time. The fact that the width of the intensity distribution in the $ttcf$ image for the PAR phase increases with time, indicating increased correlation times with time, i.e., the slowing down of the dynamics of this phase with time (For a system in a dynamic equilibrium, the width of the two-time correlation image is expected to be a constant, i.e., no time evolution). This indicates formation of larger and larger domains of the PAR phase at the expense of small domains, consistent with models for growth processes [@growMod]. In the potential energy landscape approach [@ELS], the dynamics of PAR phase suggests that the system finds itself in a configuration space with deep energy basin and evolving towards a deeper and deeper local energy minima.
Unlike the PAR phase, the AFM phase shows a very peculiar dynamics. Indeed, the two-time correlation of the AFM phase (Fig. 2(d)) reveals dramatic decorrelation events characterized by sudden narrowing of the intensity distribution profile appearing intermittently with time. This temporal intermittence indicates an avalanche like atomic dynamics in the majority AFM phase. In fact, such an intermittent dynamics describes rearrangements to localized micro-collapses of groups of particles, which trigger subsequent collapses in the neighboring regions through the formation of stress dipoles. Therefore, the intermittent progression of the AFM phase can be identified as an incubation time effect, i.e., silent growth and explosions in sequences. This avalanche like dynamics has been found in a number of physical phenomena including martensitic transformations [@mart31; @mart_san], deformation of metallic glasses [@mg26e32], crystallization of a hardsphere glass [@glass33], and shear flow of droplet emulsions through a thin opening [@flow34]. It is likely that these events of microscopic rearrangement act as important mediators in the particular phase via the cooperative relief of atomic-level strain between the coexisting AFM and PAR phases. This particular phase is the interface (INT) phase identified in micro-diffraction study on the same material [@RicciPRB16]. Therefore, the AFM and PAR phases are separated by a well defined INT phase. Thus, the intermittent dynamics of the AFM texture in K$_x$Fe$_{2-y}$Se$_2$ is intrinsic and indicative of a complex energy landscape with numerous minima (different equilibrium states) in which the system stays for long periods of time in stable configurations, reflecting both localized and cascade relaxation dynamics.
After the study of the non-equilibrium dynamics in which the sample temperature was varied sharply by 1 K, the sample was kept at constant temperature (at 518 K) for a long time (more than one hour). Assuming the sample to be in the equilibrim state, we measured a second time series, collecting speckle patterns for 500 seconds. The instantaneous autocorrelation function $g_2(t)$ [@xpcs1; @xpcs2; @xpcs3] was calculated using this time series revealing characteristic correlation times ($\tau$) to be $\sim$550 seconds and $\sim$400 seconds respectively for the PAR and AFM phases (see supplimental material for a detailed description [@SuppMat] and reference [@Sup_ref2] therein).
![**Nanodomain dynamics across superconducting transition temperature (T$_c$).** (a-b) Two-temperature correlation function ($TTcf$) images calculated for a time series collected during a linear temperature ramp. The $TTcf$ of the PAR phase (a) shows a normal cone shape indicating a linear acceleration of domains dynamics with increasing temperature. Instead the $TTcf$ of AFM phase (b) shows a clear anomaly around T$_c$. (c) Temperature evolution of the speckle contrast (upper panel) for PAR (dotted blue line) and AFM (solid red line) phases. The lower panel shows the autocorrelation function C($\Delta$T,T) at T=T$_c$ for PAR (blue dots) and AFM (red squares) phases. The C($\Delta$T,T) curves have been extracted selecting horizontal cuts from the $TTcf$ images of the two phases and normalized them to the speckle-contrast. C($\Delta$T,T) curves are fitted using the stretched exponential behavior (eq. 2). (d) The activation energy (E) extracted from the fits is plotted as a function of temperature for the PAR (blue dots) and AFM (red squares) phases. The activation energy shows a bump at T= T$_c$ for the AFM phase.[]{data-label="Fig3"}](Fig3.jpg){width="8"}
The fact that the AFM phase in K$_{x}$Fe$_{2-y}$Se$_2$ shows avalanche like and intermittent temporal fluctuations in the collective dynamics, this poses a question if such a dynamics has any relationship with the superconductivity in K$_{x}$Fe$_{2-y}$Se$_2$. To search for a possible connection between the dynamics and the superconductivity, we have studied the speckles evolution for the two phases while the sample is cooled down across the superconducting transition temperature T$_c$. For the purpose, we have varied the temperature with a constant rate. The temperature evolution of the mean diffraction profiles is shown in Fig. 4 of supplemental material [@SuppMat]. Here, the dynamics has been studied by evaluating the temperature-temperature correlation function ($TTcf$). The $TTcf$ has been calculated using the same equation used for the calculations of the $ttcf$ (eq. 1) in which the time is replaced by temperature. The procedure has been commonly used to explore response of atomic dynamicsacross temperature dependent transitions [@Gutt]. Figure 3(a) and 3(b) show the $TTcf$ for PAR and AFM phases. Apparently, both phases display similar dynamics upon cooling, however, a clear anomaly for the AFM phase around T$_c$ can be seen. At this temperature the intensity distribution is sharper before the spreading out, i.e., the correlation time at T$_c$ is much smaller for the AFM phase. This indicates large fluctuations near T$_c$ followed by a slowing of the AFM phase below the transition temperature. On the other hand, the PAR phase seems to evolve normally. Therefore, the AFM phase shows anomalous dynamic correlations across T$_c$ in the phase separated K$_{x}$Fe$_{2-y}$Se$_2$.
To have a detailed insight further analysis of the temperature dependent speckle patterns was done. The autocorrelation function C($\Delta$T,T) was calculated at different temperatures in the shown interval around T$_c$. The following equation was used to describe the calculated autocorrelation functions: $$C(\Delta T,T) = A~exp \Big [- \bigg ( \frac{\Delta T}{E}\bigg)^{\beta} \Big ]$$ where $A$ represents the speckles contrast, $\beta$ is the shape parameter of the stretched exponential function, $E$ is the activation energy. The evaluated speckles contrast around T$_c$ for the two phases is displayed in Fig. 3(c) (upper). In the lower panel of Fig. 3(c) we have shown the C($\Delta$T,T) at T$_c$ and is normalized with respect to the speckle contrast. The activation energy ($E$) around T$_c$ is also shown (in Fig. 3(d)). It is evident from Fig. 3(c)(top) that the speckle contrast around T$_c$ is much smaller for the AFM phase than for the PAR phase implying the existence of faster fluctuations in the AFM phase at T$_c$. The speckles contrast at T$_c$ does not drop to zero meaning that the dynamics is not fully decorrelating the speckle patterns of the AFM phase. On the other hand, the autocorrelation C($\Delta$T,T) at T$_c$ reveals that the AFM phase has slower relaxation than the PAR phase (see Fig. 3(c), bottom) but with an anomalously increased activation energy (Fig. 3(d)). These observations indicate that there should be some other much slower processes actively incorporated within the AFM phase. Therefore, the dynamics of the AFM phase is indeed more complex with at least two relaxation channels present which are well separated in activation energy. The underlying correlation function could resemble that of glasses or other disordered systems with coexisting fast and slow relaxations.
Conclusions {#S:conclusions}
===========
In summary, we have studied the dynamics of nano-domains in phase separated K$_{x}$Fe$_{2-y}$Se$_2$ system. While the minority PAR phase reveals commonly known steady slowing down with time, the majority AFM phase shows intermittent non-equilibrim dynamics as in martensites involving cooperative atomic rearrangements with avalanches. This complex dynamics of the AFM phase may have some direct correlation with the superconductivity in K$_{x}$Fe$_{2-y}$Se$_2$. Indeed, the measurements across the superconducting transition temperature show that the AFM phase goes through an anomalous atomic dynamics across the superconducting transition temperature reflecting involvement of complex energy landscape to establish the superconducting quantum state. It is worth recalling that superconductivity is accompanied by a hardening in local atomic modes, that has been seen in a series of superconducting families [@exafs1; @exafs2; @exafs3]. Therefore, the behavior of the AFM phase could be a result of the superconducting transition that can be mediated by lattice fluctuations (Fe-Fe lattice) or spin fluctuations in the AFM phase. The fact that local magnetic moment decreases sharply at T$_c$ (shown by XES [@xes]) as well the AFM order tends to suppress at T$_c$ [@neutron], it is plausible to think that the superconductivity in these materials may have some exotic mechanism involving collective mode of lattice and spin characterized by slow dynamics. It should be mentioned that, in addition to the PAR and AFM phases, the system is characterized by the INT phase which forms out of the AFM phase [@RicciPRB16; @xes]. Therefore, it is likely that the superconductivity appears in the INT phase as argued earlier. However, more efforts are required to clarify issues on the dynamics of the INT phase.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank PETRA staff for the assistance during the measurements. Y.T would like to acknowledge hospitality at the Sapienza University of Rome. The authors acknowledge stimulating and motivating discussions with A. Bianconi.
[40]{}
F.-C. Hsu, J.-Y. Luo, K.-W. Yeh, T.-K. Chen, T.-W. Huang, P.M. Wu, Y.-C. Lee, Y.-L. Huang, Y.-Y. Chu, D.-C. Yan, M.-K. Wu, Proc. Nat. Acad. Sci. [**105**]{}, 14262 (2008).
P. Dai, Rev. Mod. Phys. [**87**]{}, (2015) 855.
$Stripes ~and ~Related ~Phenomena$; Bianconi, A., Saini, N.L., Eds.; Kluwer/Plenum: New York, 2000.
G. Campi, A. Bianconi, N. Poccia, G. Bianconi, L. Barba, G. Arrighetti, D. Innocenti, J. Karpinski, N.D. Zhigadlo, S.M. Kazakov, M. Burghammer, M. Zimmermann, M. Sprung, A. Ricci, Nature, [**525**]{}, 359 (2015).
A. Bianconi, N. Poccia, A.O. Sboychakov, A.L. Rakhmanov, K.I. Kugel, Supercond. Sci. Technol., [**28**]{}, 024005 (2015).
N. Poccia, M. Chorro, A. Ricci, W. Xu, A. Marcelli, G. Campi, A. Bianconi, Applied Physics Letters [**104**]{}, 221903 (2014).
N. Poccia, A.. Ricci, G. Campi, A. Bianconi, Supercond. Sci. Technol., [**30**]{}, 035016 (2017).
J. Guo, S. Jin, G. Wang, S. Wang, K. Zhu, T. Zhou, M. He, X. Chen, Phys. Rev. B [**82**]{}, 180520 (2010).
Y. Mizuguchi, H. Takeya, Y. Kawasaki, T. Ozaki, S. Tsuda, T. Yamaguchi, and Y. Takano, Appl. Phys. Lett. [**98**]{}, 042511 (2011).
J.J. Ying, X. F. Wang, X. G. Luo, A. F. Wang, M. Zhang, Y. J. Yan, Z. J. Xiang, R. H. Liu, P. Cheng, G. J. Ye, and X. H. Chen, Phys. Rev. B [**83**]{}, 212502 (2011).
M.H. Fang, H.D. Wang, C.H. Dong, Z.J. Li, C.M. Feng, J. Chen, and H. Q. Yuan, EPL [**94**]{}, 27009 (2011).
A. Ricci, N. Poccia, G. Campi, B. Joseph, G. Arrighetti, L. Barba, M. Reynolds, M. Burghammer, H. Takeya, Y. Mizuguchi, Y. Takano, M. Colapietro, N. L. Saini, and A. Bianconi, Phys. Rev. B [**84**]{}, 060511(R) (2011).
W. Li, H. Ding, P. Deng, K. Chang, C. Song, K. He, L. Wang, X. Ma, J.-P. Hu, X. Chen, Q.-K. Xue, Nature Physics [**8**]{}, 126 (2012).
R.H. Yuan, T. Dong, Y.J. Song, P. Zheng, G.F. Chen, J.P. Hu J.Q. Li, N.L. Wang, Sci. Rep. [**2**]{}, 221 (2012).
Z. Wang, Y. J. Song, H. L. Shi, Z. W. Wang, Z. Chen, H. F. Tian, G. F. Chen, J. G. Guo, H. X. Yang, and J. Q. Li, Phys. Rev B [**83**]{}, 140505(R) (2011).
F. Chen, M. Xu, Q. Q. Ge, Y. Zhang, Z. R. Ye, L. X. Yang, Juan Jiang, B. P. Xie, R. C. Che, M. Zhang, A.F. Wang, X. H. Chen, D. W. Shen, J. P. Hu, and D. L. Feng, Phys. Rev. X [**1**]{}, 021020 (2011).
M. Bendele, A. Barinov, B. Joseph, D. Innocenti, A. Iadecola, H. Takeya, Y. Mizuguchi, T. Takano, T. Noji, T. Hatakeda, Y. Koike, M. Horio, A. Fujimori, D. Ootsuki, T. Mizokawa, and N. L. Saini, Sci. Rep. [**4**]{}, 5592 (2014).
M. Tanaka, H. Takeya, Y. Takano, Appl. Phys. Express [**10**]{}, 023101 (2017).
A. Ricci, N. Poccia, B. Joseph, G. Arrighetti, L. Barba, J. Plaisier, G. Campi, Y. Mizuguchi, H. Takeya, Y. Takano, N.L. Saini and A. Bianconi, Supercond. Sci. Technol. [**24**]{} 082002 (2011).
A. Madsen, R. L. Leheny, H. Guo, M. Sprung and O. Czakkel, New J. Phys. [**12**]{} 055001 (2010).
O. G. Shpyrko, J. Synchrotron Rad. [**21**]{}, 1057 (2014).
S.K. Sinha, Z. Jiang, L.B. Lurio, Adv. Mater. [**26**]{}, 7764 (2014).
A.V. Zozulya, S. Bondarenko, A. Schavkan, F. Westermeier, G. Gr$\ddot{u}$bel, and M. Sprung, Optics Express [**20**]{}, 18967 (2012).
See Supplemental Material at \[URL will be inserted by the production group\] for more information on the XPCS setup used for the measurements. Supplemental Material also includes evolution of (004) reflection mean profiles during time series measurements together with more information on equilibrium state dynamics of K$_{x}$Fe$_{2-y}$Se$_2$ (see text).
http://photon-science.desy.de/facilities/petra$\_$iii/ beamlines/p10$\_$coherence$\_$applications/index$\_$eng.html
G. Brown, P.A. Rikvold, M. Sutton, M. Grant, Phys. Rev. E [**56**]{}, 6601 (1997).
M. Sutton, K. Laaziri, F. Livet, F. Bley, Optics Express [**11**]{} 2268 (2003).
B. Ruta, G. Baldi, G. Monaco, Y. Chushkin, J. Chem. Phys. [**138**]{}, 054508 (2013).
B. Ruta, Y. Chushkin, G. Monaco, L. Cipelletti, E. Pineda, P. Bruna, V.M. Giordano, M. Gonzalez-Silveira, Phys. Rev. Lett. [**109**]{}, 165701 (2012).
A. Fluerasu, M. Sutton, E.M. Dufresne, Phys. Rev. Lett. [**94**]{}, 055501 (2005).
A. Malik, A. R. Sandy, L. B. Lurio, G. B. Stephenson, S. G. J. Mochrie, I. McNulty, and M. Sutton, Phys. Rev. Lett. [**81**]{}, 5832 (1998).
S. Gorfman, A.A. Bokov, A. Davtyan, M. Reiser, Y. Xie, Z.-G. Ye, A.V. Zozulya, M. Sprung, U. Pietsch, C. Gutt, Proc. Nat. Acad. Sci. [**115**]{}, E6680 (2018).
M. Goldstein, J. Chem. Phys. [**51**]{}, 3728 (1969). P. K.Gupta, W. Kob, J. Non-Crystalline Sol: X, [**3**]{}, 100031 (2019).
Z. Evenson, B. Ruta, S. Hechler, M. Stolpe, E. Pineda, I. Gallino, R. Busch, Phys. Rev. Lett. [**115**]{}, 175701 (2015).
L. Mueller, M. Waldorf, C. Gutt, G. Gruebel, A. Madsen, T. R. Finlayson, and U. Klemradt, Klemradt, Phys. Rev. Lett. 107, 105701 (2011).
C. Sanborn, K.F. Ludwig, M.C. Rogers, M. Sutton, Phys. Rev. Lett. [**107**]{}, 015702 (2011).
E. Sanz, C. Valeriania, E. Zaccarelli, W.C.K. Poon, M.E. Cates, P.N. Pusey, Proc. Nat. Acad. Sci. [**111**]{}, 75 (2014).
D. Chen, K.W. Desmond, E.R. Weeks, Soft Matter [**8**]{}, 10486 (2012).
B.J. Berne, $ \&$ R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics, (Dover Publications, 2000).
A. Ricci, N. Poccia, B. Joseph, D. Innocenti, G. Campi, A. Zozulya, F. Westermeier, A. Schavkan, F. Coneri, A. Bianconi, H. Takeya, Y. Mizuguchi, Y. Takano, T. Mizokawa, M. Sprung, N.L. Saini, Phys. Rev. B [**91**]{}, 020503 (2015).
N.L. Saini, A. Bianconi, H. Oyanagi, J. Phys. Soc. Jpn. [**70**]{}, 2092 (2001).
N.L. Saini, M. Filippi, H. Oyanagi, H. Ihara, A. Iyo and A. Bianconi, Phys. Rev. B [**68**]{}, 104507 (2003).
E. Paris, L. Simonelli, T. Wakita, C. Marini, J.-H. Lee, W. Olszewski, K. Terashima,T. Kakuto, N. Nishimoto, T. Kimura, K. Kudo, T. Kambe, M. Nohara, T. Yokoya, N. L. Saini, Sci. Rep. [**6**]{}, 27646 (2016).
L. Simonelli, T. Mizokawa, M. Moretti Sala, H. Takeya, Y. Mizuguchi, Y. Takano, G. Garbarino, G. Monaco, and N. L. Saini, Phys. Rev. B [**90**]{}, 214516 (2014).
S. Li, Y. Gan, J. Wang, R. Zhong, J. A. Schneeloch, Z. Xu, W. Tian, M. B. Stone, S. Chi, M. Matsuda, Y. Sidis, Ph. Bourges, Q. Li, G. Gu, J. M. Tranquada, G. Xu, R. J. Birgeneau, and J. Wen, Phys. Rev. B [**96**]{}, 094503 (2017).
|
---
author:
- 'Aftab Alam$^{\ast}$ and Mohammad Imdad'
title: 'Relation-Theoretic Metrical Coincidence Theorems'
---
[^1]
[Department of Mathematics\
Aligarh Muslim University\
Aligarh-202002, India.\
Email addresses: aafu.amu@gmail.com, mhimdad@gmail.com\
]{}
[ In this article, we generalize some frequently used metrical notions such as: completeness, closedness, continuity, $g$-continuity and compatibility to relation-theoretic setting and utilize these relatively weaker notions to prove results on the existence and uniqueness of coincidence points involving a pair of mappings defined on a metric space endowed with an arbitrary binary relation. Particularly, under universal relation our results deduce the classical coincidence point theorems of Goebel (Bull. Acad. Pol. Sci. S$\acute{\rm e}$r. Sci. Math. Astron. Phys. 16 (1968) 733-735) and Jungck (Int. J. Math. Math. Sci. 9 (4) (1986) 771-779). In process our results generalize, extend, modify and unify several well-known results especially those obtained in Alam and Imdad (J. Fixed Point Theory Appl. 17 (4) (2015) 693-702), Karapinar $et\;al.$ (Fixed Point Theory Appl. 2014:92 (2014) 16 pp), Alam $et\;al.$ (Fixed Point Theory Appl. 2014:216 (2014) 30 pp), Alam and Imdad (Fixed Point Theory, in press) and Berzig (J. Fixed Point Theory Appl. 12 (1-2) (2012) 221-238.\
[**Keywords**]{}: Binary relations; $\mathcal{R}$-completeness; $\mathcal{R}$-continuity; $\mathcal{R}$-connected sets; $d$-self-closedness.\
[**AMS Subject Classification**]{}: 47H10, 54H25.]{}
Introduction and Preliminaries {#SC:Introduction and Preliminaries}
==============================
Throughout this manuscript, $\mathbb{N}$, $\mathbb{N}_0$, $\mathbb{Q}$ and $\mathbb{R}$ denote the sets of natural numbers, whole numbers, rational numbers and real numbers respectively ($i.e.$ $\mathbb{N}_0=\mathbb{N} \cup \{0
\}$). For the sake of completeness, firstly we recall some known relevant definitions.\
**Definition 1 [@CP1; @CP2].** Let $X$ be a nonempty set and $f$ and $g$ two self-mappings on $X$. Then
1. an element $x\in X$ is called a coincidence point of $f$ and $g$ if $$g(x)=f(x),$$
2. if $x\in X$ is a coincidence point of $f$ and $g$ and $\overline{x}\in X$ such that $\overline{x}=g(x)=f(x)$, then $\overline{x}$ is called a point of coincidence of $f$ and $g$,
3. if $x\in X$ is a coincidence point of $f$ and $g$ such that $x=g(x)=f(x)$, then $x$ is called a common fixed point of $f$ and $g$,
4. $f$ and $g$ are called commuting if $$g(fx)=f(gx)\hspace{0.5cm}\forall~x\in X\;{\rm and}$$
5. $f$ and $g$ are called weakly compatible (or partially commuting or coincidentally commuting) if $f$ and $g$ commute at their coincidence points, $i.e.$, for any $x\in X,$ $$g(x)=f(x)\Rightarrow g(fx)=f(gx).$$
**Definition 2 [@CP3; @CP4; @g-C].** Let $(X,d)$ be a metric space and $f$ and $g$ two self-mappings on $X$. Then
1. $f$ and $g$ are called weakly commuting if for all $x\in X,$ $$d(gfx,fgx)\leq d(gx,fx)\;{\rm and}$$
2. $f$ and $g$ are called compatible if $$\lim\limits_{n\to \infty}d(gfx_n,fgx_n)=0$$ whenever $\{x_n\}$ is a sequence in $X$ such that $$\lim\limits_{n\to \infty}g(x_n)=\lim\limits_{n\to \infty}f(x_n),$$
3. $f$ is called $g$-continuous at some $x\in X$ if for all sequences $\{x_n\}\subset X$, $$g(x_n)\stackrel{d}{\longrightarrow} g(x)\Rightarrow f(x_n)\stackrel{d}{\longrightarrow} f(x).$$ Moreover, $f$ is called $g$-continuous if it is $g$-continuous at each point of $X$.
Recently, Alam and Imdad [@RT1] extended the classical Banach contraction principle to complete metric space endowed with a binary relation and observed that the partial order (see Nieto and Rodríguez-López [@PF2]), preorder (see Turinici [@T-LCP]), transitive relation (see Ben-El-Mechaiekh [@BR3]), tolerance (see Turinici [@T-RRF; @T-NLF]), strict order (see Ghods $et\;al.$ [@C-SO]), symmetric closure (see Samet and Turinici [@BR1]) $etc$ utilized in several well-known metrical fixed point theorems are not necessary and we can extend such results for an arbitrary binary relation. In this context, the contraction condition is relatively weaker than usual contraction as it is required to hold for only those elements which are related in the underlying relation rather than for every pair of elements.\
The aim of this paper is to extend our results proved in [@RT1] to prove some existence and uniqueness results on coincidence points in metric space endowed with an arbitrary binary relation. In proving the results, we employ our newly introduced notions such as: $\mathcal{R}$-completeness, $\mathcal{R}$-closedness, $\mathcal{R}$-continuity, $(g,\mathcal{R})$-continuity, $\mathcal{R}$-compatibility, $\mathcal{R}$-connected sets $etc.$. In this course, we also observe that our results combine the idea contained in Karapinar $et\;al.$ [@FIC9] as the set [*M*]{} (utilized by Karapinar $et\;al.$ [@FIC9]) being subset of $X^2$ is infact a binary relation on $X$. As consequences of our newly proved results, we deduce several other established metrical coincidence point theorems. Finally, we furnish some illustrative examples to demonstrate our results.
Relation-Theoretic Notions and Auxiliary Results {#SC:Relation-Theoretic Notions and Auxiliary Results}
================================================
In this section, to make our exposition self-contained, we give some definitions and basic results related to our main results.\
**Definition 3 [@ST].** Let $X$ be a nonempty set. A subset $\mathcal{R}$ of $X^2$ is called a binary relation on $X$. Notice that for each pair $x,y\in X$, one of the following holds:\
(i) $(x,y)\in \mathcal{R}$; means that “$x$ is $\mathcal{R}$-related to $y$" or “$x$ relates to $y$ under $\mathcal{R}$". Sometimes, we write $x\mathcal{R}y$ instead of $(x,y)\in\mathcal{R}$.\
(ii) $(x,y)\not\in \mathcal{R}$; means that “$x$ is not $\mathcal{R}$-related to $y$" or “$x$ doesn’t relate to $y$ under $\mathcal{R}$".\
Trivially, $X^2$ and $\emptyset$ being subsets of $X^2$ are binary relations on $X$, which are respectively called the universal relation (or full relation) and empty relation.\
Throughout this paper, $\mathcal{R}$ stands for a nonempty binary relation but for the sake of simplicity, we often write ‘binary relation’ instead of ‘nonempty binary relation’.\
**Definition 4 [@RT1].** Let $\mathcal{R}$ be a binary relation on a nonempty set $X$ and $x,y\in X$. We say that $x$ and $y$ are $\mathcal{R}$-comparative if either $(x,y)\in
\mathcal{R}$ or $(y,x)\in \mathcal{R}$. We denote it by $[x,y]\in
\mathcal{R}$.\
[**Proposition 1.**]{} If $(X,d)$ is a metric space, $\mathcal{R}$ is a binary relation on $X$, $f$ and $g$ are two self-mappings on $X$ and $\alpha\in [0,1)$, then the following contractivity conditions are equivalent:\
(I) $d(fx,fy)\leq\alpha d(gx,gy)\;\;\forall~ x,y\in X$ with $(gx,gy)\in \mathcal{R}$,\
(II) $d(fx,fy)\leq\alpha d(gx,gy)\;\;\forall~ x,y\in X$ with $[gx,gy]\in \mathcal{R}$.\
[[Proof.]{}]{} The implication (II)$\Rightarrow$(I) is trivial. On the other hand, suppose that (I) holds. Take $x,y\in X$ with $[gx,gy]\in \mathcal{R}$. If $(gx,gy)\in \mathcal{R}$, then (II) is directly follow from (I). Otherwise, in case $(gy,gx)\in
\mathcal{R}$, using symmetry of $d$ and (I), we obtain $$d(fx,fy)=d(fy,fx)\leq\alpha d(gy,gx)=\alpha d(gx,gy).$$ This proves that (I)$\Rightarrow$(II).\
**Definition 5 [@ST; @RA].** A binary relation $\mathcal{R}$ on a nonempty set $X$ is called\
if $(x,x)\in \mathcal{R}~~~\forall x\in X$,\
if whenever $(x,y)\in \mathcal{R}$ then $(y,x)\in \mathcal{R}$,\
if whenever $(x,y)\in \mathcal{R}$ and $(y,x)\in \mathcal{R}$ then $x=y$,\
if whenever $(x,y)\in \mathcal{R}$ and $(y,z)\in \mathcal{R}$ then $(x,z)\in \mathcal{R}$,\
$\bullet$ or [*connected*]{} or [*dichotomous*]{} if $[x,y]\in \mathcal{R}~~~\forall~ x,y\in X$,\
$\bullet$ or [*weakly connected*]{} or [*trichotomous*]{} if $[x,y]\in \mathcal{R}$\
or $x=y~~~\forall~ x,y\in X$.\
**Definition 6 [@BR1; @ST; @RA; @BR01; @BR02; @BR03].** A binary relation $\mathcal{R}$ defined on a nonempty set $X$ is called\
$\bullet$ if $\mathcal{R}$ has no specific properties at all,\
$\bullet$ or [*sharp order*]{} if $\mathcal{R}$ is irreflexive and transitive,\
$\bullet$ if $\mathcal{R}$ is antisymmetric and transitive,\
$\bullet$ if $\mathcal{R}$ is reflexive and antisymmetric,\
$\bullet$ or [*preorder*]{} if $\mathcal{R}$ is reflexive and transitive,\
$\bullet$ if $\mathcal{R}$ is reflexive, antisymmetric and transitive,\
$\bullet$ if $\mathcal{R}$ is weakly complete strict order,\
$\bullet$ if $\mathcal{R}$ is complete preorder,\
$\bullet$ or [*linear order*]{} or [*chain*]{} if $\mathcal{R}$ is complete partial order,\
$\bullet$ if $\mathcal{R}$ is reflexive and symmetric,\
$\bullet$ if $\mathcal{R}$ is reflexive, symmetric and transitive.
Clearly, universal relation $X^2$ on a nonempty set $X$ remains a complete equivalence relation.
**Definition 7 [@ST].** Let $X$ be a nonempty set and $\mathcal{R}$ a binary relation on $X$.
1. The inverse or transpose or dual relation of $\mathcal{R}$, denoted by $\mathcal{R}^{-1}$, is defined by $\mathcal{R}^{-1}:=\{(x,y)\in X^2:(y,x)\in \mathcal{R}\}$.
2. The symmetric closure of $\mathcal{R}$, denoted by $\mathcal{R}^s$, is defined to be the set $\mathcal{R}\cup \mathcal{R}^{-1}$ ($i.e.~\mathcal{R}^s:=\mathcal{R}\cup \mathcal{R}^{-1}$). Indeed, $\mathcal{R}^s$ is the smallest symmetric relation on $X$ containing $\mathcal{R}$.
[**Proposition 2 [@RT1].**]{} For a binary relation $\mathcal{R}$ on a nonempty set $X$, $$(x,y)\in\mathcal{R}^s\Longleftrightarrow [x,y]\in\mathcal{R}.$$
**Definition 8 [@DM].** Let $X$ be a nonempty set, $E\subseteq X$ and $\mathcal{R}$ a binary relation on $X$. Then, the restriction of $\mathcal{R}$ to $E$, denoted by $\mathcal{R}|_E$, is defined to be the set $\mathcal{R}\cap E^2$ ($i.e.~\mathcal{R}|_E:=\mathcal{R}\cap E^2$). Indeed, $\mathcal{R}|_E$ is a relation on $E$ induced by $\mathcal{R}$.\
**Definition 9 [@RT1].** Let $X$ be a nonempty set and $\mathcal{R}$ a binary relation on $X$. A sequence $\{x_n\}\subset X$ is called $\mathcal{R}$-preserving if $$(x_n,x_{n+1})\in\mathcal{R}\;\;\forall~n\in \mathbb{N}_{0}.$$
**Definition 10 [@RT1].** Let $X$ be a nonempty set and $f$ a self-mapping on $X$. A binary relation $\mathcal{R}$ on $X$ is called $f$-closed if for all $x,y\in X$, $$(x,y)\in \mathcal{R}\Rightarrow (fx,fy)\in \mathcal{R}.$$
**Definition 11.** Let $X$ be a nonempty set and $f$ and $g$ two self-mappings on $X$. A binary relation $\mathcal{R}$ on $X$ is called $(f,g)$-closed if for all $x,y\in X$, $$(gx,gy)\in \mathcal{R}\Rightarrow (fx,fy)\in \mathcal{R}.$$
Notice that under the restriction $g=I,$ the identity mapping on $X,$ Definition 11 reduces to Definition 10.\
[**[Proposition 3.]{}**]{} Let $X$ be a nonempty set, $\mathcal{R}$ a binary relation on $X$ and $f$ and $g$ two self-mappings on $X$. If $\mathcal{R}$ is $(f,g)$-closed, then $\mathcal{R}^s$ is also $(f,g)$-closed.\
In the following lines, we introduce relation-theoretic variants of the metrical notions: completeness, closedness, continuity, $g$-continuity and compatibility.\
**Definition 12**. Let $(X,d)$ be a metric space and $\mathcal{R}$ a binary relation on $X$. We say that $(X,d)$ is $\mathcal{R}$-complete if every $\mathcal{R}$-preserving Cauchy sequence in $X$ converges.
Every complete metric space is $\mathcal{R}$-complete, for any binary relation $\mathcal{R}$. Particularly, under the universal relation the notion of $\mathcal{R}$-completeness coincides with usual completeness.
**Definition 13**. Let $(X,d)$ be a metric space and $\mathcal{R}$ a binary relation on $X$. A subset $E$ of $X$ is called $\mathcal{R}$-closed if every $\mathcal{R}$-preserving convergent sequence in $E$ converges to a point of $E$.
Every closed subset of a metric space is $\mathcal{R}$-closed, for any binary relation $\mathcal{R}$. Particularly, under the universal relation the notion of $\mathcal{R}$-closedness coincides with usual closedness.
**Proposition 4**. An $\mathcal{R}$-complete subspace of a metric space is $\mathcal{R}$-closed.\
[**Proof.**]{} Let $(X,d)$ be a metric space. Suppose that $Y$ is an $\mathcal{R}$-complete subspace of $X$. Take an $\mathcal{R}$-preserving sequence $\{x_n\}\subset Y$ such that $x_n\stackrel{d}{\longrightarrow} x\in X$. As each convergent sequence is Cauchy, $\{x_n\}$ is an $\mathcal{R}$-preserving Cauchy sequence in $Y$. Hence, $\mathcal{R}$-completeness of $Y$ implies that the limit of $\{x_n\}$ must lie in $Y$, $i.e.$, $x\in Y$. Therefore, $Y$ is $\mathcal{R}$-closed.\
**Proposition 5**. An $\mathcal{R}$-closed subspace of an $\mathcal{R}$-complete metric space is $\mathcal{R}$-complete.\
[**Proof.**]{} Let $(X,d)$ be an $\mathcal{R}$-complete metric space. Suppose that $Y$ is $\mathcal{R}$-closed subspace of $X$. Let $\{x_n\}$ be an $\mathcal{R}$-preserving Cauchy sequence in $Y$. As $X$ is $\mathcal{R}$-complete, $\exists~x\in X$ such that $x_n\stackrel{d}{\longrightarrow} x$ and so $\{x_n\}$ is an $\mathcal{R}$-preserving sequence converging to $x$. Hence, $\mathcal{R}$-closeness of $Y$ implies that $x\in Y$. Therefore, $Y$ is $\mathcal{R}$-complete.\
**Definition 14.** Let $(X,d)$ be a metric space, $\mathcal{R}$ a binary relation on $X$ and $x\in X$. A mapping $f:X\rightarrow X$ is called $\mathcal{R}$-continuous at $x$ if for any $\mathcal{R}$-preserving sequence $\{x_n\}$ such that $x_n\stackrel{d}{\longrightarrow} x$, we have $f(x_n)\stackrel{d}{\longrightarrow} f(x)$. $f$ is called $\mathcal{R}$-continuous if it is $\mathcal{R}$-continuous at each point of $X$.
Every continuous mapping is $\mathcal{R}$-continuous, for any binary relation $\mathcal{R}$. Particularly, under the universal relation the notion of $\mathcal{R}$-continuity coincides with usual continuity.
**Definition 15.** Let $(X,d)$ be a metric space, $\mathcal{R}$ a binary relation on $X$, $g$ a self-mapping on $X$ and $x\in X$. A mapping $f:X\rightarrow X$ is called $(g,\mathcal{R})$-continuous at $x$ if for any sequence $\{x_n\}$ such that $\{gx_n\}$ is $\mathcal{R}$-preserving and $g(x_n)\stackrel{d}{\longrightarrow} g(x)$, we have $f(x_n)\stackrel{d}{\longrightarrow} f(x)$. Moreover, $f$ is called $(g,\mathcal{R})$-continuous if it is $(g,\mathcal{R})$-continuous at each point of $X$.\
Notice that under the restriction $g=I,$ the identity mapping on $X,$ Definition 15 reduces to Definition 14.
Every $g$-continuous mapping is $(g,\mathcal{R})$-continuous, for any binary relation $\mathcal{R}$. Particularly, under the universal relation the notion of $(g,\mathcal{R})$-continuity coincides with usual $g$-continuity.
**Definition 16.** Let $(X,d)$ be a metric space, $\mathcal{R}$ a binary relation on $X$ and $f$ and $g$ two self-mappings on $X$. We say that $f$ and $g$ are $\mathcal{R}$-compatible if for any sequence $\{x_n\}\subset X$ such that $\{fx_n\}$ and $\{gx_n\}$ are $\mathcal{R}$-preserving and $\lim\limits_{n\to \infty}g(x_n)=\lim\limits_{n\to \infty}f(x_n)$, we have $$\lim\limits_{n\to \infty}d(gfx_n,fgx_n)=0.$$
In a metric space $(X,d)$ endowed with a binary relation $\mathcal{R}$, commutativity $\Rightarrow$ weak commutativity $\Rightarrow$ compatibility $\Rightarrow$ $\mathcal{R}$-compatibility $\Rightarrow$ weak compatibility. Particularly, under the universal relation the notion of $\mathcal{R}$-compatibility coincides with usual compatibility .
The following notion is a generalization of $d$-self-closeness of a partial order relation $(\preceq)$ defined by Turinici [@T-LCP].\
**Definition 17 [@RT1].** Let $(X,d)$ be a metric space. A binary relation $\mathcal{R}$ on $X$ is called $d$-self-closed if for any $\mathcal{R}$-preserving sequence $\{x_n\}$ such that $x_n\stackrel{d}{\longrightarrow} x$, there exists a subsequence $\{x_{n_k}\}{\rm \;of\;} \{x_n\} \;{\rm
with}\;\;[x_{n_k},x]\in\mathcal{R}~~~\forall~k\in \mathbb{N}_{0}.$\
**Definition 18.** Let $(X,d)$ be a metric space and $g$ a self-mapping on $X$. A binary relation $\mathcal{R}$ on $X$ is called $(g,d)$-self-closed if for any $\mathcal{R}$-preserving sequence $\{x_n\}$ such that $x_n\stackrel{d}{\longrightarrow} x$, there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ with $[gx_{n_k},gx]\in\mathcal{R}~~~\forall~k\in \mathbb{N}_{0}.$\
Notice that under the restriction $g=I,$ the identity mapping on $X,$ Definition 18 reduces to Definition 17.\
**Definition 19 [@BR1].** Let $X$ be a nonempty set and $\mathcal{R}$ a binary relation on $X$. A subset $E$ of $X$ is called $\mathcal{R}$-directed if for each pair $x,y\in E$, there exists $z\in X$ such that $(x,z)\in\mathcal{R}$ and $(y,z)\in\mathcal{R}$.\
**Definition 20 [@DM].** Let $X$ be a nonempty set and $\mathcal{R}$ a binary relation on $X$. For $x,y\in X$, a path of length $k$ (where $k$ is a natural number) in $\mathcal{R}$ from $x$ to $y$ is a finite sequence $\{z_0,z_1,z_2,...,z_{k}\}\subset X$ satisfying the following conditions:\
(i) $z_0=x~{\rm and}~z_k=y$,\
(ii) $(z_i,z_{i+1})\in\mathcal{R}$ for each $i~(0\leq i\leq k-1)$.\
Notice that a path of length $k$ involves $k+1$ elements of $X$, although they are not necessarily distinct.\
**Definition 21.** Let $X$ be a nonempty set and $\mathcal{R}$ a binary relation on $X$. A subset $E$ of $X$ is called $\mathcal{R}$-connected if for each pair $x,y\in E$, there exists a path in $\mathcal{R}$ from $x$ to $y$.\
Given a binary relation $\mathcal{R}$ and two self-mappings $f$ and $g$ defined on a nonempty set $X$, we use the following notations:
1. [C]{}$(f,g):=\{x\in X:gx=fx\},~i.e.,$ the set of all coincidence points of $f$ and $g$,
2. $\overline{{\rm C}}(f,g):=\{\overline{x}\in X:\overline{x}=gx=fx,\;x\in X\},~i.e.,$ the set of all points of coincidence of $f$ and $g$,
3. $X(f,\mathcal{R}):=\{x\in X:(x,fx)\in \mathcal{R}\}$,
4. $X(f,g,\mathcal{R}):=\{x\in X:(gx,fx)\in \mathcal{R}\}$.
The main result of Alam and Imdad [@RT1] which is indeed the relation-theoretic version of Banach contraction principle runs as follows:\
[**Theorem 1 [@RT1].**]{} Let $(X,d)$ be a complete metric space, $\mathcal{R}$ a binary relation on $X$ and $f$ a self-mapping on $X$. Suppose that the following conditions hold:\
(i) $\mathcal{R}$ is $f$-closed,\
(ii) either $f$ is continuous or $\mathcal{R}$ is $d$-self-closed,\
(iii) $X(f,\mathcal{R})$ is nonempty,\
(iv) there exists $\alpha\in [0,1)$ such that\
$d(fx,fy)\leq\alpha d(x,y)\;\;\forall~ x,y\in X$ with $(x,y)\in \mathcal{R}$.\
Then $f$ has a fixed point. Moreover, if\
(v) $X$ is $\mathcal{R}^s$-connected,\
then $f$ has a unique fixed point.\
Finally, we record the following known results, which are needed in the proof of our main results.\
[**Lemma 1 [@L2].**]{} Let $X$ be a nonempty set and $g$ a self-mapping on $X$. Then there exists a subset $E\subseteq X$ such that $g(E)=g(X)$ and $g:E\rightarrow X$ is one-one.\
[**Lemma 2 [@PGF13].**]{} Let $X$ be a nonempty set and $f$ and $g$ two self-mappings on $X$. If $f$ and $g$ are weakly compatible, then every point of coincidence of $f$ and $g$ is also a coincidence point of $f$ and $g$.\
Main Results {#SC:Main Results}
============
Now, we are equipped to prove our main result on the existence of coincidence points which runs as follows:\
[**Theorem 2.**]{} Let $(X,d)$ be a metric space, $\mathcal{R}$ a binary relation on $X$ and $Y$ an $\mathcal{R}$-complete subspace of $X$. Let $f$ and $g$ be two self-mappings on $X$. Suppose that the following conditions hold:\
$(a)$ $f(X)\subseteq g(X)\cap Y$,\
$(b)$ $\mathcal{R}$ is $(f,g)$-closed,\
$(c)$ $X(f,g,\mathcal{R})$ is nonempty,\
$(d)$ there exists $\alpha\in [0,1)$ such that\
$d(fx,fy)\leq\alpha d(gx,gy)\;\;\forall~ x,y\in X$ with $(gx,gy)\in \mathcal{R}$,\
$(e)$ $(e1)$ $f$ and $g$ are $\mathcal{R}$-compatible,\
$(e2)$ $g$ is $\mathcal{R}$-continuous,\
$(e3)$ either $f$ is $\mathcal{R}$-continuous or $\mathcal{R}$ is $(g,d)$-self-closed,\
or alternately\
$(e^\prime)$ $(e^{\prime}1)$ $Y \subseteq g(X)$,\
$(e^{\prime}2)$ either $f$ is $(g,\mathcal{R})$-continuous or $f$ and $g$ are continuous or $\mathcal{R}|_Y$ is $d$-self-\
closed.\
Then $f$ and $g$ have a coincidence point.\
[Proof.]{} Assumption $(a)$ is equivalent to saying that $f(X)\subseteq g(X)$ and $f(X)\subseteq Y$. In view of assumption $(c)$, let $x_0$ be an arbitrary element of $X(f,g,\mathcal{R})$, then $(gx_0,fx_0)\in \mathcal{R}$. If $g(x_0)=f(x_0)$, then $x_0$ is a coincidence point of $f$ and $g$ and hence we are through. Otherwise, if $g(x_0)\neq f(x_0)$, then from $f(X)\subseteq g(X)$, we can choose $x_1\in X$ such that $g(x_1)=f(x_0)$. Again from $f(X)\subseteq g(X)$, we can choose $x_2\in X$ such that $g(x_2)=f(x_1)$. Continuing this process, we construct a sequence $\{x_n\}\subset X$ (of joint iterates) such that $$g(x_{n+1})=f(x_n)\;\;\forall ~n \in \mathbb{N}_{0}.\eqno (1)$$ Now, we claim that $\{gx_n\}$ is $\mathcal{R}$-preserving sequence, $i.e.,$ $$(gx_n,gx_{n+1})\in \mathcal{R}\;\;\forall~ n \in \mathbb{N}_{0}.\eqno (2)$$ We prove this fact by mathematical induction. On using equation (1) (with $n=0$) and the fact that $x_0\in X(f,g,\mathcal{R})$, we have $$(gx_{0},gx_1)\in \mathcal{R},$$ which shows that (2) holds for $n=0.$ Suppose that (2) holds for $n=r>0$, $i.e.,$ $$(gx_r,gx_{r+1})\in \mathcal{R}.$$ As $\mathcal{R}$ is $(f,g)$-closed, we have $$(fx_r,fx_{r+1})\in \mathcal{R},$$ which, on using (1), yields that $$(gx_{r+1},gx_{r+2})\in \mathcal{R},$$ $i.e.$, (2) holds for $n=r+1$. Hence, by induction, (2) holds for all $n \in \mathbb{N}_{0}$.\
In view of (1) and (2), the sequence $\{fx_n\}$ is also an $\mathcal{R}$-preserving, $i.e.,$ $$(fx_n,fx_{n+1})\in \mathcal{R}\;\;\forall~ n \in \mathbb{N}_0.\eqno (3)$$ On using (1), (2) and assumption $(d)$, we obtain $$d(gx_{n},gx_{n+1})=d(fx_{n-1},fx_{n})\leq \alpha d(gx_{n-1},gx_{n})\;\;\;\forall~n\in \mathbb{N}.$$ By induction, we have $$d(gx_{n},gx_{n+1})\leq \alpha d(gx_{n-1},gx_{n})\leq \alpha^2 d(gx_{n-2},gx_{n-1})\leq\cdots\leq \alpha^n d(gx_{0},gx_{1})\;\;\forall~n\in\mathbb{N}$$ so that $$d(gx_{n},gx_{n+1})\leq \alpha^n d(gx_{0},gx_{1})\;\;\;\forall~n\in \mathbb{N}.\eqno(4)$$ For $n<m$, using (4), we obtain $$\begin{aligned}
d(gx_{n},gx_{m})&\leq&
d(gx_{n},gx_{n+1})+d(gx_{n+1},gx_{n+2})+\cdots+d(gx_{m-1},gx_{m})\\
&\leq& (\alpha^n+\alpha^{n+1}+\cdots+\alpha^{m-1})d(gx_{0},gx_{1})\\
&=& \frac{\alpha^n-\alpha^m}{1-\alpha}d(gx_{0},gx_{1})\\
&\leq& \frac{\alpha^n}{1-\alpha}d(gx_{0},gx_{1})\\
&\rightarrow& 0\;{\rm as}\;m,n\rightarrow \infty.\end{aligned}$$ Therefore $\{gx_n\}$ is a Cauchy sequence.\
Owing to (1), $\{gx_n\}\subset f(X)\subseteq Y$ so that $\{gx_n\}$ is $\mathcal{R}$-preserving Cauchy sequence in $Y$. As $Y$ is $\mathcal{R}$-complete, there exists $z\in Y$ such that $$\lim\limits_{n\to\infty} g(x_n)=z.\eqno(5)$$ On using (1) and (5), we obtain $$\lim\limits_{n\to\infty} f(x_n)=z.\eqno(6)$$
Now, we use assumptions $(e)$ and $(e^\prime)$ to accomplish the proof. Assume that $(e)$ holds. Using (2), (5) and assumption $(e2)$ ($i.e.$ $\mathcal{R}$-continuity of $g$), we have $$\lim\limits_{n\to\infty} g(gx_n)=g(\lim\limits_{n\to\infty}gx_n)=g(z).\eqno(7)$$ Now, using (3), (6) and assumption $(e2)$ ($i.e.$ $\mathcal{R}$-continuity of $g$), we have $$\lim\limits_{n\to\infty} g(fx_n)=g(\lim\limits_{n\to\infty}fx_n)=g(z).\eqno(8)$$ As $\{fx_n\}$ and $\{gx_n\}$ are $\mathcal{R}$-preserving (due to (2) and (3)) and $\lim\limits_{n\to\infty}
f(x_n)=\lim\limits_{n\to\infty} g(x_n)=z$ (due to (5) and (6)), on using assumption $(e1)$($i.e.$ $\mathcal{R}$-compatibility of $f$ and $g$), we obtain $$\lim\limits_{n\to\infty}d(gfx_n,fgx_n)=0.\eqno(9)$$
Now, we show that $z$ is a coincidence point of $f$ and $g$. To accomplish this, we use assumption $(e)$. Firstly, suppose that $f$ is $\mathcal{R}$-continuous. On using (2), (5) and $\mathcal{R}$-continuity of $f$, we obtain $$\lim\limits_{n\to\infty} f(gx_n)=f(\lim\limits_{n\to\infty} gx_n)=f(z).\eqno(10)$$ On using (8), (9), (10) and continuity of $d$, we obtain $$\begin{aligned}
d(gz,fz)&=&d(\lim\limits_{n\to\infty}gfx_{n},\lim\limits_{n\to\infty}fgx_{n})\\
&=&\lim\limits_{n\to\infty}d(gfx_n,fgx_n)\\
&=&0\end{aligned}$$ so that $$g(z)=f(z).$$ Hence we are through. Alternately, suppose that $\mathcal{R}$ is $(g,d)$-self-closed. As $\{gx_n\}$ is $\mathcal{R}$-preserving (due to (2)) and $g(x_n)\stackrel{d}{\longrightarrow} z$ (due to (5)), by using $(g,d)$-self-closedness of $\mathcal{R}$, there exists a subsequence $\{gx_{n_k}\}$ of $\{gx_n\}$ such that $$[ggx_{n_k},gz]\in \mathcal{R}\;\;\forall~ k\in \mathbb{N}_{0}.\eqno(11)$$ Since $g(x_{n_k})\stackrel{d}{\longrightarrow} z,$ so equations (5)-(9) hold for also $\{x_{n_k}\}$ instead of $\{x_n\}$. On using (11), assumption $(d)$ and Proposition 1, we obtain $$d(fgx_{n_k},fz)\leq\alpha d(ggx_{n_k},gz)\;\;\forall~ k\in \mathbb{N}_0.\eqno(12)$$ On using triangular inequality, (7), (8), (9) and (12), we get $$\begin{aligned}
\nonumber d(gz,fz)&\leq& d(gz,gfx_{n_k})+d(gfx_{n_k},fgx_{n_k})+d(fgx_{n_k},fz)
\\&\leq&d(gz,gfx_{n_k})+d(gfx_{n_k},fgx_{n_k})+\alpha d(ggx_{n_k},gz)\\
&\rightarrow& 0\;{\rm as}\; k\rightarrow \infty\end{aligned}$$ so that $$g(z)=f(z).$$
Thus $z\in X$ is a coincidence point of $f$ and $g$ and hence we are through.\
Now, assume that $(e^\prime)$ holds. Owing to assumption $(e^\prime1)$ ($i.e.$, $Y\subseteq g(X)$), we can find some $u\in X$ such that $z=g(u).$ Hence, (5) and (6) respectively reduce to $$\lim\limits_{n\to\infty} g(x_n)=g(u).\eqno(13)$$ $$\lim\limits_{n\to\infty} f(x_n)=g(u).\eqno(14)$$
Now, we show that $u$ is a coincidence point of $f$ and $g$. To accomplish this, we use assumption $(e^{\prime}2)$. Firstly, suppose that $f$ is $(g,\mathcal{R})$-continuous, then using (2) and (13), we get $$\lim\limits_{n\to\infty} f(x_n)=f(u).\eqno(15)$$ On using (14) and (15), we get $$g(u)=f(u).$$ Hence, we are done. Secondly, suppose that $f$ and $g$ are continuous. Owing to Lemma 1, there exists a subset $E\subseteq X$ such that $g(E)=g(X)$ and $g:E \rightarrow X$ is one-one. Now, define $T: g(E) \rightarrow g(X)$ by $$T(ga)=f(a)\;\;\forall\; g(a)\in g(E)\; {\rm where}\; a\in E.\eqno(16)$$ As $g:E \rightarrow X$ is one-one and $f(X)\subseteq g(X)$, $T$ is well defined. Again since $f$ and $g$ are continuous, it follows that $T$ is continuous. Using the fact $g(X)=g(E)$, assumptions $(a)$ and $(e^\prime1)$ reduce to respectively $f(X)\subseteq
g(E)\cap Y$ and $Y\subseteq g(E)$, which follows that, without loss of generality, we are able to construct $\{x_n\}_{n=1}^\infty\subset
E$ satisfying (1) and to choose $u\in E$. On using (13), (14), (16) and continuity of $T$, we get $$f(u)=T(gu)=T(\lim\limits_{n\to\infty} gx_n)=\lim\limits_{n\to\infty} T(gx_n)=\lim\limits_{n\to\infty} f(x_n)=g(u).$$ Thus $u\in X$ is a coincidence point of $f$ and $g$ and hence we are through. Finally, suppose that $\mathcal{R}|_Y$ is $d$-self-closed. As $\{gx_n\}$ is $\mathcal{R}|_Y$-preserving (due to (2)) and $g(x_n)\stackrel{d}{\longrightarrow} g(u)\in Y$ (due to (13)), using $d$-self-closedness of $\mathcal{R}|_Y$, there exists a subsequence $\{gx_{n_k}\}$ of $\{gx_n\}$ such that $$[gx_{n_k},gu]\in \mathcal{R}|_Y\;\;\forall~ k\in \mathbb{N}_{0}.\eqno(17)$$ On using (13), (17), assumption $(d)$ and Proposition 1, we obtain $$\begin{aligned}
\nonumber d(fx_{n_k},fu)&\leq&\alpha d(gx_{n_k},gu)\\
&\rightarrow& 0~{\rm as~} k\rightarrow \infty\end{aligned}$$ so that $$\lim\limits_{k\to\infty} f(x_{n_k})=f(u).\eqno(18)$$ Using (14) and (18), we get $$g(u)=f(u).$$\
Thus, we are done. This completes the proof.\
Now, as a consequence, we particularize Theorem 2 by assuming the $\mathcal{R}$-completeness of whole space $X$.\
[**Corollary 1.**]{} Let $X$ be a nonempty set equipped with a binary relation $\mathcal{R}$ and a metric $d$ such that $(X,d)$ is an $\mathcal{R}$-complete metric space. Let $f$ and $g$ be two self-mappings on $X$. Suppose that the following conditions hold:\
$(a)$ $f(X)\subseteq g(X)$,\
$(b)$ $\mathcal{R}$ is $(f,g)$-closed,\
$(c)$ $X(f,g,\mathcal{R})$ is nonempty,\
$(d)$ there exists $\alpha\in [0,1)$ such that\
$d(fx,fy)\leq\alpha d(gx,gy)\;\;\forall~ x,y\in X$ with $(gx,gy)\in \mathcal{R}$,\
$(e)$ $(e1)$ $f$ and $g$ are $\mathcal{R}$-compatible,\
$(e2)$ $g$ is $\mathcal{R}$-continuous,\
$(e3)$ either $f$ is $\mathcal{R}$-continuous or $\mathcal{R}$ is $(g,d)$-self-closed,\
or alternately\
$(e^\prime)$ $(e^{\prime}1)$ there exists an $\mathcal{R}$-closed subspace $Y$ of $X$ such that $f(X)\subseteq Y\subseteq g(X)$,\
$(e^{\prime}2)$ either $f$ is $(g,\mathcal{R})$-continuous or $f$ and $g$ are continuous or $\mathcal{R}|_{Y}$ is $d$-self-\
closed.\
Then $f$ and $g$ have a coincidence point.\
[**Proof**]{}. The result corresponding to part $(e)$ follows easily on setting $Y=X$ in Theorem 2, while the same (result) in the presence of part $(e^{\prime})$ follows using Proposition 5.
If $g$ is onto in Corollary 1, then we can drop assumption $(a)$ as in this case it trivially holds. Also, we can remove assumption $(e^{\prime}1)$ as it trivially holds for $Y=g(X)=X$ using Proposition 4. Whenever, $f$ is onto, owing to assumption $(a)$, $g$ must be onto and hence again same conclusion is immediate.
On using Remarks 2-6, we obtain the more natural version of Theorem 2 in the form of the following consequence.\
[**Corollary 2**]{}. Theorem 2 (also Corollary 1) remains true if the usual metrical terms namely: completeness, closedness, compatibility (or commutativity/weak commutativity), continuity and $g$-continuity are used instead of their respective $\mathcal{R}$-analogous.\
Now, we present certain results enunciating the uniqueness of a point of coincidence, coincidence point and common fixed point corresponding to Theorem 2.\
[**Theorem 3.**]{} In addition to the hypotheses of Theorem 2, suppose that the following condition holds:\
$(u_1)$: $f(X)$ is $\mathcal{R}|_{g(X)}^s$-connected.\
Then $f$ and $g$ have a unique point of coincidence.\
[[Proof.]{}]{} In view of Theorem 2, $\overline{{\rm
C}}(f,g)\neq\emptyset$. Take $\overline{x},\overline{y}\in\overline{{\rm C}}(f,g)$, then $\exists~x,y\in X$ such that $$\overline{x}=g(x)=f(x)\;{\rm and}\; \overline{y}=g(y)=f(y).\eqno(19)$$ Now, we show that $\overline{x}=\overline{y}.$ As $f(x),f(y)\in
f(X)\subseteq g(X)$, by assumption $(u_1)$, there exists a path (say $\{gz_0,gz_1,gz_2,...,gz_{k}\}$) of some finite length $k$ in $\mathcal{R}|_{g(X)}^s$ from $f(x)$ to $f(y)$ (where $z_0,z_1,z_2,...,z_{k}\in X$). Owing to (19), without loss of generality, we may choose $z_0=x$ and $z_k=y$. Thus, we have $$[gz_i, gz_{i+1}]\in \mathcal{R}|_{g(X)} \;{\rm for~each}\;i\;(0\leq i\leq k-1).\eqno(20)$$ Define the constant sequences $z_n^0=x$ and $z_n^k=y$, then using (19), we have $g(z^0_{n+1})=f(z^0_n)=\overline{x}~{\rm and~}
g(z^k_{n+1})=f(z^k_n)=\overline{y}\;\;\forall~ n\in \mathbb{N}_{0}$. Put $z_0^1=z_1,z_0^2=z_2,..., z_0^{k-1}=z_{k-1}$. Since $f(X)\subseteq g(X)$, on the lines similar to that of Theorem 2, we can define sequences $\{z_n^1\},\{z_n^2\},...,\{z_n^{k-1}\}$ in $X$ such that $g(z^1_{n+1})=f(z^1_n),g(z^2_{n+1})=f(z^2_n),...,
g(z^{k-1}_{n+1})=f(z^{k-1}_n)\;\forall~ n\in \mathbb{N}_{0}$. Hence, we have $$g(z^i_{n+1})=f(z^i_n)\;\;\forall~ n\in \mathbb{N}_{0}\;{\rm and~for~each}\;i\;(0\leq i\leq k).\eqno(21)$$ Now, we claim that $$[gz_n^i,gz_n^{i+1}]\in \mathcal{R}\;\;\forall~ n\in \mathbb{N}_{0}\;{\rm and~for~each}\;i\;(0\leq i\leq k-1).\eqno(22)$$ We prove this fact by the method of mathematical induction. It follows from (20) that (22) holds for $n=0.$ Suppose that (22) holds for $n=r>0$, $i.e.,$ $$[gz_r^i, gz_r^{i+1}]\in \mathcal{R}\;\;{\rm for~each}\;i\;(0\leq i\leq k-1).$$ As $\mathcal{R}$ is $(f,g)$-closed, using Proposition 3, we obtain $$[fz_r^i, fz_r^{i+1}]\in \mathcal{R}\;\;{\rm for~each}\;i\;(0\leq i\leq k-1),$$ which on using (22), gives rise $$[gz_{r+1}^i, gz_{r+1}^{i+1}]\in \mathcal{R}\;\;{\rm for~each}\;i\;(0\leq i\leq k-1).$$ It follows that (22) holds for $n=r+1$. Thus, by induction, (22) holds for all $n \in \mathbb{N}_0$. Now for all $n \in \mathbb{N}_0$ and for each $i\;(0\leq i\leq k-1)$, define $t_n^i=:d(gz_n^i,gz_n^{i+1})$. Then, we claim that $$\lim\limits_{n\to\infty}t_n^i=0.\eqno(23)$$ On using (21), (22), assumption $(d)$ and Proposition 1, for each $i\;(0\leq i\leq k-1)$ and for all $n \in \mathbb{N}_0$, we obtain $$\begin{aligned}
t_{n+1}^i&=& d(gz_{n+1}^i,gz_{n+1}^{i+1})\\
&=& d(fz_{n}^i,fz_{n}^{i+1})\\
&\leq& \alpha d(gz_{n}^i,z_{n}^{i+1})\\
&=&\alpha t_{n}^i.\end{aligned}$$ By induction, we have $$t_{n+1}^i\leq \alpha t_{n}^i\leq \alpha^2 t_{n-1}^i\leq... \leq\alpha^{n+1} t_{0}^i$$ so that $$t_{n+1}^i\leq \alpha^{n+1} t_{0}^i,$$ yielding thereby $$\lim\limits_{n\to\infty}t_{n}^i=0\;{\rm for~each}\;i\;(0\leq i\leq
k-1).$$ Thus, (23) is proved for each $i\;(0\leq i\leq k-1)$. On using triangular inequality and (23), we obtain $$d(\overline{x},\overline{y})\leq t_n^0+t_n^1+\cdots+t_n^{k-1}
\to 0\;\; as \;\; n\to\infty$$ $\Longrightarrow \indent\hspace{4cm}\overline{x}=\overline{y}.$\
[**Corollary 3.**]{} Theorem 3 remains true if we replace the condition $(u_1)$ by one of the following conditions:\
$(u_1^\prime)$ $\mathcal{R}|_{f(X)}$ is complete,\
$(u_1^{\prime\prime})$ $f(X)$ is $\mathcal{R}|_{g(X)}^s$-directed.\
[[Proof.]{}]{} If $(u_1^\prime)$ holds, then for each $u,v\in f(X)$, $[u,v]\in\mathcal{R}|_{f(X)}\subseteq\mathcal{R}|_{g(X)}$ (owing to assumption $f(X)\subseteq g(X)$), which amounts to say that $\{u,v\}$ is a path of length 1 in $\mathcal{R}|_{g(X)}^s$ from $u$ to $v$. Hence $f(X)$ is $\mathcal{R}|_{g(X)}^s$-connected consequently Theorem 3 gives rise the conclusion.\
Otherwise, if $(u_1^{\prime\prime})$ holds then for each $u,v\in f(X)$, $\exists~w\in g(X)$ such that $[u,w]\in\mathcal{R}|_{g(X)}$ and $[v,w]\in\mathcal{R}|_{g(X)}$ (owing to assumption $f(X)\subseteq g(X)$), which amounts to say that $\{u,w,v\}$ is a path of length 2 in $\mathcal{R}|_{g(X)}^s$ from $u$ to $v$. Hence $f(X)$ is $\mathcal{R}|_{g(X)}^s$-connected and again by Theorem 3 conclusion is immediate.\
[**Theorem 4.**]{} In addition to the hypotheses of Theorem 3, suppose that the following condition holds:\
$(u_2)$: one of $f$ and $g$ is one-one.\
Then $f$ and $g$ have a unique coincidence point.\
[[Proof.]{}]{} In view of Theorem 2, ${\rm
C}(f,g)\neq\emptyset$. Take $x,y\in {\rm C}(f,g)$, then in view of Theorem 3, we have $$g(x)=f(x)=f(y)=g(y).$$ As $f$ or $g$ is one-one, we have $$x=y.$$
[**Theorem 5.**]{} In addition to the hypotheses embodied in condition $(e^\prime)$ of Theorem 3, suppose that the following condition holds:\
$(e^\prime3)$: $f$ and $g$ are weakly compatible.\
Then $f$ and $g$ have a unique common fixed point.\
[[Proof.]{}]{} Owing to Remark 6 as well as assumption $(e^\prime3)$, the mappings $f$ and $g$ are weakly compatible. Take $x\in {\rm C}(f,g)$ and denote $g(x)=f(x)=\overline{x}$. Then in view of Lemma 2, $\overline{x}\in {\rm C}(f,g)$. It follows from Theorem 3 with $y=\overline{x}$ that $g(x)=g(\overline{x}),$ $i.e.$, $\overline{x}=g(\overline{x})$, which yields that $$\overline{x}=g(\overline{x})=f(\overline{x}).$$ Hence, $\overline{x}$ is a common fixed point of $f$ and $g$. To prove uniqueness, assume that $x^*$ is another common fixed point of $f$ and $g$. Then again from Theorem 3, we have $$x^*=g(x^*)=g(\overline{x})=\overline{x}.$$ Hence we are through.\
On setting $g = I$, the identity mapping on $X$, in Theorems 2-5, we get respectively the following corresponding fixed point result.\
[**Corollary 4.**]{} Let $(X,d)$ be a metric space, $\mathcal{R}$ a binary relation on $X$ and $f$ a self-mapping on $X$. Let $Y$ be an $\mathcal{R}$-complete subspace of $X$ such that $f(X)\subseteq Y$. Suppose that the following conditions hold:\
(i) $\mathcal{R}$ is $f$-closed,\
(ii) either $f$ is $\mathcal{R}$-continuous or $\mathcal{R}|_{Y}$ is $d$-self-closed,\
(iii) $X(f,\mathcal{R})$ is nonempty,\
(iv) there exists $\alpha\in [0,1)$ such that\
$d(fx,fy)\leq\alpha d(x,y)\;\;\forall~ x,y\in X$ with $(x,y)\in \mathcal{R}$.\
Then $f$ has a fixed point. Moreover, if\
(v) $f(X)$ is $\mathcal{R}^s$-connected,\
then $f$ has a unique fixed point.\
Notice that Corollary 4 is an improvement of Theorem 1 in the following respects:\
$\bullet$ Usual notions of completeness and continuity are not necessary. Alternately, they can be replaced by their respective $\mathcal{R}$-analogues.\
$\bullet$ $\mathcal{R}$-completeness of whole space $X$ and $d$-self-closedness of whole relation $\mathcal{R}$ are not necessary as they can be respectively replaced by $\mathcal{R}$-completeness of any subspace $Y$ and $d$-self-closedness of $\mathcal{R}|_{Y}$, where$f(X)\subseteq Y\subseteq X$.\
$\bullet$ For uniqueness part, $\mathcal{R}^s$-connectedness of whole space $X$ is not required but it suffices to take the same merely of the subset $f(X)$.\
[**Corollary 5.**]{} Corollary 4 remains true if we replace assumption (v) by one of the following conditions:\
(v)$^\prime$ $\mathcal{R}|_{f(X)}$ is complete,\
(v)$^{\prime\prime}$ $f(X)$ is $\mathcal{R}^s$-directed.\
Some Consequences {#SC:Some Consequences}
=================
In this section, we derive several results of the existing literature as consequences of our newly proved results presented in the earlier sections.\
Coincidence theorems in abstract metric spaces
----------------------------------------------
. Under the universal relation ($i.e.$ $\mathcal{R}=X^2$), Theorems 2-5 reduce to the following coincidence point theorems:\
[**Corollary 6**]{}. Let $(X,d)$ be a metric space and $Y$ a complete subspace of $X$. Let $f$ and $g$ be two self-mappings on $X$. Suppose that the following conditions hold:\
$(a)$ $f(X)\subseteq g(X)\cap Y$,\
$(b)$ there exists $\alpha\in [0,1)$ such that\
$d(fx,fy)\leq\alpha d(gx,gy)\;\;\forall~ x,y\in X$ with $(gx,gy)\in \mathcal{R}$,\
$(e)$ $(e1)$ $f$ and $g$ are compatible,\
$(e2)$ $g$ is continuous,\
or alternately\
$(e^\prime)$ $Y \subseteq g(X)$.\
Then $f$ and $g$ have a unique point of coincidence.\
[**Corollary 7.**]{} In addition to the hypotheses of Corollary 6, if one of $f$ and $g$ is one-one, then $f$ and $g$ have a unique coincidence point.\
[**Corollary 8.**]{} In addition to the hypothesis $(e^\prime)$ of Corollary 6, if $f$ and $g$ are weakly compatible, then $f$ and $g$ have a unique common fixed point.\
Notice that Corollaries 6, 7 and 8 improve the well-known coincidence theorems of Goebel [@CP0] and Jungck [@CP4].
Coincidence theorems under $(f,g)$-closed sets
----------------------------------------------
Samet and Vetro [@NFI] introduced the notion of $F$-invariant sets and utilized the same to prove some coupled fixed point results for generalized linear contractions on metric spaces without any partial order. Recently, Kutbi $et\;al.$ [@FIC8] weakened the notion of $F$-invariant sets by introducing the notion of $F$-closed sets. Most recently, Karapinar $et\;al.$ [@FIC9] proved some unidimensional versions of some earlier coupled fixed point results involving $F$-closed sets. To describe such results, we need to recall the following notions:\
**Definition 21 [@FIC9].** Let $f,g:X\rightarrow
X$ be two mappings and [*M*]{}$\subseteq X^2$ a subset. We say that[*M*]{} is:
1. $(f,g)$-closed if $(fx,fy)\in{\it
M}$ for all $x,y\in X$ implies that $(gx,gy)\in {\it M}$,
2. $(f,g)$-compatible if $f(x)=f(y)$ for all $x,y\in X$ implies that $g(x)=g(y)$.\
**Definition 22 [@FIC9].** We say that a subset [*M*]{} of $X^2$ is transitive if $(x,y),(y,z)\in {\it M}$ implies that $(x,z)\in {\it M}$.\
**Definition 23 [@FIC9].** Let $(X, d)$ be a metric space and [*M*]{}$\subseteq X^2$ a subset. We say that $(X,d,{\it M})$ is regular if for all sequence $\{x_n\}\subseteq X$ such that $x_n\stackrel{d}{\longrightarrow} x$ and $(x_n, x_{n+1})\in {\it M}$ for all $n$, we have $(x_n,x)\in {\it M}$ for all $n$.\
**Definition 24 [@FIC9].** Let $(X,d)$ be a metric space, [*M*]{}$\subseteq X^2$ a subset and $x\in X$. A mapping $f:X\rightarrow X$ is said to be [*M*]{}-continuous at $x$ if for all sequence $\{x_n\}\subseteq X$ such that $x_n\stackrel{d}{\longrightarrow}
x$ and $(x_n, x_{n+1})\in {\it M}$ for all $n$, we have $f(x_n)\stackrel{d}{\longrightarrow} f(x)$. Moreover, $f$ is called [*M*]{}-continuous if it is [*M*]{}-continuous at each $x\in X$.\
The following notion is introduced in order to improve the commutativity condition of the pair of mappings $f$ and $g$, which is inspired by the notion of $O$-compatibility of Luong and Thuan [@CP5] in ordered metric spaces.\
**Definition 25 [@FIC9].** Let $(X,d)$ be a metric space and [*M*]{}$\subseteq X^2$. Two mappings $f,g:X\rightarrow X$ are said to be [*M*]{}-compatible if $$\lim\limits_{n\to \infty}d(gfx_n,fgx_n)=0$$ whenever $\{x_n\}$ is a sequence in $X$ such that $(gx_n,
gx_{n+1})\in {\it M}$ for all $n$ and $\lim\limits_{n\to
\infty}f(x_n)=\lim\limits_{n\to \infty}g(x_n)\in X$.\
Notice that Karapinar $et\;al.$ [@FIC9] (inspired by the notion of $O$-compatibility in [@CP5]) preferred to call“[*(O,M)*]{}-compatible" instead of “[*M*]{}-compatible“. Here the symbol ”$O$" has no pertinence as Luong and Thuan [@CP5] used the term “$O$-compatible" due to available partial ordering on the underlying metric space ($i.e.$ $O$ means order relation). But in above context, Karapinar $et\;al.$ [@FIC9] used a nonempty subset [*M*]{} without partial ordering, so it is appropriate to use the term “[*M*]{}-compatible".\
Here, it can be point out that the involved set [*M*]{} being a subset of $X^2$ is indeed a binary relation on $X$. Therefore, the concept of $(f,g)$-closed subset of $X^2$ can be interpreted as $(f,g)$-closed binary relation on $X$. Obviously, Definitions 16 and 17 are weaker than Definitions 25 and 23 respectively. Taking $\mathcal{R}=$[*M*]{} in Corollary 1, we get an improved version of the following result of Karapinar $et\;al.$ [@FIC9].\
[**Corollary 9**]{} (see Corollary 34 [@FIC9]). Let $(X,d)$ be a complete metric space, let $f,g:X\rightarrow X$ be two mappings and let [*M*]{}$\subseteq X^2$ be a subset such that\
(i) $f(X)\subseteq g(X)$,\
(ii) [*M*]{} is $(f,g)$-compatible and $(f,g)$-closed,\
(iii) there exists $x_{0}\in X$ such that $(gx_{0},fx_{0})\in {\it M}$,\
(iv) there exits $\alpha\in [0,1)$ such that\
$d(fx,fy)\leq\alpha d(gx,gy)\;\;\forall~ x,y\in X$ with $(gx,gy)\in {\it M}$.\
Also assume that, at least, one of the following conditions holds:\
$(a)$ $f$ and $g$ are [*M*]{}-continuous and [*M*]{}-compatible,\
$(b)$ $f$ and $g$ are continuous and commuting,\
$(c)$ $(X,d,{\it M})$ is regular and $g(X)$ is closed.\
Then $f$ and $g$ have, at least, a coincidence point.\
Observe that [*M*]{}-compatibility of $(f,g)$(see assumption (ii)) is unnecessary.\
Coincidence theorems in ordered metric spaces via increasing mappings
---------------------------------------------------------------------
Indeed the present trend was initiated by Turinici [@P15; @P16], Ran and Reurings [@PF1] and Nieto and Rodríguez-López [@PF2] which was was later generalized by many authors (e.g. [@PGF2; @PGF3; @PGF13]). In this subsection as well as in succeeding subsection, $X$ denotes a nonempty set endowed with a partial order $\preceq$. In what follows, we write $\succeq:=\preceq^{-1}$ and $\prec\succ:=\preceq^{s}$. On the lines of O’Regan and Petruşel [@PGF2], the triple $(X,d,\preceq)$ is called ordered metric space wherein $X$ denotes a nonempty set endowed with a metric $d$ and a partial order $\preceq$.\
**Definition 26 [@PGF3].** Let $(X,\preceq)$ be an ordered set and $f$ and $g$ two self-mappings on $X$. We say that $f$ is $g$-increasing if for any $x,y\in X$, $g(x)\preceq
g(y)\Rightarrow f(x)\preceq f(y)$.
It is clear that $f$ is $g$-increasing iff $\preceq$ is $(f,g)$-closed.
**Definition 27 [@PGF13].** Given a mapping $g:X\rightarrow X$, we say that an ordered metric space $(X,d,\preceq)$ has [*g-ICU*]{}(increasing-convergence-upper bound) property if $g$-image of every increasing sequence $\{x_n\}$ in $X$ such that $x_n\stackrel{d}{\longrightarrow} x$, is bounded above by $g$-image of its limit (as an upper bound), $i.e.,$ $g(x_n)\preceq g(x)\;\;\forall~ n\in \mathbb{N}_{0}.$\
Notice that under the restriction $g=I,$ the identity mapping on $X,$ Definition 27 transforms to the notion of [*ICU*]{} property.
It is clear that if $(X,d,\preceq)$ has [*ICU*]{} property (resp. [*g-ICU*]{} property), then $\preceq$ is $d$-self-closed (resp. $(g,d)$-self-closed).
On taking $\mathcal{R}=\preceq$ in Corollary 2 and using Remarks 8 and 9, we obtain the following result, which is an improved version of Corollary 3 of Alam $et\;al.$ [@PGF13].\
[**Corollary 10.**]{} Let $(X,d,\preceq)$ be an ordered metric space and $Y$ a complete subspace of $X$. Let $f$ and $g$ be two self-mappings on $X$. Suppose that the following conditions hold:\
$(a)$ $f(X)\subseteq g(X)\cap Y$,\
$(b)$ $f$ is $g$-increasing,\
$(c)$ there exists $x_{0}\in X$ such that $g(x_{0})\preceq f(x_{0})$,\
$(d)$ there exists $\alpha\in [0,1)$ such that\
$d(fx,fy)\leq\alpha d(gx,gy)\;\;\forall~ x,y\in X$ with $g(x)\preceq g(y)$,\
$(e)$ $(e1)$ $f$ and $g$ are compatible,\
$(e2)$ $g$ is continuous,\
$(e3)$ either $f$ is continuous or $(Y,d,\preceq)$ has [*g-ICU*]{} property,\
or alternately\
$(e^\prime)$ $(e^{\prime}1)$ $Y \subseteq g(X)$,\
$(e^{\prime}2)$ either $f$ is $g$-continuous or $f$ and $g$ are continuous or $(Y,d,\preceq)$ has\
property.\
Then $f$ and $g$ have a coincidence point.\
Coincidence points in ordered metric spaces via comparable mappings
-------------------------------------------------------------------
The core results involving comparable mappings are contained in Nieto and Rodríguez-López [@PF3], Turinici [@T-RRF; @T-NLF], Dorić $et\;al.$ [@PF-C0] and Alam and Imdad [@PGF15].\
**Definition 28 [@PGF15].** Let $(X,\preceq)$ be an ordered set and $f$ and $g$ two self-mappings on $X$. We say that $f$ is $g$-comparable if for any $x,y\in X,$ $$g(x)\prec\succ g(y)\Rightarrow f(x)\prec\succ f(y).$$
It is clear that $f$ is $g$-comparable iff $\prec\succ$ is $(f,g)$-closed.
**Definition 29 [@PGF14].** Let $(X,\preceq)$ be an ordered set and $\{x_n\}\subset X.$
1. the sequence $\{x_n\}$ is said to be termwise bounded if there is an element $z\in X$ such that each term of $\{x_n\}$ is comparable with $z,$ $i.e.$, $$x_n\prec\succ z\;\;\;\;\;\;\;\forall~ n\in \mathbb{N}_0$$ so that $z$ is a c-bound of $\{x_n\}$ and
2. the sequence $\{x_n\}$ is said to termwise monotone if consecutive terms of $\{x_n\}$ are comparable, $i.e.$, $$x_n\prec\succ x_{n+1}\;\;\forall~ n\in \mathbb{N}_0.$$
Clearly, $\{x_n\}$ is termwise monotone iff it is $\prec\succ$-preserving.
**Definition 30 [@PGF14].** Given a mapping $g:X\rightarrow X$, we say that an ordered metric space $(X,d,\preceq)$ has [*g-TCC*]{}(termwise monotone-convergence-c-bound) property if every termwise monotone sequence $\{x_n\}$ in $X$ such that $x_n\stackrel{d}{\longrightarrow} x$ has a subsequence, whose $g$-image is termwise bounded by $g$-image of limit (of the sequence) as a c-bound, $i.e.,$ $g(x_{n_k})\prec\succ g(x)\;\forall~
k\in \mathbb{N}_{0}.$\
Notice that under the restriction $g=I,$ the identity mapping on $X,$ Definition 30 transforms to the notion of [*TCC*]{} property.
Clearly, $(X,d,\preceq)$ has [*TCC*]{} property (resp. [*g-TCC*]{} property) iff $\prec\succ$ is $d$-self-closed (resp. $(g,d)$-self-closed).
On taking $\mathcal{R}=\prec\succ$ in Corollary 2 and using Remarks 10 and 12, we obtain the following result, which is an improved version of Theorem 3.7 of Alam and Imdad [@PGF15].\
[**Corollary 11.**]{} Let $(X,d,\preceq)$ be an ordered metric space and $Y$ a complete subspace of $X$. Let $f$ and $g$ be two self-mappings on $X$. Suppose that the following conditions hold:\
$(a)$ $f(X)\subseteq g(X)\cap Y$,\
$(b)$ $f$ is $g$-comparable,\
$(c)$ there exists $x_{0}\in X$ such that $g(x_{0})\prec\succ f(x_{0})$,\
$(d)$ there exists $\alpha\in [0,1)$ such that\
$d(fx,fy)\leq\alpha d(gx,gy)\;\;\forall~ x,y\in X$ with $g(x)\prec\succ g(y)$,\
$(e)$ $(e1)$ $f$ and $g$ are compatible,\
$(e2)$ $g$ is continuous,\
$(e3)$ either $f$ is continuous or $(Y,d,\preceq)$ has [*g-TCC*]{} property,\
or alternately\
$(e^\prime)$ $(e^{\prime}1)$ $Y \subseteq g(X)$,\
$(e^{\prime}2)$ either $f$ is $g$-continuous or $f$ and $g$ are continuous or $(Y,d,\preceq)$ has\
property.\
Then $f$ and $g$ have a coincidence point.\
Coincidence theorems under symmetric closure of a binary relation
-----------------------------------------------------------------
The origin of such results can be traced back to Samet and Turinici [@BR1] which is also pursued in Berzig [@BR2]. In this context, $\mathcal{R}$ stands for an arbitrary binary relation on a nonempty set $X$ and $\mathcal{S}:=\mathcal{R}^s$.\
**Definition 31 [@BR2].** Let $f$ and $g$ be two self-mappings on $X$. We say that $f$ is $g$-comparative if for any $x,y\in X,$ $$(gx,gy)\in \mathcal{S}\Rightarrow (fx,fy)\in \mathcal{S}.$$
It is clear that $f$ is $g$-comparative iff $\mathcal{S}$ is $(f,g)$-closed.
**Definition 32 [@BR1].** We say that $(X,d,\mathcal{S})$ is regular if the following condition holds: if the sequence $\{x_n\}$ in $X$ and the point $x\in X$ are such that $$(x_n,x_{n+1})\in \mathcal{S}\;{\rm for~all~} n\; {\rm and~}\lim\limits_{n\to\infty} d(x_n,x)=0,$$ then there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that $(x_{n_k},x)\in \mathcal{S}$ for all $k$.
Clearly, $(X,d,\mathcal{S})$ is regular iff $\mathcal{S}$ is $d$-self-closed.
Taking the symmetric closure $\mathcal{S}$ of an arbitrary relation $\mathcal{R}$ in Corollary 2 and using Remarks 13 and 14, we obtain an improved version of the following result of Berzig [@BR2].\
[**Corollary 12**]{} (see Corollary 4.5 [@BR2]). Let $(X,d)$ be a metric space, $\mathcal{R}$ a binary relation on $X$ and $f$ and $g$ two self-mappings on $X$. Suppose that the following conditions hold:\
$(a)$ $f(X)\subseteq g(X)$,\
$(b)$ $f$ is $g$-comparative,\
$(c)$ there exists $x_0\in X$ such that $(gx_0,fx_0)\in \mathcal{S}$,\
$(d)$ there exists $\alpha\in [0,1)$ such that\
$d(fx,fy)\leq\alpha d(gx,gy)\;\;\forall~ x,y\in X$ with $(gx,gy)\in \mathcal{S}$,\
$(e)$ $(X,d)$ is complete and $g(X)$ is closed,\
$(f)$ $(X,d,\mathcal{S})$ is regular.\
Then $f$ and $g$ have a coincidence point.\
Examples {#SC:Examples}
========
In this section, we provide two examples establishing the utility of Theorems 2-5.
Consider $X=\mathbb{R}$ equipped with usual metric and also define a binary relation $\mathcal{R}=\{(x,y)\in \mathbb{R}^2:|x|-|y|\geq 0\}$. Then $(X,d)$ is an $\mathcal{R}$-complete metric space. Consider the mappings $f,g:X\to X$ defined by $f(x)=\frac{x^2}{3}\;{\rm and}\;g(x)=\frac{x^2}{2}\;\forall~ x\in
X.$ Clearly, $\mathcal{R}$ is $(f,g)$-closed. Now, for $x,y\in X$ with $(gx,gy)\in \mathcal{R}$, we have $$d(fx,fy)=\left|\frac{x^2}{3}-\frac{y^2}{3}\right|=\frac{2}{3}\left|\frac{x^2}{2}-\frac{y^2}{2}\right|=\frac{2}{3}d(gx,gy)< \frac{3}{4}d(gx,gy).$$ Thus, $f$ and $g$ satisfy assumption $(d)$ of Theorem 2 with $\alpha=\frac{3}{4}$. By a routine calculation, one can verify all the conditions mentioned in $(e)$ of Theorem 2. Hence all the conditions of Theorem 2 are satisfied for $Y=X$, which guarantees that $f$ and $g$ have a coincidence point in $X$. Moreover, observe that $(u_1)$ holds and henceforth in view of Theorem 3, $f$ and $g$ have a unique point of coincidence (namely: $\overline{x}=0$), which remains also a unique common fixed point (in view of Theorem 5).\
Observe that the underlying binary relation $\mathcal{R}$ is a preorder which is not antisymmetric and henceforth not a partial order. Thus, in all, our results are genuine extension of several corresponding results proved under partial ordering.
Consider $X=\mathbb{R}$ equipped with usual metric and also define a binary relation $\mathcal{R}=\{(x,y)\in \mathbb{R}^2:x\geq 0,\;
y\in \mathbb{Q}\}$. Consider the mappings $f,g:X\to X$ defined by $f(x)=1\;{\rm and}\;g(x)=x^2-3\;\forall~ x\in
X.$ Clearly, $\mathcal{R}$ is $(f,g)$-closed. Now, for $x,y\in X$ with $(gx,gy)\in \mathcal{R}$, we have $$d(fx,fy)=|1-1|=0\leq\alpha|x^2-y^2|=\alpha d(gx,gy).$$ Thus, $f$ and $g$ satisfy assumption $(d)$ of Theorem 2 for any arbitrary $\alpha\in [0,1)$. Also, the mappings $f$ and $g$ are not $\mathcal{R}$-compatible and hence $(e)$ does not hold. But the subspace $Y:=g(X)=[-3,\infty)$ is $\mathcal{R}$-complete and $f$ and $g$ are continuous, $i.e.$, all the conditions mentioned in $(e^{\prime})$ are satisfied. Hence, in view of Theorem 2, $f$ and $g$ have a coincidence point in $X$. Further, in this example $(u_1)$ holds and henceforth, owing to Theorem 3, $f$ and $g$ have a unique point of coincidence (namely: $\overline{x}=1$). Notice that neither $f$ nor $g$ is one-one, $i.e.$, $(u_2)$ does not hold and hence, we can not apply Theorem 4, which guarantees the uniqueness of coincidence point. Observe that, in the present example, there are two coincidence points (namely: x=2 and x=-2). Also, $f$ and $g$ are not weakly compatible , $i.e.$, $(e^\prime3)$ does not hold and hence, we can not apply Theorem 5, which ensures the uniqueness of common fixed point. Notice that there is no common fixed point of $f$ and $g$.\
Observe that the underlying binary relation $\mathcal{R}$ is a transitive relation. Indeed, $\mathcal{R}$ is non-reflexive, non-irreflexive, non-symmetric as well as non-antisymmetric and hence it is not a preorder, partial order, near order, strict order, tolerance or equivalence and also never turns out to be a symmetric closure of any binary relation.
Here, it can be point out that corresponding results contained in Section 4 cannot be used in the context of present example, which substantiate the utility of our newly proved coincidence theorems over corresponding several relevant results.
[99]{}
G. Jungck, [*Commuting maps and fixed points,*]{} Amer. Math. Monthly 83 (4) (1976) 261-263.
G. Jungck, [*Common fixed points for noncontinuous nonself maps on non-metric spaces,*]{} Far East J. Math. Sci. 4 (1996) 199-215.
S. Sessa, [*On a weak commutativity condition of mappings in fixed point considerations,*]{} Publ. Inst. Math. Soc. 32 (1982) 149-153.
G. Jungck, [*Compatible mappings and common fixed points,*]{} Int. J. Math. Math. Sci. 9 (4) (1986) 771-779.
K. P. R. Sastry and I. S. R. Krishna Murthy, [*Common fixed points of two partially commuting tangential selfmaps on a metric space,*]{} J. Math. Anal. Appl. 250 (2) (2000) 731-734.
A. Alam and M. Imdad, [*Relation-theoretic contraction principle,*]{} J. Fixed Point Theory Appl. 17 (4) (2015) 693-702.
J. J. Nieto and R. Rodríguez-López, [*Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,*]{} Order 22 (3) (2005) 223-239.
M. Turinici, [*Linear contractions in product ordered metric spaces,*]{} Ann Univ Ferrara 59 (2013) 187-198.
H. Ben-El-Mechaiekh, [*The Ran-Reurings fixed point theorem without partial order: A simple proof,*]{} J. Fixed Point Theory Appl. 16 (2014) 373-383
M. Turinici, [*Ran-Reurings fixed point results in ordered metric spaces,*]{} Libertas Math. 31 (2011) 49-55.
M. Turinici, [*Nieto-Lopez theorems in ordered metric spaces,*]{} Math. Student 81 (1-4) (2012) 219-229.
S. Ghods, M. E. Gordji, M. Ghods and M. Hadian, [*Comment on “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces" \[Lakshmikantham and Ćirić, Nonlinear Anal. TMA 70 (2009) 4341-4349\],*]{} J. Comput. Anal. Appl. 14 (5) (2012) 958-966.
B. Samet and M. Turinici, [*Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications,*]{} Commun. Math. Anal. 13 (2) (2012) 82-97.
E. Karapinar, A. Rold$\acute{\rm a}$n, N. Shahzad and W. Sintunavarat, [*Discussion of coupled and tripled coincidence point theorems for $\varphi$-contractive mappings without the mixed g-monotone property,*]{} Fixed Point Theory Appl. 2014:92 (2014) 16 pp.
S. Lipschutz, [*Schaum’s outlines of theory and problems of set theory and related topics,*]{} McGraw-Hill, New York (1964).
R. D. Maddux, [*Relation algebras*]{}, Studies in Logic and the Foundations of Mathematics, vol. 150, Elsevier B. V., Amsterdam (2006).
V. Flaška, J. Ježek, T. Kepka and J. Kortelainen, [*Transitive closures of binary relations I,*]{} Acta Univ. Carolin. Math. Phys. 48 (1) (2007) 55-69.
H. L. Skala, [*Trellis theory,*]{} Algebra Universalis 1 (1971) 218-233.
A. Stouti and A. Maaden, [*Fixed points and common fixed points theorems in pseudo-ordered sets,*]{} Proyecciones 32 (4) (2013) 409-418.
B. Kolman, R. C. Busby and S. Ross, [*Discrete mathematical structures,*]{} Third Edition, PHI Pvt. Ltd., New Delhi (2000).
M. Berzig, [*Coincidence and common fixed point results on metric spaces endowed with an arbitrary binary relation and applications,*]{} J. Fixed Point Theory Appl. 12 (1-2) (2012) 221-238.
R. H. Haghi, Sh. Rezapour and N. Shahzad, [*Some fixed point generalizations are not real generalizations,*]{} Nonlinear Anal. 74 (2011) 1799-1803.
A. Alam, A. R. Khan and M. Imdad, [*Some coincidence theorems for generalized nonlinear contractions in ordered metric spaces with applications,*]{} Fixed Point Theory Appl. 2014:216 (2014) 30 pp.
K. Goebel, [*A coincidence theorem,*]{} Bull. Acad. Pol. Sci. S$\acute{\rm e}$r. Sci. Math. Astron. Phys. 16 (1968) 733-735.
B. Samet and C. Vetro, [*Coupled fixed point, $F$-invariant set and fixed point of N-order,*]{} Ann. Funct. Anal. 1 (2) (2010) 46-56.
M. A. Kutbi, A. Rold$\acute{\rm a}$n, W. Sintunavarat, J. Martinez-Moreno and C. Roldan, [*$F$-closed sets and coupled fixed point theorems without the mixed monotone property,*]{} Fixed Point Theory Appl. 2013:330 (2013) 11 pp.
N. V. Luong and N. X. Thuan, [*Coupled points in ordered generalized metric spaces and application to integro differential equations,*]{} An. $\c{S}$t. Univ. Ovidius Constan$\c{t}$a 21 (3) (2013) 155-180.
M. Turinici, [*Abstract comparison principles and multivariable Gronwall-Bellman inequalities,*]{} J. Math. Anal. Appl. 117 (1) (1986) 100-127.
M. Turinici, [*Fixed points for monotone iteratively local contractions,*]{} Dem. Math. 19 (1) (1986) 171-180.
A. C. M. Ran and M. C. B. Reurings, [*A fixed point theorem in partially ordered sets and some applications to matrix equations,*]{} Proc. Amer. Math. Soc. 132 (5) (2004) 1435-1443.
D. O’Regan and A. Petruşel, [*Fixed point theorems for generalized contractions in ordered metric spaces,*]{} J. Math. Anal. Appl. 341 (2) (2008) 1241-1252.
L. Ćirić, N. Cakic, M. Rajovic and J. S. Ume, [*Monotone generalized nonlinear contractions in partially ordered metric spaces,*]{} Fixed Point Theory Appl. 2008:131294 (2008) 11 pp.
J. J. Nieto and R. Rodríguez-López, [*Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equation,*]{} Acta Math. Sin. (Engl. Ser.) 23 (12) (2007) 2205-2212.
D. Dorić, Z. Kadelburg, S. Radenović and P. Kumam, [*A note on fixed point results without monotone property in partially ordered metric space,*]{} Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 108 (2) (2014) 503-510.
A. Alam and M. Imdad, [*Comparable linear contractions in ordered metric spaces,*]{} Fixed Point Theory, in press, (accepted-2015)/arXiv:1507.08987, 2015.
A. Alam and M. Imdad, [*Monotone generalized contractions in ordered metric spaces,*]{} Bull. Korean Math. Soc. 53 (1) (2016) 61-81.
[^1]: $^\ast$Correspondence:
|
---
abstract: 'Graphene thermionic electron emission across high-interface-barrier involves energetic electrons residing far away from the Dirac point where the Dirac cone approximation of the band structure breaks down. Here we construct a full-band model beyond the simple Dirac cone approximation for the thermionic injection of high-energy electrons in graphene. We show that the thermionic emission model based on the Dirac cone approximation is valid only in the graphene/semiconductor Schottky interface operating near room temperature, but breaks down in the cases involving high-energy electrons, such as graphene/vacuum interface or heterojunction in the presence of photon absorption, where the full-band model is required to account for the band structure nonlinearity at high electron energy. We identify a critical barrier height, $\Phi_B^{(\text{c})} \approx 3.5$ eV, beyond which the Dirac cone approximation crosses over from underestimation to overestimation. In the high-temperature thermionic emission regime at graphene/vacuum interface, the Dirac cone approximation severely overestimates the electrical and heat current densities by more than 50% compared to the more accurate full-band model. The large discrepancies between the two models are demonstrated using a graphene-based thermionic cooler. These findings reveal the fallacy of Dirac cone approximation in the thermionic injection of high-energy electrons in graphene. The full-band model developed here can be readily generalized to other 2D materials, and shall provide an improved theoretical avenue for the accurate analysis, modeling and design of graphene-based thermionic energy devices.'
author:
- Yee Sin Ang
- Yueyi Chen
- Chuan Tan
- 'L. K. Ang'
title: |
\
Generalized High-Energy Thermionic Electron Injection at Graphene Interface
---
Introduction
============
Recent theoretical and experimental developments have revealed graphene’s extraordinary potential in various thermionic-based energy applications [@yuan; @massicotte]. In thermionic emission process, electrons are thermally excited to overcome the interface potential barrier and emitted across the interface. Collection of these emitted electrons at the anode forms an electricity through an external load, thus achieving thermionic-based heat-to-electricity conversion. Recent experiment has demonstrated graphene monolayer as a highly efficient anode for direct heat-to-electricity energy conversion with high conversion efficiency reaching 9.8%, which can be further optimized by electrostatic gating and cathode-anode gap reduction [@yuan]. Thermionic emission of electrons over an insulating barrier can also be harnessed to achieve electronic cooling effect [@shakouri; @mahan; @shakouri2; @vanshee]. The performance is, however, fundamentally limited by thermal backflow directed from hot to cold electrodes. Recent advancements of 2D material van der Waals heterostructure [@geim; @novoselov; @liu] (VDWH) offer new opportunities in solid-state thermionic cooler. The layered structure of VDWH strongly impedes phonon propagation and effectively diminishes the thermal backflow effect that is detrimental to the efficiency of thermionic cooler [@liang; @wang; @wang2]. Photon-enhanced thermionic injection across graphene/WSe$_2$/graphene VDWH represents another novel route towards the efficient broadband photodetection and harvesting of light energy in a compact solid-state platform [@massicotte].
Previous theoretical models [@nieva; @misra; @misra2; @zhang; @zhang2; @SL; @liang2; @ang; @ang2] describes the out-of-plane thermionic electron emission from the surface of graphene via two key assumptions: (i) electron wavevector component lying in the plane of graphene, denoted as $k_\parallel$, is conserved during the out-of-plane emission process [@liang2]; and (ii) electrons undergoing thermionic emission are described by a conic energy band structure, commonly known as the *Dirac cone approximation* [@SL; @liang2; @ang3; @trushin; @trushin2]. These assumptions break down in the case of thermionic emission of high-energy electrons due to the following reasons. Firstly, the conservation of $\varepsilon_\parallel$ is violated due to the presence of electron-electron, electron-impurity, and electron-phonon scattering effect [@meshkov; @russell; @perebeinos; @vdovin; @liu_acs]. Clear experimental evidence of $k_\parallel$-nonconserving vertical electron transport due to electron-phonon scattering has been observed in graphene heterostructures [@vdovin]. At typical device operated at room temperature and above, electrons in graphene are expected to undergo strong scatterings with phonons and impurities. Thus, at the high-temperature thermionic emission regime, $k_\parallel$-conservation is expected to be strongly violated due to phonon [@vdovin] and impurity scattering [@liu_acs] effects, which immediately implies the breakdown of assumption (i). Secondly, as the interface potential exceeds $\Phi_B \approx 1$ eV, the thermionic emission should be dominantly contributed by energetic carriers with $\varepsilon_\parallel > 1$ eV, where the band structure deviates significantly from the simple Dirac cone approximation. Barrier-lowering mechanisms, such as surface coating which lowers the vacuum level by about 1 eV, and electrostatic doping via gate voltage which raises the Fermi level by up to 1 eV [@voss; @yu; @yuan2], can typically reduce $\Phi_B$ by up to $\sim 1$ eV. In this case, electrons undergoing thermionic emission are still energetically beyond the Dirac cone regime. Furthermore, in the presence of photon absorption [@massicotte; @misra2], electrons are energetically promoted to high-energy state beyond the Dirac cone regime. The Dirac cone approximation is thus expected to produce inaccurate results for high-temperature, high-barrier and photon-enhanced thermionic injections in graphene-based heterointerface. For this reason, the assumption (ii) should also be revised to account for the band structure nonlinearity at high energy.
In this paper, we construct an improved graphene thermionic emission model that relaxes the two assumptions discussed above by explicitly including the effects of: (i) $k_\parallel$-nonconservation [@ang3]; and (ii) band structure nonlinearity beyond the simple Dirac cone approximation by using the full-band tight-binding energy dispersion [@reich]. We found that the simple Dirac-based model is only valid for solid-state graphene/semiconductor Schottky contact with low interface-barrier ($\Phi_B < 1$ eV) [@tongay] operating near room temperature ($T \approx 300$ K). At high-barrier ($\Phi_B \gg 1$ eV) and high-temperature ($T >1000$ K) regimes, the Dirac-based model deviates significantly from the full-band model developed here, thus revealing the shortcoming of Dirac cone approximation in the thermionic emission of graphene/vacuum interface. We identify the existence of a *critical potential barrier height*, $\Phi_B^{(c)} \approx 3.5$ eV, beyond which the Dirac cone approximation crosses over from underestimation to overestimation in comparison with the more accurate full-band model. We calculate the efficiency of a graphene-based thermionic cooler device using both Dirac approximation and the improved full-band model, and a large discrepancy between the two models is revealed. These findings prompt an urgent need in replacing the Dirac cone approximation with the full-band model developed here so to achieve a more reliable modeling and understanding of graphene-based high-temperature thermionic energy devices.
Theory
======
The thermionic electrical and heat current densities from the surface of a 2D electronic system are \[see Fig. 1(a) for the energy band diagram\],
$$\label{current}
\mathcal{J} = \frac{g_{s,v} e}{(2\pi)^2} \sum_i \frac{v_\perp^{(i)}(k_\perp^{(i)})}{L_\perp} \int \text{d}^2\mathbf{k}_\parallel \mathcal{T}^{(i)}(\mathbf{k}_\parallel, k_\perp) f(\varepsilon_{\mathbf{k}_\parallel}),$$
$$\begin{aligned}
\label{heat}
\mathcal{Q} &=& \frac{g_{s,v} }{(2\pi)^2} \sum_i \frac{v_\perp^{(i)}(k_\perp^{(i)})}{L_\perp} \nonumber \\
&& \times \int \text{d}^2\mathbf{k}_\parallel (\varepsilon_\parallel -\varepsilon_F) \mathcal{T}^{(i)}(\mathbf{k}_\parallel, k_\perp) f(\varepsilon_{\mathbf{k}_\parallel}),\end{aligned}$$
where $g_{s,v}=4$ is the spin-valley degeneracy, $L_\perp$ is the 2D material thickness ($L_\perp = 0.335$ nm for graphene [@ni]), $\varepsilon_F$ is the Fermi level, $f(\varepsilon_{\mathbf{k}_\parallel})$ is the Fermi-Dirac distribution function, $\mathbf{k}_\parallel = (k_x, k_y)$ is the electron wave vector component lying in the 2D plane, $\perp$ denotes the direction orthogonal to the $x$-$y$ plane of the 2D system, $k_\perp^{(i)}$ is the quantized out-of-plane wave vector component of $i$-th subband, $v_\perp^{(i)}(k_\perp^{(i)}) = \sqrt{2m\varepsilon_\perp^{(i)}}$ is the cross-plane electron group velocity, $m$ is the free electron mass, $\varepsilon_\perp^{(i)}$ is the discrete bound state energy level, and the summation, $\sum_i \cdots$, runs over all of the $i$-th quantized subbands. The non-conservation of $\mathbf{k}_\parallel$ during the out-of-plane thermionic emission process leads to the coupling between $\mathbf{k}_\parallel$ and $k_\perp$. Accordingly, the $i$-th subband transmission probability becomes $\mathcal{T}^{(i)}(k_\perp, \mathbf{k}_\parallel)$, i.e. the cross-plane electron tunneling is dependent on both $k_\perp$ and $\mathbf{k}_\parallel$. The $\mathbf{k}_\parallel$-nonconserving model has been extensively studied in previous theoretical works [@vanshee; @ang3; @vashaee2; @dwyer; @kim] and has been successfully employed in the analysis of thermionic transport experiments in graphene-based devices [@massicotte; @ma; @SL].
For thermionic emission, the transmission probability can be written as $\mathcal{T}^{(i)}(k_\perp, \mathbf{k}_\parallel) = \lambda \Theta\left( \varepsilon_\parallel + \varepsilon_\perp^{(i)}- \Phi_{B} \right)$, i.e. $\varepsilon^{(i)}_\perp$ and $\varepsilon_\parallel$ are combined to overcome the interface barrier $\Phi_{B}$. Here $\lambda$ is a parameter representing the strength of $\mathbf{k}_\parallel$-non-conserving scattering processes. The term $\Theta(x)$ denotes the Heaviside step-function. Equations (\[current\]) and (\[heat\]) can be simplified into a single-subband thermionic emission electrical and heat current densities for graphene, respectively, as
\[flux\] $$\label{current_sim}
\mathcal{J} = \frac{g_{s,v}ev_\perp}{(2\pi)^2} \int \text{d}^2 \mathbf{k}_\parallel \mathcal{T}(\varepsilon_\parallel) \xi_T(\varepsilon_\parallel),$$ $$\label{heat_sim}
\mathcal{Q} = \frac{g_{s,v}v_\perp}{(2\pi)^2} \int \text{d}^2 \mathbf{k}_\parallel (\varepsilon_\parallel - \varepsilon_F) \mathcal{T}(\varepsilon_\parallel) \xi_T(\varepsilon_\parallel),$$
where $\xi_T(x) \equiv \exp\left( -\frac{x - \varepsilon_F}{k_BT} \right)$. In writing $\mathcal{T}(\varepsilon_\parallel) = \lambda \Theta\left(\varepsilon_\parallel(\mathbf{k}_\parallel) - \Phi_{B}\right)$, we have used the fact that the quantized subband energy level, $\varepsilon_\perp^{(1)}$, can be absorbed into $\varepsilon_\parallel$ by setting $\varepsilon_\perp^{(1)}$ as the zero-reference of $\varepsilon_\parallel$ [@ang3], and denoted $v_\perp \equiv v_\perp^{(1)}(k_\perp^{(1)})$. The Fermi-Dirac distribution function approaches the semiclassical Maxwell-Boltzmann distribution function, $\xi_T(\varepsilon_\parallel)$, since the emitted electrons are in the non-degenerate regime, i.e. $\varepsilon_\parallel \approx \Phi_{B}$, and $\Phi_{B} \gg \varepsilon_F$ for typical values of $\varepsilon_F < 1$ eV and graphene/vacuum interface barrier of $\Phi_{B} = 4.5$ eV.
The electronic properties of graphene enter Eq. (\[current\_sim\]) via the $\mathbf{k}_\parallel$-integration. The $\mathbf{k}_\parallel$-integral is transformed rewritten as $\text{d}^2 \mathbf{k}_\parallel/(2\pi)^2 = D(\varepsilon_\parallel) d\varepsilon_\parallel$, where $D(\varepsilon_\parallel)$ is the electronic density of states (DOS). In general, the $\mathbf{k}_\parallel$-nonconserving thermionic emission model in Eq. (\[flux\]) can be solved for any 2D materials using the appropriate DOS. Here we shall focus on solving Eq. (\[flux\]) for graphene. The full-band tight-binding model of graphene yields, $$\label{full_DOS}
D_{\text{FB}}(\varepsilon_\parallel) =
\begin{cases}
\frac{D_0}{\sqrt{F\left(\varepsilon_\parallel/t'\right)}} \mathbb{K} \left( \frac{4\varepsilon_\parallel/t'}{F\left(\varepsilon_\parallel/t'\right)} \right) & 0 <\varepsilon_\parallel < t' \\
\frac{D_0}{\sqrt{4\varepsilon_\parallel/t'}} \mathbb{K} \left( \frac{F\left(\varepsilon_\parallel/t'\right)}{ 4\varepsilon_\parallel/t' } \right) & t' <\varepsilon_\parallel < 3t'
\end{cases},$$ where $D_0 \equiv \frac{1}{A_c} \frac{g_{s,v}}{\pi^2} \frac{\varepsilon_\parallel}{t'^2}$, $t' = 2.8$ eV, $a = 0.142 $ nm, $A_c = 3a^2\sqrt{3}/2$, $F(x) \equiv (1+x)^2 - (x^2 - 1)^2/4$, and $\mathbb{K}(m) \equiv \int_0^1 dx \left[ (1-x^2)(1-mx^2) \right]^{-1/2}$ is the complete Elliptic integral of the first kind [@abramowitz].
The conduction and valence band touches at the $K$ and $K'$ points in the first Brillouin zone, commonly known as the *Dirac cone* \[Figs. 1(b) and (c)\]. At the vicinity of Dirac cone, $\varepsilon_\parallel$ can be expanded up to the first order in $|\mathbf{k}_\parallel|$ to yields a pseudo-relativistic relation, $\varepsilon_\parallel = \hbar v_F |\mathbf{k}_\parallel|$, where $v_F = 10^6$ m/s. The corresponding DOS is $$\label{DOS_Dirac}
D_{\text{D}}(\varepsilon_\parallel) = \frac{g_{s,v} \varepsilon_\parallel}{2\pi\hbar^2 v_F^2},$$ which exhibits a monotonous linear relation with $\varepsilon_\parallel$. This linear energy dispersion and the corresponding density of states have led to many unusual physical phenomena in graphene, such as Klein tunnelling effect [@klein], room-temperature quantum Hall effect [@hall], exceptionally large electron mobility [mobility]{}, gate-tunable optical and plasmonic responses [@plasmonics], strong optical nonlinearity [@NOR] and the emergence of new electromagnetic modes [@em]. It should be noted that the $D_{\text{D}}(\varepsilon_\parallel)$ is in good agreement with $D_{\text{FB}}(\varepsilon_\parallel)$ only for $\varepsilon_\parallel < 1$ eV. For $\varepsilon_\parallel > 1$ eV, the ever-increasing $D_{\text{D}}(\varepsilon_\parallel)$ severely overestimates the actual DOS calculated from the full-band model \[see Fig. 1(d)\]. The question of how the high-energy discrepancy between $D_{\text{D}}(\varepsilon_\parallel)$ and $D_{\text{FB}}(\varepsilon_\parallel)$ affect the thermionic emission of energetic electrons with $\varepsilon_\parallel > 1$ eV remains unanswered thus far. As demonstrated below, we found that the band nonlinearity effect at high electron energy directly leads to the large discrepancy between the Dirac cone approximation and the full-band thermionic emission model of energetic electrons with $\varepsilon_\parallel > 1$ eV.
By combining Eqs. (\[flux\]) and (\[DOS\_Dirac\]), the electrical and heat current under Dirac cone approximation can be analytically solved as,
\[dirac\] $$\begin{aligned}
\label{Jd}
\mathcal{J}_{ \text{D}} &=& \frac{ \lambda v_\perp}{L_\perp} \frac{g_{s,v}e\left(k_BT\right)^2}{2\pi\hbar^2v_F^2} \left( 1 + \frac{\Phi_{B}}{k_BT} \right) \xi_T(\Phi_B),
\end{aligned}$$ $$\begin{aligned}
\mathcal{Q}_{\text{D}} &=& \frac{ \lambda v_\perp }{L_\perp} \frac{g_{s,v} (k_BT)^3}{2\pi\hbar^2 v_F^2} \Lambda
\xi_T(\Phi_B),
\end{aligned}$$
where $\Lambda \equiv \left( \Phi_{B}/k_BT\right)^2 + \left(2 - \varepsilon_F / k_BT \right) \left(1 + \Phi_{B} / k_BT\right)$. On the other hand, using the DOS in Eq. (\[full\_DOS\]), the full-band equivalence of Eq. (\[dirac\]) is
\[J\] $$\label{phi_small}
\mathcal{J}_{ \text{FB}}= \lambda \frac{v_\perp}{L_\perp} \frac{g_{s,v}e}{\pi^2 t'^2} \mathcal{I}_0(\Phi_B),$$ $$\label{phi_big}
\mathcal{Q}_{ \text{FB}}= \lambda \frac{v_\perp}{L_\perp} \frac{g_{s,v}}{\pi^2 t'^2} \mathcal{I}_1(\Phi_B),$$
where $\mathcal{I}_\mu(\Phi_B)$ can be numerically solved from
$$\label{integral}
\mathcal{I}_\mu(\Phi_B) \equiv \Theta\left( t'-\Phi_B \right)\int_{\Phi_B}^{t'} \frac{ \varepsilon_\parallel (\varepsilon_\parallel - \varepsilon_F)^\mu d\varepsilon_\parallel }{\sqrt{F\left(\varepsilon_\parallel/t'\right)}} \mathbb{K} \left( \frac{4\varepsilon_\parallel/t'}{ F\left(\varepsilon_\parallel/t'\right)} \right) \xi_T(\varepsilon_\parallel) + \int_{t}^{3t'} \frac{ \varepsilon_\parallel (\varepsilon_\parallel - \varepsilon_F)^\mu d\varepsilon_\parallel }{\sqrt{4\varepsilon_\parallel/t'}} \mathbb{K} \left( \frac{F\left(\varepsilon_\parallel/t'\right)}{ 4\varepsilon_\parallel/t' } \right)\xi_T(\varepsilon_\parallel) .$$
The second term of Eq. (\[integral\]) is set to $3t'$ as $D_{\text{Gr}}(\varepsilon_\parallel)$ reaches the maximum of the conduction band at $\varepsilon_\parallel \leq3t'$. It should be noted that $\mathcal{J}_{\text{D}}$ in Eq. (\[Jd\]) has been rigorously studied in graphene-based Schottky contacts and a good agreement between theory and experimental data is demonstrated [@ma; @SL]. Such good agreement immediately suggests the need to extend Eq. (\[dirac\]) via the full-band model in Eq. (\[full\_DOS\]), so to obtain a *generalized* theoretical framework that encompasses both the low-energy thermionic emission in graphene-based Schottky contact and the high-energy counterpart in graphene/vacuum interface.
Results and Discussions
=======================
In Fig. 2, the analytical results of the Dirac cone approximation, $\mathcal{J}_{\text{D}}$ and $\mathcal{Q}_{\text{D}}$, and the numerical results of the full-band model, $\mathcal{J}_{\text{FB}}$ and $\mathcal{Q}_{\text{FB}}$, are plotted for two values of $\Phi_{B}$ which are typical for graphene-based Schottky contact [@SL; @tongay] and for graphene/vacuum thermionic emitter [@liang2]: (i) the low-barrier graphene/semiconductor Schottky diode regime ($\Phi_{B} = 0.5$ eV) around room temperature; and (ii) the high-barrier regime ($\Phi_{B} = 4.5$ eV) operating at $T>1000$ K. We have used $v_\perp = 3.7 \times 10^6$ m/s and $L_\perp = 0.335$ nm for graphene [@ang3], and scattering strength [@russell] of $\lambda = 10^{-4}$. In the low-barrier regime, both Dirac and full-band models produce nearly identical electrical \[Fig. 2(a)\] and heat \[Fig. 2(c)\] current densities in the typical room temperature operating regime for a Schottky diode. The Dirac model slightly underestimates the electrical and heat current densities by approximately 5% \[see insets of Figs. 2(a) and (c)\]. Conversely, in the high-barrier graphene/vacuum regime, the Dirac model severely overestimates the electrical and heat current densities by $\sim 60\%$ in the high-temperature range from 1000 K to 1800 K \[see Figs. 2(b) and (d)\]. This rather sizable discrepancy immediately reveals the incompatibility of Dirac cone approximation in the thermionic emission of high-energy electrons occurring at graphene/vacuum interface [@liang; @wang; @wang2] or in the presence of photon absorption [@massicotte; @ma]. This fallacy of Dirac cone approximation arises from the fact that graphene band structure becomes highly nonlinear at high electron energy $\varepsilon_\parallel>1$ eV at which the Dirac cone approximation fails to capture this band nonlinearity. We thus arrive at the following key finding: the simple analytical Dirac model in Eq. (\[dirac\]) is only well-suited for the modeling of thermionic emission mediated by electron with energy $< 1$ eV, such as graphene-based Schottky diode [@tongay; @SL], while the full-band model in Eq. (\[J\]) must be used for the thermionic emission of energetic electrons ($> 1$ eV) in high-barrier graphene interface and photon-enhanced thermionic devices so to produce a more accurate modeling results.
In Figs. 2(e) and (f), we further investigate the difference between the Dirac cone approximation and the full-band model in the intermediate regime between $\Phi_B = 0.5$ eV and $\Phi_B = 4.5$ eV. In general, the electrical and heat current density ratios, i.e. $\mathcal{J}_{\text{D}}/\mathcal{J}_{\text{FB}}$ \[Fig. 2(e)\] and $\mathcal{Q}_{\text{D}}/\mathcal{Q}_{\text{FB}}$ \[Fig. 2(f)\], are weakly dependent on temperature, and both ratios are approximately equal, i.e. $\mathcal{J}_{\text{D}}/\mathcal{J}_{\text{FB}} \approx \mathcal{Q}_{\text{D}}/\mathcal{Q}_{\text{FB}}$. For $\Phi_B$ lying approximately between $0.5$ eV and $3.5$ eV, we found that $\mathcal{J}_{\text{D}}/\mathcal{J}_{\text{FB}}\approx\mathcal{Q}_{\text{D}}/\mathcal{Q}_{\text{FB}} < 1$, which signifies the underestimation of the thermionic emission current densities by the Dirac cone approximation in comparison with the full-band model. Such underestimation peaks at $\Phi_B \approx 2.7$ eV with $\mathcal{J}_{\text{D}}/\mathcal{J}_{\text{FB}} \approx \mathcal{Q}_{\text{D}}/\mathcal{Q}_{\text{FB}} \approx 0.35$. Interestingly, there exists a critical barrier height, $\Phi_B^{(\text{c})} \approx 3.5$ eV, beyond which the current density ratios switch from $\mathcal{J}_{\text{D}}/\mathcal{J}_{\text{FB}} \approx \mathcal{Q}_{\text{D}}/\mathcal{Q}_{\text{FB}} < 1$ to $\mathcal{J}_{\text{D}}/\mathcal{J}_{\text{FB}} \approx \mathcal{Q}_{\text{D}}/\mathcal{Q}_{\text{FB}} > 1$, thus signifying the transition from underestimation to overestimation of the thermionic emission due to the Dirac cone approximation. This critical $\Phi_B^{(\text{c})}$ corresponds to an *accidental DOS averaging effect* at which the underestimation of the electron population available for thermionic emission due to the Dirac cone approximation at energy slightly above $\Phi_B$ is *exactly compensated* by the overestimation of that at higher energy. Beyond this critical barrier height, the overestimation of the electron population available for thermionic emission due to Dirac cone approximation becomes increasingly severe, which directly leads to the monotonously increasing trend in both $\mathcal{J}_{\text{D}}/\mathcal{J}_{\text{FB}}$ and $\mathcal{Q}_{\text{D}}/\mathcal{Q}_{\text{FB}}$ as $\Phi_B > \Phi_B^{(\text{c})}$.
We now investigate the current-temperature scaling of the full-band thermionic emission model. For 2D materials, the current-temperature scaling of thermionic emission in the out-of-plane direction follows the universal scaling law, $\ln{\left(\mathcal{J}/T\right)} \propto -1/T$ [@ang3], rather than the classic Richardson-Dushman scaling law of $\ln{\left(\mathcal{J}/T^2\right)} \propto -1/T$ for 3D materials [@p_zhang; @shinozaki]. In Fig. 4, the numerical value of $\ln{\left(\mathcal{J}_{\text{FD}}/T\right)}$ is plotted against $1/T$. The numerical results fit excellently into a straight line for both low-barrier \[Fig. 3(a)\] and high-temperature \[Fig. 3(b)\] regimes, thus confirming the expected universal scaling behavior in the full-band model of graphene.
To further investigate the impact of the full-band model on the modeling of graphene-based thermionic energy device, we calculate the thermionic cooling efficiency of a graphene thermionic cooler (see inset of Fig. 4(c) for a schematic drawing of the energy band diagram). Here, the thermionic cooler is composed of two graphene electrodes in parallel-plate configurations where a bias voltage, $V$, is used to modulate the net emitted electrical and heat current to achieve cooling [@liang2]. The hot (cold) graphene electrode temperature is denoted as $T_H$ ($T_C$). In Figs. 4(a) and (b), we see that the Dirac model overestimates both electrical and heat current densities immediately after the onset of cooling effect where $\Delta \mathcal{Q} >0$. The coefficient of performance (COP) of the thermionic cooler is calculated as $\eta(V, T_c, T_h) = \Delta Q/V\Delta J$, $\Delta Q = Q(T_c) - Q(T_H)$ and $\Delta \mathcal{J} \equiv \mathcal{J}(T_c) - \mathcal{J}(T_H)$ are the net heat and electrical current densities, respectively. In Fig. 4(c), the COP, normalized by Carnot efficiency $\eta_c\equiv T_C/(T_H - T_C)$, is plotted as a function of $V$ with $T_c = 1400$ K, $T_H = 1600$ K and $\varepsilon_F = 0.2$ eV, which reveals an especially large discrepancy between the two models near the onset of cooling at $V \approx 0.65$ V. Around the maximal efficiency point $\eta_{\text{max}}$, the full-band model yields $\eta_{\text{max}}$ = 0.75 at 0.7 V, compared to $\eta_{\text{max}}$ = 0.7 at 0.8 V as predicted by the less accurate Dirac model. These discrepancies can have a significant impact on the practical design of graphene-based thermionic cooler as it can affect multiple values, such as net transported heat current density and the optimal bias voltage, that are crucially important for the optimization of device figures of merit. Finally, we remark that the simplistic graphene thermionic cooler model reported in Fig. 4 is aimed to illustrate the discrepancy between the Dirac cone approximation and the full-band model. Realistic modeling of graphene-based thermionic energy device should include important effects, such as image potential lowering [@liang2], blackbody radiation [@zhang], space charge [@ang4; @p_zhang; @shinozaki; @ang_QSCL], electric-field-induced Fermi level shifting [@meric], secondary electron emission [@ueda; @ueda2], and carrier scattering effects [@meshkov; @russell; @vdovin; @liu_acs]. Such detailed modeling is beyond the scope of this work and shall form the subjects of future works. Importantly, the generalized 2D thermionic emission model of graphene developed here shall provide a theoretical foundation that may be directly useful for both the theoretical and experimental studies of the above-mentioned effects.
Finally, for completeness, we compare the model developed above with the widely-used classic Richardson-Dushman (RD) model [@RD], i.e. $\mathcal{J}_{\text{RD}} = \mathcal{A}_{\text{RD}} T^2 \exp\left( - \Phi_B / k_BT \right)$ where $\mathcal{A}_{\text{RD}} \approx 120$ Acm$^{-2}$K$^{-2}$ is the RD constant. Due to the strong dominance of the exponential term at the typical operating regime of $\Phi_B \gg k_BT$, the experimental data of graphene thermionic emission could still be fitted using the classic RD model [@zhu; @starodub], especially through the RD scaling law $\ln\left( \mathcal{J}_{\text{RD}} / T^2 \right) \propto -1/T$, without yielding significant errors in the extraction of $\Phi_B$ [@ang]. However, the *magnitude* of current density can deviate by several orders of magnitude when an inappropriate model is used [@ang]. Previous experiments have demonstrated that the extracted pre-exponential term can differ by nearly two orders of magnitude between the classic RD model and the 2D graphene thermionic emission model with Dirac cone approximation [@SL; @liang3]. Such large deviation can severely impact applications of the model in cases where the magnitude of the emission current density is important. We quantitatively compare the classic RD model and the generalized full-band model developed in this work by defining the ratio, $\mathcal{J}_{\text{RD}} / \mathcal{J}_{\text{FB}}$. For the two cases studied above, i.e. $\Phi_B = 0.5$ eV at $T = 300$ K and $\Phi_B = 4.5$ eV at $T = 1200$ K for high-temperature graphene/vacuum field emitter, we obtain $\mathcal{J}_{\text{RD}} / \mathcal{J}_{\text{FB}} \approx 1.28\times 10^3$ and $\mathcal{J}_{\text{RD}} / \mathcal{J}_{\text{FB}} \approx 11.6$, respectively. This exceedingly large ratios of $\mathcal{J}_{\text{RD}} / \mathcal{J}_{\text{FB}} \gg 1$ suggests that the RD model can severely overestimate the thermionic emission current densities in graphene. Thus, the generalized model proposed here, which rigorously captures the detailed high-energy features of the band structure and the two-dimensionality of graphene, shall be more advantageous than both the classic RD model and the simplified model based on Dirac cone approximation, especially for purposes where the magnitude of thermionic emission current density is important, such as the modelling, computational design and parametric optimization of graphene-based thermionic devices [@liang; @misra; @misra2; @zhang; @zhang2; @yang_z].
Conclusion
==========
In conclusion, we have revealed the fallacy of Dirac cone approximation in the modeling of high-barrier high-temperature thermionic emission in graphene. While the classic Richardson-Dushman and the Dirac approximation models [@SL; @liang2; @ang3] remain usable for the simple analysis of experimental data [@zhu; @starodub; @liang3], the full-band model developed here should be used in the case of high-energy thermionic electron emission in graphene. The proposed full-band model is especially critical for the computational design and modeling of graphene-based thermionic energy devices where the magnitude of the emission current densities is required to be determined accurately. As the 2D thermionic emission formalism developed above can be readily generalized to other 2D materials, our findings shall provide an important theoretical foundation for the understanding of thermionic emission physics in 2D materials.
This work is supported by A\*STAR AME IRG (A1783c0011) and AFOSR AOARD (FA2386-17-1-4020). Y. C. is supported by SUTD Undergraduate Research Opportunity Program (UROP).
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H. Yuan, D. C. Riley, Z.-X. Shen, P. A. Pianetta, N. A. Melosh, and R. T. Howe, ‘Back-gated graphene anode for more efficient thermionic energy converters’, Nano Energy 32, 67 (2017).
M. Massicotte, P. Schmidt, F. Vialla, K. Watanabe, T. Taniguchi, K. J. Tielrooij, and F. H. L. Koppens, ‘Photo-thermionic effect in vertical graphene heterostructures’, Nat. Commun. 7, 12174 (2016).
A. Shakouri, and John E. Bowers, ‘Heterostructure integrated thermionic coolers’, Appl. Phys. Lett. 71, 1234 (1997).
G. D. Mahan and L. M. Woods, ‘Multilayer thermionic refrigeration’, Phys. Rev. Lett. 80, 4016 (1998).
A. Shakouri, ‘Thermionic emission cooling in single barrier heterostructures’, Appl. Phys. Lett. 74, 88 (1999).
D. Vashaee and A. Shakouri, ‘Improved thermoelectric power factor in metal-based superlattices’, Phys. Rev. Lett. 92, 106103 (2004).
A. K. Geim, and I. V. Grigorieva, ‘Van der Waals heterostructures’, Nature 499, 419 (2013).
Y. Liu, N. O. Weiss, X. Duan, H.-C. Cheng, Y. Huang, and X. Duan, ‘Van der Waals heterostructures and devices’, Nat. Rev. Mater. 1, 16042 (2016).
K. S. Novoselov, A. Mishchenko, A. Carvalho, and A. H. Castro Neto, ‘2D materials and van der Waals heterostructures’, Science 353, aac9439 (2016).
S.-J. Liang, B. Liu, W. Hu, K. Zhou, and L. K. Ang, ‘Thermionic energy conversion based on graphene van der Waals heterostructures’, Sci. Rep. 7, 46211 (2017).
X. Wang, M. Zebarjadi, and K. Esfarjani, ‘First principles calculations of solid-state thermionic transport in layered van der Waals heterostructures’, Nanoscale 8, 14695 (2016).
X. Wang, M. Zebarjadi, and. K. Esfarjani, ‘High-Performance Solid-State Thermionic Energy Conversion Based on 2D van der Waals Heterostructures: A First-Principles Study’, Sci. Rep. 8, 9303 (2018).
Q. Ma et al, ‘Tuning ultrafast electron thermalization pathways in a van der Waals heterostructure’, Nat. Phys. 12, 455 (2016).
J. F. Rodriguez-Nieva, M. S. Dresselhaus, and J. C. W. Song, ‘Enhanced Thermionic-Dominated Photoresponse in Graphene Schottky Junctions’, Nano Lett. 16, 6036 (2016).
S. Misra, M. Upadhyay Kahaly, and S. K. Mishra, ‘Thermionic emission from monolayer graphene, sheath formation and its feasibility towards thermionic converters’, J. Appl. Phys. 121, 065102 (2017).
S. Misra, M. Upadhyay Kahaly, and S. K. Mishra, ‘Photo-assisted electron emission from illuminated monolayer graphene’, J. Appl. Phys. 121, 205110 (2017).
X. Zhang, Y. Pan, and J. Chen, ‘Parametric Optimum Design of a Graphene-Based Thermionic Energy Converter’, IEEE Trans. Electron. Dev. 64, 4594 (2017).
X. Zhang, Y. Zhang, Z. Ye, W. Li, T. Liao, and J. Chen, ‘Graphene-based thermionic solar cells’, IEEE Electron. Dev. Lett. 39, 383 (2018).
D. Sinha, and J.-U. Lee, ‘Ideal graphene/silicon Schottky junction Diodes’, Nano Lett. 14, 4660 (2014).
S.-J. Liang, and L. K. Ang, ‘Electron Thermionic Emission from Graphene and a Thermionic Energy Converter’, Phys. Rev. Appl. 3, 014002 (2015).
Y. S. Ang, and L. K. Ang, ‘Current-Temperature Scaling for a Schottky Interface with Nonparabolic Energy Dispersion’, Phys. Rev. Appl. 6, 034013 (2016).
Y. S. Ang, S.-J. Liang, and L. K. Ang, ‘Theoretical modeling of electron emission from graphene’, MRS Bullet. 42, 505 (2017).
M. Trushin, ‘Theory of thermionic emission from a two-dimensional conductor and its application to a graphene-semiconductor Schottky junction’, Appl. Phys. Lett. 112, 171109 (2018).
M. Trushin, ‘Theory of photoexcited and thermionic emission across a two-dimensional graphene-semiconductor Schottky junction’, Phys. Rev. B 97, 195447 (2018).
Y. S. Ang, H. Y. Yang, and L. K. Ang, ‘Universal scaling laws in Schottky heterostructures based on two-dimensional materials’, Phys. Rev. Lett. 121, 056802 (2018).
S. V. Meshkov, ‘Tunneling of electrons from a two-dimensional channel into the bulk ’, Zh. Eksp. Teor. Fiz. 91,2252 (1986).
K. J. Russell, F. Capasso, V. Narayanamurti, H. Lu, J. M. O. Zide, and A. C. Gossard, ‘Scattering-assisted tunneling: Energy dependence, magnetic field dependence, and use as an external probe of two-dimensional transport’, Phys. Rev. B 82, 115322 (2010).
V. Perebeinos, J. Tersoff, and Ph. Avouris, ‘Phonon-mediated interlayer conductance in twisted graphene bilayers’, Phys. Rev. Lett. 109, 236604 (2012).
E. E. Vdovin, A. Mishchenko, M. T. Greenaway, M. J. Zhu, D. Ghazaryan, A. Misra, Y. Cao, S. V. Morozov, O. Makarovsky, et al, ‘Phonon-assisted resonant tunneling of electrons in graphene–boron nitride transistors’, Phys. Rev. Lett. 116, 186603 (2016).
Y. Liu, Z. Gao, Y. Tan, and F. Chen, ‘Enhancement of out-of-plane charge transport in a vertically stacked two-dimensional heterostructure using point defects’, ACS Nano 12, 10529 (2018).
J. Voss, A. Vojvodic, S. H. Chou, R. T. Howe, and F. Abild-Pedersen, ‘Inherent enhancement of electronic emission from hexaboride heterostructure’, Phys. Rev. Applied 2, 024004 (2014).
Y.-J. Yu, Y. Zhao, S. Ryu, L. E. Brus, K. S. Kim, and P. Kim, ‘Tuning the graphene work function by electric field effect’, Nano Lett. 9, 3430 (2009).
H. Yuan et al, ‘Engineering ultra-low work function of graphene’, Nano Lett. 15, 6475 (2015).
S. Reich, J. Maultzsch, C. Thomsen, and P. Ordejon, ‘Tight-binding description of graphene’, Phys. Rev. B 66, 035412 (2002).
S. Tongay, M. Lemaitre, X. Miao, B. Gila, B. R. Appleton, and A. F. Hebard, ‘Rectification at graphene-semiconductor interfaces: Zero-gap semiconductor-based diodes’, Phys. Rev. X 2, 011002 (2012). Z. H. Ni et al, ‘Graphene thickness determination using reflection and contrast spectroscopy’, Nano Lett. 7, 2758 (2007).
D. Vashaee, and A. Shakouri, ‘Electronic and thermoelectric transport in semiconductor and metallic superlattices’, J. Appl. Phys. 95, 1233 (2004).
M. F. O’Dwyer, R. A. Lewis, C. Zhang, and T. E. Humphrey, ‘Efficiency in nanostructured thermionic and thermoelectric devices’, Phys. Rev. B 72, 205330 (2005). R. Kim, C. Jeong, and M. S. Lundstrom, ‘On momentum conservation and thermionic emission cooling’, J. Appl. Phys. 107, 054502 (2010).
M. Abramowitz, and I. A. Stegun, ‘Handbook of mathematical functions: with formulas, graphs, and mathematical tables’, Dover Publications (USA, 1965). M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, ‘Chiral tunnelling and the Klein paradox in graphene’, Nat. Phys. 2, 620 (2006). K. S. Novoselov et al, ‘Room-temperature quantum Hall effect n graphene’, Science 315, 1379 (2007). X. Du, I. Skachko, A. Barker, and E. Y. Andrei, ‘ Approaching ballistic transport in suspended graphene’, Nat. Nanotechnol. 3, 491 (2008). A. N. Grigorenko, M. Polini, and K. S. Novoselov, ‘Graphene Plasmonics’, Nat. Photon. 6, 749 (2012). E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A. Mikhailov, ‘Coherent nonlinear optical response of graphene’, Phys. Rev. Lett. 105, 097401 (2010). S. A. Mikhailov, and K. Ziegler, ‘New electromagnetic mode in graphene’, Phys. Rev. Lett. 99, 016803 (2007).
Y. S. Ang, M. Zubair, and L. K. Ang, ‘Relativistic space-charge-limited current for massive Dirac fermions’, Phys. Rev. B 95, 165409 (2017).
P. Zhang, A. Valfells, L. K. Ang, J. W. Luginsland, and Y. Y. Lau, ‘100 years of the physics of diodes’, Appl. Phys. Rev. 4, 011304 (2017).
T. Shinozaki, S. Hagiwara, N. Morioka, Y. Kimura, and K. Watanabe, ‘Real-time first-principles simulations of thermionic emission from N-doped diamond surfaces’, Appl. Phys. Expr. 11, 064301 (2018).
L. K. Ang, T. J. T. Kwan, Y. Y. Lau, ‘New scaling of Child-Langmuir law in the quantum regime’, Phys. Rev. Lett. 91, 208303 (2003).
I. Meric, M. Y. Han, A. F. Young, B. Ozyilmaz, P. Kim, and K. L. Shepard, ‘Current saturation in zero-bandgap, top-gated graphene field-effect transistors’, Nat. Nanotech. 3, 654 (2008).
Y. Ueda, Y. Suzuki, and K. Watanabe, ‘Time-dependent first-principles study of angle-resolved secondary electron emission from atomic sheets’, Phys. Rev. B 97, 075406 (2018).
Y. Ueda, Y. Suzuki, and K. Watanabe, ‘Secondary-electron emission from multi-layer graphene: time-dependent first-principles study’, Appl. Phys. Expr. 11 105101 (2018).
O. W. Richardson, ‘Some applications of the electron theory of matter’, Phil. Mag. 23, 594 (1912); S. Dushman, ‘Electron emission from metals as a function of temperature’, Phys. Rev. 21, 623 (1923).
F. Zhu et al, ‘Heating graphene to incandescene and the measurement of its work function by the thermionic emission method’, Nano Res. 7, 553 (2014).
E. Starodub, N. C. Bartelt, and K. F. McCarty, ‘Viable thermionic emission from graphene-covered metals’, Appl. Phys. Lett. 100, 181604 (2012).
S.-J. Liang, W. Hu, A. Di Bartolomeo, S. Adam, and L. K. Ang, ‘A modified Schottky model for graphene-semiconductor (3D/2D) contact: A combined theoretical and experimental study’, 2016 IEEE International Electron Devices Meeting (IEDM), San Francisco, CA, 14.4.1 (2016).
Z. Yang, W. Peng, W. Li, S. Su, G. Lin, and J. Chen, ‘Using a multi-layer graphene-based emitter to improve the performance of a concentrated solar thermionic converter’, J. Appl. Phys. 124, 154501 (2018).
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abstract: 'We study energy flow between two resistors coupled by an arbitrary linear and lossless electric circuit. We show that the fluctuations of energy transferred between the resistors are determined by random scattering of photons on an effective barrier with frequency dependent transmission probability $\tau(\omega)$. We express the latter in terms of the circuit parameters. Our results are valid in both quantum and classical regimes and for non-equilibrium electron distribution functions in the resistors. Our theory is in good agreement with recent experiment performed in the classical regime.'
address: 'Low Temperature Laboratory, Department of Applied Physics, Aalto University School of Science, P.O. Box 13500, 00076 AALTO, Finland'
author:
- 'D.S. Golubev and J.P. Pekola'
title: Statistics of heat exchange between two resistors
---
Introduction
============
The problem of energy exchange between two resistors has been first analyzed by Nyquist [@Nyquist] on the way towards his famous formula for the current noise of a resistor, $$\begin{aligned}
S_I = 4k_BT/R.
\label{Nyquist}\end{aligned}$$ Here $S_I$ is the spectral density of noise at low frequencies $|\omega|\ll k_BT/\hbar$, $k_B$ is the Boltzmann constant, $T$ is the temperature and $R$ is the resistance. Equation (\[Nyquist\]) has been confirmed by Johnson [@Johnson] and by numerous subsequent experiments. For a long time afterwards transport of heat in electric circuits has been considered well understood. Recently, however, it has attracted renewed attention due to advances both in theory and in technology. On the theoretical side, the discovery of the fluctuation theorem [@BK; @Evans; @Crooks; @Campisi] has triggered the interest in the statistics of heat transport. Statistics of effective electron temperature fluctuations in small metallic grains is also under discussion [@Nazarov1; @Nazarov2]. The experiments have recently advanced in two directions. First, quantum transport of heat between two resistors coupled by superconducting wires and separated by up to 50 $\mu$m distance has been demonstrated at sub-kelvin temperatures [@Meshke; @Timofeev]. Second, utilizing low noise amplifiers Ciliberto [*et al.*]{} have recently measured the full statistical distribution of heat transferred between two resistors kept at temperatures 88 K and 296 K respectively [@Ciliberto1; @Ciliberto2]. They have verified the validity of the fluctuation theorem and worked out a theoretical model based on Nyquist’s formula (\[Nyquist\]).
{width="8.5cm"}
Motivated by these developments, in this letter we propose a theory of full counting statistics of photon mediated heat exchange between two metallic resistors valid both at high and at low temperatures, where the classical formula for the noise (\[Nyquist\]) can no longer be used. We consider two resistors, $R_1$ and $R_2$ shunted by impedances $Z_1(\omega)$ and $Z_2(\omega)$, and coupled by a linear element (e.g. transmission line, capacitor, etc.) having the impedance $Z_0(\omega)$ (see Fig. 1a). The impedances $Z_j(\omega)$, ($j=0,1,2$) are purely reactive and do not generate noise. The average photonic heat current flowing from the resistor 1 to the resistor 2 reads $$\begin{aligned}
J_Q=\int_0^\infty \frac{d\omega}{2\pi}\,\omega\tau(\omega)\big[n_1(\omega)-n_2(\omega)\big],
\label{P}\end{aligned}$$ where $\tau(\omega)$ is the effective transmission, which we will specify later, $n_j(\omega)$ are photon distribution functions (here and below we put $k_B=\hbar =1$). Typically $\tau(\omega)$ drops at certain cutoff frequency $\omega_c$. Assuming that $n_1(\omega),n_2(\omega)$ have equilibrium Bose form with the temperatures $T_1$ and $T_2$, one finds that at high temperatures, $T_1,T_2\gtrsim \omega_c$ (Fig. 1b), $J_Q\approx \tau(0)\omega_c(T_1-T_2)$ in agreement with experimental findings of Refs. [@Ciliberto1; @Ciliberto2]. In this classical regime Nyquist’s formula (\[Nyquist\]) may be used to derive the heat current. In this letter we will be mostly interested in the opposite, quantum, limit $T_1,T_2\lesssim \omega_c$ (Fig. 1c), which is relevant for typical low temperature experiments [@Meshke; @Timofeev]. Indeed, the cutoff frequency may be estimated as $\omega_c\sim \min\{{1/R_jC_j, R_j/L_j}\}$, where $C_j\sim \epsilon\epsilon_0 l$ are stray capacitances, $L_j\sim \mu_0 l$ are inductances of the wires (Fig. 1d), $\epsilon_0$ and $\mu_0$ are vacuum permittivity and permeability, $\epsilon$ is the dielectric constant, and $l$ is the characteristic size of the sample. For the parameters of the low temperature experiments [@Meshke; @Timofeev], namely $T\sim 100$ mK, $R\sim 1$ k$\Omega$ and $l\sim 10$ $\mu$m, one finds $T/\omega_c\sim 10^{-3}\ll 1$. Thus the circuit is in the quantum regime. In contrast, for the experiments by Ciliberto [*et al*]{} [@Ciliberto1; @Ciliberto2] with $T\sim 100$ K, $R=10$ M$\Omega$ and $l\sim 1$ cm one finds $T/\omega_c\sim 10^{10}\gg 1$, which corresponds to strongly classical regime.
Model
=====
Our goal is to find the distribution of the energy $Q$ transferred from the resistor 1 to the resistor 2 during the time $t$, which we denote as $P(t,Q)$. It is more convenient to work with the cumulant generating function (CGF), $F(t,\lambda)$, which depends on the counting field $\lambda$ and defined as $$\begin{aligned}
e^{F(t,\lambda)}=\int dQ\, e^{i\lambda Q} P(t,Q).
\label{PQ}\end{aligned}$$ We describe the system by a Hamiltonian $$\begin{aligned}
\hat H=\hat H_0 + \hat H_{\rm em} + \hat H_{\rm int},\end{aligned}$$ where $\hat H_0=\sum_{k\sigma} \epsilon_k \hat a^\dagger_{k\sigma} \hat a_{k\sigma}$ is the Hamiltonian of non-interacting electrons moving in the combined potential of ion lattice and impurities, $\hat a_{k\sigma}$ is an annihilation operator of an electron in the eigenstate $|\psi_{k\sigma}\rangle$ ($\sigma$ is the spin index) and $\epsilon_k$ is the corresponding eigen-energy; $
\hat H_{\rm em}=\int d^3{\bm r}(\hat{\bm E}^2+\hat{\bm H}^2)/8\pi
$ is the Hamiltonian of electro-magnetic field; $\hat{\bm E}$ and $\hat{\bm H}$ are the operators of the electric and magnetic fields respectively; $
\hat H_{\rm int}=-\sum_{kn,\sigma} e\hat V_{kn}\hat a^\dagger_{k\sigma}\hat a_{n\sigma}
$ is the interaction Hamiltonian; and $\hat V_{kn}=\langle \psi_k|\hat V(\bm{r})|\psi_n\rangle$ are the matrix elements of the electric potential operator between two eigenfunctions of the non-interacting electron Hamiltonian $\hat H_0$. The Hamiltonian $\hat H_0$ describes the two resistors, the wires connecting them and the leads attached to them if they present.
An important point is the definition of the transferred energy $Q$. Here we have in mind the detection scheme based on normal metal - superconductor tunnel junctions attached to the resistors [@Meshke; @Timofeev]. Such a junction allows one to measure the effective temperature of a resistor or, more generally, the distribution function, $f(E,{\bm r})$, of electrons in it [@Pothier]. The latter can be converted into the total electron energy of the resistor $j$ ($j=1,2$) as ${\cal E}_j=2\int_{\Omega_j} d^3{\bm r}\int dE\, E\nu_j(E)f(E,{\bm r})$ (here $\Omega_j$ in the volume of the resistor $j$ and $\nu_j(E)$ is the density of states). Within this approach it is natural to define the transferred energy as the drop in the electronic energy of the resistor 1 during the time $t$, $Q=-{\cal E}_1(t)+{\cal E}_1(0)$. The corresponding quantum expression for the CGF reads [@Campisi]: $$\begin{aligned}
e^{F(t,\lambda)} = \,{\rm tr}\,\left[ e^{-i\lambda \hat H_{1}} e^{-i\hat Ht} e^{i\lambda \hat H_{1}} \hat\rho_0 e^{i\hat Ht} \right],
\label{F}\end{aligned}$$ where $\hat\rho_0$ is the initial density matrix and $\hat H_1$ is the free electron part of the Hamiltonian of the resistor 1.
The trace in Eq. (\[F\]) can be expressed as a path integral over the fluctuating potentials $V^{F},V^B,{\bm A}^{F},{\bm A}^{B}$ defined on the forward ($F$) and backward ($B$) branches of the Keldysh contour, and over the Grassman fields $a^F_{k\sigma},a^{F*}_{k\sigma},a_{k\sigma}^B,a^{B*}_{k\sigma}$ describing electrons. Performing the Gaussian integral over the latter, we get $$\begin{aligned}
e^{F} = \int{\cal D}V^{F,B}{\cal D}{\bm A}^{F,B} \,e^{iS^\lambda[V^{F,B},{\bm A}^{F,B}]},
\label{F2}\end{aligned}$$ where the effective action $iS^\lambda[V^{F,B},{\bm A}^{F,B}]$ is the sum of the electronic and electromagnetic contributions, $$\begin{aligned}
iS^\lambda &=& iS_{\rm el}^\lambda +iS_{\rm em},
\label{S}
\\
iS_{\rm el}^\lambda &=& 2\ln[\det (\check G^{-1}[V^F,V^B])],
\label{Sel}
\\
iS_{\rm em}&=& i\int_0^t dt'\int d^3{\bm r}\frac{E_F^2-E_B^2-H_F^2+H_B^2}{8\pi}.
\label{Sem}\end{aligned}$$ Here we introduced the inverse Keldysh Green function of electrons $\check G^{-1}_{kn} = \check G^{-1}_{0,kn} + \delta\check G^{-1}_{kn}$, where $$\begin{aligned}
\check G^{-1}_{0,kn} &=& \delta_{kn}\left(\begin{array}{cc}
i\partial_t-\epsilon_k & 0 \\
0 & -i\partial_t+\epsilon_k \end{array}\right),
\nonumber\\
\delta\check G^{-1}_{kn} &=& \left(\begin{array}{cc}
e V^F_{kn}e^{-i\lambda_k\epsilon_k+i\lambda_n\epsilon_n} & 0 \\
0 & - e V^B_{kn}\end{array}\right).\end{aligned}$$ At this stage we retain the information about occupation numbers of all energy levels keeping the dependence of the counting filed $\lambda_k$ on the level index $k$. Below we will only consider linear circuits free of highly resistive junctions or quantum dots in the Coulomb blockade regime. Then one can expand the action (\[Sel\]) to the second order in $V^F,V^B$, $$\begin{aligned}
iS_{\rm el}^\lambda \to 2\ln[\det\check G^{-1}_0]
+ {\rm tr}\left[2 \check G_0\delta\check G^{-1}-\left(\check G_0\delta\check G^{-1}\right)^2\right].
\label{expansion}\end{aligned}$$ This expression contains the Green function of non-interacting electrons, $\check G_0$. It is defined as $$\begin{aligned}
&&\check G_{0,kn}(t_1,t_2)=-i\delta_{kn}e^{-i\epsilon_k(t_1-t_2)}
\nonumber\\ &&\times\,
\left(\begin{array}{cc}
\theta_{12}(1-f_k)-\theta_{21}f_k & -f_k \\ 1-f_k & -\theta_{12}f_k +\theta_{21}(1-f_k)
\end{array}\right),
\nonumber\end{aligned}$$ where $\theta_{ij}=\theta(t_i-t_j)$ are Heaviside functions and $f_k=\langle\hat a^\dagger_{k\sigma}\hat a_{k\sigma}\rangle$ are the occupation numbers of the energy levels. The first term in the expansion (\[expansion\]) does not depend on $\lambda_k$ and may be omitted. The second term, ${\rm tr}\left[2 \check G_0\delta\check G^{-1}\right]$, is canceled by a similar contribution coming from positively charged ion background. Thus, only the last term of Eq. (\[expansion\]) matters. We transform it to the from $$\begin{aligned}
iS_{\rm el}^\lambda &=& e^2\int_0^t dt'dt'' \sum_{kn}\sum_{\alpha,\beta=\pm} e^{-i(\epsilon_k-\epsilon_n)(t'-t'')}
\nonumber\\ &&\times\,
\chi_{kn}^{\alpha\beta} \, V_{nk}^\beta(t')V_{kn}^\alpha(t'').
\label{action}\end{aligned}$$ Here we have introduced the potentials $V^+=(V^F+V^B)/2$ and $V^-=V^F-V^B$, as well as dimensionless combinations $\chi_{kn}^{\alpha\beta}$ containing electronic distribution functions $f_k$ and counting fields $\lambda_k$: $$\begin{aligned}
\chi_{kn}^{++} &=& f_k(1-f_n)\left(e^{-i\lambda_k\epsilon_k+i\lambda_n\epsilon_n}-1\right)
\nonumber\\ &&
+\, (1-f_k)f_n\left(e^{i\lambda_k\epsilon_k-i\lambda_n\epsilon_n}-1\right),
\nonumber\\
\chi_{kn}^{+-} &=& (\theta_{12}-\theta_{21})(f_k-f_n)+f_k(1-f_n)e^{-i\lambda_k\epsilon_k+i\lambda_n\epsilon_n}
\nonumber\\ &&
-\, (1-f_k)f_ne^{i\lambda_k\epsilon_k-i\lambda_n\epsilon_n},
\nonumber\\
\chi_{kn}^{-+} &=& 0,
\nonumber\\
\chi_{kn}^{--} &=& -f_k(1-f_n)\left(e^{-i\lambda_k\epsilon_k+i\lambda_n\epsilon_n}+1\right)/4
\nonumber\\ &&
-\,(1-f_k)f_n\left(e^{i\lambda_k\epsilon_k-i\lambda_n\epsilon_n}+1\right)/4.\end{aligned}$$
Next we perform disorder averaging of the matrix elements $V_{kn}^\alpha$ in Eq. (\[action\]) inside the metallic parts of the system ignoring weak localization and utilizing the rule of averaging for the product of electronic wave functions [@ABG] $$\begin{aligned}
\sum_{kn}\left\langle \psi_k^*({\bm r}_2)\psi_n({\bm r}_2)\psi_k({\bm r}_1)\psi_n^*({\bm r}_1) \delta(E_1-\epsilon_k)\delta(E_2-\epsilon_n)\right\rangle
\nonumber\\
=({\nu}/{\pi})\,{\rm Re}\,{\cal D}(E_1-E_2,{\bm r}_1,{\bm r}_2).\hspace{0.5cm}\end{aligned}$$ Here $\nu$ is the density of states and ${\cal D}(E,{\bm r}_1,{\bm r}_2)$ is the solution of the diffusion equation $
(-iE-D({\bm r})\nabla^2){\cal D} = \delta({\bm r}_1-{\bm r}_2),
$ where $D({\bm r})$ is the diffusion constant. In good metals with local current-field relation, ${\bm j}=\sigma({\bm r}){\bm E}$, where $\sigma({\bm r})=2e^2\nu_0({\bm r})D({\bm r})$ is the conductivity, one can approximate ${\rm Re}\,{\cal D}(E,{\bm r}_1,{\bm r}_2)\to -D({\bm r})\nabla^2/E^2$, and the action (\[action\]) acquires the form $$\begin{aligned}
iS_{\rm el}^\lambda &=& -\int_0^t dt'dt''\int d^3{\bm r}\sigma({\bm r})
\int\frac{d\omega}{2\pi} \frac{e^{-i\omega(t'-t'')}}{\omega}
\nonumber\\ &&\times\,
\sum_{\alpha,\beta=\pm} \eta^{\alpha\beta}_{\omega,{\bm r}} \nabla V^\beta(t',{\bm r})\nabla V^\alpha(t'',{\bm r}).
\label{action1}\end{aligned}$$ Here $$\begin{aligned}
\eta^{++}_{\omega,{\bm r}}&=& -n_{\omega,{\bm r}}(e^{i\lambda_{\bm r}\omega}-1)
-(n_{\omega,{\bm r}}+1)(e^{-i\lambda_{\bm r}\omega}-1),
\nonumber\\
\eta^{+-}_{\omega,{\bm r}}&=&1-
n_{\omega,{\bm r}}(e^{i\lambda_{\bm r}\omega}-1)
+(n_{\omega,{\bm r}}+1)(e^{-i\lambda_{\bm r}\omega}-1),
\nonumber\\
\eta^{-+}_{\omega,{\bm r}} &=& 0,
\nonumber\\
\eta^{--}_{\omega,{\bm r}}&=&\frac{n_{\omega,{\bm r}}(e^{i\lambda_{\bm r}\omega}+1)+(n_{\omega,{\bm r}}+1)(e^{-i\lambda_{\bm r}\omega}+1)}{4},
\label{eta}\end{aligned}$$ and $n_{\omega,{\bm r}}$ is the effective photon distribution function, $$\begin{aligned}
n_{\omega,{\bm r}}=\frac{1}{\omega}\int dE f\left(E+\frac{\omega}{2},{\bm r}\right)\left[1-f\left(E-\frac{\omega}{2},{\bm r}\right)\right].
\label{n_photon}\end{aligned}$$ It satisfies $n_{-\omega,{\bm r}}=-1-n_{\omega,{\bm r}}$ and in local equilibrium, i.e. for momentum isotropic electron distribution function of the form $f(E,{\bm r}) =1/(e^{E/T({\bm r})}+1)$, where $T(\bm{r})$ is the local electron temperature, it reduces to Bose function $1/(e^{\omega/T({\bm r})}-1)$. However, $n_{\omega,{\bm r}}$ may deviate from simple Bose form if the electron distribution function is driven out of equilibrium by, for example, bias voltage applied to a resistor [@Pothier]. In Eq. (\[action1\]) we have also assumed that the counting field $\lambda_{\bm r}$ is the same for all energy levels with wave functions localized in the vicinity of the point ${\bm r}$ and that it slowly varies in space at distances exceeding the spatial extension of these wave functions.
We are now in position to write down the action of two coupled resistors depicted in Fig. 1a. We put $\lambda({\bm r})=\lambda_j$, $\sigma({\bm r})=\sigma_j$ $(j=1,2)$ inside each resistor. Considering low frequency modes, we also put $\nabla V({\bm r})=V_j/L_j$, where $V_j$ is the instantaneous voltage drop across the $j-$th resistor, and $L_j$ is its length. We also define the resistances $R_j=L_j/\sigma_j{\cal A}_j$, where ${\cal A}_j$ are the cross-sectional areas of the resistors. With these approximations we get $$\begin{aligned}
&& iS_{\rm el}^\lambda = - \sum_{j=1,2} \int_0^t dt'dt''
\int\frac{d\omega}{2\pi} \frac{e^{-i\omega(t'-t'')}}{\omega R_j}
\nonumber\\ &&\times\,
\big[\eta_{j}^{++}(\omega) V_j^+(t') V_j^+(t'') + \eta_{j}^{+-}(\omega) V_j^-(t') V_j^+(t'')
\nonumber\\ &&
+\, \eta_{j}^{--}(\omega) V_j^-(t') V_j^-(t'') \big],
\label{action_res}\end{aligned}$$ where the functions $\eta_{j}^{\alpha\beta}(\omega)$ are given by Eqs. (\[eta\]) with photon distribution functions averaged over the volume of the resistors, $n_j(\omega)=\int_{\Omega_j} d^3{\bm r}n_{\omega,\bm{r}}/\Omega_j$, and with $\lambda_{\bm r}$ replaced by $\lambda_j$. The fields ${\bm E}$ and ${\bm H}$ in 3d space around the resistors and other circuit elements can be expressed via the voltages $V_j$ by solving linear Maxwell equations with proper boundary conditions. In this way one finds $$\begin{aligned}
{\bm E}_{F,B}(t,{\bm r})&=&\int_{-\infty}^t dt'\big[ {\bm e}_1(t-t',{\bm r})V_1^{F,B}(t')
\nonumber\\ &&
+\,{\bm e}_2(t-t',{\bm r})V_2^{F,B}(t')\big],
\label{E}
\\
{\bm H}_{F,B}(t,{\bm r})&=& \int_{-\infty}^t dt'\big[{\bm h}_1(t-t',{\bm r})V_1^{F,B}(t')
\nonumber\\ &&
+\,{\bm h}_2(t-t',{\bm r})V_2^{F,B}(t')\big],
\label{H}\end{aligned}$$ where ${\bm e}_j(t,{\bm r})$ and ${\bm h}_j(t,{\bm r})$ are the fundamental solutions for electric and magnetic fields, which depend on the sample geometry. The solutions (\[E\],\[H\]) should be substituted into the electro-magnetic part of the action (\[Sem\]). After the integration over coordinates, this action becomes quadratic in the potentials $V_j$. Moreover, since $E_F^2-E_B^2-H_F^2+H_B^2=2E^-E^+ - 2H^-H^+$ only the combinations $V_i^- V_j^+$ appear in it. The coefficients in front of these combinations are expressed in terms of the functions ${\bm e}_j(t,{\bm r})$, ${\bm h}_j(t,{\bm r})$ and determine the impedances $Z_j(\omega)$, shown in Fig. 1a, for a given sample. Finally the electro-magnetic part of the action acquires the form $$\begin{aligned}
&& iS_{\rm em} = -\int_0^t dt'dt''
\int\frac{d\omega}{2\pi}
\frac{e^{-i\omega(t'-t'')}}{\omega}
\nonumber\\&&\times\,
\bigg[\sum_{j=1,2}\frac{V_j^-(t') V_j^+(t'')}{Z_j(\omega)}+\frac{V_{12}^-(t')V_{12}^+(t'')}{Z_0(\omega)}\bigg],
\label{action_em}\end{aligned}$$ where $V_{12}^\pm=V_1^\pm-V_2^\pm$. According to our assumptions the impedances $Z_j(\omega)$ are purely imaginary, i.e. ${\rm Re}\,(1/Z_j)=0$. That is why the terms $\propto V^-(t')V^-(t'')$ do not appear in $iS_{\rm em}$. In contrast, such terms present in the action of the resistors (\[action\_res\]) even if one puts $\lambda_1=\lambda_2=0$. These terms are related to dissipation in the resistors and describe the current noise associated with it.
At long observation time, $t\gg 1/T_j,1/\omega_c$, the full action (\[S\]) acquires the form $$\begin{aligned}
iS^\lambda=\frac{i t}{2}\sum_n \vec V^T(-\omega_n) \frac{{\bm M}_\lambda(\omega_n)}{i\omega_n}\vec V(\omega_n),\end{aligned}$$ where $\omega_n=2\pi n/t$ are discrete frequencies, $\vec V^T(\omega_n)=(V^+_1(\omega_n),V^-_1(\omega_n),V^+_2(\omega_n),V^-_2(\omega_n))$ is the vector of Fourier transformed voltages, and
$$\begin{aligned}
{\bm M}_\lambda(\omega_n)=\left(
\begin{array}{cccc}
-\frac{2\eta^{++}_1(\omega_n)}{R_1} & \frac{1}{Z_1^*}+\frac{1}{Z^*_0} - \frac{\eta_1^{+-}(-\omega_n)}{R_1} & 0 & -\frac{1}{Z^*_0}\\
-\frac{1}{Z_1}-\frac{1}{Z_0} - \frac{\eta_1^{+-}(\omega_n)}{R_1} & -\frac{2\eta^{--}_1(\omega_n)}{R_1} & \frac{1}{Z_0} & 0 \\
0 & -\frac{1}{Z^*_0} & -\frac{2\eta^{++}_2(\omega_n)}{R_2} & \frac{1}{Z_2^*}+\frac{1}{Z^*_0} - \frac{\eta_2^{+-}(-\omega_n)}{R_2}\\
\frac{1}{Z_0} & 0 & -\frac{1}{Z_2}-\frac{1}{Z_0} - \frac{\eta_1^{+-}(\omega_n)}{R_2} & -\frac{2\eta^{--}_1(\omega_n)}{R_2}
\end{array}
\right).\end{aligned}$$
The Gaussian path integral (\[F2\]) over $\vec V_n$ is evaluated exactly. Utilizing the property ${\bm M}_\lambda(\omega)=-{\bm M}_\lambda^T(-\omega)$ in the long time limit we find CGF in the form $
F(t,\lambda) = -t\int_0^\infty \frac{d\omega}{2\pi}
\ln\big[{\det {\bm M}(\omega)}/{\det {\bm M}_{\lambda=0}(\omega)}\big].
$ Evaluating the determinants, and keeping in mind that $Z^*_j=-Z_j$ for reactive elements, we find $$\begin{aligned}
&& F(t,\lambda) = -t\int_0^\infty \frac{d\omega}{2\pi}
\ln\big[ 1-\tau(\omega)\big\{ n_1(\omega)[1+n_2(\omega)]
\nonumber\\ &&\times\,
\left(e^{i\lambda\omega}-1\right)
+[1+n_1(\omega)]n_2(\omega)\left(e^{-i\lambda\omega}-1\right)\big\}\big].\;\;\;\;
\label{FCS}\end{aligned}$$ Here $\lambda=\lambda_1-\lambda_2$, $$\begin{aligned}
\tau(\omega)=\frac{4}{R_1R_2\left|G_1+G_2+Z_0G_1G_2\right|^2},
\label{transmission1}\end{aligned}$$ is the effective transmission probability, and $G_j={1}/{R_j}+{1}/{Z_j(\omega)}$.
Equation (\[FCS\]) is the main result of our paper. It is the CGF of photons which are scattered by a barrier with the transparency $\tau(\omega)$ and carry the energy $\omega$ each. It is consistent with standard results of quantum optics [@Glauber] and closely resembles the CGF of scattered electrons [@Levitov1], which are fermions. In the context of photon scattering by a cavity similar expression has been derived by Beenakker [@Beenakker], and in the context of phonon heat conductance — by Saito and Dhar [@Saito1]. If both $n_1(\omega)$ and $n_2(\omega)$ have the equilibrium Bose form, CGF (\[FCS\]) acquires the property $F(\lambda)=F(-\lambda + i(T_1^{-1}-T_2^{-1}) )$, which translates into the fluctuation theorem $P(Q)=P(-Q)\exp[Q(T_1^{-1}-T_2^{-1})]$. We remind that the Eq. (\[FCS\]) has been derived assuming Gaussian fluctuations of currents and voltages in the electric circuit. That implies, in particular, that the resistors $R_1$ and $R_2$ are linear elements, which do not exhibit Coulomb blockade or other types of non-linearities. Besides that we have assumed that the real parts of the impedances $Z_j(\omega)$ are equal to zero and they correspond to purely reactive elements like inductors, capacitors or their arbitrary combinations.
![Distribution of energy transmitted between the resistors during time $t$ for different transmission probabilities $\tau(\omega)$. (a) $\tau(\omega)=$const, $T_1=300$ mK, $T_2=100$ mK, the observation time is $t=10$ ns. (b) $\tau(\omega)$ has the Lorentzian shape, $T_1=300$ mK, $T_2=100$ mK, $t=1$ ms, CGF is given by Eq. (\[Fnarrow\]). Discrete number of transferred photons $n$ is shown on the horizontal axis. []{data-label="Fig_distribution"}](graphs.eps){width="8cm"}
Results and discussion
======================
Let us now consider some limiting cases. First we assume that the transmission probability, $\tau$, is constant and the photon distribution functions have equilibrium Bose form. In this case the heat current acquires familiar form $
J_Q=-(i/t)dF/d\lambda|_{\lambda=0}=\pi\tau\left(T_1^2-T_2^2\right)/12.
$ The simplest example of such a system is given by two directly connected resistors (Fig. 1e), in which case $\tau=4R_1R_2/(R_1+R_2)^2$. In Fig. \[Fig\_distribution\]a we show the distribution $P(t,Q)$ for three different values of $\tau$. The distribution becomes Gaussian at sufficiently long observation time such that $J_Qt\gg T_1$. The low frequency noise of the heat current is given by the expression $$\begin{aligned}
&& S_Q=-\frac{1}{t}\frac{d^2F}{d\lambda^2}\bigg|_{\lambda=0}=
\left( \frac{\zeta(3)}{\pi}\tau(1-\tau) +\frac{\pi}{6}\tau^2 \right)(T_1^3+T_2^3)
\nonumber\\&&
+\, 2\tau(1-\tau)\int_0^\infty \frac{d\omega}{2\pi}\frac{\omega^2}{\left(e^{\omega/T_1}-1\right)\left(e^{\omega/T_2}-1\right)}.\end{aligned}$$
Another interesting limit is transmission within a narrow Lorentzian with $\tau(\omega)=\tau_{\max}\Gamma^2/[(\omega-\omega_0)^2+\Gamma^2]$ and $\Gamma\ll\omega_0,T_1,T_2$. In this case $$\begin{aligned}
F = -\Gamma t\left(\sqrt{1-\tau_{\max} f(\omega_0)}-1\right),
\label{Fnarrow}\end{aligned}$$ where $
f(\omega_0) = n_1(\omega_0)[1+n_2(\omega_0)]\left(e^{i\lambda\omega_0}-1\right)
+[1+n_1(\omega_0)]n_2(\omega_0)\left(e^{-i\lambda\omega_0}-1\right).
$ Since $F(\lambda)$ becomes a periodic function of $\lambda$ in this approximation, we get $
P(t,Q)=\sum_n p_n\delta(Q-n\omega_0)
$ with $
p_n=\frac{\omega_0}{2\pi}\int_{-\pi/\omega_0}^{\pi/\omega_0} d\lambda \,e^{in\lambda\omega_0}e^{F(\lambda)}
$ being the probability to transmit $n$ photons with one frequency $\omega_0$. The distributions $p_n$ for three different values of $\tau_{\max}$ are shown in Fig. \[Fig\_distribution\]b. Due to the suppression of the average heat current between the resistors the distributions $p_n$ significantly deviate from the Gaussian form even though the observation time is long, $t=1$ ms. It is obvious from Eq. (\[Fnarrow\]) that the distribution $p_n$ becomes Poissonian in the limit $T_1\gg T_2$ and $\tau_{\max}\ll 1$. At higher transparencies it deviates from the Poissonian form similarly to what has been predicted in Ref. , where the statistics of photons emitted by a coherent conductor has been studied and rectangular shape of the transmission line has been assumed. The average heat current and the noise corresponding to CGF (\[Fnarrow\]) are (here $n_j\equiv n_j(\omega_0)$) $$\begin{aligned}
&& J_Q=\frac{\tau_{\max}}{2}\Gamma\omega_0[n_1-n_2],\;\; S_Q = \frac{\tau_{\max}}{2}\Gamma\omega_0^2\big( n_1[1+n_2]
\nonumber\\ &&
+\,[1+n_1]n_2 + \tau_{\max} [n_1-n_2]^2/2\big).\end{aligned}$$
![ (a) Bias current $I$ is applied to the resistor 1 in order to drive it out of equilibrium. Two capacitors $C$, which shield the detector resistor 2 at low frequencies, are big enough to become fully transparent at frequencies $\omega\sim\max\{T_1,T_2,eV\}$, where $V=IR_1$. In this case the barrier transmission $\tau$ may be approximately treated as frequency independent constant. (b) Distribution of transmitted energy during the observation time $t=100/eV$ for three different values of $\tau$. $Q$ and $P(Q)$ are scaled with the characteristic photon energy $eV$. []{data-label="resistor-biased"}](resistors_biased.eps){width="8cm"}
Next we assume that leads are attached to the resistor 1 and bias current $I$ is applied to it (see Fig. \[resistor-biased\]a). The electron distribution function inside it acquires a non-equilibrium double step form [@Nagaev], $f(E,x)=(x/L_1)f(E-eV)+(1-x/L_1)f(E)$, where $V=IR_1$ is the voltage drop. We also assume that the temperatures of the resistor 2 and of the outer leads are much lower than $eV$. In this case one can put $n_2(\omega)=0$ and from the Eq. (\[n\_photon\]) we find $n_1(\omega)=(eV-\omega)\theta(eV-\omega)/6\omega$. Thus the CGF (\[FCS\]) takes the form $$\begin{aligned}
F = -t\int_0^{eV} \frac{d\omega}{2\pi} \ln\left[ 1-\frac{\tau}{6}\frac{eV-\omega}{\omega} \left(e^{i\lambda\omega}-1\right)\right].
\label{FeV}\end{aligned}$$ The corresponding distribution $P(Q)$ is shown in Fig. \[resistor-biased\]b. It is strongly asymmetric with $P(Q)=0$ for $Q<0$, i.e. over long intervals of time, $eVt\gtrsim 1$, the energy flows from the biased resistor to the unbiased one, but never in the opposite direction. A somewhat similar system, namely a biased resistor coupled to an open transmission line, has been earlier considered in Ref. , where the average value of the heat current and its noise have been derived. From CGF (\[FeV\]) we find these parameters in our model $$\begin{aligned}
J_Q=\frac{\tau(eV)^2}{24\pi},\;\;
S_Q=\left(1+\frac{\tau}{3}\right)\frac{\tau(eV)^3}{72\pi}.\end{aligned}$$
![ (a) Setup of the experiment [@Ciliberto1; @Ciliberto2]. The circuit parameters are: $R_1=R_2=10$ M$\Omega$, $C_0=100$ pF, $C_1=680$ pF, $C_2=420$ pF. The parameters defined in the text take the values $\alpha=2.134$, $\beta=0.0506$, and $t_0=6.29$ ms. (b) Distribution of energy transmitted during the time $t=0.1$ sec and for resistor temperatures $T_1=296$ K, $T_2=88$ K. Circles – experimental points[@Ciliberto1; @Ciliberto2]; blue line – Eq. (\[Pclassical\]); red dashed line – Gaussian approximation $P(t,Q)=\exp[-(Q-J_Qt)^2/2S_Qt]/\sqrt{2\pi S_Qt}$, where $J_Q$ and $S_Q$ are defined by Eqs. (\[SPcl\]).[]{data-label="Ciliberto_fig"}](ciliberto.eps){width="8cm"}
In the classical limit $T_j\gg\omega_c$ CGF (\[FCS\]) reduces to [@Saito2] $$\begin{aligned}
F=-t\int_0^\infty \frac{d\omega}{2\pi}\ln\left[1-\tau(\omega)\left( i\lambda\Delta T_{12}-\lambda^2T_1T_2 \right)\right],
\label{Fclassical}\end{aligned}$$ where $\Delta T_{12}=T_1-T_2$. It is interesting to compare this result with the experiment[@Ciliberto1; @Ciliberto2]. In that experiment capacitors have been used, which implies $Z_j(\omega)=1/(-i\omega C_j)$ (see Fig. \[Ciliberto\_fig\]a). Accordingly, $\tau(\omega)$ (\[transmission1\]) takes the form $$\begin{aligned}
\tau(\omega)=\frac{2\beta (\omega t_0)^2}{1+2(\alpha-1) (\omega t_0)^2 + (\omega t_0)^4},
\label{transmission2}\end{aligned}$$ with $
t_0=\sqrt{R_1R_2(C_1C_2+C_0C_1+C_0C_2)}
$, $$\begin{aligned}
\alpha=1+[R_1^2(C_0+C_1)^2+R_2^2(C_0+C_2)^2+2R_1R_2C_0^2]/2t_0^2,
\nonumber\end{aligned}$$ and $\beta={2C_0^2}/(C_1C_2+C_0C_1+C_0C_2)$. For this model one can exactly evaluate CGF (\[Fclassical\]), $$\begin{aligned}
F=\frac{t}{t_0}\left(\sqrt{\frac{\alpha}{2}}-\sqrt{\frac{\alpha}{2}-\frac{\beta(i\lambda\Delta T_{12}-\lambda^2T_1T_2)}{2}}\right),\end{aligned}$$ and the distribution of the transferred heat $P(t,Q)=\int\frac{d\lambda}{2\pi}e^{-i\lambda Q + F(t,\lambda)}$, which reads $$\begin{aligned}
P(t,Q) &=& \frac{t}{\pi}\sqrt{\frac{2a}{\beta T_1T_2t^2+2Q^2t_0^2}}\,
e^{\frac{t}{t_0}\sqrt{\frac{\alpha}{2}}+\frac{T_1-T_2}{2T_1T_2}Q}
\nonumber\\ &&
K_1\left(\sqrt{a\left(\frac{t^2}{t_0^2}+\frac{2Q^2}{\beta T_1T_2}\right)}\right).
\label{Pclassical}\end{aligned}$$ Here $K_1(x)$ is the modified Bessel function of the second kind, and $
a=\alpha/2+\beta(T_1-T_2)^2/8T_1T_2.
$ One should bear in mind that the expression (\[Pclassical\]) is valid in the long time limit $t\gtrsim t_0$. The average heat current from the resistor 1 to the resistor 2 and the corresponding noise in this model have the form $$\begin{aligned}
J_Q=\frac{\beta(T_1-T_2)}{2\sqrt{2\alpha} t_0},\;\;
S_Q = \frac{\beta T_1T_2}{\sqrt{2\alpha}\, t_0} + \frac{\beta^2(T_1-T_2)^2}{4\sqrt{2}\alpha^{3/2} t_0}.
\label{SPcl}\end{aligned}$$ We compare the distribution (\[Pclassical\]) with the experimental one [@Ciliberto1; @Ciliberto2] in Fig. \[Ciliberto\_fig\]b. The agreement between the two is quite good. In particular, one can see the deviations from Gaussian form at the tails of the distribution. The subtle point of the measurements [@Ciliberto1; @Ciliberto2] was the difference between the heat $Q_1$, i.e. the change of the energy of the resistor 1, and the work $W_1$, which also includes the change of the electrostatic energy of the capacitor $C_1$. We have verified that in the long time limit both $Q_1$ and $W_1$ should have the same distribution (\[Pclassical\]). On the qualitative level this can be understood from the relation $W_1=Q_1 + C_1[V_1^2(t)-V_1^2(0)]/2$. Indeed, the average value of the last term, i.e. of the change in the energy of the capacitor $C_1$ during the observation time $t$, equals to zero because $\langle V_1^2(t)\rangle$ is finite and does not grow in time. Since both $Q_1$ and $W_1$ grow in time linearly, one can put $W_1\approx Q_1$ at sufficiently long $t$ even without averaging. Experimentally, however, the work distribution has approached the long time limit form faster than the heat distribution. That is why in Fig. \[Ciliberto\_fig\]b we plot the experimental work distribution $P(W_1)$. Further analysis is required in order to understand the origin of this behavior.
We propose the distribution of heat in the low temperature quantum regime to be measured in the setup similar to the one used in the experiments \[,\]. Namely, one would monitor the temperature of the detector resistor 2 in real time with the time resolution of the order of $t_0\approx 12\hbar/\pi\tau k_BT_1$, that is the time interval during which an average energy $k_BT_1$ is transferred from the resistor 1 to the resistor 2. Assuming $T_1=100$ mK and $\tau=3\times 10^{-4}$ one finds $t_0\approx 1$ $\mu$s, which is within the reach of current technology[@Simone]. The expected magnitude of temperature fluctuations in the second resistor caused by fluctuations of heat flow may be estimated as $\delta T_2^2 \approx 3\tau T_2 t/\pi^3\hbar k_B\nu^2\Omega_2^2$, where $t$ is the observation time. For a resistor with the volume $\Omega_2=0.001$ $\mu$m$^3$ made of copper (density of states $\nu\approx 10^{29}$ J$^{-1}$ $\mu$m$^{-3}$) and for $T_2=50$ mK and $t=100 t_0$ one finds $\delta T_2\sim 15$ mK, which is measurable with currently available thermometers based on normal metal – superconductor tunnel junctions[@Simone; @Klara]. One can further optimize the system by, for example, designing the coupling circuit with narrow line transmission spectrum, or by using other types of temperature sensors like, e.g., recently proposed sensor based on an SNS Josephson junction [@Tero; @Jonas].
In summary, we have developed a theory of full counting statistics of heat exchange between two metallic resistors, which is valid both at high and at low temperatures, where the classical formula for the noise (\[Nyquist\]) can no longer be used. Fluctuations of the heat current in this system can be interpreted as scattering of photons by an effective potential barrier. In high temperature limit our results are in good agreement with recent experiment[@Ciliberto1; @Ciliberto2]. We acknowledge very useful discussions with S. Ciliberto, G. Lesovik, O. Saira and Y. Utsumi. We are grateful to S. Ciliberto for providing us with the experimental data. This work has been supported in part by the Academy of Finland (projects no. 272218 and 284594), and the European Union Seventh Framework Programme INFERNOS (FP7/2007- 2013) under grant agreement no. 308850.
[99]{} H. Nyquist, Phys. Rev. [**32**]{}, 110, (1928). J. Johnson, Phys. Rev. [**32**]{}, 97 (1928). G.N. Bochkov, and Y.E. Kuzovlev, Sov. Phys. JETP [**45**]{}, 125 (1977). D.J. Evans, E.G.D. Cohen, and G.P. Morriss, Phys. Rev. Lett. [**71**]{}, 2401 (1993). G.E. Crooks, Phys. Rev. E [**60**]{}, 2721 (1999). M. Campisi, P. Hänggi, and P. Talkner, Rev. Mod. Phys. [**83**]{}, 771 (2011). M. A. Laakso, T. T. Heikkilä, and Y. V. Nazarov, Phys. Rev. Lett. [**104**]{}, 196805 (2010). M. A. Laakso, T. T. Heikkilä, Yuli V. Nazarov, Phys. Rev. Lett. [**108**]{}, 067002 (2012). M. Meschke, W. Guichard, and J.P. Pekola, Nature [**444**]{}, 187 (2006). A.V. Timofeev, M. Helle, M. Meschke, M. Möttönen, and J.P. Pekola, Phys. Rev. Lett. 102, 200801 (2009). S. Ciliberto, A. Imparato, A. Naert, and M. Tanase, Phys. Rev. Lett. [**110**]{}, 180601 (2013). S. Ciliberto, A. Imparato, A. Naert and M. Tanase, J. Stat. Mech., P12014 (2013). H. Pothier, S. Gueron, Norman O. Birge, D. Esteve, and M. H. Devoret, Phys. Rev. Lett. [**79**]{}, 3490 (1997). I.L. Aleiner, P.W. Brouwer, L.I. Glazman, Phys. Rep. [**358**]{}, 309 (2002). L.S. Levitov, H. Lee, and G.B. Lesovik, J. Math. Phys. [**37**]{}, 4845 (1996). R.J. Glauber, Rev. Mod. Phys. [**78**]{}, 1267 (2006). C.W.J. Beenakker, Phys. Rev. Lett. [**81**]{}, 1829 (1998). K. Saito and A. Dhar, Phys. Rev. Lett. [**99**]{}, 180601 (2007). K.E. Nagaev, Phys. Rev. B [**52**]{}, 4740 (1995). T. Ojanen and T.T. Heikkilä, Phys. Rev. B [**76**]{}, 073414 (2007). C.W. J. Beenakker and H. Schomerus, Phys. Rev. Lett. [**86**]{}, 700 (2001). K. Saito and A. Dhar, Phys. Rev. E [**83**]{}, 041121 (2011). S. Gasparinetti, K. L. Viisanen, O.-P. Saira, T. Faivre, M. Arzeo, M. Meschke, and J. P. Pekola, Phys. Rev. Applied [**3**]{}, 014007 (2015). K. Viisanen, S. Suomela, S. Gasparinetti, O.P. Saira, J. Ankerhold, and J.P. Pekola, arXiv:1412.7322, accepted to New Journal of Physics (2015). J. Voutilainen, P. Virtanen, and T.T. Heikkilä, Phys. Rev. Lett. [**109**]{}, 067002 (2012). J. Govenius, R. E. Lake, K. Y. Tan, V. Pietilä, J. K. Julin, I. J. Maasilta, P. Virtanen, and M. Möttönen, Phys. Rev. B [**90**]{}, 064505 (2014).
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abstract: 'We present a study of the decay $B^-{\to}D^0_{(\CP)}K^-$ and its charge conjugate, where $D^0_{(\CP)}$ is reconstructed in -even, -odd, and non-flavor eigenstates, based on a sample of 232 million $\Upsilon(4S){\to}\BB$ decays collected with the detector at the PEP-II $e^+e^-$ storage ring. We measure the partial-rate charge asymmetries $A_{\CP\pm}$ and the ratios $R_{\CP\pm}$ of the $B{\to}D^0K$ decay branching fractions as measured in $\CP\pm$ and non-$\CP$ $D^0$ decays: $A_{\CPp} = 0.35\pm 0.13{\ensuremath{\mathrm{(stat)}}\xspace}\pm 0.04{\ensuremath{\mathrm{(syst)}}\xspace}$, $A_{\CPm} =-0.06\pm 0.13{\ensuremath{\mathrm{(stat)}}\xspace}\pm 0.04{\ensuremath{\mathrm{(syst)}}\xspace}$, $R_{\CP+} = 0.90\pm 0.12{\ensuremath{\mathrm{(stat)}}\xspace}\pm 0.04{\ensuremath{\mathrm{(syst)}}\xspace}$, $R_{\CP-} = 0.86\pm 0.10{\ensuremath{\mathrm{(stat)}}\xspace}\pm 0.05{\ensuremath{\mathrm{(syst)}}\xspace}$.'
title: ' [ **Measurements of the branching fractions and -asymmetries of decays** ]{} '
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SLAC-PUB-[11610]{}0.4cm
authors\_nov2005
A stringent test of the flavor and sector of the Standard Model can be obtained from the measurements, in $B$ meson decays, of the sides and angles of the unitarity triangle, which are related to the elements of the Cabibbo-Kobayashi-Maskawa matrix $V$. A theoretically clean measurement of the angle $\gamma=\arg(-V_{ud}V_{ub}^*/V_{cd}V_{cb}^*)$ can be obtained from the study of $B^-{\to}D^{(*)0}K^{(*)-}$ decays [@chargeconj] by exploiting the interference between the $b\ra c\bar{u}s$ and $b\ra u\bar{c}s$ decay amplitudes [@gronau1991; @others_btdk]. Among the proposed methods, the one originally suggested by Gronau, London, and Wyler (GLW) exploits the interference between $B^-{\to} D^0K^-$ and $B^-{\to} \Dzb K^-$ when the $D^0$ and $\Dzb$ mesons decay to the same eigenstate.
The results of the GLW analyses are usually expressed in terms of the ratios $R_{\CP\pm}$ of charge-averaged partial rates and of the partial-rate charge asymmetries $A_{\CP\pm}$, $$\begin{aligned}
&&R_{\CP\pm} = \frac{\Gamma(B^-{\to}\Dz_{\CP\pm}K^-) +
\Gamma(B^+{\to}\Dz_{\CP\pm}K^+)} {\left[\Gamma(B^-{\to}\Dz K^-)+\Gamma(B^+{\to}\Dzb K^+)\right]/2}\,,\ \ \ \ \\
&&A_{\CP\pm}=\frac{\Gamma(B^-{\to}\Dz_{\CP\pm}K^-)-\Gamma(B^+{\to}\Dz_{\CP\pm}K^+)}{\Gamma(B^-{\to}\Dz_{\CP\pm}K^-)+\Gamma(B^+{\to}\Dz_{\CP\pm}K^+)}\,.\ \ \ \\end{aligned}$$ Here, $D^0_{\CP\pm} = (\Dz \pm \Dzb)/\sqrt{2}$ are the eigenstates of the neutral $D$ meson system, and we have followed the notation used in [@gronau1998]. Neglecting $D^0{-}\Dzb$ mixing [@dmixing], the observables $R_{\CP\pm}$ and $A_{\CP\pm}$ are related to the angle $\gamma$, the magnitude $r$ of the ratio of the amplitudes for the processes $B^-{\to} \Dzb K^-$ and $B^-{\to} D^0 K^-$, and the relative strong phase $\delta$ between these two amplitudes, through the relations $R_{\CP\pm}=1+r^2\pm 2r\cos\delta\cos\gamma$ and $A_{\CP\pm}=\pm 2r\sin\delta\sin\gamma/R_{\CP\pm}$ [@gronau1991]. Theoretical expectations for $r$ are in the range $\approx
0.1-0.2$ [@gronau1991; @gronau2003], in agreement with the 90% C.L. upper limits on $r$ set by ($r<0.23$) and Belle ($r<0.18$) through the study of $B^-{\to} DK^-, D{\to} K^+\pi^-$ decays [@babar_ads].
In this paper we present the measurements of $R_{\CP\pm}$ and $A_{\CP\pm}$. The ratios $R_{\CP\pm}$ are computed using the relations $R_{\CP\pm}\simeq R_{\pm}/R$, where the quantities $R$ and $R_{\pm}$ are defined as: $$\begin{aligned}
&& R\ = \frac{\BR(B^-{\to}\Dz K^-)+\BR(B^+{\to}\Dzb
K^+)}{\BR(B^-{\to}\Dz\pi^-)+\BR(B^+{\to}\Dzb\pi^+)}\,,\label{eq:r}\\
&& R_{\pm}
=\frac{\BR(B^-{\to}\Dz_{\CP\pm}K^-)+\BR(B^+{\to}\Dz_{\CP\pm}K^+)}{\BR(B^-{\to}\Dz_{\CP\pm}\pi^-)+\BR(B^+{\to}\Dz_{\CP\pm}\pi^+)}\,.\ \ \label{eq:rpm}\end{aligned}$$ Several systematic uncertainties cancel out in the measurement of these double ratios. We also express the $\CP$-sensitive observables in terms of three independent quantities: $$\begin{aligned}
&&x_\pm=\frac{R_{\CP+}(1\mp A_{\CP+})-R_{\CP-}(1\mp A_{\CP-})}{4}\,,\\
&&r^2=x_\pm^2+y_\pm^2=\frac{R_{\CP+}+R_{\CP-}-2}{2}\,,\end{aligned}$$ where $x_\pm=r\cos(\delta\pm\gamma)$ and $y_\pm=r\sin(\delta\pm\gamma)$ are the same $\CP$ parameters as were measured by the Collaboration with $B^-{\to} DK^-, D{\to}
K^0_S\pi^-\pi^+$ decays [@babar_dalitz]. This choice allows the results of the two measurements to be expressed in a consistent manner.
The measurements use a sample of 232 million decays into $B\overline{B}$ pairs, corresponding to an integrated luminosity of 211 , collected with the detector at the 2 asymmetric-energy $B$ factory. Since the detector is described in detail elsewhere [@detector], only the components that are crucial to this analysis are summarized here. Charged-particle tracking is provided by a five-layer silicon vertex tracker (SVT) and a 40-layer drift chamber (DCH). For charged-particle identification, ionization energy loss in the DCH and SVT, and Cherenkov radiation detected in a ring-imaging device (DIRC) are used. Photons are identified by the electromagnetic calorimeter (EMC), consisting of 6580 thallium-doped CsI crystals. These systems are mounted inside a 1.5-T solenoidal superconducting magnet. We use the GEANT [@geant] software to simulate interactions of particles traversing the detector, taking into account the varying accelerator and detector conditions.
We reconstruct decays, where the prompt track $h^-$ is a kaon or a pion. $D^0$ candidates are reconstructed in the -even eigenstates $\pi^-\pi^+$ and $K^-K^+$ ($D^0_{\CP+}$), in the -odd eigenstates $K^0_S\pi^0$, $K^0_S\phi$ and $K^0_S\omega$ ($D^0_{\CP-}$), and in the non-, flavor eigenstate $K^-\pi^+$. $\phi$ candidates are reconstructed in the $K^-K^+$ channel and $\omega$ candidates in the $\pi^-\pi^+\pi^0$ channel. We optimize our event selection to minimize the statistical error on the $B^-{\to} D^0_{(\CP)}K^-$ signal yield, determined for each $\Dz$ decay channel using simulated signal and background events.
The prompt particle $h$ is required to have a momentum greater than 1.4 [${\mathrm{\,Ge\kern -0.1em V\!/}c}$]{}and the number of photons associated to its Cherenkov ring is required to be greater than four to improve the quality of the reconstruction. We reject a candidate track if its Cherenkov angle does not agree within four standard deviations ($\sigma$) with either the pion or kaon hypothesis, or if it is identified as an electron by the DCH and the EMC. Particle identification (PID) information from the drift chamber and, when available, from the DIRC, must be consistent with the kaon hypothesis for the $K$ meson candidate in $\Dz{\to} \Km\pip$, $\Dz{\to}
\Km\Kp$, and $\phi{\to}\Km\Kp$ decays and with the pion hypothesis for the $\pi^\pm$ meson candidates in $D^0{\to}\pi^-\pi^+$ and $\omega{\to}\pip\pim\piz$ decays.
Neutral pions are reconstructed by combining pairs of photon candidates with energy deposits larger than 70 [$\mathrm{\,Me\kern -0.1em V}$]{}that are not matched to charged tracks. The $\gamma\gamma$ invariant mass is required to be in the range 115–150 [${\mathrm{\,Me\kern -0.1em V\!/}c^2}$]{} and the total energy must be greater than 200 [$\mathrm{\,Me\kern -0.1em V}$]{}. When $\pi^0$’s are combined with other particles to form composite particles, the mass is constrained to the nominal mass [@PDG2004].
Neutral kaons are reconstructed from pairs of oppositely charged tracks with invariant mass within 7.8 [${\mathrm{\,Me\kern -0.1em V\!/}c^2}$]{}($\sim 3\sigma$) of the nominal $K^0$ mass. We also require that the ratio between the flight length in the plane transverse to the beam direction and its error be greater than 2. The $\phi$ mesons are reconstructed from two oppositely charged kaons with invariant mass in the range $1.008<M(K^+K^-)<1.032$ [${\mathrm{\,Ge\kern -0.1em V\!/}c^2}$]{}. We also require $|\cos\theta_{\rm hel}(\phi)|>0.4$, where $\theta_{\rm hel}(\phi)$ is the angle between the flight direction of one of the $\phi$ daughters and the $D^0$ flight direction, in the $\phi$ rest frame. The $\omega$ mesons are reconstructed from $\pi^+\pi^-\pi^0$ combinations with invariant mass in the range $0.763<M(\pi^+\pi^-\pi^0)<0.799$ [${\mathrm{\,Ge\kern -0.1em V\!/}c^2}$]{}. We define $\theta_N$ as the angle between the normal to the $\omega$ decay plane and the $D^0$ momentum in the $\omega$ rest frame, and $\theta_{\pi\pi}$ as the angle between the flight direction of one of the three pions in the $\omega$ rest frame and the flight direction of one of the other two pions in their center-of-mass (CM) frame. The quantities $\cos\theta_N$ and $\cos\theta_{\pi\pi}$ follow $\cos^2\theta_N$ and $\sin^2\theta_{\pi\pi}$ distributions for the signal and are almost flat for wrongly reconstructed or false $\omega$ candidates. We require the product $\cos^2\theta_N\sin^2\theta_{\pi\pi}>0.08 $. The invariant mass of a candidate, $M(D^0)$, must be within 2.5$\sigma$ of the mean fitted mass, with resolution $\sigma$ ranging from 4 to 20[${\mathrm{\,Me\kern -0.1em V\!/}c^2}$]{}depending on the $D^0$ decay mode. For , the invariant mass of the $(h^-\pi^+)$ system, where $\pi^+$ is the pion from $D^0$, and $h^-$ is the prompt track from $B^-$ taken with the kaon mass hypothesis, must be greater than $1.9\ {\ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c^2}}\xspace}$ to reject background from $B^-{\to}D^0\pi^-, D^0{\to}K^-\pi^+$ and $B^-{\to}K^{*0}\pi^-,
K^{*0}{\to}K^-\pi^+$ decays. When reconstructing $B$ mesons, for all $D^0$ decay channels the $D^0$ candidate invariant mass is constrained to the nominal $D^0$ mass [@PDG2004].
We reconstruct $B$ meson candidates by combining a candidate with a track $h$. For the $D^0{\to}K^-\pi^+$ mode, the charge of the track $h$ must match that of the kaon from the $D^0$ meson decay. We select $B$ meson candidates using the beam-energy-substituted mass $\mes = \sqrt{(E_0^{*2}/2 + \mathbf{p}_0\cdot\mathbf{p}_B)^2/E_0^2-p_B^2}$ and the energy difference $\Delta E=E^*_B-E_0^*/2$, where the subscripts $0$ and $B$ refer to the initial $e^+e^-$ system and the $B$ candidate respectively, and the asterisk denotes the CM ($\Upsilon(4S)$) frame. The distributions for signals are Gaussian functions centered at the $B$ mass with a resolution of $2.6 {\ensuremath{{\mathrm{\,Me\kern -0.1em V\!/}c^2}}\xspace}$, which do not depend on the decay mode or on the nature of the prompt track. In contrast, the distributions depend on the mass assigned to the prompt track and on the momentum resolution. We evaluate $\Delta E$ with the kaon mass hypothesis so that the distributions are Gaussian and centered near zero for events and shifted by approximately $50 {\ensuremath{\mathrm{\,Me\kern -0.1em V}}\xspace}$ for events. The resolution is about $17{\ensuremath{\mathrm{\,Me\kern -0.1em V}}\xspace}$ for all the decay modes. All $B$ candidates are selected with within 3$\sigma$ of the peak value and with in the range $-0.16<\Delta E<0.23{\ensuremath{\mathrm{\,Ge\kern -0.1em V}}\xspace}$.
To reduce background from continuum production of light quarks, we construct a linear Fisher discriminant [@fisher] based on the following quantities: (i) $L_0=\sum_i p_i$ and $L_2=\sum_i p_i\cos^2\theta_i$, evaluated in the CM frame, where $p_i$ is the momentum, and $\theta_i$ is the angle with respect to the thrust axis of the $B$ candidate of charged tracks and neutral clusters not used to reconstruct the $B$; (ii) $|\cos\theta_T|$, where $\theta_T$ is the angle between the thrust axes of the $B$ candidate and of the remaining tracks and clusters, evaluated in the CM frame; (iii) $|\cos\theta_B|$, where $\theta_B$ is the polar angle of the $B$ candidate in the CM frame.
For events with multiple candidates (1-7% of the selected events, depending on the decay mode), we choose that with the smallest $\chi^2$ formed from the differences of the measured and true masses of the candidate $B$, $D^0$, $\pi^0$ (only for $\Dz{\to}
K^0_S\pi^0, K^0_S\omega$), $\phi$ ($\Dz{\to} K^0_S\phi$), and $\omega$ ($\Dz{\to}K^0_S\omega$), compared to the appropriate reconstructed mass resolutions. The total reconstruction efficiencies, based on simulated signal events, are 39% ($K^-\pi^+$), 31% ($K^-K^+$), 30% ($\pi^-\pi^+$), 17% ($K^0_S\pi^0$), 20% ($K^0_S\phi$), and 7% ($K^0_S\omega$).
The main contributions to the background from events come from the processes $B{\to}D^{*}h$ ($h=\pi,K$), $B^-{\to}D^0\rho^-$, mis-reconstructed , and from charmless $B$ decays to the same final state as the signal: for instance, the process $B^-{\to}K^-K^+K^-$ is a background for $B^-{\to}D^0K^-, D^0{\to}K^-K^+$. These charmless backgrounds have similar and distribution as the $D^0 K^-$ signal and we call them “peaking backgrounds”.
For each decay mode an extended unbinned maximum likelihood fit to the selected data events determines yields for two signal channels, and , and four kinds of backgrounds: candidates selected either from continuum or from events, in which the prompt track is either a pion or a kaon. The fit uses as input and a particle identification probability for the prompt track based on the Cherenkov angle $\theta_C$, the momentum $p$, and the polar angle $\theta$ of the track.
The extended likelihood function $\cal L$ is defined as $${\cal L}= \exp\left(-\sum_{i=1}^6 n_i\right)\, \prod_{j=1}^N
\left[\sum_{i=1}^6 n_i {\cal P}_i\left(\vec{x}_j;
\vec{\alpha}_i\right) \right]\,,$$ where $N$ is the total number of observed events and $n_i$ is the yield of the $i^{th}$ event category. The six functions ${\cal P}_i(\vec{x}_j;\vec{\alpha}_i)$ are the probability density functions (PDFs) for the variables $\vec{x}_j$, given the set of parameters $\vec{\alpha}_i$. They are evaluated as a product $\mathcal{P}_i=\mathcal{P}_{1i}(\DeltaE)\times\mathcal{P}_{2i}({\theta_C})$.
The distribution for signal events is parametrized with a Gaussian function. The distribution for is parametrized with the same Gaussian function used for with an additional shift, computed event by event as a function of the prompt track momentum, arising from the wrong mass assignment to the prompt track. The offset and width of the Gaussian functions are determined from data together with the yields.
The distribution for the continuum background is parametrized with a linear function whose slope is determined from off-resonance data. The distribution for the non-peaking background is empirically parametrized with the sum of a Gaussian function and an exponential function when the prompt track is a pion, and with an exponential function when the prompt track is a kaon. The parameters are determined from simulated events. The distribution for the peaking charmless background is parametrized with the same Gaussian function used for the signal. The yield of the peaking background is estimated from the sidebands of the $D^0$ invariant mass distribution and fixed in the fit.
The parametrization of the particle identification PDF is performed by fitting with two Gaussian functions the background-subtracted distribution of the difference between the reconstructed and expected Cherenkov angles of kaon and pion samples. The parametrization is performed as a function of the momentum and polar angle of the track. Pions and kaons are selected from a pure $D^{*+}\to\Dz\pip$, $\Dz{\to}\Km\pip$ control sample.
The results of the fit are summarized in Table \[tab:fitresults\]. Figure \[fig:fit\_kaons\] shows the distributions of for the $K^-\pi^+$, and modes after enhancing the $B{\to}D^0K$ purity by requiring that the prompt track be consistent with the kaon hypothesis. The total PDF, normalized by the fitted signal and background yields, integrated over the Cherenkov angle variable and modified to take into account the tighter selection criteria, is overlaid in the figure.
$D^0$ mode $N(D\pi^+)$ $N(D\pi^-)$ $N(DK^+)$ $N(DK^-)$
--------------- ---------------- ----------------- ------------ ------------
$K^-\pi^+$ $8151\pm 95$ $7899 \pm 93$ $649\pm29$ $611\pm28$
$K^-K^+$ $705\pm 28$ $690 \pm 28$ $26\pm 9$ $70\pm 10$
$\pi^-\pi^+$ $256\pm 18$ $219 \pm 17$ $18\pm 7$ $17\pm 7$
$K^0_S\pi^0$ $707\pm 29$ $677 \pm 29$ $39\pm 9$ $42\pm 9$
$K^0_S\phi$ $176\pm 14$ $157 \pm 13$ $15\pm 5$ $13\pm 4$
$K^0_S\omega$ $235\pm 17$ $230 \pm 17$ $25\pm 7$ $14\pm 6$
: Yields from the maximum likelihood fit.[]{data-label="tab:fitresults"}
![Distributions of $\Delta E$ for events enhanced in $B^-{\to}D^0K^-$ signal. Top: $B^-{\to}D^0K^-, D^0{\to}K^-\pi^+$; middle: $B^-{\to}D^0_{\CPp}K^-$; bottom: $B^-{\to}D^0_{\CPm}K^-$. Solid curves represent projections of the maximum likelihood fit; dashed, dashed-dotted and dotted curves represent the $B{\to}D^0K$, $B{\to}D^0\pi$ and background contributions.[]{data-label="fig:fit_kaons"}](fit_noncp_thccut.eps "fig:"){width="7.5cm" height="3.6cm"} ![Distributions of $\Delta E$ for events enhanced in $B^-{\to}D^0K^-$ signal. Top: $B^-{\to}D^0K^-, D^0{\to}K^-\pi^+$; middle: $B^-{\to}D^0_{\CPp}K^-$; bottom: $B^-{\to}D^0_{\CPm}K^-$. Solid curves represent projections of the maximum likelihood fit; dashed, dashed-dotted and dotted curves represent the $B{\to}D^0K$, $B{\to}D^0\pi$ and background contributions.[]{data-label="fig:fit_kaons"}](fit_cpeven_thccut.eps "fig:"){width="7.5cm" height="3.6cm"} ![Distributions of $\Delta E$ for events enhanced in $B^-{\to}D^0K^-$ signal. Top: $B^-{\to}D^0K^-, D^0{\to}K^-\pi^+$; middle: $B^-{\to}D^0_{\CPp}K^-$; bottom: $B^-{\to}D^0_{\CPm}K^-$. Solid curves represent projections of the maximum likelihood fit; dashed, dashed-dotted and dotted curves represent the $B{\to}D^0K$, $B{\to}D^0\pi$ and background contributions.[]{data-label="fig:fit_kaons"}](fit_cpodd_thccut.eps "fig:"){width="7.5cm" height="3.6cm"}
The ratios $R_{\CP\pm}$ are computed for the five $\CP$ modes using the relations in Eqs. (\[eq:r\]) and (\[eq:rpm\]). A number of systematic uncertainties, such as the uncertainty from the tracking efficiency and the uncertainty on the $D^0$ decay branching fractions, cancel out in the measurement of the double ratio. The relations $R_{\CP\pm}=R_\pm/R$ hold under the assumption that the magnitude $r_\pi$ of the ratio of the amplitudes of the $B^-{\to}
\Dzb\pi^-$ and $B^-{\to} D^0\pi^-$ processes can be neglected [@gronau2003] ($r_\pi \sim r
\frac{\lambda^2}{1-\lambda^2} \lesssim 0.012$, where $\lambda \approx 0.22$ [@PDG2004] is the sine of the Cabibbo angle). This assumption is considered further when we discuss the systematic uncertainties. The quantities $R_{\pm}/R$ are computed from the ratios of the $B{\to}DK$ and $B{\to}D\pi$ yields in Table \[tab:fitresults\], scaled by correction factors taking into account small differences in the selection efficiency between $B{\to}DK$ and $B{\to}D\pi$. These correction factors are evaluated from simulated events and range between $0.982\pm 0.018$ and $1.020\pm 0.031$ depending on the decay mode. The results for the -even and -odd combinations are listed in Table \[tab:final\_ratio\].
The partial-rate charge asymmetries $A_{\CP\pm}$ are calculated from the measured yields of positive and negative $B{\to}DK$ decays in Table \[tab:fitresults\]. The results for the -even and -odd combinations are reported in Table \[tab:final\_ratio\].
$D^0$ mode $R_{\CP}$ $A_{\CP}$
------------ -------------------------------- ---------------------------------
$\CP+$ $0.90\pm 0.12 \pm 0.04$ $0.35\pm 0.13 \pm 0.04$
$\CP-$ $0.86\pm 0.10 \pm 0.05$ $-0.06\pm 0.13 \pm 0.04$
: Measured ratios $R_{\CP\pm}$ and $A_{\CP\pm}$ for $\CP$-even and $\CP$-odd $D$ decay modes. The first error is statistical, the second is systematic. $R_{\CP-}$ and $A_{\CP-}$ are corrected for the -even dilution described in the text.[]{data-label="tab:final_ratio"}
In the case of $D^0{\to}\KS \phi$, $\phi{\to}K^+K^-$, and $D^0{\to}\KS \omega$, $\omega{\to}\pi^+\pi^-\pi^0$, the values of $R_{\CP-}$ and $A_{\CP-}$ quoted in Table \[tab:final\_ratio\] are obtained after correcting the measured values to take into account the dilution from a -even background arising from $B^-{\to}\Dz h^-$, $\Dz{\to}K^0_S(K^-K^+)_{\textrm{non}-\phi}$ and $\Dz{\to}K^0_S(\pi^-\pi^+\pi^0)_{\textrm{non}-\omega}$ decays. For the $K^0_S \phi$ channel we exploit the investigation performed by of the $\Dz{\to}\KS K^+ K^-$ Dalitz plot [@babar_dztokskk] to estimate the level of the -even background ($0.160\pm0.006$ relative to the $\KS\phi$ signal) and the corresponding $R_{\CP-}$ and $A_{\CP-}$ dilution. For the $K^0_S \omega$ channel there is little information on this background. We estimate the amount of $\Dz{\to}
K^0_S(\pi^+\pi^-\pi^0)_{\textrm{non}-\omega}$ background ($0.25\pm 0.05$ relative to the $\KS \omega$ signal) from the $\cos\theta_N$ distribution of $B^-{\to}D^0\pi^-$, $\Dz{\to}\KS\pi^+\pi^-\pi^0$ candidates, and assume the -even content of this background to be $(50\pm 29)\%$.
Systematic uncertainties in the ratios $R_{\CP\pm}$ and in the asymmetries $A_{\CP\pm}$ are listed in Table \[tab:syst\]. They arise both from the uncertainties on the signal yields, extracted through the maximum likelihood fit, and from the assumptions used to compute $R_{\CP\pm}$ and $A_{\CP\pm}$. The correlations between the different sources of systematic errors, when non-negligible, are considered when combining the two -even or the three -odd modes.
------------------------------------------------- ------------------- ------------------- ------------------- -------------------
Source $\Delta R_{\CP+}$ $\Delta R_{\CP-}$ $\Delta A_{\CP+}$ $\Delta A_{\CP-}$
$(\%)$ $(\%)$ $(\%)$ $(\%)$
bkg. $\DeltaE$ PDF 1.3 1.1 1.1 0.4
PID PDF 0.1 0.1 0.2 0.2
peaking bkg. yields 3.0 4.2 2.6 2.2
opposite-bkg. - 1.3 - 1.0
detector charge asym. - - 2.7 2.7
$\varepsilon^{K/\pi}_{\pm}/\varepsilon^{K/\pi}$ 1.0 1.1 - -
$r_\pi$ 2.2 2.1 - -
[**Total**]{} 4.1 5.1 3.9 3.7
------------------------------------------------- ------------------- ------------------- ------------------- -------------------
: Systematic uncertainties on the observables $R_{\CP\pm}$ and $A_{\CP\pm}$ after combination of the two -even and the three -odd decay modes.[]{data-label="tab:syst"}
The uncertainties on the fitted signal yields are due to the imperfect knowledge of the $\DeltaE$ and PID PDFs and of the peaking background yields, and are evaluated by varying the parameters of the PDFs and the peaking background yields by $\pm1\sigma$ and taking the difference in the signal yields. The uncertainties in the branching fractions used in the simulation of the $B$ decays that contribute to the background are also taken into account. The yields of the and continuum backgrounds found in data are consistent with what is expected from the simulation. In the $K^0_S\phi$ and $K^0_S\omega$ channels we also take into account the uncertainties in the dilution factors due to the imperfect knowledge of the levels of the -even backgrounds from $B^-{\to}D^0K^-$, $\Dz{\to}K^0_S(K^-K^+)_{\textrm{non}-\phi}$ and $\Dz{\to}K^0_S(\pi^-\pi^+\pi^0)_{\textrm{non}-\omega}$ decays.
A possible bias in the measured $A_{\CP\pm}$ may come from an intrinsic detector charge asymmetry due to asymmetries in acceptance or tracking and particle identification efficiencies. An upper limit on this bias has been obtained from the measured asymmetries in the processes $B^-{\to}D^0h^-, D^0{\to}K^-\pi^+$ and $B^-{\to}D^0_{\CP\pm}\pi^-$, where violation is expected to be negligible. From the average asymmetry, ($-1.8\pm 0.9)\%$, we obtain the limit $\pm 2.7\%$ for the bias. This has been added in quadrature to the total systematic uncertainty on the asymmetry.
For the branching fraction ratios $R_{\CP\pm}$ two additional sources of uncertainty are the correction factors used to scale the yield ratios, and the assumption that $R_{\CP\pm} = R_{\pm}/R$. The scaling factor, estimated from simulated events, is a double ratio of efficiencies, $\varepsilon^{K/\pi}_{\pm}/\varepsilon^{K/\pi}$, where $\varepsilon^{K/\pi}_{(\pm)}$ denotes the ratio between the selection efficiencies of $B{\to}D^0_{(\CP\pm)}K$ and $B{\to}D^0_{(\CP\pm)}\pi$. In the double ratio the systematic uncertainties arising from possible discrepancies between data and simulation are negligible, and only the contribution from the limited statistics of the simulated samples remains. The assumption $R_{\CP\pm} = R_{\pm}/R$ introduces a relative uncertainty $\pm 2 r_\pi
\cos\delta_\pi \cos\gamma$ on $R_{\CP\pm}$, where $\delta_\pi$ is the relative strong phase between the amplitudes $A(B^-{\to}\Dzb \pi^-)$ and $A(B^-{\to}\Dz \pi^-)$. Since $|\cos\delta_\pi \cos\gamma|\le 1$ and $r_\pi \lesssim 0.012$, we assign a relative uncertainty $\pm 2.4\%$ to $R_{\CP\pm}$, which is completely anti-correlated between $R_{\CP+}$ and $R_{\CP-}$.
We quote the measurements in terms of $x_\pm$ and $r^2$, $$\begin{aligned}
&&x_+=-0.082\pm 0.053{\ensuremath{\mathrm{(stat)}}\xspace}\pm0.018{\ensuremath{\mathrm{(syst)}}\xspace}\,,\\
&&x_-=+0.102\pm 0.062{\ensuremath{\mathrm{(stat)}}\xspace}\pm0.022{\ensuremath{\mathrm{(syst)}}\xspace}\,,\\
&&r^2=-0.12\pm0.08{\ensuremath{\mathrm{(stat)}}\xspace}\pm 0.03{\ensuremath{\mathrm{(syst)}}\xspace}.\end{aligned}$$ The measured values of $x_\pm$ are consistent with those found, on a slightly smaller data sample, with the $B^-{\to}DK^-$, $D{\to}K^0_S\pi^-\pi^+$ decays, and the precision is comparable [@babar_dalitz]. The measured value of $r^2$ is consistent with the upper limits on $r$ from and Belle [@babar_ads].
In conclusion, we have reconstructed decays with $D^0$ mesons decaying to non-, -even and -odd eigenstates. We have improved the previous measurements of $R_{\CP\pm}$ and $A_{\CP\pm}$ [@babarbtdk; @bellebtdk], and we have also expressed the results in terms of the same $x_\pm$ parameters as were measured with $B^-{\to}DK^-$, $D{\to}K^0_S\pi^-\pi^+$ through a Dalitz plot analysis of the $D$ final state [@babar_dalitz], with a comparable precision. These measurements, combined with the existing measurements of the $B{\to}DK$ decays, will improve the knowledge of the angle $\gamma$ and the parameter $r$.
We are grateful for the excellent luminosity and machine conditions provided by our 2 colleagues, and for the substantial dedicated effort from the computing organizations that support . The collaborating institutions wish to thank SLAC for its support and kind hospitality. This work is supported by DOE and NSF (USA), NSERC (Canada), IHEP (China), CEA and CNRS-IN2P3 (France), BMBF and DFG (Germany), INFN (Italy), FOM (the Netherlands), NFR (Norway), MIST (Russia), and PPARC (United Kingdom). Individuals have received support from the A. P. Sloan Foundation, Research Corporation, and Alexander von Humboldt Foundation.
[99]{}
Reference to the charge-conjugate state is implied here and throughout the text unless otherwise stated.
M. Gronau and D. Wyler, [[Phys. Lett.]{} [**B265**]{}]{}, 172 (1991); M. Gronau and D. London, [[Phys. Lett.]{} [**B253**]{}]{}, 483 (1991).
D. Atwood, I. Dunietz and A. Soni, [[Phys. Rev. Lett.]{} [**78**]{}]{}, 3257 (1997); A. Giri, Y. Grossman, A. Soffer, J. Zupan, [[Phys. Rev.]{} [**D68**]{}]{}, 054018 (2003).
M. Gronau, [[Phys. Rev.]{} [**D58**]{}]{}, 037031 (1998).
Y. Grossman, A. Soffer, J. Zupan, [[Phys. Rev.]{} [**D72**]{}]{}, 031501 (2005).
M. Gronau, [[Phys. Lett.]{} [**B557**]{}]{}, 198 (2003).
Collaboration, B. Aubert [*et al.*]{}, [[Phys. Rev.]{} [**D72**]{}]{}, 032004 (2005); Belle Collaboration, K. Abe [*et al.*]{}, hep-ex/0508048.
Collaboration, B. Aubert [*et al.*]{}, [[Phys. Rev. Lett.]{} [**95**]{}]{}, 121802 (2005).
Collaboration, B. Aubert [*et al.*]{}, [[Nucl. Instr. and Methods]{} [**A479**]{}]{}, 1 (2002).
GEANT4 Collaboration, S. Agostinelli [*et al.*]{}, [[Nucl. Instr. and Methods]{} [**A506**]{}]{}, 250 (2003).
Particle Data Group, S. Eidelman [*et al.*]{}, [[Phys. Lett.]{} [**B592**]{}]{}, 1 (2004).
R. A. Fisher, Annals Eugen. [**7**]{}, 179 (1936).
Collaboration, B. Aubert [*et al.*]{}, [[Phys. Rev.]{} [**D72**]{}]{}, 052008 (2005).
Collaboration, B. Aubert [*et al.*]{}, [[Phys. Rev. Lett.]{} [**92**]{}]{}, 202002 (2004). Note that $R_{\CP}$ in this reference is defined as $R_+$ in this paper.
Belle Collaboration, S. K. Swain [*et al.*]{}, [[Phys. Rev.]{} [**D68**]{}]{}, 051101 (2003).
|
---
abstract: 'We study skew product lifts and overlap numbers for equilibrium measures $\mu_\psi$ of Hölder continuous potentials $\psi$ on such lifts. We find computable formulas and estimates for the overlap numbers in several concrete significant cases of systems with overlaps. In particular we obtain iterated systems which are asymptotically irrational-to-1 and absolutely continuous on their limit sets. Then we look into the general structure of the Rokhlin conditional measures of $\mu_\psi$ with respect to different fiber partitions associated to the lift $\Phi$, and find relations between them. Moreover we prove an estimate on the box dimension of a certain associated invariant measure $\nu_\psi$ on the limit set $\Lambda$ by using the overlap number of $\mu_\psi$.'
author:
- Eugen Mihailescu
title: Ergodic lifts and overlap numbers
---
**Mathematics Subject Classification 2000:** 37D20, 37D35, 37A35, 37C70.
**Keywords:** Equilibrium measures on lifts; overlap numbers of measures; conditional measures; conditional entropy.
Introduction.
=============
In this paper we study and give several formulas and applications of overlap numbers of equilibrium measures over iterated systems. These overlap numbers were introduced in [@MU-JSP2016], and represent asymptotic averages of the numbers of generic preimages in the limit set.
Consider thus a finite iterated function system (IFS) $\mathcal S = \{\phi_i, i \in I\}$, where the contractions $\phi_i$ are conformal and injective on an open set $U \subset \mathbb R^d$. Denote by $\Sigma_I^+$ the one-sided symbolic space $\{\omega = (\omega_1, \omega_2, \ldots), \omega_i \in I, i \ge 1\}$, with the canonical metric and topology, and with the shift map $\sigma: \Sigma_I^+ \to \Sigma_I^+$. Denote also by $[\omega_1 \ldots \omega_n]$ the cylinder $\{\eta \in \Sigma_I^+, \eta_1 = \omega_1, \ldots, \eta_n = \omega_n\}$. In general we denote by $\phi_{i_1\ldots i_p} := \phi_{i_1} \circ \phi_{i_2} \circ \ldots \circ \phi_{i_p}$ for any $p \ge 1, i_1, \ldots, i_p \in I$, and where $\phi_{i_1 i_2 \ldots}$ is the point given as the intersection of the descending sequence of sets $\phi_{i_1\ldots i_p}(U)$, when $p \to \infty$. We denote by $\Lambda$ the *limit set* of $\mathcal S$, and consequently $$\Lambda = \pi(\Sigma_I^+),$$ where $\pi: \Sigma_I^+ \to \Lambda, \ \pi(\omega) = \phi_{\omega_1 \omega_2 \ldots}, \ \omega \in \Sigma_I^+$, is the canonical projection to the limit set. We then consider the following skew product map, which we call *the lift of* $\mathcal S$, namely: $$\Phi: \Sigma_I^+ \times \Lambda \to \Sigma_I^+ \times \Lambda, \ \Phi(\omega, x) = (\sigma \omega, \phi_{\omega_1}(x)) \ \text{for} \ (\omega, x) \in \Sigma_I^+ \times \Lambda$$ In general, for any $n \ge 1$, the $n$-th iterate of $\Phi$ looks like: $$\Phi^n(\omega, x) = (\sigma^n(\omega), \phi_{\omega_n\ldots \omega_1}(x)), \ (\omega, x) \in \Sigma_I^+ \times \Lambda$$ It is clear that, due to the expansion of the shift map $\sigma$ and the contraction of $\phi_i, i \in I$, the skew product $\Phi$ has a hyperbolic character. Consider now a Hölder continuous potential $\psi: \Sigma_I^+ \times \Lambda \to \mathbb R$. Then there exists a unique *equilibrium measure* $\mu_\psi$ on $\Sigma_I^+ \times \Lambda$, i.e a measure which maximizes in the Variational Principle for Pressure (for eg [@KH], [@KS], [@Pe], [@Wa]). More precisely, if $P_\Phi: \mathcal C(\Sigma_I^+\times \Lambda, \mathbb R) \longrightarrow \mathbb R$ is the pressure functional for $\Phi$ on $\Sigma_I^+ \times \Lambda$, then $\mu_\psi$ is the unique $\Phi$-invariant probability for which $$P_\Phi(\psi) = h_\Phi(\mu_\psi) + \int_{\Sigma_I^+\times \Lambda}\psi \ d\mu_\psi = \sup \{h_\Phi(\mu) + \int_{\Sigma_I^+ \times \Lambda}\psi \ d\mu, \ \mu \ \Phi-\text{invariant probability on} \ \Sigma_I^+ \times \Lambda\},$$ where $h_\Phi(\mu)$ is the measure-theoretic entropy of $\mu$ with respect to $\Phi$. It is known that $\mu_\psi$ is an ergodic measure. Denote also by $$\nu_\psi:= \pi_{2*} \mu_\psi,$$ the projection of $\mu_\psi$ on the second coordinate. Then $\nu_\psi$ is a probability measure on the limit set $\Lambda$, and we want to study the metric properties of this measure. Notice that, in general, $\nu_\psi$ is not equal to the classical projection $ \pi_*(\pi_{1*}(\mu_\psi))$ of the measure $\mu_\psi$ from $\Sigma_I^+\times \Lambda$ to the limit set $\Lambda$.
For a $\Phi$-invariant probability measure $\mu$ on $\Sigma_I^+ \times \Lambda$, we define as usual its *Lyapunov exponent*, $$\chi(\mu) = \int_{\Sigma_I^+ \times \Lambda} - \log|\phi'_{\omega_1}(x)| \ d\mu(\omega, x)$$ Also let us notice that since the skew product $\Phi$ is contracting in the second coordinate, we have that the entropy of $\mu$ is actually equal to the entropy of its projection on the first coordinate, $$h_\Phi(\mu) = h_\sigma(\pi_{1*}\mu)$$ In [@Ru-folding], [@Ru-survey99] was introduced a notion of folding entropy of a measure, denoted in our case by $F_\Phi(\mu)$ which is defined as the conditional entropy $H_{\mu}(\epsilon|\Phi^{-1}\epsilon)$. If $\Phi^{-1}(\epsilon)$ is the measurable partition of $\Sigma_I^+ \times \Lambda$ with the fibers of $\Phi$, and if $\mu$ is an $\Phi$-invariant probability measure on $\Sigma_I^+\times \Lambda$, then we obtain a system of conditional measures of $\mu$ denoted by $(\mu_{(\omega, x)})_{(\omega, x) \in \Sigma_I^+\times \Lambda}$, where $\mu_{(\omega, x)}$ is a probability supported on the finite fiber $\Phi^{-1}(\omega, x)$.
Also let us recall that the *Jacobian* of an invariant measure introduced in [@Pa], as the local Radon-Nikodym derivative of the push-forward with respect to the measure. If $\mu$ is a $\Phi$-invariant measure on $\Sigma_I^+ \times \Lambda$, then we denote by $J_\Phi(\mu)$ its Jacobian; from definition, $J_\Phi(\mu) \ge 1$. In our case it follows that the folding entropy and Jacobian are related by
$$\label{foldingentropy}
F_\Phi(\mu_\psi) = \int_{\Sigma_I^+ \times \Lambda} \log J_\Phi(\mu)(\omega, x) \ d\mu(\omega, x)$$
In [@MU-JSP2016] we introduced a notion of *overlap number* $o(\mathcal S, \mu_\psi)$ for an equilibrium measure $\mu_\psi$ of a Hölder continuous potential on the lift $\Sigma_I^+ \times \Lambda$. This overlap number is an average asymptotic number of the $\mu_\psi$-generic preimages in $\Lambda$ (since the points in $\Lambda$ can be covered multiple times by the images $\phi_{i_1\ldots i_m}(\Lambda)$ if the system $\mathcal S$ has overlaps). More precisely, for an arbitrary number $\tau>0$ denote the set of $\mu_\psi$-generic preimages having the same iterates as $(\omega, x)$ by $$\Delta_n((\omega, x), \tau, \mu_\psi) = \{(\eta_1, \ldots, \eta_n) \in I^n, \exists y \in \Lambda, \phi_{\omega_n\ldots \omega_1}(x) = \phi_{\eta_n\ldots \eta_1}(y), \ |\frac{S_n\psi(\eta, y)}{n} - \int_{\Sigma_I^+ \times \Lambda} \psi\ d\mu_\psi| < \tau\},$$ where $(\omega, x) \in \Sigma_I^+ \times \Lambda$ and $S_n\psi(\eta, y)$ is the consecutive sum of $\psi$ with respect to the skew product $\Phi$. Denote by $$b_n((\omega, x), \tau, \mu_\psi):= Card\Delta_n((\omega, x), \tau, \mu_\psi)$$ Then, in [@MU-JSP2016] we proved that the following limit exists and defines the *overlap number* of $\mu_\psi$, $$o(\mathcal S, \mu_\psi) = \mathop{\lim}\limits_{\tau \to 0} \mathop{\lim}\limits_{n \to \infty} \frac 1n \int_{\Sigma_I^+ \times \Lambda} \log b_n((\omega, x), \tau, \mu_\psi) \ d\mu_\psi(\omega, x)$$ Moreover in the same paper [@MU-JSP2016] we proved a connection between the overlap number and the folding entropy of $\mu_\psi$, namely, $$\label{oS}
o(\mathcal S, \mu_\psi) = \exp(F_\Phi(\mu_\psi))$$ We found the following estimate for the Hausdorff dimension of the projection $\nu_\psi$ of $\mu_\psi$ on the second coordinate. Recall that $\pi_2: \Sigma_I^+ \times \Lambda \to \Lambda, \pi_2(\omega, x) = x$. The measure $\nu_\psi$ is not usually equal to the other projection $ \pi_*(\pi_{1*}(\mu_\psi))$ of $\mu_\psi$ from $\Sigma_I^+\times \Lambda$ onto the limit set $\Lambda$.
If $\mathcal S$ is a finite conformal iterated function system as above, and if $\psi: \Sigma_I^+ \times \Lambda \to \mathbb R$ is Hölder continuous with its equilibrium measure $\mu_\psi$, and if $\nu_\psi := \pi_{2*}(\mu_\psi)$, then $$HD(\nu_\psi) \le t(\mathcal S, \psi),$$ where $t(\mathcal S, \psi)$ is the unique zero of the pressure function $t \to P_\sigma(t\log|\phi_{\omega_1}(\pi(\sigma\omega))| - \log o(\mathcal S, \mu_\psi))$.
In the current paper, in Theorems \[garsia\] and \[pisot\], we compute/estimate overlap numbers in several concrete significant cases, namely for Bernoulli convolution systems associated to reciprocals of Garsia and Pisot numbers (see [@Ga], [@PU]). In particular we obtain examples of systems which asymptotically are *irrational*-to-1 on their limit sets. More precisely, for any $n \ge 1$, we obtain systems with overlaps which asymptotically are $\sqrt[n]{2^{n-1}}$ -to-1 and absolutely continuous on their limit sets.
Then in Proposition \[o\] and Corollaries \[levelp\] and \[partial\], we compute the overlap numbers for systems with eventual exact overlaps, and estimate the overlap numbers for systems with partial overlaps.
Next, for general systems and equilibrium measures, we compare in Theorem \[Hfold\] the conditional measures obtained from $\mu_\psi$ by taking certain special measurable partitions of the skew product into fibers. We apply this to find a formula for overlap numbers, by using families of conditional measures which may be easier to find in certain cases (for example for Bernoulli measures).
Then in Theorem \[mthm\] we find an upper bound for the lower box dimension of $\nu_\psi$, with the help of the overlap number of $\mu_\psi$, and using the Bounded Distortion Property for conformal systems of contractions. We give a *constructive* method to find a set of large $\nu_\psi$-measure in $\Lambda$ whose lower box dimension is bounded with the help of overlap numbers, namely is less than $$\frac{h_\sigma(\pi_{1*}(\mu_\psi)) - \log o(\mathcal S, \mu_\psi))}{|\chi(\mu_\psi)|}$$ This is done by careful estimates of the proportion of the measure of generic points within the measure of balls, by Jacobians of iterates, and employing the distribution of regular points from the Borel Density Lemma. In general for estimates of box dimensions one needs covers with balls of same radii (see [@Ba], [@Pe]), unlike for Hausdorff dimension; thus generic points are important. We then give in Corollaries \[o1\] and \[o2\] applications to estimates for box dimensions for projection measures, which work in particular for Bernoulli convolutions.
Formulas and estimates for overlap numbers. {#formulas}
===========================================
First we study the topological overlap number for various systems with overlaps. The *topological overlap number* of a conformal iterated system $\mathcal S = \{\phi_i, i \in I\}$ is defined (see [@MU-JSP2016]) as the overlap number of the measure of maximal entropy $\mu_{max}$ for $\Phi$ on $\Sigma_I^+ \times \Lambda$, and denoted by $o(\mathcal S)$. Thus $$o(\mathcal S) = o(\mathcal S, \mu_{max})$$
Consider now a probabilistic vector $\bf p = (p_1, \ldots, p_{|I|})$ and its associated Bernoulli measure $\mu_{\bf p}^+$ on $\Sigma_I^+$. Then the classical projection of $\mu_{\bf p}^+$ on the limit set $\Lambda$ of $\mathcal S$ is $\pi_*\mu_{\bf p}^+$. The Bernoulli measure $\mu_{\bf p}^+$ is the equilibrium measure with respect to $\sigma$ of the potential $g:\Sigma_I^+ \to \mathbb R, \ g(\omega) = \log p_{\omega_1}, \ \omega \in \Sigma_I^+$. Let $\psi:= g \circ \pi_1: \Sigma_I^+ \times \Lambda \to \mathbb R$, and $\mu_\psi$ be its equilibrium measures with respect to $\Phi$. Then we proved in [@MU-JSP2016] that for this choice of $\psi$, $\pi_{2*} \mu_\psi = \pi_*\pi_{1*}\mu_\psi$. On the other hand, notice that from estimates of equilibrium measures on Bowen balls, it follows that for some constant $r_0$, $\mu_\psi([\omega_1\ldots \omega_n]\times B(x, r_0)) \approx e^{S_n \psi(\omega, x) - nP_\Phi(\psi)},$ where the comparability constant is independent of $n, x, \omega$. Thus by summing up, $$\mu_\psi([\omega_1\ldots \omega_n]\times \Lambda) \approx e^{S_ng(\omega)-nP_\sigma(g)},$$ since $\Phi$ is contracting in the second coordinate and since $\psi$ depends only on $\omega$. Denote $\mu_{g\circ\pi_1}$ by $\mu_{\bf p}$, which can be considered a lift of $\mu_{\bf p}^+$ to $\Sigma_I^+ \times \Lambda$. So $\pi_{1*}\mu_{\bf p}$ satisfies the same estimates on cylinders as the Bernoulli measure $\mu_{\bf p}^+$, and thus from above, we obtain $\pi_{1*}\mu_{\bf p} = \mu_{\bf p}^+$. Therefore, $$\label{bernoulli}
\pi_{2*}\mu_{\bf p} = \pi_*\mu_{\bf p}^+$$ In particular, if $\mu_{max}^+$ denotes the measure of maximal entropy for the shift on $\Sigma_I^+$, i.e the Bernoulli measure associated to the probability vector $(\frac{1}{|I|}, \ldots, \frac{1}{|I|})$), we obtain $$\label{max}
\pi_{2*}\mu_{max} = \pi_*\mu_{max}^+$$ We showed in [@MU-JSP2016] that, if $\pi: \Sigma_I^+ \to \Lambda$ is the canonical projection to the limit set of $\mathcal S$ and if $$\beta_n(x):= Card\{(\eta_1, \ldots, \eta_n) \in I^n, x \in \phi_{\eta_1\ldots \eta_n}(\Lambda)\}, \ n \ge 1,$$ then the topological overlap number of $\mathcal S$ is given by the formula: $$\label{otop}
o(\mathcal S) = \exp\big(\mathop{\lim}\limits_{n \to \infty} \frac 1n \int_{\Sigma_I^+} \log \beta_n(\pi\omega) \ d\mu_{max}^+(\omega)\big)$$ **3.1.** Consider the IFS $\mathcal S_\lambda = \{\phi_{-1}, \phi_1\}$, where $\phi_{-1}(x) = \lambda x -1, \ \phi_1(x) = \lambda x+1$. When $\lambda \in (\frac 12, 1)$ this system has overlaps, and its limit set is the interval $I_\lambda = [-\frac{1}{1-\lambda}, \frac{1}{1-\lambda}]$. When there is no confusion about $\lambda$, this limit set will also be denoted by $\Lambda$. We consider then the measure of maximal entropy $\mu_{max}$ for $\Phi$ on $\Sigma_2^+\times \Lambda$.
**3.1a.** Let us look first at reciprocals of *Garsia numbers*. A number $\gamma$ is called a *Garsia number* if it is an algebraic integer in $(1, 2)$ whose minimal polynomial has constant coefficient $\pm 2$ and so that $\gamma$ and all of its conjugates have absolute value strictly greater than 1 (see [@Ga]). Examples of such minimal polynomials are $x^{n+p} - x^n -2$ for $n, p \ge 1$, with $\max\{p, n\} \ge 2$. For instance $2^{\frac 1n}, n \ge 2,$ are Garsia numbers. We prove the following:
\[garsia\] The topological overlap number $o(\mathcal S_\lambda)$ of the system $\mathcal S_\lambda$ for $\lambda \in (\frac 12, 1)$ with $\frac 1\lambda$ a Garsia number, is equal to $ 2\lambda$.
Recall that the limit set of $\mathcal S_\lambda$ is the interval $I_\lambda = [-\frac{1}{1-\lambda}, \frac{1}{1-\lambda}]$. From [@Ga] it follows that, if $\lambda$ is the reciprocal of a Garsia number, then all $2^n$ sums of type $\mathop{\sum}\limits_{0}^{n-1} \pm \lambda^k$ are distinct and at least $\frac{C}{2^n}$ apart, for some constant $C>0$. Let us order increasingly these $2^n$ numbers $\mathop{\sum}\limits_{0}^{n-1} \pm \lambda^k$, and denote them by $\zeta_1, \ldots, \zeta_{2^n}$. Hence these points $\zeta_i$ are distinct, and $$\label{di}
|\zeta_i - \zeta_j| \ge \frac{C}{2^n}, i \ne j$$ Now the numbers of type $\zeta_j + \mathop{\sum}\limits_{k \ge n} r_k\lambda^k$, where $\zeta_j = \mathop{\sum}\limits_{0 \le k \le n-1} \omega_k\lambda^k$ and $\omega_k \in \{-1, 1\}$, form the interval $I_j:= \pi([\omega_0, \ldots \omega_{n-1}]$. The length of $I_j$ is $C_1 \lambda^n$, for some fixed constant $C_1>0$. Since $\lambda > \frac 12$, it follows from (\[di\]) that any interval $I_j$ contains at least $C_2(2\lambda)^n$ points $\zeta_j$ and at most $C_3 (2\lambda)^n$ points $\zeta_j$, for some constants $C_3 > C_2 >0$. With the possible exception of an interval $J_1$ of length $C_4\lambda^n$ with left endpoint $-\frac{1}{1-\lambda}$ (i.e the left endpoint of $I_\lambda$), and an interval $J_2$ of same length with right endpoint $\frac{1}{1-\lambda}$ (i.e the right endpoint of $I_\lambda$), we see that any point $x$ belongs to at least $C_5(2\lambda)^n$ intervals $I_j$ and to at most $C_6(2\lambda)^n$ intervals $I_j$, where the constants $C_1, \ldots, C_6$ do not depend on $n$. Recall that $I_j = \pi([\omega_0, \ldots, \omega_{n-1}]$ for some $\omega_k \in \{-1, 1\}, 0 \le k \le n-1$, and that $\mu_{max}^+([\omega_0, \ldots, \omega_{n-1}]) = \frac{1}{2^n}$, where $\mu_{max}^+$ is the measure of maximal entropy on $\Sigma_2^+$. From above (\[otop\]) we know that, $$o(\mathcal S_\lambda) = \exp (\mathop{\lim}\limits_n \frac 1n\int_{\Sigma_2^+} \log \beta_n(\pi\omega) \ d\mu_{max}^+(\omega)),$$ where $\beta_n(x):= Card\{(\eta_0, \ldots, \eta_{n-1}) \in \{-1, 1\}^n, x \in \phi_{\eta_0\ldots\eta_{n-1}}(\Lambda_\lambda)\}$ for $x \in \Lambda_\lambda$ and $n \ge 1$. But from above, we see that for $x$ outside the intervals $J_1, J_2$ of length $C_4\lambda^n$ at the endpoints of $I_\lambda$, $$C_5 (2\lambda)^n \le \beta_n(x) \le C_6(2\lambda)^n$$ Thus from the last estimate on $\beta_n(x)$ on the complement of $J_1 \cup J_2$, and using that $\mu_{max}([\omega_0, \ldots, \omega_{n-1}]) = \frac{1}{2^n}$, we obtain that for some constant $C_7>0$ (independent of $n$), $$(2^n-C_7(2\lambda)^n) \cdot n\log(2\lambda) \frac{1}{2^n} \le \int_{\Sigma_2^+} \log \beta_n(\pi \omega) \ d\mu_{max}^+(\omega) \le 2^n\cdot n \log (2\lambda) \cdot \frac{1}{2^n} = n \log (2\lambda)$$ Therefore $o(\mathcal S_\lambda) = 2\lambda$, since from the last displayed inequalities it follows that, $$\mathop{\lim}\limits_{n \to \infty} \frac{1}{n} \int_{\Sigma_2^+} \log \beta_n(\pi\omega) \ d\mu_{max}^+(\omega) = \log (2\lambda)$$
Since for any $n \ge 1$, $2^{\frac 1n}$ is a Garsia number (see [@Ga]), we then obtain from Theorem \[garsia\] a system which asymptotically is $\sqrt[n] {2^{n-1}}$ -to-1; for these examples the projection $\pi_*\mu_{max}^+$ is absolutely continuous, and $\pi_*\mu_{max}^+ = \pi_{2*}\mu_{max}$ from (\[max\]), hence:
For the system $\mathcal S_\lambda$ with $\lambda = 2^{-\frac 1n}$, the topological overlap number is $o(\mathcal S_\lambda) = \sqrt[n] {2^{n-1}}$, and the measure $\pi_{2*}\mu_{\max}$ is absolutely continuous.
**3.1b.** The second example is of Bernoulli convolutions with $\lambda$ being the reciprocal of a *Pisot number*. A Pisot number is by definition an algebraic integer all of whose conjugates are strictly less than 1 in absolute value (for eg [@Ga], [@PU], etc). We prove the following.
\[pisot\] The topological overlap number of $\mathcal S_\lambda$ for $\lambda\in (\frac 12, 1)$ with $\frac 1\lambda$ a Pisot number, satisfies $$o(\mathcal S_\lambda) \ge 2\lambda >1$$
If $\frac 1\lambda$ is a Pisot number, the distance between any two different polynomial sums of type $P(\omega, \lambda, n)= \mathop{\sum}\limits_{i = 0}^{n-1}\omega_i\lambda^i$ for $\omega \in \Sigma_2^+ = \{-1, 1\}^\infty$, is at least $C\lambda^n$, for some constant $C>0$, which follows from the algebraic properties of $\frac 1\lambda$ (see [@Ga], [@PU]). Then the number $q(n)$ of all possible values of such polynomials $P(\omega, \lambda, n)$, when $n, \lambda$ are fixed, satisfies $$\label{qn}
q(n) \le C_1 \lambda^{-n},$$ for some constant $C_1$ independent of $n$. Since there are $2^n$ tuples $(\omega_0, \ldots, \omega_{n-1}) \in \{-1, 1\}^n$, but only at most $C_1\lambda^{-n}$ values for polynomials $P(\omega, \lambda, n)$, and since $\lambda > \frac 12$, there must be many equalities between such values. Denote by $V_n(\lambda)$ the set of values of polynomials $P(\omega, \lambda, n)$, $$\label{vn}
V_n(\lambda) = \{\alpha_1, \ldots, \alpha_{q(n)}\}, \ \text{where} \ \alpha_1 < \ldots < \alpha_{q(n)},$$ where $q(n)$ satisfies (\[qn\]). We know that $$\pi([\omega_0, \ldots, \omega_{n-1}]) = \{P(\omega, \lambda, n) + \mathop{\sum}\limits_{i = n}^\infty \omega_i \lambda^i, \ \omega_i \in \{-1, 1\}, i \ge n\},$$ so $\pi([\omega_0, \ldots, \omega_{n-1}])$ is an interval in $\Lambda_\lambda$ of length between $\lambda^n$ and $2\lambda^n$ (depending on its location). Denote by $$N_i:= Card\{(\omega_0, \ldots, \omega_{n-1}) \in \{-1, 1\}^n, P(\omega, \lambda, n) = \alpha_i\}, \ 1 \le i \le q(n)$$ From (\[qn\]) recall that $|\alpha_i - \alpha_j| \ge C_1\lambda^n$ if $i \ne j$. Since each value $\alpha_i$ is taken $N_i$ times by polynomials $P(\omega, \lambda, n)$, $1 \le i, j \le q(n)$, it follows that there exists a constant $C_2>0$ so that for all $n \ge 1$, $$\label{betan}
\beta_n(\pi\omega) \ge C_2 N_i, \ \text{whenever} \ P(\omega, \lambda, n) = \alpha_i, 1 \le i \le q(n)$$ But for the measure of maximal entropy $\mu_{max}^+$ on $\Sigma_2^+$ we have $\mu_{max}^+([\omega_0, \ldots, \omega_{n-1}]) = \frac {1}{2^n}$, so from (\[betan\]), $$\label{ni}
\int_{\Sigma_2^+} \log \beta_n(\pi \omega) \ d\mu_{max}^+(\omega) \ge \mathop{\sum}\limits_{j=1}^{q(n)} (\log C_2 N_j) \cdot \frac{N_j}{2^n} = \log 2^n + \mathop{\sum}\limits_{j=1}^{q(n)} \frac{N_j}{2^n}\log\frac{N_j}{2^n} + \log C_2$$ However in general for any probability vector $(p_1, \ldots, p_{m})$, one has the upper bound (for eg [@Wa]), $$-\mathop{\sum}\limits_{i = 1 }^{m} p_i\log p_i \le \log m$$ From (\[vn\]), we know $N_1 + \ldots N_{q(n)} = 2^n$, so we can take the probability vector $(\frac{N_1}{2^n}, \ldots, \frac{N_{q(n)}}{2^n})$, and from (\[ni\]) it follows that: $$\frac 1n \log \int_{\Sigma_2^+} \log \beta_n(\pi \omega) \ d\mu_{max}^+(\omega) \ge \log 2 - \frac{\log C_1\lambda^{-n}}{n} + \frac{\log C_2}{n}$$ This implies then from (\[otop\]) that $o(\mathcal S_\lambda) \ge 2\lambda$, hence $o(\mathcal S_\lambda) > 1$ since $\lambda > \frac 12$.
**3.2.** We now look at examples with eventual exact or at least substantial overlaps, in which case the overlap number will be estimated, or even computed exactly. We look at the case when there are *exact overlaps*, i.e. when we have $$\phi_{i_1\ldots i_p}(\Lambda) = \phi_{j_1\ldots j_p}(\Lambda),$$ for certain maximal tuples $(i_1, \ldots, i_p), (j_1, \ldots, j_p)$. Exact overlaps may appear after certain number of iterates, however for simplicity we look firstly at the case when $p=1$; the generalization is straightforward.
So, consider the system $\mathcal S = \{\phi_i, 1 \le i \le m\}$ of conformal injective contractions, and assume we have the blocks $$\label{blocks}
\phi_1 = \ldots = \phi_{k_1}, \ \phi_{k_1+1} = \ldots = \phi_{k_2}, \ldots, \ \phi_{k_p} = \phi_m,$$ where there are no overlaps between the different blocks, i.e the system $\{\phi_{k_i}, 1 \le i \le p\}$ satisfies the Open Set Condition.
Let $\mu_{max}^+$ be the measure of maximal entropy on $\Sigma_m^+$, and denote the measure of maximal entropy for $\Phi$ on $\Sigma_m^+ \times \Lambda$ by $\mu_{max}$; then the overlap number $o(\mathcal S):= o(\mathcal S,\mu_{max})$ is in fact the topological overlap number which takes in consideration all preimages, and we proved in [@MU-JSP2016] that $$\label{8}
o(\mathcal S) = \exp\big(\mathop{\lim}\limits_{n \to \infty} \frac 1n \int_{\Sigma_m^+} \log \beta_n(\pi \omega) \ d\mu_{max}^+(\omega)\big),$$ where $\beta_n(x):= \text{Card}\{(\eta_1, \ldots, \eta_n) \in I^n, \ x\in \phi_{\eta_1\ldots \eta_n}(\Lambda)\}$. In this case, if $x \in \phi_{j_1\ldots j_n}(\Lambda)$ and if $k_{i_\ell -1}+1 \le j_\ell \le k_{i_\ell}$, then for $x = \pi\omega$ and $\omega= (j_1j_2 \ldots)$, we have: $$\label{8}
\beta_n(x) = (k_{i_1} - k_{i_1 -1}) \cdot \ldots \cdot (k_{i_n} - k_{i_n-1}),$$ where if $i_\ell = 1$, then the factor $(k_{i_\ell}-k_{i_\ell -1})$ is replaced by $k_1$. Let us take the function $\Psi: \Sigma_m^+ \to \mathbb R$, $\Psi(\omega):= \log k_1$ for $1 \le \omega_1 \le k_1$, and $\Psi(\omega):= \log(k_i - k_{i-1})$ for $k_{i-1}+1 \le \omega_1 \le k_i$. If $\omega, \eta$ are close enough in $\Sigma_m^+$, then $\omega_1 = \eta_1$, hence $\Psi$ is Hölder continuous on $\Sigma_m^+$. Notice that, if $\omega \in [j_1\ldots j_n]$ and $k_{i_s-1}+1 \le j_s \le k_{i_s}$ if $i_s > 1$, or $1 \le j_1 \le k_1$ if $i_s = 1$, then $$\Psi(\omega) = \log(k_{i_1} - k_{i_1-1}), \ \Psi(\sigma \omega) = \log(k_{i_2} - k_{i_2-1}), \ldots$$ However from above,$$\int_{\Sigma_m^+} \log \beta_n(\pi \omega) \ d\mu_{max}^+(\omega) = \mathop{\sum}\limits_{s = 1, \ldots n} \mathop{\sum}\limits_{k_{i_s-1} +1 \le j_s \le k_{i_s}} \int_{[j_1 \ldots j_n]} \log(k_{i_1} - k_{i_1-1}) + \ldots \log (k_{i_n}-k_{i_n-1}) \ d\mu_{max}^+(\omega)$$ Thus, if $S_n\Psi$ denotes the consecutive sum of $\Psi$ with respect to $\sigma$, we obtain $$\label{9}
\int_{\Sigma_m^+} \log \beta_n(\pi\omega) \ d\mu_{max}^+(\omega) = \int_{\Sigma_m^+} S_n\Psi(\omega) \ d\mu_{max}^+(\omega)$$ Hence from (\[9\]), by Birkhoff Egodic Theorem for the measure of maximal entropy $\mu_{max}^+$ on $\Sigma_m^+$, $$\frac{1}{n} \int\log\beta_n(\pi\omega) \ d\mu_{max}^+(\omega) = \frac 1n\int_{\Sigma_m^+} S_n\Psi(\omega) \ d\mu_{max}^+(\omega) \mathop{\longrightarrow}\limits_{n \to \infty} \int_{\Sigma_m^+} \Psi(\omega) \ d\mu_{max}^+(\omega)$$ We have thus proved the following
\[o\] In the above setting from (\[blocks\]), the topological overlap of the system $\mathcal S$ is given by $$o(\mathcal S) = o(\mathcal S, \mu_{max}) = \exp\big(\frac{k_1\log k_1 + (k_2 - k_1) \log(k_2 - k_1) + \ldots + (k_p-k_{p-1}) \log(k_p -k_{p-1})}{m}\big)$$
As in Corollary \[o1\], the above estimates can be extended for the $p$-iterated system $\mathcal S^p = \{\phi_{i_1\ldots i_p}, \ i_j \in I, 1 \le j \le p\}$, and thus we obtain:
\[levelp\] Assume we have the system of conformal injective contractions $\mathcal S = \{\phi_i, i \in I\}$ with $|I| = m$, and let $\Lambda$ be its limit set. Assume also that there exists a family $\mathcal F \subset I^p$ of $p$-tuples such that $\phi_{i_p\ldots i_1}(\Lambda) = \phi_{j_p\ldots j_1}(\Lambda)$ for $(i_1, \ldots, i_p), (j_1, \ldots, j_p) \in \mathcal F$, and denote $Card(\mathcal F) = N(\mathcal F)$. Then $$o(\mathcal S) \ge \exp\big(\frac{N(\mathcal F)\log N(\mathcal F)}{m^p}\big)$$
However, á priori there may exist only *partial overlaps* at the level of $p$-iterates, which comprise a positive proportion of the measure. In particular the next Corollaries apply well for Bernoulli convolutions systems $\mathcal S_\lambda$, since in this case the limit set is an interval $\Lambda = I_\lambda$ and we can numerically estimate the proportion of overlaps at some iterate $p$. As above we obtain.
\[partial\] In the above setting assume that there is a family $\mathcal F\subset I^p$ of $p$-tuples and $k \ge 1$ so that for any $(i_1, \ldots, i_p) \in \mathcal F$, there exists $(j_1 \ldots j_k) \in I^k$ such that $$\phi_{i_1\ldots i_pj_1\ldots j_k}(\Lambda) \subset \mathop{\cap}\limits_{(\ell_1, \ldots \ell_p) \in \mathcal F} \phi_{\ell_1 \ldots \ell_p}(\Lambda)$$ Then if $N(\mathcal F)$ denotes the cardinality of $\mathcal F$, we obtain: $$o(\mathcal S) \ge \exp\big(\frac{N(\mathcal F) \log N(\mathcal F)}{m^{p+k}}\big)$$
More generally we have the following:
\[partial2\] In the above setting assume that there are families $\mathcal F_1, \ldots, \mathcal F_s \subset I^p$ of $p$-tuples and positive integers $k_1, \ldots, k_s$ such that, for any $1 \le j \le s$ and for any $(i_{j1}, \ldots, i_{jp}) \in \mathcal F_j$ there exists some $k_j$-tuple $(j_{1}, \ldots, j_{k_j}) \in I^{k_j}$ with $$\phi_{i_{j1}\ldots i_{jp}j_{1}\ldots j_{k_j}}(\Lambda) \subset \mathop{\cap}\limits_{(\ell_1, \ldots, \ell_p) \in \mathcal F_j} \phi_{\ell_1\ldots \ell_p}(\Lambda)$$ Then if $N(\mathcal F_j) := Card \mathcal F_j, \ 1\le j \le s$, we obtain: $$o(\mathcal S) \ge \exp\big(\frac{N(\mathcal F_1)\log N(\mathcal F_1)}{m^{p+k_1}} + \ldots + \frac{N(\mathcal F_s)\log N(\mathcal F_s)}{m^{p+k_s}}\big)$$
In Corollaries \[o1\] and \[o2\] below, we will apply these formulas to box dimension estimates for the measure $\pi_{*}\mu_{max}^+$.
**3.3. Conditional measures associated to the lift in the general case.**
We now study several families of conditional measures associated to the lift $\Phi$ and to the equilibrium state $\mu:= \mu_\psi$ and various fiber partitions. We look at the relations between them, and find in particular a formula for the folding entropy.Thus let the following measurable partitions:
**i)** Consider the skew product map $\Phi: \Sigma_I^+ \times \Lambda \to \Sigma_I^+ \times \Lambda$ and its fibers $\Phi^{-1}(\omega, x)$ for $(\omega, x) \in \Sigma_I^+ \times \Lambda$. They form a partition which is clearly measurable, and according to Rokhlin ([@Ro]) there exists a canonical family of conditional measures of $\mu:=\mu_\psi$, so for $\mu$-a.e $(\omega, x) \in \Sigma_I^+ \times \Lambda$, the conditional measure $\mu_{(\omega, x)}$ is supported on the finite set $\Phi^{-1}(\omega, x)$. Notice that $$\Phi^{-1}(\omega, x) = \{(i\omega, \phi_i^{-1}x), \ i \in I, \ \text{if} \ x \in \phi_i(\Lambda)\},$$ where we denote $\mu_{(\omega, x)}(i) := \mu_{(\omega, x)}(i\omega, \phi_i^{-1}x)$ if $x \in \phi_i(\Lambda)$, and $\mu_{(\omega, x)}(i\omega, \phi_i^{-1}(x)) = 0$ if $x \notin \phi_i(\Lambda)$.
**ii)** Denote by $\mu^+:= \pi_{1*} \mu$ on $\Sigma_I^+$, where $\pi_1: \Sigma_I^+ \times \Lambda \to \Sigma_I^+$ is the projection on the first coordinate, $\pi_1(\omega, x) = \omega$. Consider the partition of $\Sigma_I^+$ with the fibers of $\sigma$, and the associated family of conditional measures $\mu^+_\omega$ on the finite set $\sigma^{-1}\omega$, for $\mu^+$-a.e $\omega \in \Sigma_I^+$. We also denote $\mu^+_\omega(i\omega)$ by $\mu^+_\omega(i)$.
**iii)** Consider the partition of $\Sigma_I^+ \times \Lambda$ with the fibers of $\pi_1: \Sigma_I^+ \times \Lambda \to \Sigma_I^+$, and the associated family of conditional measures of $\mu$, namely $\mu_\omega$ on $\pi_1^{-1}(\omega) = \{\omega\} \times \Lambda$, for $\mu$-a.e $\omega \in \Sigma_I^+$. So $\mu_\omega$ is actually a probability measure on $\Lambda$.
From [@MU-JSP2016] we know that, for an equilibrium measure $\mu_\psi$ on $\Sigma_I^+ \times \Lambda$, the overlap number is $$o(\mathcal S, \mu_\psi) = \exp(F_\Phi(\mu_\psi))$$ We prove now a formula, which gives $F_\Phi(\mu)$ (and thus $o(\mathcal S, \mu_\psi))$ in terms of the conditional measures $\mu_\omega$ and $\mu_\omega^+$:
\[Hfold\] The overlap number $o(\mathcal S, \mu)$ of the equilibrium measure $\mu := \mu_\psi$ of a Hölder continuous potential on $\Sigma_I^+ \times \Lambda$, is determined by the corresponding conditional families $(\mu_\omega)_\omega, (\mu_\omega^+)_\omega$ by, $$\log o(\mathcal S, \mu) = -\mathop{\sum}\limits_{i \in I} \int_{\Sigma_I^+ \times \Lambda} \frac{\mu^+_\omega(i)}{\mathop{\sum}\limits_{j \in I} \mu^+_\omega(j) \cdot \mathop{\lim}\limits_{A_2 \to x} \frac{\mu_{j\omega}(\phi_j^{-1}\phi_i A_2)}{\mu_{i\omega}(A_2)}} \cdot \log \big(\frac{\mu^+_\omega(i)}{\mathop{\sum}\limits_{j \in I} \mu^+_\omega(j) \cdot \mathop{\lim}\limits_{A_2 \to x} \frac{\mu_{j\omega}(\phi_j^{-1}\phi_i A_2)}{\mu_{i\omega}(A_2)}}\Large) \ d\mu(\omega, x)$$
From the properties of conditional measures, if $\tilde g: \Sigma_I^+ \times \Lambda \to \mathbb R$ is $\mu$-integrable, then $$\label{1}
\aligned
\int_{\Sigma_I^+ \times \Lambda} &\tilde g(\omega, x) \ d\mu(\omega, x) = \int_{\Sigma_I^+ \times \Lambda} \int_{\Phi^{-1}(\omega, x)} \tilde g(\omega', x') d\mu_{(\omega, x)}(\omega', x') \ d\mu(\omega, x) \\
&= \mathop{\sum}\limits_{i \in I} \int_{\Sigma_I^+ \times \Lambda} \tilde g(i\omega, \phi_i^{-1}x) \cdot \mu_{(\omega, x)}(i) \ d\mu(\omega, x)
\endaligned$$ Notice that since our IFS has overlaps, a point $x\in \Lambda$ may belong to several sets of type $\phi_i(\Lambda)$. But $\mu$ also decomposes after the fibers of $\pi_1$, so for any real-valued function $\tilde g$ $\mu$-integrable on $\Sigma_I^+ \times \Lambda$, $$\label{2}
\aligned
\int_{\Sigma_I^+ \times \Lambda} \tilde g(\omega, x) &\ d\mu(\omega, x) = \int_{\Sigma_I^+} \int_{\{\omega\} \times \Lambda} \tilde g(\omega, x) d\mu_\omega(x) \ d\mu^+(\omega) =
\int_{\Sigma_I^+} \Gamma(\omega) d\mu^+(\omega) \\
&= \int_{\Sigma_I^+} \int_{\sigma^{-1}\omega}\Gamma(\omega') d\mu^+_\omega(\omega') \ d\mu^+(\omega) = \mathop{\sum}\limits_{i \in I} \int_{\Sigma_I^+} \Gamma(i\omega) \mu^+_\omega(i) \ d\mu^+(\omega)\\
&= \mathop{\sum}\limits_{i \in I} \int_{\Sigma_I^+ \times \Lambda} \mu^+_\omega(i) \cdot \int_{\{i\omega\}\times \Lambda} \tilde g(i\omega, x) \ d\mu_{i\omega}(x) \ d\mu(\omega, x),
\endaligned$$ where $\Gamma(\omega):= \int_{\{\omega\}\times \Lambda}\tilde g(\omega, x) d\mu_\omega(x)$. By taking $\tilde g$ such that $\tilde g|_{[j]} = 0$ for $j \ne i$, we obtain from (\[1\]) and (\[2\]) that: $$\label{3}
\int_{\Sigma_I^+ \times \Lambda} \tilde g(i\omega, \phi_i^{-1}x) \mu_{(\omega, x)}(i) \ d\mu(\omega, x) = \int_{\Sigma_I^+\times \Lambda} \mu^+_\omega(i) \cdot \int_{\{i\omega\}\times \Lambda} \tilde g(i\omega, x) d\mu_{i\omega}(x) \ d\mu(\omega, x)$$
Let us take now $\tilde g = \chi_A$, where $A = A_1 \times A_2$ is the product of two Borelian sets, and $A_1 \subset [i] \subset \Sigma_I^+$. Then if $i\omega \in A_1$, we have $$\int_{\{i\omega\} \times \Lambda} \tilde g(i\omega, x) d\mu_{i\omega}(x) = \mu_{i\omega}(A_2)$$ Let us denote $A_1(i):= \{\omega \in \Sigma_I^+, \ i\omega \in A_1\}$. Thus, with the above choice of $\tilde g$, $$\int_{\Sigma_I^+ \times \Lambda} \tilde g(i\omega, \phi_i^{-1}x) \mu_{(\omega, x)}(i) \ d\mu(\omega, x) = \int_{A_1(i) \times \phi_i(A_2)} \mu_{(\omega, x)}(i) \ d\mu(\omega, x)$$ So from the last two displayed equalities and (\[3\]), it follows that $$\label{4}
\int_{A_1(i) \times \phi_i(A_2)} \mu_{(\omega, x)}(i) \ d\mu(\omega, x) = \int_{A_1(i) \times \Lambda} \mu_{i\omega}(A_2) \cdot \mu_\omega^+(i) \ d\mu(\omega, x)$$ Since $\mu = \mu_\psi$ is the equilibrium measure of a Hölder continuous potential, and since the Bowen balls in $\Sigma_I^+ \times \Lambda$ are of type $[\omega_1\ldots \omega_n] \times B(x, r_0)$, it follows that $\mu^+$ is a doubling measure on $\Sigma_I^+$. Hence from Borel Density Lemma ([@Pe]), if $A_1(i)$ is a ball around some fixed $\bar \omega$ in $\Sigma_I^+$, we obtain: $$\label{5}
\aligned
\frac{1}{\mu^+(A_1(i))} \int_{A_1(i)}\int_{\phi_i A_2} \mu_{(\omega, x)}(i) d\mu_\omega(x) &\ d\mu(\omega) = \frac{1}{\mu^+(A_1(i))} \int_{A_1(i)\times \phi_iA_2}\mu_{(\omega, x)}(i) \ d\mu(\omega, x) \\
&\mathop{\longrightarrow}\limits_{A_1(i) \to \bar\omega}\ \int_{\phi_i (A_2)}\mu_{(\bar\omega, x)}(i) \ d\mu_{\bar \omega}(x)
\endaligned$$ On the other hand, $\int_{A_1(i)\times \Lambda} \mu_{i\omega}(A_2) \cdot \mu^+_\omega(i) d\mu(\omega, x) = \int_{A_1(i)}\mu_{i\omega}(A_2) \mu^+_\omega(i) \ d\mu^+(\omega)$. Hence from Borel Density Lemma, for $\mu^+$-a.e $\bar \omega \in \Sigma_I^+$, $$\frac{1}{\mu^+(A_1(i))}\int_{A_1(i)} \mu_{i\omega}(A_2) \cdot \mu_\omega^+(i) \ d\mu^+(\omega) \mathop{\longrightarrow}\limits_{A_1(i) \to \bar \omega}\mu_{i\bar\omega}(A_2) \cdot \mu_{\bar \omega}^+(i)$$ Therefore from (\[4\]) and (\[5\]) it follows that, for $\mu^+$-a.e $\bar\omega \in \Sigma_I^+$, $$\label{6}
\int_{\phi_i(A_2)} \mu_{(\bar \omega, x)}(i) \ d\mu_{\bar \omega}(x) = \mu_{i\bar\omega}(A_2) \cdot \mu_{\bar \omega}^+(i)$$
On the other hand from the $\Phi$-invariance of $\mu$, it follows that $$\int_{\Sigma_I^+\times \Lambda}\tilde g(\omega, x) \ d\mu(\omega, x) = \int_{\Sigma_I^+\times \Lambda} \tilde g\circ \Phi(\omega, x) d\mu(\omega, x) = \int_{\Sigma_I^+\times \Lambda} \tilde g(\sigma\omega, \phi_{\omega_1} x) d\mu(\omega, x)$$ Hence using the conditional decomposition of $\mu$ along the fibers of $\pi_1$, $$\int_{\Sigma_I^+}\int_{\{\omega\}\times \Lambda} \tilde g(\omega, x) d\mu_\omega(x) \ d\mu^+(\omega) = \int_{\Sigma_I^+} \int_{\{\omega\}\times \Lambda} \tilde g(\sigma\omega, \phi_{\omega_1}x) d\mu_\omega(x) \ d\mu^+(\omega)$$ Let us take now again $\tilde g = \chi_{A_1 \times A_2}$, and notice that $\sigma \omega \in A_1$ and $\phi_{\omega_1}x \in A_2$, if and only if $\omega \in \sigma^{-1}A_1$ and $ x \in \phi_{\omega_1}^{-1}A_2$. So from above, $$\int_{A_1}\mu_\omega(A_2) \ d\mu^+(\omega) = \int_{\sigma^{-1}A_1} \mu_\omega(\phi_{\omega_1}^{-1}A_2) \ d\mu^+(\omega)$$ Since $\mu^+$ is $\sigma$-invariant on $\Sigma_I^+$, it follows then that: $$\int_{\sigma^{-1}A_1}\mu_{\sigma\omega}(A_2) \ d\mu^+(\omega) = \int_{\sigma^{-1}A_1} \mu_\omega(\phi_{\omega_1}^{-1}A_2) \ d\mu^+(\omega)$$ Taking $A_1 \to \omega$, we obtain from above that, for any Borelian set $A_2\subset \Lambda, \ i \in I$ and $\mu^+$-a.e $\omega \in \Sigma_I^+$, $$\label{6'}
\mu_\omega(\phi_i A_2) = \mathop{\sum}\limits_{j \in I} \mu_{j\omega}(\phi_j^{-1}\phi_i(A_2)) \cdot \mu^+_{\omega}(j)$$ But we can apply Borel Density Lemma for the measure $\phi_*\mu_\omega$ on $\phi_i(\Lambda)$ in (\[6\]), and we see that for any $x\in \phi_i(\Lambda)$ and any $r>0$ small, $B(x, r) \cap \phi_i\Lambda = \phi_i(B(\phi_i^{-1}x, r') \cap \Lambda)$ for some $r'>0$ since $\phi_i$ is injective. Thus by taking $A_2$ to be a neighbourhood of $x$, we obtain from (\[5\]), (\[6\]), (\[6’\]), that $ \mathop{\lim}\limits_{A_2 \to x} \frac{\mu_{j\omega}(\phi_j^{-1}\phi_i A_2)}{\mu_{i\omega}(A_2)}$ exist, and that for $\mu$-a.e $(\omega, x) \in \Sigma_I^+ \times \Lambda$ and any $i\in I$, $$\label{7}
\mu_{(\omega, x)}(i) = \frac{\mu^+_\omega(i)}{\mathop{\sum}\limits_{j \in I} \mu^+_\omega(j) \cdot \mathop{\lim}\limits_{A_2 \to x} \frac{\mu_{j\omega}(\phi_j^{-1}\phi_i A_2)}{\mu_{i\omega}(A_2)}}$$ So from (\[7\]) and the fact that $F_\Phi(\mu) = -\int_{\Sigma_I^+\times \Lambda} \mu_{(\omega, x)} \log \mu_{(\omega, x)}d\mu(\omega, x)$, we obtain the formula for the folding entropy $F_\Phi(\mu)$, and thus from (\[otop\]) the formula for the overlap number $o(\mathcal S, \mu)$.
Box dimension estimates.
========================
For $\mu$ a Borel finite measure on $\mathbb R^d$, recall ([@Pe]) that the *lower box dimension* of $\mu$ is: $$\underline{dim}_B(\mu) = \mathop{\lim}\limits_{\delta \to 0} \inf\{\underline{dim}_B(Z), \mu(Z) \ge 1-\delta\}$$ Also denote the Hausdorff dimension of $\mu$ by $HD(\mu)$. The following inequality holds (see [@Pe]), $$HD(\mu) \le \underline{dim}_B(\mu)$$
In the sequel denote by $\chi(\mu_\psi)$ the Lyapunov exponent of the measure $\mu_\psi$ on $\Sigma_I^+ \times \Lambda$. Some aspects of dimensions and measures for various other cases were studied in [@M-MZ], [@M-ETDS11], etc. We are now ready to prove the estimate for the lower box dimension of the projection $\nu_\psi:= \pi_{2*}(\mu_\psi)$; recall that $\nu_\psi$ is *not* the usual projection measure $\pi_*\pi_{1*}\mu_\psi$. The following Theorem gives a *constructive method* to obtain sets $Z$ of large $\nu_\psi$-measure whose box dimensions is estimated using overlap numbers, and an estimate of the number of balls needed to cover such sets $Z$.
\[mthm\] Consider the conformal IFS $\mathcal S = \{\phi_i, i \in I\}$ wih limit set $\Lambda$, and the Hölder continuous potential $\psi: \Sigma_I^+ \times \Lambda \to \mathbb R$, with its equilibrium measure $\mu_\psi$, and let $\nu_\psi:= \pi_{2*}\mu_\psi$. Then, $$\underline{dim}_B(\nu_\psi) \le \frac{h_\sigma(\pi_{1*}(\mu_\psi)) - \log o(\mathcal S, \mu_\psi))}{|\chi(\mu_\psi)|}$$
For $n \ge 1$, let $S_n\psi(\omega, x):= \psi(\omega, x) + \psi(\Phi(\omega, x)) + \ldots + \psi(\Phi^{n-1}(\omega, x))$. For all $(\omega, x) \in \Sigma_I^+ \times \Lambda$, $\Phi^n(\omega, x) = (\sigma^n(\omega), \phi_{\omega_n\ldots \omega_1}(x))$. From Chain Rule, $J_{\Phi^n}(\mu_\psi)(\omega, x) = J_\Phi(\mu_\psi)(\omega, x) \ldots J_\Phi(\mu_\psi)(\Phi^{n-1}(\omega, x))$. We know from the Birkhoff Ergodic Theorem, from the formula for folding entropy (\[foldingentropy\]) and the fact that $\mu_\psi$ is ergodic that, $$\frac{1}{n}\log|\phi'_{\omega_n\ldots\omega_1}(x)| \mathop{\to}\limits_{n \to \infty} \int_{\Sigma_I^+ \times \Lambda} \log|\phi'_{\omega_1}(x)| d\mu_\psi(\omega, x), \ \text{and} \ \frac 1n \log J_{\Phi^n}(\mu_\psi)(\omega, x) \mathop{\to}\limits_{n \to \infty} F_\Phi(\mu_\psi), \ \text{and}$$ $$\frac{1}{n}S_n\psi(\omega, x) \mathop{\to}\limits_{n \to \infty} \int_{\Sigma_I^+ \times \Lambda} \psi(\omega, x) d\mu_\psi(\omega, x)$$ For an integer $n \ge 1$ and an arbitrary number $\tau >0$, consider therefore the Borelian set $$\aligned
D_n(\tau) := \{&(\omega, x) \in \Sigma_I^+ \times \Lambda, \text{with} \ |\frac{1}{p}\log J_{\Phi^p}(\mu_\psi)(\omega, x)- F_\Phi(\mu_\psi)| < \tau, \ \text{and} \\
&|\frac 1p \log|\phi'_{\omega_p\ldots\omega_1}(x)| - \int\log|\phi'_{\omega_1}|(x) d\mu_\psi(\omega, x)| < \tau,
\ \ |\frac 1p S_p\psi(\omega, x) - \int\psi d\mu_\psi | < \tau, \ \forall p \ge n\}
\endaligned$$ From above, $\mu_\psi(D_n(\tau)) \mathop{\to}\limits_{n\to \infty} 1$ for all $\tau >0$, and moreover, $$\label{Dn}
D_1(\tau) \subset \ldots \subset D_n(\tau) \subset D_{n+1}(\tau) \subset \ldots$$ On the other hand, notice that a Bowen ball in $\Sigma_I^+ \times \Lambda$ has the form $[\omega_1\ldots \omega_n] \times B(x, r_0)$, and from the estimates of equilibrium measures on Bowen balls (for eg [@KH]), we have: $$\mu_\psi([\omega_1 \ldots \omega_n] \times B(x, r_0)) \approx \exp(S_n\psi(\omega, x) - n P_\Phi(\psi)), n \ge 1,$$ where $\approx$ means that the two quantities are comparable with a comparability constant which depends only on $\psi$ and is independent of $n, x, \omega$. Now, if $\omega' \in [\omega_1 \ldots \omega_n]$ and if $(\eta, y) \in \Phi^{-n}\Phi^n(\omega, x)$, then $(\eta, y) \in \Phi^{-n}\Phi^n(\omega', x)$, and viceversa. But we proved in [@MU-JSP2016] that for $\mu_\psi$-a.e $(\omega, x) \in \Sigma_I^+ \times \Lambda$, $$\label{J}
J_{\Phi^n}(\mu_\psi)(\omega, x) \approx \frac{\mathop{\sum}\limits_{(\eta, y) \in \Phi^{-n}\Phi^n(\omega, x)} e^{S_n\psi(\eta, y)}}{e^{S_n\psi(\omega, x)}},$$ with comparability constant independent of $\omega, x, n$. Therefore, if $\omega' \in [\omega_1 \ldots\omega_n]$, it follows from (\[J\]) that there exists a constant $C>0$ so that for $\mu_\psi$-a.e $(\omega, x)$ and all $n \ge 1$,
$$\label{omega'}
\frac 1C J_{\Phi^n}(\mu_\psi)(\omega', x) \le J_{\Phi^n}(\mu_\psi)(\omega, x) \le C J_{\Phi^n}(\mu_\psi)(\omega', x)$$
This means that $D_n(\tau)$ is basically a product set, or more precisely that there exists a set $E_n(\tau)\subset \Lambda$ such that $D_n(\tau/2) \subset [\omega_1\ldots\omega_n]\times E_n(\tau) \subset D_n(\tau)$. Notice now that the map $\Phi^n$ is injective on the set $[\omega_1 \ldots \omega_n] \times B(x, r_0)$, for some fixed $r_0$, since the composition map $\phi_{\omega_n\ldots\omega_1}$ is injective on $U$. Thus from the properties of Jacobians of measures on sets of injectivity, we get $$\label{iteratemu}
\aligned
\mu_\psi(\Phi^n([\omega_1\ldots\omega_n]&\times B(x, r_0)\cap D_n(\tau))) = \int_{[\omega_1\ldots \omega_n]\times B(x, r_0)\cap D_n(\tau)} J_{\Phi^n}(\mu_\psi)(\eta, y) \ d\mu_\psi(\eta, y) \\
&\ge C e^{n(F_\Phi(\mu_\psi)-\tau)}\cdot \mu_\psi\big([\omega_1\ldots \omega_n]\times B(x, r_0) \cap D_n(\tau)\big)
\endaligned$$
We now want to estimate $\mu_\psi\big([\omega_1\ldots \omega_n]\times B(x, r_0) \cap D_n(\tau)\big)$. First notice that, since $\psi$ is Hölder continuous, the consecutive sum $S_n\psi(\omega, x)$ with respect to $\Phi$, does not really depend on $x$, but only on $\omega$. So there exists a constant $C>0$ such that for any $x, y \in \Lambda, \ \omega \in \Sigma_I^+$, $$|S_n\psi(\omega, x) - S_n\psi(\omega, y)| \le C$$ Thus one can fix $ y = x_0$ above in $\Lambda$. We want to show that for any Borel set $A \subset \Lambda$ and any $n$, $$\label{prod}
\mu_\psi([\omega_1\ldots\omega_n]\times A) \approx e^{S_n\psi(\omega, x_0)-nP_\Phi(\psi)}\cdot \nu_\psi(A),$$ with comparability constants independent of $\omega, n , A$. Since $\mu_\psi$ is a Borel measure, it is enough to show (\[prod\]) for open balls $A = B(y, r)$. Let us also recall that all the contractions $\phi_i$ are conformal, thus we have a Bounded Distortion property on Bowen balls of $\Phi$, namely there exists constants $C>0, 0 < r_0 < 1$, such that for any $x, y \in \Lambda$ with $d(x, y) < r_0$, any integer $n$ and any sequence $\underline i \in \Sigma_I^+$, then, $$\label{BDPR}
C^{-1} \phi_{i_1\ldots i_n}'(x) \le \phi_{i_1\ldots i_n}'(y) \le C \phi_{i_1\ldots i_n}'(x)$$ From the $\Psi$-invariance of $\mu_\psi$, we know that $\mu_\psi([\omega_1\ldots\omega_n]\times B(y, r)) = \sum \mu_\psi([\eta_1\ldots\eta_p\omega_1\ldots\omega_n] \times \phi_{\eta_1}^{-1}\ldots \phi_{\eta_p}^{-1}(B(y, r)))$, and using the above Bounded Distortion property, we take these backward iterates of $B(y, r)$ until we reach diameter $r_0$. The Bowen balls for the map $\Phi$ are sets of type $[\omega_1 \ldots \omega_n] \times B(z, r_0)$. Then from the properties of equilibrium measures on Bowen balls (see [@KH]), $$\mu_\psi([\eta_1\ldots \eta_p\omega_1\ldots \omega_n]\times B(z, r_0)) \approx e^{S_{n+p}\psi(\eta_1\ldots \eta_p\omega_1\ldots \omega_n, z) - (n+p) P_\Phi(\psi)},$$ where the comparability constants do not depend on $z, \omega, \eta, n, p$. We will write also $S_n\psi(\omega_1\ldots\omega_n)$ for $S_n\psi(\omega, x_0)$, since from the above it does not matter (up to a constant) which $x_0$ we take. But $$\begin{aligned}
e^{S_{n+p}\psi(\eta_1\ldots \eta_p\omega_1\ldots \omega_n, z) - (n+p) P_\Phi(\psi)} &= e^{S_p\psi(\eta_1\ldots\eta_p) + S_n\psi(\omega_1\ldots \omega_n) - (n+p) P_\Phi(\psi)} = \\
&= e^{S_p\psi(\eta_1\ldots \eta_p) - p P_\Phi(\psi)} \cdot e^{S_n\psi(\omega_1\ldots \omega_n) - nP_\Phi(\psi)}
\end{aligned}$$ So when we take the above sum we obtain $e^{S_n\psi(\omega_1\ldots \omega_n) - nP_\Phi(\psi)} \cdot \nu_\psi(B(y, r))$. Thus relation (\[prod\]) holds, i.e. there exists a constant $C>0$ independent of $n, x, y, r, \omega$, such that $$\frac 1C \nu_\psi(B(y, r)) e^{S_n\psi(\omega, x_0) - nP_\Phi(\psi)} \le \mu_\psi([\omega_1\ldots \omega_n] \times B(y, r)) \le C \nu_\psi(B(y, r)) e^{S_n\psi(\omega, x_0) - nP_\Phi(\psi)}$$
Now recall that $\mu_\psi(D_n(\tau)) \to 1$ when $n \to \infty$; hence for any $\delta>0$ small, there exists $n(\delta)\ge 1$ such that $\mu_\psi(D_n(\tau)) \ge 1-\delta$ for all $n \ge n(\delta)$; hence from the $\Phi$-invariance of $\mu_\psi$, $\mu_\psi(\Phi^{n}(D_n(\tau))) \ge 1-\delta$. Moreover there exists a strictly increasing sequence of integers $(k_n)_n$, with $k_n \ge n$, such that, $$\label{kn}
\mu_\psi(D_{k_n}) \ge 1-\alpha_n, \ \text{and} \ \mathop{\sum}\limits_{n\ge 1} \alpha_n < \infty$$
Denote now by $Y_n(\tau):= \pi_2D_n(\tau) \subset \Lambda$. We want to apply a version of Borel Density Lemma ([@Pe] pg 293), in order to estimate the portion of the $\nu_\psi$-measure of the intersection between a ball and $ Y_n(\tau)$. Indeed for any $\delta>0$ it follows that for any $n \ge n(\delta)$, there exists a borelian subset $\tilde Y_n(\tau) \subset Y_n(\tau)$ and $\rho_n>0$, such that $\nu_\psi(\tilde Y_n(\tau)) \ge 1-2\delta$, and for any $x \in \tilde Y_n(\tau)$ and any $r \le \rho_n$, $$\label{tilde}
\nu_\psi(B(x, r) \cap Y_n(\tau)) \ge \frac 12 \nu_\psi(B(x, r))$$ Let $Z_n(\tau):= \pi_2\Phi^n(D_n(\tau))$ and $\tilde Z_n(\tau):= \mathop{\bigcap}\limits_{\ell \ge n} Z_{k_\ell}(\tau)$, for $n \ge 1$. Then, since $\mu_\psi(\Phi^n(D_n(\tau))) \ge \mu_\psi(D_n(\tau))$, it follows from (\[kn\]) that $$\nu_\psi(\tilde Z_{n}(\tau)) \ge 1-\mathop{\sum}\limits_{m \ge n}\alpha_m, \ \text{and} \ \nu_\psi(\tilde Z_n(\tau)) \mathop{\to}\limits_{n\to \infty} 1$$ Given the radius $\rho_n$ above, we can find an integer $s_n \ge n$, such that any ball $B(y, \frac{\rho_n}{2})$ with $y \in \Lambda$, intersects the set $\tilde Y_{s_n}(\tau)$. This is true since $\nu_\psi(\tilde Y_n(\tau)) \to 1$, and since $\mu_\psi$ is the equilibrium measure of a Hölder continuous potential, thus it is positive on balls of radius $\rho_n/2$. Denote now $$r_n:= e^{n(\chi(\mu_\psi) + \tau)}, \ n \ge 1$$
Consider an arbitrary family $\mathcal F_{k_\ell}$ of mutually disjoint balls of radii $\rho_n r_{k_\ell}$ with centers in $\pi_2\Phi^{k_\ell}(D_{k_\ell}(\tau))$, for $\ell \ge s_n$, and assume the balls in $\mathcal F_{k_\ell}$ contain images of type $\phi_{i_{k_\ell}\ldots i_1}(B(z, \rho_n))$ for $z$ in a family of centers $F_{k_\ell}$. But from above, for all $\ell\ge s_n$ and $z \in F_{k_\ell}$, the ball $B(z, \rho_n/2)$ must contain a point $\xi_z \in \tilde Y_{s_n}(\tau)$. Hence $B(\xi_z, \rho_n/2) \subset B(z, \rho_n)$, and thus $\phi_{i_{k_\ell}\ldots i_1}(B(\xi_z, \rho_n/2)) \subset \phi_{i_{k_\ell}\ldots i_1}(B(z, \rho_n))$ for all $z \in F_{k_\ell}$. So we obtain a family $\mathcal G_{k_\ell}$ of disjoint sets $\phi_{i_{k_\ell}\ldots i_1}(B(\xi_z, \rho_n/2)), \ z\in F_{k_\ell}$. From our construction, $$N(\mathcal G_{k_\ell}):= Card(\mathcal G_{k_\ell}) = N(\mathcal F_{k_\ell}):= Card(\mathcal F_{k_\ell})$$ However $\tilde Y_{s_n}(\tau) \subset Y_{s_n}(\tau) \subset \pi_2 D_{k_\ell}(\tau)$, if $\ell \ge s_n \ge n$, so from the above properties of the set $\tilde Y_{s_n}(\tau)$ and (\[tilde\]), it follows that $\nu_\psi(\tilde Y_{s_n}(\tau)) \ge 1-2\delta$ and, $$\nu_\psi(B(\xi_z, \rho_n/2) \cap Y_{s_n}(\tau)) \ge \frac 12 \nu_\psi(B(\xi_z, \rho_n/2))$$ But now from (\[Dn\]), $Y_{s_n}(\tau) \subset Y_k(\tau) = \pi_2D_{k}(\tau)$ for all $k \ge s_n$, and recall $\ell \ge s_n \ge n$; hence from the last inequality, $$\label{meas}
\nu_\psi(B(\xi_z, \rho_n/2) \cap Y_{k_\ell}(\tau)) \ge \frac 12 \nu_\psi(B(\xi_z, \rho_n/2))$$
Let us estimate now the $\nu_\psi$-measure of a set from $\mathcal G_{k_\ell}$, for $\ell \ge s_n$. Since $\Phi^{k_\ell}$ is injective on $[i_1\ldots i_{k_\ell}]\ \times \Lambda$, we obtain from (\[prod\]) and (\[meas\]), $$\label{mare}
\aligned
\nu_\psi(\phi_{i_{k_\ell}\ldots i_1}&B(\xi_z, \rho_n/2)\cap Y_{k_\ell}(\tau)) = \mu_\psi(\Phi^{k_\ell}([i_1\ldots i_{k_\ell}] \times B(\xi_z, \rho/2) \cap D_{k_\ell})) = \\
& =\int_{[i_1\ldots i_{k_\ell}]\times (B(\xi_z, \rho_n/2) \cap Y_{k_\ell}(\tau))} J_{\Phi^{k_\ell}}(\mu_\psi)(\omega, x) \ d\mu_\psi(\omega, x)\\
& \ge C \exp(k_\ell(F_\Phi(\mu_\psi) - \tau)) \cdot \exp(S_{k_\ell}\psi(\omega, x_0) - k_\ell P_\Phi(\psi))\cdot \nu_\psi(B(\xi_z, \rho_n/2) \cap Y_{k_\ell})\\
& \ge \tilde C_n\exp(k_\ell(F_\Phi(\mu_\psi)-\tau) \cdot \exp(k_\ell(-h_\Phi(\mu_\psi) - \tau)) = \tilde C_n\exp(k_\ell(F_\Phi(\mu_\psi)-h_\Phi(\mu_\psi) -2\tau)),
\endaligned$$ for some constants $C_n, \tilde C_n>0$, where we used the estimate on the Jacobian of $\Phi^{k_\ell}$ on $D_{k_\ell}$, the estimate on the equilibrium measure $\mu_\psi$ of a Bowen ball $[i_1\ldots i_{k_\ell}]\times B(\xi_z, \rho_n/2)$, and the behaviour of $S_{k_\ell}\psi$ on the generic points from $D_{k_\ell}$. Since the balls in $\mathcal F_{k_\ell}$ are disjoint, and each of them contains a set of type $\phi_{i_{k_\ell}\ldots i_1}B(\xi_z, \rho_n/2)\cap Y_{k_\ell}$, it follows that for all integers $\ell \ge s_n$, $$\mathop{\sum}\limits_{\xi_z\in G_{k_\ell}} \nu_\psi(\phi_{i_{k_\ell}\ldots i_1}B(\xi_z, \rho_n/2)\cap Y_{k_\ell}(\tau)) \le 1$$ Thus, using (\[mare\]) and the fact that $N(\mathcal G_{k_\ell}) = N(\mathcal F_{k_\ell})$, we obtain for any family $\mathcal F_{k_\ell}$ as above, $$\label{NF}
N(\mathcal F_{k_\ell}) \le C_n^{-1}\exp(-k_\ell(F_\Phi(\mu_\psi)- h_\Phi(\mu_\psi) -2\tau))$$ So if for some $\ell \ge s_n$ we take a disjointed family $\mathcal W$ of balls of radii $\rho_n\cdot r_{k_\ell}$ with centers in $\tilde Z_{s_n}(\tau) = \mathop{\bigcap}\limits_{\ell \ge s_n} \pi_2\Phi^{k_\ell}D_{k_\ell}(\tau)$, then its cardinality $N(\mathcal W)$ is less than the cardinality of some family $\mathcal F_{k_\ell}$ from above, hence from (\[NF\]) we obtain an estimate for the lower box dimension, $$\underline{dim}_B(\tilde Z_{s_n}(\tau)) \le \frac{h_\Phi(\mu_\psi) - F_\Phi(\mu_\psi) + 2\tau}{|\chi(\mu_\psi) + \tau|}$$
But on the other hand, we know from construction that $\nu_\psi(\tilde Z_{s_n}(\tau)) \ge 1- \mathop{\sum}\limits_{j\ge s_n} \alpha_j \to 1$, when $n \to \infty$. So from the above, using the definition of lower box dimension of a measure, it follows $$\underline{dim}_B(\nu_\psi) \le \frac{h_\Phi(\mu_\psi) - F_\Phi(\mu_\psi) + 2\tau}{|\chi(\mu_\psi) + \tau|},$$ for any small number $\tau >0$, and thus the conclusion follows, namely $$\underline{dim}_B(\nu_\psi) \le \frac{h_\Phi(\mu_\psi) - F_\Phi(\mu_\psi)}{|\chi(\mu_\psi)|}$$
Recall now from (\[bernoulli\]) that for Bernoulli measures we have the equality of the two projectional measures, i.e $\pi_{2*} \mu_{\bf p} = \pi_* \mu_{\bf p}^+$. Also recall that $\mu_{max}$ is the measure of maximal entropy for $\Phi$ on $\Sigma_I^+ \times \Lambda$, and $\mu_{max}^+$ is the measure of maximal entropy for the shift on $\Sigma_I^+$.
Then from Proposition \[o\], Theorem \[mthm\] and Corollaries \[levelp\] and \[partial\], we obtain the following estimates. In particular, these can be applied to Bernoulli convolutions (for which the limit set $\Lambda$ is the whole interval $I_\lambda$), to get *numerical estimates* of the box dimension of the projection measure, based on how many overlaps we count at level $p$ and on how large are these overlaps.
\[o1\] Assume we have the system of conformal injective contractions $\mathcal S = \{\phi_i, i \in I\}$ with $|I| = m$, and let $\Lambda$ be its limit set, and denote by $\mu_{max}$ the measure of maximal entropy on $\Sigma_I^+ \times \Lambda$. Assume also that there exists a family $\mathcal F$ of $p$-tuples such that $\phi_{i_p\ldots i_1}(\Lambda) = \phi_{j_p\ldots j_1}(\Lambda)$ for $(i_1, \ldots, i_p), (j_1, \ldots, j_p) \in \mathcal F$, and denote $Card(\mathcal F) = N(\mathcal F)$. Then $o(\mathcal S) \ge \exp\big(\frac{N(\mathcal F)\log N(\mathcal F)}{m^p}\big)$, and $$\underline{dim}_B(\pi_{2*}\mu_{max}) = \underline{dim}_B(\pi_*\mu^+_{max}) \le
\frac{p \cdot h_\sigma(\mu_{max}^+) - \frac{N(\mathcal F)\log N(\mathcal F)}{m^p}}{p\cdot \chi(\mu_{max})}$$
\[o2\]
In the above setting assume that there are families $\mathcal F_1, \ldots, \mathcal F_s \subset I^p$ of $p$-tuples and positive integers $k_1, \ldots, k_s$ such that, for any $1 \le j \le s$ and any $(i_{j1}, \ldots, i_{jp}) \in \mathcal F_j$ there exists some $k_j$-tuple $(j_{1}, \ldots, j_{k_j}) \in I^{k_j}$, with $$\phi_{i_{j1}\ldots i_{jp}j_{1}\ldots j_{k_j}}(\Lambda) \subset \mathop{\cap}\limits_{(\ell_1, \ldots, \ell_p) \in \mathcal F_j} \phi_{\ell_1\ldots \ell_p}(\Lambda)$$ Then if $N(\mathcal F_j) := Card \mathcal F_j, \ 1\le j \le s$, we obtain: $$\underline{dim}_B(\pi_{2*}\mu_{max}) = \underline{dim}_B(\pi_*\mu^+_{max}) \le
\frac{p \cdot h_\sigma(\mu_{max}^+) - \frac{N(\mathcal F_1)\log N(\mathcal F_1)}{m^{p+k_1}} - \ldots - \frac{N(\mathcal F_s)\log N(\mathcal F_s)}{m^{p+k_s}}}{{p\cdot \chi(\mu_{max})}}$$
**Acknowledgements:** This work was supported by grant PN-III-P4-ID-PCE-2016-0823 from CNCS - UEFISCDI.
[99]{}
L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics, Birkhäuser, Basel, 2008.
A. Garsia, Arithmetic properties of Bernoulli convolutions, Trans. AMS, 102, 1962, 409-432.
A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, London-New York, 1995.
I.P. Kornfeld, Ya.G. Sinai, Basic Notions of Ergodic Theory and Examples of Dynamical Systems, in: Ya.G. Sinai (Ed.), Dynamical Systems, Ergodic Theory and Applications, in: Encyclopaedia Math. Sci., vol.100, Springer Verlag, 2000.
E. Mihailescu, Unstable directions and fractal dimension for a class of skew products with overlaps in fibers, Math Zeitschrift, 269, 2011, 733-750.
E. Mihailescu, On a class of stable conditional measures, Ergod Th Dyn Syst 31, 2011, 1499-15.
E. Mihailescu, M. Urbański, Overlap functions for measures in conformal iterated function systems, J. Statistical Physics, 162, 2016, 43-62.
W. Parry, Entropy and generators in ergodic theory, W. A Benjamin, New York, 1969.
Y. Pesin, Dimension Theory in Dynamical Systems, Chicago Lectures in Mathematics, 1997.
F. Przytycki, M. Urbański, On Hausdorff dimension of some fractal sets. Studia Math. 93, 155-186, 1989.
V. A. Rokhlin, Lectures on the theory of entropy of transformations with invariant measures, Russian Math. Surveys, **22**, 1967, 1-54.
D. Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics, J. Statistical Physics **95**, 1999, 393-468.
D. Ruelle, Positivity of entropy production in nonequilibrium statistical mechanics, J. Statistical Physics **85**, 1/2, 1996, 1-23.
P. Walters, An introduction to ergodic theory (2nd edition), Springer New York, 2000.
Address: Institute of Mathematics “Simion Stoilow“ of the Romanian Academy,
Calea Grivitei 21, P.O. Box 1-764, RO 014700, Bucharest, Romania.
Eugen.Mihailescu@imar.ro
www.imar.ro/$\sim$mihailes
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Vrije Universiteit Amsterdam\
emiel.van.miltenburg@vu.nl
bibliography:
- 'image\_bias\_refs.bib'
title: Stereotyping and Bias in the Flickr30K Dataset
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Introduction
============
The Flickr30K dataset [@young2014image] is a collection of over 30,000 images with 5 crowdsourced descriptions each. It is commonly used to train and evaluate neural network models that generate image descriptions (e.g. [@vinyals2015show]). An untested assumption behind the dataset is that the descriptions are based on the images, and nothing else. Here are the authors (about the Flickr8K dataset, a subset of Flickr30K):
> “By asking people to describe the people, objects, scenes and activities that are shown in a picture without giving them any further information about the context in which the picture was taken, we were able to obtain conceptual descriptions that focus only on the information that can be obtained from the image alone.” [@hodosh2013framing p. 859]
What this assumption overlooks is the amount of *interpretation* or *recontextualization* carried out by the annotators. Let us take a concrete example. Figure \[fig:girlboss\] shows an image from the Flickr30K dataset.
![Image 8063007 from the Flickr30K dataset.[]{data-label="fig:girlboss"}](8063007){width="35.00000%"}
This image comes with the five descriptions below. All but the first one contain information that cannot come from the image alone. Relevant parts are highlighted in **bold**:
1. A blond girl and a bald man with his arms crossed are standing inside looking at each other.
2. A **worker** is **being scolded** by her **boss** in a **stern lecture**.
3. A **manager talks to an employee about job performance**.
4. A hot, blond girl **getting criticized by her boss**.
5. Sonic employees **talking about work**.
We need to understand that the descriptions in the Flickr30K dataset are subjective descriptions of events. This can be a good thing: the descriptions tell us what are the salient parts of each image to the average human annotator. So the two humans in Figure \[fig:girlboss\] are relevant, but the two soap dispensers are not. But subjectivity can also result in *stereotypical* descriptions, in this case suggesting that the male is more likely to be the manager, and the female is more likely to be the subordinate. do note that some descriptions are speculative in nature, which they say hurts the accuracy and the consistency of the descriptions. But the problem is not with the lack of consistency here. Quite the contrary: the problem is that stereotypes may be pervasive enough for the data to be consistently biased. And so language models trained on this data may propagate harmful stereotypes, such as the idea that women are less suited for leadership positions.
This paper aims to give an overview of linguistic bias and unwarranted inferences resulting from stereotypes and prejudices. I will build on earlier work on linguistic bias in general [@beukeboom2014mechanisms], providing examples from the Flickr30K data, and present a taxonomy of unwarranted inferences. Finally, I will discuss several methods to analyze the data in order to detect biases.[^1]
Stereotype-driven descriptions
==============================
Stereotypes are ideas about how other (groups of) people commonly behave and what they are likely to do. These ideas guide the way we talk about the world. I distinguish two kinds of verbal behavior that result from stereotypes: (i) linguistic bias, and (ii) unwarranted inferences. The former is discussed in more detail by , who defines linguistic bias as “a systematic asymmetry in word choice as a function of the social category to which the target belongs.” So this bias becomes visible through the *distribution* of terms used to describe entities in a particular category. Unwarranted inferences are the result of speculation about the image; here, the annotator goes beyond what can be glanced from the image and makes use of their knowledge and expectations about the world to provide an overly specific description. Such descriptions are directly identifiable as such, and in fact we have already seen four of them (descriptions 2–5) discussed earlier.
Linguistic bias
---------------
Generally speaking, people tend to use more concrete or specific language when they have to describe a person that does not meet their expectations. lists several linguistic ‘tools’ that people use to mark individuals who deviate from the norm. I will mention two of them.[^2]
Adjectives
: One well-studied example [@stahlberg2007representation; @romaine2001corpus] is sexist language, where the sex of a person tends to be mentioned more frequently if their role or occupation is inconsistent with ‘traditional’ gender roles (e.g. *female surgeon, male nurse*). Beukeboom also notes that adjectives are used to create “more narrow labels \[or subtypes\] for individuals who do not fit with general social category expectations” (p. 3). E.g. *tough woman* makes an exception to the ‘rule’ that women aren’t considered to be tough.
Negation
: can be used when prior beliefs about a particular social category are violated, e.g. *The garbage man was not stupid.* See also [@beukeboom2010negation].
These examples are similar in that the speaker has to put in additional effort to mark the subject for being unusual. But they differ in what *we* can conclude about the speaker, especially in the context of the Flickr30K data. Negations are much more overtly displaying the annotator’s prior beliefs. When one annotator writes that *A little boy is eating pie **without** utensils* (image 2659046789), this immediately reveals the annotator’s normative beliefs about the world: pie should be eaten *with* utensils. But when another annotator talks about *a girls basketball game* (image 8245366095), this cannot be taken as an indication that the annotator is biased about the gender of basketball players; they might just be helpful by providing a detailed description. In section 3 I will discuss how to establish whether or not there is any bias in the data regarding the use of adjectives.
Unwarranted inferences
----------------------
Unwarranted inferences are statements about the subject(s) of an image that go beyond what the visual data alone can tell us. They are based on additional assumptions about the world. After inspecting a subset of the Flickr30K data, I have grouped these inferences into six categories (image examples between parentheses):
Activity
: We’ve seen an example of this in the introduction, where the ‘manager’ was said to be *talking about job performance* and *scolding \[a worker\] in a stern lecture* (8063007).
Ethnicity
: Many dark-skinned individuals are called *African-American* regardless of whether the picture has been taken in the USA or not (4280272). And people who look Asian are called Chinese (1434151732) or Japanese (4834664666).
Event
: In image 4183120 (Figure \[fig:watchingthegame\]), people sitting at a gym are said to be watching a game, even though there could be any sort of event going on. But since the location is so strongly associated with sports, crowdworkers readily make the assumption.
![Image 4183120 from the Flickr30K dataset.[]{data-label="fig:watchingthegame"}](4183120.jpg){width="35.00000%"}
Goal
: Quite a few annotations focus on explaining the *why* of the situation. For example, in image 3963038375 a man is fastening his climbing harness *in order to have some fun*. And in an extreme case, one annotator writes about a picture of a dancing woman that *the school is having a special event in order to show the american culture on how other cultures are dealt with in parties* (3636329461). This is reminiscent of the Stereotypic Explanatory Bias [@sekaquaptewa2003stereotypic SEB], which refers to “the tendency to provide relatively more explanations in descriptions of stereotype inconsistent, compared to consistent behavior” [@beukeboom2010negation p. 5]. So in theory, odd or surprising situations should receive more explanations, since a description alone may not make enough sense in those cases, but it is beyond the scope of this paper to test whether or not the Flickr30K data suffers from the SEB.
Relation
: Older people with children around them are commonly seen as parents (5287405), small children as siblings (205842), men and women as lovers (4429660), groups of young people as friends (36979).
Status/occupation
: Annotators will often guess the status or occupation of people in an image. Sometimes these guesses are relatively general (e.g. college-aged people being called *students* in image 36979), but other times these are very specific (e.g. a man in a workshop being called a *graphics designer*, 5867606).
Detecting stereotype-driven descriptions
========================================
In order to get an idea of the kinds of stereotype-driven descriptions that are in the Flickr30K dataset, I made a browser-based annotation tool that shows both the images and their associated descriptions.[^3] You can simply leaf through the images by clicking ‘Next’ or ‘Random’ until you find an interesting pattern.
Ethnicity/race
--------------
One interesting pattern is that the ethnicity/race of babies doesn’t seem to be mentioned *unless* the baby is black or asian. In other words: white seems to be the default, and others seem to be marked. How can we tell whether or not the data is actually biased?
We don’t know whether or not an entity belongs to a particular social class (in this case: ethnic group) until it is marked as such. But we can approximate the proportion by looking at all the images where the annotators have used a marker (in this case: adjectives like *black, white, asian*), and for those images count how many descriptions (out of five) contain a marker. This gives us an *upper bound* that tells us how often ethnicity is indicated by the annotators. Note that this upper bound lies somewhere between 20% (one description) and 100% (5 descriptions). Figure \[table:babies\] presents count data for the ethnic marking of babies. It includes two false positives (talking about a *white baby stroller* rather than a *white baby*). In the Asian group there is an additional complication: sometimes the mother gets marked rather than the baby. E.g. *An Asian woman holds a baby girl.* I have counted these occurrences as well.
**Asian**
------------ --------------------------------------- ----
2339632913 Asian child/baby 2
3208987435 Asian baby, Asian/oriental woman 3
7327356514 Asian girl/baby, Asian/oriental woman 4
**Black**
1319788022 African-American (AA)/black baby 3
149057633 African/AA child, black baby 3
3217909454 Dark-skinned baby 1
3614582606 AA baby 1
**White**
11034843 White baby boy 1
176230509 White baby boy 1
2058947638 White baby 1
3991342877 White baby 1
4592281294 White baby stroller FP
661546153 White baby stroller FP
442983801 Fair-skinned baby 1
: Number of times ethnicity/race was mentioned per category, per image. The average is expressed as a percentage of the number of descriptions. Counts in the last column correspond to the number of descriptions containing an ethnic/racial marker. Images were found by looking for descriptions matching `(asian|white|black|African-American|skinned) baby`. I found two false positives, indicated with FP.[]{data-label="table:babies"}
The numbers in Table \[table:babies\] are striking: there seems to be a real, systematic difference in ethnicity marking between the groups. We can take one step further and look at all the 697 pictures with the word ‘baby’ in it. If there turn out to be disproportionately many white babies, this strengthens the conclusion that the dataset is biased.[^4]
I have manually categorized each of the baby images. There are 504 white, 66 asian, and 36 black babies. 73 images do not contain a baby, and 18 images do not fall into any of the other categories. While this does bring down the average number of times each category was marked, it also increases the contrast between white babies (who get marked in less than 1% of the images) and asian/black babies (who get marked much more often). A next step would be to see whether these observations also hold for other age groups, i.e. children and adults.$^{\ref{imageviewer}}$
Other methods
-------------
It may be difficult to spot patterns by just looking at a collection of images. Another method is to tag all descriptions with part-of-speech information, so that it becomes possible to see e.g. which adjectives are most commonly used for particular nouns. One method readers may find particularly useful is to leverage the structure of Flickr30K Entities [@plummer2015flickr30k]. This dataset enriches Flickr30K by adding coreference annotations, i.e. which phrase in each description refers to the same entity in the corresponding image. I have used this data to create a coreference graph by linking all phrases that refer to the same entity. Following this, I applied Louvain clustering [@blondel2008fast] to the coreference graph, resulting in clusters of expressions that refer to similar entities. Looking at those clusters helps to get a sense of the enormous variation in referring expressions. To get an idea of the richness of this data, here is a small sample of the phrases used to describe beards (cluster 268): *a scruffy beard; a thick beard; large white beard; a bubble beard; red facial hair; a braided beard; a flaming red beard.* In this case, ‘red facial hair’ really stands out as a description; why not choose the simpler ‘beard’ instead?[^5]
Discussion
==========
In the previous section, I have outlined several methods to manually detect stereotypes, biases, and odd phrases. Because there are many ways in which a phrase can be biased, it is difficult to automatically detect bias from the data. So how should we deal with stereotype-driven descriptions?
Neutralizing stereotypes for production
: One way to move forward might be to work with multilingual data. propose a model that generates image descriptions given data from multiple languages, in their case German and English. Multilingual, or better: multicultural data might force models to put less emphasis on features that are only salient to annotators from one particular country.
Stereotypes and interpretation
: While stereotypes might be a problem for production, further study of cultural stereotyping might be beneficial to systems that have to interpret human descriptions and determine likely referents of those descriptions. E.g. knowing that *baseball player* probably refers to a *male* baseball player is very useful.
Levels of describing an image
: There is a large body of work in art, information science, library science and related fields dedicated to the description and categorization of images [@shatford1986analyzing; @jaimes1999conceptual]. A common thread is that we can divide image description into multiple levels or stages, starting from concrete physical attributes up to abstract contextual information. These levels build on each other; we first have to recognize separate entities before we can reason about their relation. But recent neural network models like [@vinyals2015show] do not match this procedure. Rather, they are trained to create a direct mapping between images and their descriptions. With this paper, I hope to have shown that the Flickr30K dataset is *layered*, reflecting not only the physical contents of the images, but also whether the images match the everyday expectations of the crowd. An interesting challenge would be for image description models to learn separate representations for both layers: the perceptual and the contextual.
Representativeness
: My argument here is not that we should explicitly remove bias from crowdsourced descriptions of images. This may result in normalising the data into a form that is less representative of actual human descriptions. I do, however, contend that we should accept that crowdsourced descriptions of images *are* biased. Acknowledging this fact is an important step towards designing models that can accommodate data based on a mixture of facts *and stereotypes* about the world.
Conclusion
==========
This paper provided a taxonomy of stereotype-driven descriptions in the Flickr30K dataset. I have divided these descriptions into two classes: linguistic bias and unwarranted inferences. The former corresponds to the annotators’ choice of words when confronted with an image that may or may not match their stereotypical expectancies. The latter corresponds to the tendency of annotators to go beyond what the physical data can tell us, and expand their descriptions based on their past experiences and knowledge of the world. Acknowledging these phenomena is important, because on the one hand it helps us think about what is learnable from the data, and on the other hand it serves as a warning: if we train and evaluate language models on this data, we are effectively teaching them to be biased.
I have also looked at methods to detect stereotype-driven descriptions, but due to the richness of language it is difficult to find an automated measure. Depending on whether your goal is production or interpretation, it may either be useful to suppress or to emphasize biases in human language. Finally, I have discussed stereotyping behavior as the addition of a contextual layer on top of a more basic description. This raises the question what kind of descriptions we would like our models to produce.
Acknowledgments
===============
Thanks to Piek Vossen and Antske Fokkens for discussion, and to Desmond Elliott and an anonymous reviewer for comments on an earlier version of this paper. This research was supported by the Netherlands Organization for Scientific Research (NWO) via the Spinoza-prize awarded to Piek Vossen (SPI 30-673, 2014-2019).
Bibliographical references
==========================
[^1]: The Flickr30K data also contains examples where annotators judge the subjects of the images on their looks. E.g. description \#4 above calling the girl in the image *hot*. Analyzing this judgmental language goes beyond the scope of this paper.
[^2]: Examples given are also due to [@beukeboom2014mechanisms].
[^3]: \[imageviewer\]Code and data is available on GitHub: <https://github.com/evanmiltenburg/Flickr30K-Image-Viewer>
[^4]: Of course this extra step does constitute an additional annotation effort, and it is fairly difficult to automate; one would have to train a classifier for each group that needs to be checked.
[^5]: Code and data is available on GitHub: <https://github.com/evanmiltenburg/Flickr30k-clusters>
|
---
abstract: 'Let $C$ be a smooth curve with gonality $k\ge 6$ and genus $g\ge 2k^2+5k-6$. We prove that $W^1_d({C})$ has the expected dimension and that the general element of any irreducible component of $W^1_d({C})$ is primitive if either $g-k+4\le d\le g-2$ or $d=g-k+3$ and either $k$ is odd or $C$ is not a double covering of a curve of gonality $k/2$ and genus $k-3$. Even in the latter case we prove the existence of a complete and primitive $g^1_{g-k+3}$.'
address: |
Dept. of Mathematics\
University of Trento\
38123 Povo (TN), Italy
author:
- 'E. Ballico'
title: On the existence of primitive pencils for smooth curves
---
[^1]
A line bundle $L$ on a smooth curve $C$ of genus $g\ge 4$ is said to be [*primitive*]{} if it is spanned and both $L$ and $\omega _C\otimes L^\vee$ are spanned, i.e. if it is spanned and $h^0(L(q)) =h^0(L)$ for all $q\in C$ (sometimes one also imposes that $L\ne \mathcal {O} _C$ and $L\ne \omega _C$) ([@ckm], [@ckm1]). Since $L$ is primitive if and only if $\omega _C\otimes L^\vee$ is primitive, to study primitive line bundles on $C$ it is sufficient to know the ones with $0 < \deg (L) \le g-1$. Let $C$ be a smooth curves of genus $g\ge 4$ with gonality $k$. If either $g\ne k(k-1)/2$ or $C$ is not isomorphic to a smooth plane curve, then $C$ has a complete and primitive $g^1_k$. For very low $k$ or for a general smooth curve of genus $g$ the Brill-Noether theory of $C$ is well-known and it gives a complete description of the complete and primitive $g^r_d$ on $C$ ([@ckm], [@ckm1], [@ckm2], [@cm]). If $g\ge 11$ and $k\ge 5$, then a general element of any irreducible component of $W^1_{g-2}({C})$ is primitive ([@v Proposition II.0]). In this note we consider the existence of complete and primitive $g^1_d$ for all $d$ near $g-1$ and prove the following result.
\[or1\] Fix an integer $k\ge 6$ and set $g(k):=2k^2+5k-6$. Fix any integer $g\ge g(k)$, any smooth curve $C$ with gonality $k$ and any integer $d$ with $g-k+3 \le d\le g-2$.
\(a) $C$ has a complete and primitive $g^1_d$ and every irreducible component of $W^1_d({C})$ has dimension $2d-g-2$.
\(b) Assume that either $k$ is odd or $d>g-k+3$ or $d=g-k+3$, $k$ is even, but $C$ is not a double covering of a smooth curve of genus $k-3$ and gonality $k/2$. Then a general element of every irreducible component of $W^1_d({C})$ is primitive.
([c]{}) Assume that $k$ is even and that $C$ is a double covering of a smooth curve of genus $k-3$ and gonality $k/2$.
(c1) There exist an irreducible component of $W^1_{g-k+3}({C})$ whose general member is base point free and an irreducible component of $W^1_{g-k+3}({C})$ whose general member has $g-2k+3$ base points.
(c2) Let $\Gamma$ be any irreducible component of $W^1_{g-k+3}({C})$; if the general element of $\Gamma$ is base point free, then it is primitive.
\[or2\] Fix an integer $x\ge 3$. Set $w:= \lfloor x/2\rfloor$, $z:= \lfloor (x-1)/2\rfloor$, $g_1(x):= 2x(2x-z)-4x+z+2$ and $g_2(x):= 4x^2-2wx+w$. Fix an integer $g\ge \max \{g_1(x),g_2(x)\}$. Let $C$ be a smooth curve of genus $g$ and gonality at least $x+3$. The interested reader may reformulate an analogous of Theorem \[or1\] with $d =g-x$ and prove it following verbatim the proof of Theorem \[or1\]. In the case $g-d=4$ of Theorem \[or1\] it is sufficient to assume that $g\ge 64$ (see Theorem \[s1.0\]).
\[or3\] In the exceptional cases of Theorem \[or1\] we have a description of the irreducible components of $W^1_{g-k+3}({C})$ whose general element has base points. We have $\dim (W^1_k({C}))=1$ and each element of these components is obtained from some $R\in W^1_k({C})$ adding a base locus of degree $g-2k+3$. Let $Y$ be any smooth curve of genus $g$ and gonality $k$ with $\dim (W^1_k({Y}))=1$. If $W^1_{g-k+3}(Y)$ has pure dimension $g-2k+4$, then it has at least one component formed by pencils with a base locus of degree $g-2k+3$. Steps (a) and (b) of the proof of Theorems \[or1\] show that if $g\gg k$, then $k$ is even and $Y$ is a double covering of a smooth curve of genus $k-3$ and gonality $k/2$.
Many thanks are due to E. Sernesi for stimulating and interesting conversations.
The proofs {#Sp}
==========
\[s1\] Fix integers $g, x$ such that $x\ge 2$ and $g\ge 4x+3$. Let $C$ be a smooth curve of genus $g$. Let $T$ be an irreducible component of $W^1_{g-x}({C})$.
\(a) If $\dim (T) > g-2x-2$, then $\dim (W^1_{2x}({C})) \ge x-1$.
\(b) If a general element of $T$ has a base point, then $\dim (W^1_{2x+2}({C})) \ge x$.
Since $g \ge 2x+2$, Brill-Noether theory gives $W^1_{g-x}({C})\ne \emptyset$ and that each irreducible component of $W^1_{g-x}({C})$ has dimension at least $g-2x-2$ ([@acgh Ch. IV]).
First assume $\dim (T)\ge g-2x-1$. Set $d: =g-x$ and $j=x-1$. We have $g-2x-1 =d-2-j$, $d \ge j+2$ and $d \le g-1-j$ (the latter inequality is an equality). By [@m] or [@h Theorem 1] we have $\dim (W^1_{2x}({C})) \ge x-1$.
Now assume that a general element of $T$ has at least one base point. We get an irreducible component of $W^1_{g-x-1}({C})$ with dimension at least $g-2x-3$. Apply part (a) with the integer $x':= x+1$ instead of the integer $x$.
Since $2d-g-2\ge 0$, Brill-Noether theory says that $W^1_d({C}) \ne \emptyset$ and that each irreducible component of $W^1_d({C})$ has dimension at least $2d-g-2$. Let $T$ be an irreducible component of $W^1_d({C})$ and let $R$ be a general element of $T$. As in the case of the general member of any irreducible component of any $W^1_y({C})$ with $y\le g-1$ we have $h^0({R}) =2$. To prove Theorem \[or1\] it is sufficient to prove that $\dim (T)=2d-g-2$, that $R$ is base point free and that $h^0(R({p}))=1$ for all $p\in C$. Let $f: X\to \mathbb {P}^1$ be any degree $k$ morphism.
\(a) Assume $\dim (T) > 2d-g-2$. The case $x=g-d$ of part (a) of Lemma \[s1\] gives $\dim (W^1_{2g-2d}({C})) \ge g-d-1$. Let $\Gamma$ be any irreducible component of $W^1_{2g-2d}({C})$ with $\dim (\Gamma)
\ge g-d-1$. Let $R'$ be a general element of $\Gamma $. Since $R'$ is general in an irreducible component of some $W^1_y({C})$, $y\le g+1$, we have $h^0(R') =2$. Let $s\ge 0$ be the degree of the base locus $B$ of $R'$. Varying $R'$ we get an irreducible family $\Gamma '\subseteq W^1_{2g-2d-s}({C})$ with $\dim (\Gamma ') \ge g-d-1$. Set $R'':= R'(-B)$. Let $u: C\to \mathbb {P}^1$ be the morphism associated to $|R''|$ and let $\alpha : C\to \mathbb {P}^1\times \mathbb {P}^1$ be the morphism associated to $(f,u)$. If $\alpha$ is birational onto its image, then $g\le k(2g-2d-s) -k-(2g-2d-s) +1 \le k(2k+6)-k-(2k+6)+1 <g(k)$, a contradiction. Hence $C$ is not birational onto its image, i.e. calling $C'$ the normalization of $\alpha ({C})$ we get a morphism $\beta ': C\to C'$ with $\beta : =\deg (\beta ')\ge 2$ and morphisms $f': C'\to \mathbb {P}^1$ and $u': C'\to \mathbb {P}^1$ such that $f = f'\circ \beta '$ and $u = u'\circ \beta '$. Since $f$ computes the gonality of $C$, we get that $C'$ has genus $q_{R'}>0$ and that $C'$ has gonality $k/\beta$.
First assume $q_{R'} \ge 2$. Since $\Gamma$ is irreducible and $C$ has only finitely many non-constant morphisms to curves of genus between $2$ and $g-1$ by a theorem of de Franchis, we get that $C'$, $\beta $ and $\beta '$ do not depend from the choice of $R'$. Since $h^0(R'')=2$, we have $h^0(C,L'')=2$, where $L''$ is the line bundle on $C'$ with $\beta ^{'\ast }(L'')\cong R''$. Since $h^0(R'')=2$, we have $h^0(L'') =2$ and hence $(2g-2d-s)/\beta \le q+1$. By [@fhl Theorem 1] we have $W^1_{(2g-2d-s)-\beta \lfloor (g-d-1)/2\rfloor}({C}) \ne \emptyset$ and hence $k\le (2g-2d-s) -\beta \lfloor (g-d-1)/2\rfloor \le (2g-2) -\beta \lfloor (g-d-1)/2\rfloor$. Since $\beta \ge 2$ and $g-d\le k-3$, we get $k \le 2g-2d -g+d+2
\le 2k-6-k+5$, a contradiction.
Now assume $q_{R'}=1$. Since $C'$ has gonality $k/\beta$, we get $\beta =k/2$. Hence $2g-2d-s$ is divisible by $k/2$. Since $d\le g-2$, we have $\dim (W^1_{2g-2d-s}({C})) \ge 3$ and hence $2g-2d-s >k$. Therefore $2g-2d-s \ge 3k/2$. Since $q_{R'}=1$, we have $h^0(C',L'') = \deg (L'') \ge 3 > h^0(R'')$, a contradiction.
\(b) In this step we prove that $R$ has no base points if one of the conditions in part (b) of the statement of Theorem \[or1\] is satisfied. If $R$ has a base point, then part (b) of Lemma \[s1\] with $x=g-d$ gives $\dim (W^1_{2g-2d+2}({C})) \ge g-d$. Let $\Gamma _1$ be any irreducible component of $W^1_{2g-2d+2}({C})$ with $\dim (\Gamma _1)
\ge g-d$. Let $R'$ be a general element of $\Gamma _1$. Since $R'$ is general in an irreducible component of some $W^1_y({C})$, $y\le g+1$, we have $h^0(R') =2$. Let $s\ge 0$ be the degree of the base locus $B$ of $R'$. Varying $R'':= R'(-B)$ we get an irreducible family $\Gamma '\subseteq W^1_{2g-2d+2-s}({C})$ with $\dim (\Gamma ') \ge g-d$ and with $R''$ as its general member. Let $u: C\to \mathbb {P}^1$ be the morphism associated to $|R''|$ and let $\alpha : C\to \mathbb {P}^1\times \mathbb {P}^1$ be the morphism associated to $(f,u)$. If $\alpha$ is birational onto its image, then $g\le k(2g-2d+2-s) -k-(2g-2d+2-s) +1 \le k(2k+8)-k-(2k+8)+1 =g(k)-1$, a contradiction. Hence $C$ is not birational onto its image, i.e. calling $C'$ the normalization of $\alpha ({C})$ we get a morphism $\beta ': C\to C'$ with $\beta : =\deg (\beta ')\ge 2$ and morphisms $f': C'\to \mathbb {P}^1$ and $u': C'\to \mathbb {P}^1$ such that $f = f'\circ \beta '$ and $u = u'\circ \beta '$. Since $f$ computes the gonality of $C$, we get that $C'$ has genus $q_{R'}>0$ and that $C'$ has gonality $k/\beta$.
First assume $q_{R'} \ge 2$. Since $\Gamma$ is irreducible and $C$ has only finitely many non-constant morphisms to curves of genus between $2$ and $g-1$ by a theorem of de Franchis, we get that $C'$, $\beta '$ and $\beta$ does not depend from the choice of $R'$. Since $h^0(R'')=2$, we have $h^0(C,L'')=2$, where $L''$ is the line bundle on $C'$ with $\beta ^{'\ast }(L'')\cong R''$. Since $h^0(R'')=2$, we have $h^0(L'') =2$ and hence $(2g-2d+2-s)/\beta \le q+1$. By [@fhl Theorem 1] we have $W^1_{2g-2d+2-2s-\deg (B_1)-\beta \lfloor (g-d)/2\rfloor}({C}) \ne \emptyset$ and hence $k\le (2g-2d+2-s) -\beta \lfloor (g-d)/2\rfloor$. Since $2 \le g-d \le k-3$, we get $k=g-d+3$, $\beta =2$, $s=0$, $g-d$ odd and $q\ge k-3$. We also get that $C'$ has gonality $k/2$. Since $\dim (W^1_{k-2}(C')) \ge \dim (\Gamma _1) \ge k-3 \ge q$, we get $q=k-3$. We are in the exceptional case allowed in the statement of Theorem \[or1\].
Now assume $q_{R'}=1$. Since $C'$ has gonality $k/\beta$, we get $\beta =k/2$. Hence $2g-2d-s$ is divisible by $k/2$. Since $d\le g-2$, we have $\dim (W^1_{2g-2d-s}({C})) \ge 3$ and hence $2g-2d-s >k$. Therefore $2g-2d-s=3k/2$. Since $q_{R'}=1$, we have $h^0(C',L'') = \deg (L'') = 3 > h^0(R'')$, a contradiction.
([c]{}) Assume the existence of $p_R\in C$ such that $h^0(R(p_R)) =3$. Since $R$ is base point free and $h^0({R}) =2$, $M:= R(p_R)$ is base point free. Let $u: C\to \mathbb {P}^2$ be the morphism induced by $|M|$. Since $R$ is general in $T$, we get $\dim (W^2_{d+1}({C})) \ge 2d-g-3$ and $\dim (W^2_{d+1}({C})) \ge 2d-g-2$, unless the same general $M$ comes from infinitely many pairs $(R_1,P_{R_1})$ with $R_1$ general in $T$. Since a smooth plane curve of degree $d+1$ has gonality $d>k$, either $\deg (u) >1$ or $u({C})$ is a singular curve. First assume $\deg (u)=1$. Taking the a linear projection from one of the finitely many singular points of $u({C})$ we get $\dim (W^1_{d-1}({C})) \ge 2d-g-3$. The case $x_1:= x+1$ of part (a) of Lemma \[s1\] gives $\dim (W^1_{2g-2d+2}({C})) \ge g-d$. We are in the exceptional case described in step (b). Now assume $\deg (u)>1$. If $\dim (W^2_{d+1}({C})) \ge 2d-g-2$, we get the same lower bound for $\dim (W^1_{d-1}({C}))$ taking a linear projection from any point of $u({C})$. Taking a linear projection we get a better estimate if either $\dim (W^2_{d+1}({C})) \le 2d-g-3$ or $\deg (u)\ge 3$ or $u({C})$ is singular. Now assume $\deg (u) =2$ and that $u({C})$ is smooth. Since $\deg (u({C})) =(d+1)/2 \ge (g-k+34)/2$, we get that $u({C})$ has genus $q' \ge (g-k+3)(g-k+2)/8$. Since $g \ge 2q'-1$ (Riemann-Hurwitz), we get a contradiction.
\(d) To conclude the proof we may assume that $k$ is even and the existence of a degree $2$ covering $\beta ':C\to C'$ with $C'$ smooth of genus $k-3$ and gonality $k/2$. By step (a) $W^1_{g-k+3}({C})$ has pure dimension $g-2k+4$. Remark \[or3\] gives the existence of an irreducible component of $W^1_{g-k+3}({C})$ whose general member has $g-2k+3$ base points. Brill-Noether theory gives $\dim (W^1_t(C'))\ge 2t-k+1$ for all $t\in \mathbb {N}$ such that $k/2 \le t \le k-2$. Since $C'$ has gonality $k/2$, [@fhl Theorem 1] first gives $\dim (W^1_{k/2}(C'))=1$ and then $\dim (W^1_t(C'))\le 2t-k+1$ for all $t\in \mathbb {N}$ such that $k/2 < t \le k-2$. Hence $C'$ has dimensionally the Brill-Noether theory for pencils of a general curve of genus $k-3$. This is enough to carry over the proof of [@bk Theorem 0.1] (see the proofs of Lemmas 1.2, 1.3 and Theorem 0.1 in [@bk]). Hence (with this observation concerning $C'$), [@bk Theorem 0.1] gives the existence of a degree $g-k+3$ morphism $f: C\to \mathbb {P}^1$ not composed with $\beta '$, i.e. such that the morphism $(\beta ',f): C\to \mathbb {P}^1\times \mathbb {P}^1$ is birational onto its image. We claim that we may take as $f$ a complete pencil. This claim is true by [@bk Lemma 1.3], which describes all the $W^1_{g-k+3}({C})$, $k-3 =p_a(C')$, whose general element has base points and the existence of at least another components of $W^1_{g-k+3}({C})$ ([@bk first line of page 155]; it is the second line of page 155, which loses the completeness statement in [@bk Theorem 0.1]). So we proved the existence of an irreducible component of $W^1_{g-k+3}({C})$ containing a base point free and complete $g^1_{g-k+3}$.
Let $\Gamma$ be any irreducible component of $W^1_{g-k+3}({C})$ containing a base point free and complete $g^1_{g-k+3}$, $\delta$. Since $\delta$ is complete, the general element of $\Gamma$ is complete and base point free. Let $R$ be a general element of $\Gamma$. Assume the existence of $p_R\in C$ such that $h^0(R(p_R)) =3$. Since $R$ is base point free and $h^0({R}) =2$, $M:= R(p_R)$ is base point free. Since $R$ is general in $\Gamma$, we get $\dim (W^2_{g-k+4}({C})) \ge g-2k+3$ and $\dim (W^2_{g-k+4}({C})) \ge g-2k+4$, unless a general $M$ comes from infinitely many pairs $(R,p_R)$ . Let $u: C\to \mathbb {P}^2$ be the morphism induced by $|M|$. First assume $\deg (u) =1$. Since $g-k+4 >k+1$ and a smooth plane curve of degree $g-k+4$ has gonality $g-k+3>k$, $u({C})$ is a singular curve. Therefore taking a linear projection from a singular point of $u({C})$ we obtain $\dim (W^1_{g-k+2}({C})) \ge g-2k+3$ (because $u({C})$ has only finitely many singular points). Write $\dim (W^1_{g-k+2}({C})) = g-k-1-j+e$ with $e\ge 0$ and $j =k-4$. By [@h Theorem 1] we have $\dim (W^1_{2k-6-2e}({C})) = k-1-e$ and (hence, even if $e>0$ by [@fhl Theorem 1]) we have $\dim (W^1_{2k-6}({C})) \ge k-4$. This is the case handled in step (a). Now assume $\deg (u) >1$ and call $C''$ the normalization of $u({C})$, $q$ its genus, and $v: C\to C''$ the morphism through which $u$ factors. Write $M = v^\ast (L)$ with $L$ base point free line bundle on $C''$ with $h^0(C,L) =3$. If $\deg (u)\ge 3$, then fixing $o\in u({C})_{\mathrm{reg}}$ and taking the linear projection from $o$ we get $\dim (W^1_{g-k+1}({C})) \ge g-2k+3$ (the same for all $M$, because $q\ge 2$ and we may apply a theorem of de Franchis). Write $\dim (W^1_{g-k+1}({C})) = g-k-1-j+e'$ with $e'\ge 0$ and $j=k-4$. We get $\dim (W^1_{2k-8}({C})) \ge k-3$ and conclude. Now assume $\deg (u) =2$. In this case $(g-k+4)/2\in \mathbb {Z}$. If $u({C})$ is singular, then a linear projection from one of its singular points gives $\dim (W^1_{g-k}({C}))\ge g-2k+3$ and so $\dim (W^1_{g-k+3}({C}))\ge g-2k+6$, contradicting step (a). Hence $u({C})$ is smooth and so it has genus $q':= (g-k+2)(g-k)/8$. Riemann-Hurwitz gives $g\ge 2q'-1$, a contradiction.
\[or4\] Fix an even integer $k\ge 6$ and set $x:= k-3$. Let $C'$ be any smooth curve of genus $x$ and gonality $k/2$. We have $\dim (W^1_{k/2}(C')) =1$. Let $u: C\to C'$ be a degree $2$ covering of genus $g \ge 3x+4$. $W^1_{g-x}({C})$ contains the $(g-2x-2)$-dimensional family of $g^1_{g-x}$ obtained from the pull-backs of the elements of $W^1_{k/2}(C')$ adding $g-2x-3$ base points. Since $g > 2x+k$, any base point free pencil on $C$ of degree $ < k$ is the pull-back of a base point free pencil on $C'$ by the Castelnuovo-Severi inequality ([@k]). Hence $C$ has gonality $k$. Hence the exceptional cases in Theorem \[or1\] arises.
See also [@bkp] (resp. [@bks]) for the existence of spanned pencils on curves which are double (resp. multiple) coverings. By [@bk2 Theorem 2.2] for every integer $d\ge g-k+2$ every $k$-gonal curves of genus $g > (3k - 6)(k - 1)$ has a base point free $g^1_d$.
\[s1.0\] Let $C$ be a smooth curve of genus $g\ge 64$ with gonality $k\ge 7$. Then $C$ has a primitive $g^1_{g-4}$, $W^1_{g-4}({C})$ has pure dimension $g-10$ and the general element of every irreducible component of $W^1_{g-4}({C})$ is primitive.
Since $2(g-4) -g -2 \ge 0$, Brill-Noether theory gives $W^1_{g-4}({C}) \ne \emptyset$ and that any irreducible component $T$ of $W^1_{g-4}({C})$ has dimension at least $2(g-4)-g-2 =g-10$. Fix a general $R\in T$. As in the case of any irreducible component of any $W^r_d({C})$ we have $\dim |R| =1$. To prove Theorem \[s1.0\] it is sufficient to prove that $R$ is base point free, that $h^0(R(q)) = 2$ for every $q\in C$, and that $\dim T =g-10$.
\(a) In this step we prove that $R$ has no base points. If $R$ has a base point, then the case $x=4$ of part (b) of Lemma \[s1\] gives $\dim (W^1_{10}({C})) \ge 4$.
(a1) Assume $k=7$. Let $h: C\to \mathbb {P}^1$ be any degree $7$ morphism. Since $7$ is a prime number and $g>7\cdot 7 -7-7+1$, the genus formula for integral curves contained in $\mathbb {P}^1\times \mathbb {P}^1$ shows that $h$ is unique. Let $m$ be the first integer $>7$ such that $C$ has a base point free $g^1_m$. Since $\dim (W^1_{10}({C}))> 10-7$ and $\dim (W^1_7({C})) =0$, we have $m\le 10$. Every integral curve of $\mathbb {P}^1\times \mathbb {P}^1$ with bidegree $(7,m)$ has arithmetic genus $7m-7-m+1 \le 70-7-m+1 <g$, a contradiction.
(a2) Assume $k=8$. First assume that $C$ has only finitely many $g^1_8$. Let $h: C\to \mathbb {P}^1$ be any degree $8$ morphism. Since $\dim (W^1_{9}({C})) \ge 2$, we get the existence of a degree $9$ morphism $f: C \to \mathbb {P}^1$. Since $8$ and $9$ are coprime, the map $(h,f): C\to \mathbb {P}^1\times \mathbb {P}^1$ is birational onto its image, $\Gamma$. Since $p_a(\Gamma ) =8\cdot 9-8-9+1$, we get $g\le 56$, a contradiction. Now assume $\dim (W^1_8({C})) \ge 1$ and hence $\dim (W^1_8({C}))=1$. Since $g > 8\cdot 9-8-9+1$, the proof of step (a1) shows that every element of $W^1_9({C})$ has one base point. Since $\dim (W^1_{10}({C})) > 3 + \dim (W^1_8({C}))$, there is a degree $10$ morphism $\ell : C\to \mathbb {P}^1$. Since $g > 8\cdot 10 -8-10+1$, we get that $(h,\ell ): C\to \mathbb {P}^1\times \mathbb {P}^1$ has degree $2$ onto its image $\Phi _{(h,\ell )}$ and $\Phi _{(h,\ell )}$ is an integral curve of bidegree $(4,5)$. Since $k>4$, the normalization $D_{(h,\ell )}$ of $\Phi _{(h,\ell )}$ has genus $>1$. Hence varying $u$ and $\ell$ we only get finitely many smooth curves $D_{(h,\ell )}$. We fix one such normalization $D = D_{(h,\ell )}$ and call $v: C\to D$ the map induced by $(u,\ell )$ and $q$ the genus of $D$. We have $q\le 4\cdot 5-4-5+1 =12$ by the Castelnuovo - Severi inequality. Since $g>2q+9$ all base point free pencils of degree $8$ (resp. $10$) pencils on $C$ are induced by a degree $4$ (resp. $5$) pencil on $D$ (again by the Castelnuovo - Severi inequality). Since $\dim (W^1_{10}({C})) \ge 4$, we get $\dim (W^1_5({D})) \ge 4$. Hence $W^1_3({C}) \ne \emptyset$ ([@fhl]). Therefore $k\le 6$, a contradiction.
\(b) Assume $\dim (T) >g-10$. The case $x=4$ of part (a) of Lemma \[s1\] gives $\dim (W^1_8({C})) \ge 3$. Hence $k\le 7$ and if $k=7$, then $\dim (W^1_7({C}))=1$. Assume $k=7$. Since $7$ is a prime number and $C$ has at least two $g^1_7$, the Castelnuovo - Severi inequality gives $g\le 7\cdot 7-7-7+1$, a contradiction.
([c]{}) Assume the existence of $q_R\in C$ such that $h^0(R(q_R)) =3$. Since $R$ is general in $T$, we get $\dim (W^2_{g-3}({C})) \ge g-11$. Since $R$ is base point free and $h^0({R}) =2$, $M:= R(q_R)$ is base point free. Let $u: C\to \mathbb {P}^2$ be the morphism induced by $|M|$. Since $g < (g-4)(g-5)/2$ either $\deg (u) >1$ or $u({C})$ is a singular curve. Therefore taking a linear projection from a point of $u({C})$ (case $\deg (u)>1$) or a singular point of $u({C})$ (case $\deg (u) =1$), we obtain $\dim (W^1_{g-5}({C})) \ge g-11$. The case $x=5$ of part (a) of Lemma \[s1\] gives $\dim (W^1_{10}({C})) \ge 4$. We are in the case excluded in step (a).
Let $C$ be a smooth curve of genus $g$ with a primitive $R\in \mathrm{Pic}^d({C})$, $d\le g-2$. Since $\omega _C\otimes R^\vee$ is primitive, $C$ has a primitive $g^{g-d}_{2g-2-d}$. In the case $k\ge 5$ and $d=g-2$ for a general $R$ the dual linear series $\omega _C\otimes R^\vee$ is birational onto it image and with image a plane nodal curve ([@v Proposition II.0]). See [@m1], [@mp], [@ckm2] for the very ampleness of some $g^3_{g+1}$ (case $d=g-3$).
[99]{}
E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, Geometry of algebraic curves. Vol. I, Springer, Berlin, 1985.
E. Ballico and C. Keem, On multiple coverings of irrational curves, Arch. Math. 61 (1995), 151–160.
E. Ballico and C. Keem, Variety of linear systems on double coverings of curves, J. Pure Appl. Algebra 128 (1998), 213–224.
E. Ballico, C. Keem and S. Park, Double coverings of curves, Proc. Amer. Math. Soc. 132 (2004), no. 11, 3153–3158.
E. Ballico, C. Keem and D. Shin, Pencils on coverings of a given curve whose degree is larger than the Castelnuovo-Severi lower bound, Bull. Inst. Math. Acad. Sin. (N.S.) 2 (2007), no. 1, 103–107.
E. Ballico and C. Keem, Clifford index of smooth algebraic curves of odd gonality with big $W^r_d({C})$, Osaka J. Math. 39 (2002), 283–292.
M. Coppens, C. Keem and G. Martens, Primitive linear series on curves, Manuscripta Math. 77 (1992), 237–264.
M. Coppens, C. Keem and G. Martens, The primitive length of a general $k$-gonal curve, Indag. Mathem., N.S. 5 (1994), no. 2, 145–159.
M. Coppens, C. Keem and G. Martens, Space models of degree $g+1$ curves of genus $g$, Manuscripta Math. 140 (2013), 237–247.
M. Coppens and G. Martens, Linear series on 4-gonal curves, Math. Nachr. 213 (2000), 35–55.
W. Fulton, J. Harris and R. Lazarsfeld, Excess linear series on an algebraic curve, Proc. Amer. Math. Soc. 92 (1984), 320–322.
R. Horiuchi, Gap orders of meromorphic functions on Riemann surfaces, J. Reine Angew. Math. 336 (1982), 213–220.
E. Kani, On Castelnuovo’s equivalence defect, J. Reine Angew. Math. 352 (1984), 24–70.
G. Martens, A remark on very ample linear series. II, Arch. Math. Arch. Math. 99 (2012), 111–124.
G. Martens and S. Park, A remark on very ample linear series, Arch. Math. 80 (2003), 611–614.
J. P. Matelski, On geometry of algebraic curves, Ph. D. Thesis, Princeton, 1978.
H. Martens, On the varieties of special divisors on a curve, J. Reine Angew. Math. 233 (1967), 111–120.
C. Voisin, Courbes tétragonal et cohomologie de Koszul, J. Reine Angew. Math. 387 (1988), 111–123.
[^1]: The author was partially supported by MIUR and GNSAGA of INdAM (Italy).
|
---
abstract: 'The article is devoted to extension groups in the category of functors from a small category to an additive category with an Abelian structure in the sense of Heller. It is proved that under additional assumptions there exists a spectral sequence which converges to the extension groups.'
title: |
On Ext in the Category of Functors to\
Preabelian Category [^1]
---
[A. A. Husainov]{}\
Komsomolsk-on-Amur State Technical University,\
Faculty of Computer Technologies,\
681013, Komsomolsk-on-Amur, prosp. Lenina, 27, Russia\
husainov@knastu.ru
[A. Pancar]{}\
Ondokuz Mayis University, Faculty of Art and Sciences,\
Department of Mathematics, 55139, Kurupelit, Samsun, Turkey\
apancar@omu.edu.tr
[M. Yapici]{}\
Ondokuz Mayis University, Faculty of Art and Sciences,\
Department of Mathematics, 55139, Kurupelit, Samsun, Turkey\
myapici@omu.edu.tr
1991 AMS Subject Classification: 18G25
[Key words : *Extension groups, cohomology of categories.*]{}
INTRODUCTION
============
Let ${{\rm{Ab}}}$ be the category of abelian groups, ${{\rm{Mod}}}_R$ the category of left $R$-modules over a ring with identity.
Jibladze and Pirashvili [@jib1986], [@jib1991] proved that if ${\cal A}={{\rm{Mod}}}_R$ then for every small category $\Bbb C$ and functors $F,G:{\Bbb C}\to {\cal A}$ there exists a first quadrant spectral sequence with $$E^{p,q}_2=H^p({\Bbb C},\{{{\rm{Ext}}}^q(F(a),G(b))\})$$ which converges to ${{\rm{Ext}}}^n(F,G)$. Here $H^n({\Bbb C},B)$ are the $n$-th Hochschild-Mitchell cohomology groups with coefficients in a bimodule $B:{\Bbb C}^{op}\times {\Bbb C}\to {{\rm{Ab}}}$ in the sense of [@bau1985], and $\{{{\rm{Ext}}}^q(F(a),G(b))\}$ is the functor ${{\rm{Ext}}}^q(F(-),G(=)): {\Bbb C}^{op}\times {\Bbb C}\to {{\rm{Ab}}}$. This spectral sequence was generalized by Khusainov [@X19921] to an Abelian category $\cal A$ with a proper class [@mac1975]. We generalize it to additive categories with Abelian structures in the sense of Heller [@hel1958].
ABELIAN STRUCTURE
=================
Let $\cal A$ be a preadditive category, $B\in \cal A$ an object. A [*direct sum decomposition*]{} $(A,A',i,p,i',p')$ of $B$ consist of objects and morphisms $$A \stackrel{i}\rightarrow B \stackrel{i'}\leftarrow{A'}, \quad
A \stackrel{p}\leftarrow B \stackrel{p'}\rightarrow{A'}$$ satisfying to the conditions: $$\begin{array}{c}
p\circ i=1_A, \quad p'\circ i'=1_{A'},\\
p'\circ i=0_{AA'},\quad p\circ i'=0_{A'A},\\
i\circ p + i'\circ p'=1_B
\end{array}$$
A morphism $\rho:B\to A$ is called a [*retraction*]{} if there exists a morphism $\nu:A\to B$ such that $\rho\circ \nu=1_A$.
An [*additive*]{} category is a preadditive category with a zero-object and with finite products. We follow to Mac Lane [@mac1975] for the notions of the kernels and cokernels.
\[l21\] Let $\cal A$ be an additive category. Then the following conditions are equivalent:
- Every retraction has a kernel;
- For every morphisms $\rho:B\to A$ and $i:A\to B$ in $\cal A$ satisfying $\rho \circ i=1_A$ there are an object $A'$ and morphisms $\rho':B\to A'$, $i':A'\to B$ such that $(A,A',i,\rho,i',\rho')$ is a direct sum decomposition of B.
The condition (2) is called Cancellation Axiom in [@hel1958]. Thus, if $\cal A$ is an additive category in which all retractions have kernels then $\cal A$ is a category with Cancellation Axiom.
The sequence of morphisms $$\begin{CD}
0 @>>> A' @>\alpha'>> A @>\alpha''>> A'' @>>> 0
\end{CD}$$ is called a [*short exact sequence*]{}, abbreviated “s.e.s”, if $(A'', \alpha'')$ is a cokernel of $\alpha'$ and $(A', \alpha')$ is a kernel of $\alpha''$. More generally, a sequence $$\begin{CD}
A_n@>f_n>> A_{n-1} @>f_{n-1}>>\dots @>f_1>> A_0
\end{CD}$$ is [*exact*]{} if there are s.e.s $0 \to B_j \stackrel{u_j}\to A_j \stackrel{\nu_j}\to B_{j-1} \to 0$ for $j=1,2,\dots,n-1$, an epimorphism $\nu_n:A_n\to B_{n-1}$ and a monomorphism $u_0:B_0\to A_0$ such that $f_j=u_{j-1}\circ \nu_j,\quad j=1,2,\dots,n$.
Let $\cal A$ be an additive category with Cancellation Axiom. An [*Abelian structure*]{} on $\cal A$ is a subclass $\cal P \subseteq {{\rm{Mor}}}{\cal A}$ whose elements are called to be [*proper maps*]{}. A short exact sequence whose maps are proper is a [*proper s.e.s*]{}.
The class $\cal P$ is to be subject to the following axioms:
- $1_A\in \cal P$ for all $A\in {{\rm{Ob}}}\cal A$.
- If $(f:B\to C)\in \cal P$, $(g:A\to B)\in \cal P$ and $g$ is an epimorphism then $f\circ g\in{\cal P}$; dually, if $f\in \cal P$, $g\in\cal P$ and $f$ is a monomorphism then $f\circ g\in \cal P$.
- If $f\circ g\in \cal P$ is a monomorphism then $g$ is proper; dually, if $f\circ g\in \cal P$ is an epimorphism then $f$ is proper.
- For every proper map $f:B\to D$ there are proper s.e.s $0 \to A \to B \to C \to 0$ and $0 \to C \to D \to E \to 0$ such that the following diagram $$\begin{CD}
0@>>> A @>>> B @>>> C @>>>0\\
@. @. @VVfV @VV1_CV\\
0@<<< E @<<< D @<<< C @<<< 0
\end{CD}$$ is commutative.
- If in the commutative diagram $$\begin{CD}
@. 0 @. 0@. 0\\
@. @VVV @VVV @VVV\\
0@>>> A' @>>> A @>>> A'' @>>> 0\\
@. @VVV @VVV @VVV\\
0@>>> B' @>>> B @>>> B'' @>>> 0\\
@. @VVV @VVV @VVV\\
0@>>> C' @>>> C @>>> C'' @>>> 0\\
@. @VVV @VVV @VVV\\
@. 0 @. 0@. 0
\end{CD}$$ all collumns and the second two rows are proper s.e.s, then the first row is a proper s.e.s.
We remark that (P3) implies the existence of kernels and cokernels for all proper maps by [@hel1958 Prop. 2.2].
Let $\cal A$ be an additive category with Cancellation Axiom, ${\Bbb C}$ a small category. We denote by ${\cal A}^{\Bbb C}$ the category of functors ${\Bbb C}\to {\cal A}$. If $\rho:F\to G$ is a retraction in ${\cal A}^{\Bbb C}$ then $\rho_c:F(c)\to G(c)$ are retractions for all $c\in {{\rm{Ob}}}{\Bbb C}$ and have kernels by Lemma 2.1. Consequently, $\rho:F\to G$ has a kernel. Therefore ${\cal A}^{\Bbb C}$ is the additive category with Cancellation Axiom.
The Abelian structure generalizes the proper class in an Abelian category in the sense of [@mac1975]. We refer to [@kuz1972] and [@hel1958] for other examples of Abelian structure.
By [@hel1958 Prop. 3.5] an additive category carries Abelian structures if and only if it satisfies the Cancellation Axiom.
Let $\cal A$ be an additive category with an Abelian structure $\cal P$, $\Bbb C$ a small category, ${\Bbb C}{\cal P}$ the class of all natural transformations $\eta:F\to G$ for which $\eta_c\in{\cal P}$ for all $c\in {{\rm{Ob}}}{\Bbb C}$. Then the class ${\Bbb C}{\cal P}$ is an Abelian structure on ${\cal A}^{\Bbb C}$.
[Proof.]{} It is clear that ${\cal A}^{\Bbb C}$ is the category with Cancellation Axiom by Lemma \[l21\]. We check the axioms (P0)-(P4) for ${\Bbb C}{\cal P}$:
- Each identity $1_F:F\to F$ consists of $1_{F(c)}$, $c\in {{\rm{Ob}}}{\Bbb C}$. Hence $1_F\in{\cal P}$.
- A morphism $\eta:F \to G$ in ${\cal A}^{\Bbb C}$ is a monomorphism if and only if $\ker \eta=0$. Hence $\eta $ is a monomorphism if and only if $\eta_c$ are monomorphisms for all $c\in{{\rm{Ob}}}{\Bbb C}$; dually for epimorphisms.
- Analogously.
- For every $f\in {\cal P}$ we choose a kernel ($\ker f$,$k_f$) of $f$. By definition of kernels [@mac1975 Ch.IX] for every commutative diagram $$\begin{CD}
A @>f>> B\\
@V\alpha VV @VV\beta V \\
A' @>f'>> B'
\end{CD}$$ with $f,f'\in {\cal P}$ and $\alpha, \beta\in {{\rm{Mor}}}{\cal A}$ there exists unique $\ker f\to \ker f'$ for which the following diagram is commutative $$\begin{CD}
\ker f@>k_f>> A @>f>> B\\
@VVV @VV\alpha V @VV\beta V\\
\ker f' @>k_{f'}>> A' @>f'>> B'
\end{CD}$$ Dually, we choose cokernels $({{\rm{coker}}}f, c_f)$. Consider the category which objects are all members of ${\cal P}$ and morphisms are commutative diagrams (1).
We see that $\ker$ and ${{\rm{coker}}}$ are functors from this category to $\cal A$. Hence for every natural transformation $\eta:F\to G$ with $\eta_c\in {\cal P}$, for all $c\in {{\rm{Ob}}}{\Bbb C}$, we have a diagram of natural transformations $$\begin{CD}
0 @>>> \ker \eta @>k_{\eta}>> F @>\varepsilon>> {{\rm{coker}}}k_{\eta} @>>> 0\\
@. @. @VV\eta V @VV\xi V \\
0 @<<< {{\rm{coker}}}\eta @<c_{\eta}<< G @<m<< \ker c_{\eta} @<<< 0
\end{CD}$$ where $\xi$ is an isomorphism of functors. By [@hel1958 Prop.3.3.] all equivalences in $\cal A$ belong to $\cal P$, therefore $\xi\circ \varepsilon$ is a proper epimorphism, and we obtain (P3) for ${\Bbb C}{\cal P}$.
- $A$ sequence of functors is exact if and only if it is exact at every $c\in {{\rm{Ob}}}{\Bbb C}$. Thus (P4) holds.
Q.E.D.
An object $P$ is called $\cal P$-projective if for every proper epimorphism $\varepsilon:A\to B$ and for a morphism $\alpha :P\to B$ there is $\beta :P\to A$ such that $\varepsilon \circ \beta= \alpha$.
Let $\cal A$ be an additive category with Cancellation Axiom, ${\cal P}\subseteq {{\rm{Mor}}}{\cal A}$ an Abelian structure. Following Heller we will say that $\cal A$ has enough $\cal P$-projectives if for each $A\in {{\rm{Ob}}}{\cal A}$ there is a $\cal P$-projective object $P(A)\in \cal A$ and a proper epimorphism $P(A) \stackrel{\pi A}\to A$.
Let $\cal A$ be an additive category with coproducts, $\cal P$ an Abelian structure. If $\cal A$ has enough $\cal P$-projectives and coproducts of proper epimorphisms are proper, then ${\cal A}^{\Bbb C}$ has enough ${\Bbb C}{\cal P}$-projectives.
[Proof.]{} We consider the set ${{\rm{Ob}}}{\Bbb C}$ as the maximal discrete subcategory of ${\Bbb C}$. Let $O:{\cal A}^{\Bbb C}\to {\cal A}^{{{\rm{Ob}}}{\Bbb C}}$ be the restriction functor, $O(F)=F|_{{{\rm{Ob}}}{\Bbb C}}$. The category $\cal A$ has coproducts, hence there is a left adjoint functor $\Lambda$ to the functor $O$. Up to a natural isomorphism, we have $\Lambda(D)(c)=\sum_{c_0\to c}D(c_o)$ for each $D\in {\cal A}^{{{\rm{Ob}}}{\Bbb C}}$. The counit of adjunction $\varepsilon_F:\Lambda O F\to F$ splits on each $c\in {{\rm{Ob}}}{\Bbb C}$, hence $\varepsilon_F$ is a proper epimorphism in ${\cal A}^{\Bbb C}$ with respect to Abelian structure ${\Bbb C}{\cal P}$. Choose a family $P=\{P(c)\}_{c\in{{\rm{Ob}}}{\Bbb C}}\in {\cal A}^{{{\rm{Ob}}}{\Bbb C}}$ of $\cal P$-projectives and a family of proper epimorphisms $\psi=\{\psi_c:P(c)\to F(c)\}_{c\in{{\rm{Ob}}}{\Bbb C}}$ and apply the functor $\Lambda$. Then $\Lambda(\psi)_c:\Lambda(P)(c)\to \Lambda (OF)(c)$ is a proper epimorphism for each $c\in {{\rm{Ob}}}{\Bbb C}$ as the coproduct of proper epimorphisms. Consequently, $\varepsilon_F\circ(\Lambda(\psi)):\Lambda(P)\to F$ is a proper epimorphism in ${\cal A}^{\Bbb C}$.
The functor ${\cal A}^{\Bbb C}(\Lambda (P),-)$ is isomorphic to the functor ${\cal A}^{{{\rm{Ob}}}{\Bbb C}}(P,O(-))$ which is exact on all proper s.e.s in ${\cal A}^{\Bbb C}$. Thus, $\Lambda (P)$ is ${\Bbb C}{\cal P}$-projective and there is an proper epimorphism $\Lambda (P)\to F$. Q.E.D.
COHOMOLOGY OF CATEGORIES
========================
Let ${\Bbb C}$ be a small category, $F:{\Bbb C}\to {{\rm{Ab}}}$ a functor.
Consider the sequence of groups $$C^n({\Bbb C}, F)=\prod_{c_0\to \dots \to c_n}F(c_n), \quad n\geq 0.$$
Let $N_n{\Bbb C}$ be the set of all sequences of morphisms $c_0\to c_1\to \dots \to c_n$ in $\Bbb C$. Regarding each $\varphi \in \prod_{c_0\to \dots \to c_n}F(c_n)$ as a function $\varphi :N_n {\Bbb C}\to \bigcup_{c\in {{\rm{Ob}}}{\Bbb C}}F(c)$ with $\varphi (c_0\to \dots \to c_n)\in F(c_n)$, we define the homomorphisms $d^n:C^n({\Bbb C}, F)\to C^{n+1}({\Bbb C}, F)$ by the formulas $$\begin{array}{c}
(d^n\varphi)(c_0
\stackrel{\alpha_1}\to c_1 \to \dots \stackrel{\alpha_{n+1}}\to
c_{n+1})=\\
\sum\limits_{i=0}^n(-1)^{i}\varphi
(c_0 \stackrel{\alpha_1}\to \dots \stackrel{\alpha_i}\to \hat{c}_i
\stackrel{\alpha_{i+1}}\to
\dots \stackrel{\alpha_{n+1}}\to
c_{n+1})+\\
(-1)^{n+1} F(\alpha_{n+1})(\varphi(c_0 \stackrel{\alpha_1}\to
\dots \stackrel{\alpha_n}\to c_n)).
\end{array}$$ Here $(c_0 \stackrel{\alpha_1}\to \dots \stackrel{\alpha_i}\to \hat{c}_i
\stackrel{\alpha_{i+1}}\to
\dots \stackrel{\alpha_{n+1}}\to c_{n+1})$ is equal to $(c_0 \stackrel{\alpha_1}\to \dots \to c_{i-1}
\stackrel{\alpha_{i+1}\circ\alpha_i}\longrightarrow c_{i+1} \to
\dots \stackrel{\alpha_{n+1}}\to
c_{n+1})
$ if $0< i< n+1$, and to $(c_1 \stackrel{\alpha_2}\to \dots \stackrel{\alpha_{n+1}}\to c_{n+1})$ if $i=0$. It is well known that $d^{n+1}\circ d^n=0$ for all $n\geq 0$. We have obtained the complex of Abelian groups and homomorphisms $$0 \to C^0({\Bbb C}, F)\stackrel{d^0}\to \dots \to C^n({\Bbb C},F)
\stackrel{d^n}\to C^{n+1}({\Bbb C},F) \to \dots$$ which is denoted by $C^*({\Bbb C}, F)$. The cohomologies $H^n(C^*({\Bbb C}, F))=\ker d^n/{{\rm{Im}}}d^{n-1}$ are isomorphic to Abelian groups $\lim\nolimits_{\Bbb C}^n F$ (see [@oli1992], for example) where $\lim\nolimits_{\Bbb C}^n:{{\rm{Ab}}}^{\Bbb C}\to {{\rm{Ab}}}$ are the $n$-th right derived functors of the limit $\lim\nolimits_{\Bbb C}:{{\rm{Ab}}}^{\Bbb C}\to {{\rm{Ab}}}$.
In particular, if ${\Bbb C}$ has the initial object then $H^n(C^*({\Bbb C},F))=0$ for $n>0$ and $H^0(C^*({\Bbb C}, F))\cong \lim\nolimits_{\Bbb C}F$.
For each $c\in {{\rm{Ob}}}{\Bbb C}$ we denote by $c/{\Bbb C}$ the [*comma-category*]{} in the sense of [@mac1971], its objects are pairs $(a,\alpha)$ of $a\in {{\rm{Ob}}}{\Bbb C}$ and $\alpha \in
{\Bbb C}(c,a)$, and for each pair of objects $(a, \alpha \in {\Bbb C}(c,a))$ and $(b,\beta\in {\Bbb C}(c,b))$ the set of morphisms from $(a, \alpha)$ to $(b, \beta)$ consists of $\gamma\in {\Bbb C}(a,b)$ for which the following diagrams $$\begin{CD}
c @>\alpha >> a\\
@VV=V @VV\gamma V \\
c @>>\beta> b
\end{CD}$$ are commutative.
Let $Q_c:c/{\Bbb C}\to {\Bbb C}$ be the functor which carries the above diagram (2) to $\gamma :a\to b$. The category $c/{\Bbb C}$ contains the initial object $(c,1_c)$. Therefore, the functor $\lim\nolimits_{c/{\Bbb C}}$ is exact and $H^n(C^*({\Bbb C},GQ_c))=0$ for every functor $F:{\Bbb C}\to {{\rm{Ab}}}$ and $n>0$.
Let ${\Bbb C}$ be a small category. For each $a\in {{\rm{Ob}}}{\Bbb C}$ we identify the morphism $1_a$ with the object a, so ${{\rm{Ob}}}{\Bbb C}\subseteq {{\rm{Mor}}}{\Bbb C}$. Objects of the [*factorization category*]{} [@bau1985] ${\Bbb C}'$ are all morphisms of ${\Bbb C}$, and the set of morphisms ${\Bbb C}'(f,g)$ for any $f,g\in {{\rm{Mor}}}{\Bbb C}$ consists of pairs $(\alpha, \beta)$ of morphisms in $\Bbb C$ for which the diagram $$\begin{CD}
b@>\beta>> d\\
@AfAA @AAgA\\
a@<<\alpha< c
\end{CD}$$ is commutative; we denote the morphisms by $(\alpha, \beta):f\to g$. The composition is defined as $(\alpha_2, \beta_2)\circ (\alpha_1,\beta_1)=(\alpha_1\circ\alpha_2,\beta_2\circ\beta_1)$. Functors $F:€{\Bbb C}'\to {{\rm{Ab}}}$ are called [*natural systems*]{}.
Let $\Bbb C$ be a small category, $F:{\Bbb C}'\to {{\rm{Ab}}}$ a natural system. For every $n\geq 0$ we consider an Abelian group $$K^n({\Bbb C},F)=
\prod_{c_0\stackrel{\alpha_1}\to c_1\stackrel{\alpha_2}\to c_2 \to \dots
\stackrel{\alpha_n}\to c_n}
F(\alpha_n\circ\alpha_{n-1}\circ \dots \circ\alpha_1)$$ We regard elements of $K^n({\Bbb C},F)$ as maps $\varphi :N_n{\Bbb C}\to \bigcup_{g\in{{\rm{Mor}}}{\Bbb C}}F(g)$ with $\varphi(\alpha_1, \dots , \alpha_n)\in F(\alpha_n\circ\alpha_{n-1}\circ \cdots\circ \alpha_1)$ and define homomorphisms $d^n:K^n({\Bbb C},F)\to K^{n+1}({\Bbb C}, F)$ by the formula $$\begin{array}{c}
(d^n\varphi)(\alpha_1, \dots, \alpha_{n+1})=F(\alpha_1,1)\varphi
(\alpha_2,\dots,\alpha_{n+1})+\\
\sum\limits_{i=1}^n(-1)^{i}\varphi(\alpha_1,\dots
\alpha_{i-1},\alpha_{i+1}\circ\alpha_{i},\alpha_{i+2},\dots,\alpha_{n+1})
+\\
(-1)^{n+1}F(1,\alpha_{n+1})\varphi(\alpha_1,\dots,\alpha_n).
\end{array}$$ Cohomology groups $H^n(K^*({\Bbb C}, F))=\ker d^n/{{\rm{Im}}}d^{n-1}$ are called $n$-th cohomology groups $H^n({\Bbb C}, F)$ of ${\Bbb C}$ with coefficients in the natural system $F$.
For any $\alpha \in {{\rm{Mor}}}{\Bbb C}$ we denote by ${{\rm{dom}}}\alpha$ and ${{\rm{cod}}}\alpha$ the domain and the codomain. We have the functor $({{\rm{dom}}}, {{\rm{cod}}}): {\Bbb C}'\rightarrow {\Bbb C}^{op}\times{\Bbb C}$ which acts as $f \mapsto ({{\rm{dom}}}f, {{\rm{cod}}}f)$ on objects, and $((\alpha, \beta): f \rightarrow g)$ $\mapsto (\alpha, \beta)$ on morphisms of ${\Bbb C}'$. Baues and Wirsching [@bau1985] proved that $H^n({\Bbb C},F)\cong\lim\nolimits_{\Bbb C'}^nF$ for all $n\geq 0$. Functors $B:{\Bbb C}^{op}\times {\Bbb C}\to {{\rm{Ab}}}$ are called [*bimodules*]{} over $\Bbb C$. For each bimodule $B$ over $\Bbb C$ we denote by $\{B({{\rm{dom}}}\alpha, {{\rm{cod}}}\alpha)\}$ the natural system $B\circ ({{\rm{dom}}},{{\rm{cod}}}):{\Bbb C}'\to {{\rm{Ab}}}$. The [*Hochschild-Mitchell cohomologies $H^n({\Bbb C},B)$ of the category*]{} $\Bbb C$ with coefficients in the bimodule $B$ are the Baues-Wirsching cohomologies of $\Bbb C$ with coefficients in the natural system $B\circ({{\rm{dom}}},{{\rm{cod}}})$. It is easy to see that $H^n({\Bbb C},B)$ isomorphic to $\lim\nolimits_{\Bbb C'}^n\{B({{\rm{dom}}}\alpha,{{\rm{cod}}}\alpha)\}$.
EXTENSION GROUPS
================
Let $\cal A$ be an additive category with an Abelian structure $\cal P$, and with enough $\cal P$-projectives. Then for every $A\in {{\rm{Ob}}}{\cal A}$ we have some object $P(A)$ and some proper epimorphism $\pi A:P(A)\to A$. We denote by $\omega A:\Omega(A)\to P(A)$ a kernel of $\pi A$.
Let $P_*(A)$ be the exact sequence of $\cal P$-projective objects and the proper morphisms which is obtained by sticking of sequences $$\begin{array}{ccccccccc}
0 & \to & \Omega(A) & \stackrel{\omega A}\to & P(A)
& \stackrel{\pi A}\to & A & \to & 0\\
0 & \to & \Omega^2(A) & \stackrel{\omega\Omega(A)}\to & P(\Omega(A))
& \stackrel{\pi\Omega(A)}\to& \Omega(A) & \to & 0\\
&&&& \cdots\\
0 & \to & \Omega^{k+1}(A) & \stackrel{\omega \Omega^k(A)}\to & P(\Omega^k(A))
& \stackrel{\pi \Omega^k(A)}\to & \Omega^k(A) & \to & 0\\
&&&& \cdots\\
\end{array}$$ That is $P_*(A)$ consists of morphisms and objects: $$\begin{array}{c}
0 \leftarrow P(A) \stackrel{\omega A\circ \pi \Omega (A)}\longleftarrow
P(\Omega(A))
\stackrel{\omega\Omega(A)\circ \pi\Omega^2(A)}\longleftarrow P(\Omega^2(A))
\leftarrow
\dots\\
\leftarrow P(\Omega^k(A))
\stackrel{\omega \Omega^k(A)\circ \pi \Omega^{k+1}(A)}\longleftarrow
P(\Omega^{k+1}(A)) \leftarrow \dots
\end{array}$$ Heller define ${{\rm{Ext}}}_{\cal P}^n(A,B)$ as the group ${{\rm{Hom}}}_{-n}(P_*(A), P_*(B))$ of homotopical classes of homogenous maps. It follows from [@hel1958 Prop. 11.6] that ${{\rm{Ext}}}_{\cal P}^n(A,B)\cong H^n({\cal A}(P_*(A),B))$, $\forall n\geq 0$.
Denote by $L:{{\rm{Set}}}\to {{\rm{Ab}}}$ the functor from the category of sets to the category of abelian groups which assigns to each set $E$ the free Abelian group generated by $E$ and to each map $f:E_1\to E_2$ the homomorphism extending $f$.
\[l41\] Let ${\Bbb C}$ be a small category, $\cal A$ an additive category with coproducts, $F:{\Bbb C}\to {\cal A}$ a functor, $D=\{D_c\}_{c\in {{\rm{Ob}}}{\Bbb C}}$ a family of objects $D_c\in {{\rm{Ob}}}{\cal A}$. Then $\lim\nolimits_{{\Bbb C}'}^n\{{\cal A}(\Lambda D({{\rm{dom}}}\alpha), F({{\rm{cod}}}\alpha))\}=0$, $\forall n>0$, where $\Lambda:{\cal A}^{{{\rm{Ob}}}{\Bbb C}}\to {\cal A}^{\Bbb C}$ is the left adjoint to the restriction functor $O:{\cal A}^{\Bbb C}\to {\cal A}^{{{\rm{Ob}}}{\Bbb C}}$.
[Proof.]{} For every $A\in {{\rm{Ob}}}{\cal A}$ and a set $E$ we denote by $\Sigma_E A$ the coproduct of the family $\{A_e\}_{e\in E}$ of objects $A_e=A$. Let $\nu_e:A\to \Sigma_E A$ be the canonical injections. Consider the isomorphism $$\omega(E,A,B):A(\Sigma_E A,B)\to {{\rm{Ab}}}(LE,{\cal A}(A,B))$$ which assigns to every morphism $\varphi:\Sigma_E A\to B$ the homomorphism $\omega(E,A,B):LE\to {\cal A}(A,B)$ acting an $e\in E$ as $e\to \varphi\circ \nu_e\in {\cal A}(A,B)$. Then we keep an arbitrary $c\in {{\rm{Ob}}}{\Bbb C}$. Consider the sets $E=h^c(a)={\Bbb C}(c,a)$ with $a\in {{\rm{Ob}}}{\Bbb C}$. Let $Lh^c$ be the composition of $L$ and $h^c$. Then there is an isomorphism $${\cal A}(\Sigma_{{\Bbb C}(c,a)}A, B)\to {{\rm{Ab}}}(Lh^c(a), h^{A}(B)),$$ which is natural in $a\in {\Bbb C}$ and $A,B\in {\cal A}$. Let $\Lambda^c:{\cal A}\to {\cal A}^{\Bbb C}$ be the left adjoint to the functor ${\cal A}^{\Bbb C}\to {\cal A}$ acting as $F\to F(c)$ for $F\in {{\rm{Ob}}}({\cal A}^{\Bbb C})$ and $\eta\to \eta_c$ for $\eta\in {{\rm{Mor}}}({\cal A}^{\Bbb C})$. It is known that $(\Lambda^cA)(a)=\Sigma_{{\Bbb C}(c,a)} A$ for all $a\in{{\rm{Ob}}}{\Bbb C}$. For each functor $F:{\Bbb C}\to {\cal A}$ we have an isomorphism of bifunctors $${\cal A}(\Lambda^cA(-),F(=))\cong{{\rm{Ab}}}(Lh^c(-),(h^{A}\circ F)(=)).$$ It leads to an isomorphism $$\label{e3}
{\cal A}(\Lambda^{c}A({{\rm{dom}}}\alpha),F({{\rm{cod}}}\alpha))\cong
{{\rm{Ab}}}(Lh^c({{\rm{dom}}}\alpha),(h^{A}\circ F)({{\rm{cod}}}\alpha))\quad$$ of the natural systems on ${\Bbb C}$ with fixed $c$. For every family $D=\{D_c\}_{c\in{{\rm{Ob}}}{\Bbb C}}$ of objects in $\cal A$ there exists an isomorphism $\Lambda D\cong \Sigma_{c\in{\Bbb C}}\Lambda^cD_c$. Hence $$\begin{array}{c}
\lim\nolimits_{{\Bbb C}'}^n\{{\cal A}(\Lambda D({{\rm{dom}}}\alpha),F({{\rm{cod}}}\alpha))\} \cong\\
\\
\lim\nolimits_{{\Bbb C}'}^n\{\prod_{c\in {\Bbb C}}{\cal A}
(\Lambda^cD_c({{\rm{dom}}}\alpha),F({{\rm{cod}}}\alpha))\}\cong\\
\\
\prod _{c\in {\Bbb C}}\lim\nolimits_{{\Bbb C}'}^n\{{\cal A}(\Lambda^cD_c
({{\rm{dom}}}\alpha),F({{\rm{cod}}}\alpha))\}
\end{array}$$\
Thus, by the isomorphism (\[e3\]) it suffices to prove that $$\lim\nolimits_{{\Bbb C}'}^n\{{{\rm{Ab}}}(Lh^c({{\rm{dom}}}\alpha),(h^{A}\circ F)
({{\rm{cod}}}\alpha))\}=0,$$ for every $A\in {{\rm{Ob}}}{\cal A}$ and $n > 0$. We will prove that $$\lim\nolimits_{{\Bbb C}'}^n\{{{\rm{Ab}}}(Lh^c({{\rm{dom}}}\alpha),G({{\rm{cod}}}\alpha))\}=0,$$ for every $c\in{{\rm{Ob}}}{\Bbb C}$ and $G\in {{\rm{Ab}}}^{\Bbb C}$.
For that purpose we consider the natural system $$M=\{{{\rm{Ab}}}(Lh^c({{\rm{dom}}}\alpha),G({{\rm{cod}}}\alpha))\}$$ and the complex $K^*({\Bbb C}, M)$. Then $$K^n({\Bbb C}, M)
=\prod_{c_0\to \dots\to c_n}{{\rm{Ab}}}(Lh^c(c_0),G(c_n))
\cong\prod_{c_0\to \dots\to c_n}\prod_{c\to c_0}G(c_n).
$$ We recall that $C^n(c/{\Bbb C},GQ_c)$ consists of functions $$g:N_n(c/{\Bbb C})\to \bigcup_{\alpha\in {{\rm{Ob}}}(c/{\Bbb C})}GQ_c(\alpha)$$ with $g(c\to c_0\to \dots \to c_n)\in GQ_c(c\to c_n)$; the homomorphisms $d^n:C^n(c/{\Bbb C},GQ_c)\to C^{n+1}(c/{\Bbb C},GQ_c)$ act as $$\begin{array}{c}
(d^ng)(c\to c_0 \to \dots \to c_{n+1})=
\sum\limits_{i=0}^n(-1)^{i} g(c\to c_0\to \dots
\to \hat{c}_{i}\to \dots \\
\to c_{n+1})+
(-1)^{n+1}G(c_n\to c_{n+1})g(c\to c_0\to \dots \to c_n).
\end{array}$$
We will prove that the complex $K^*({\Bbb C},M)$ for $$M = \{{{\rm{Ab}}}(Lh^c({{\rm{dom}}}\alpha),G({{\rm{cod}}}\alpha))\}$$ is isomorphic to $C^*(c/{\Bbb C},GQ_c)$.
It is clear that $K^n({\Bbb C},M) \cong C^n(c/{\Bbb C},GQ_c)$. The homomorphism $d^n :K^n({\Bbb C}, M)\to K^{n+1}({\Bbb C}, M)$ has the following action on functions $f$ for which $f(c_0\to \dots \to
c_n)(c\to c_0)\in G(c_n)$: $$\begin{array}{rl}
d^nf & (c_0\to c_1 \to \dots \to c_{n+1})(c\to c_0)=\\
& f(c_1 \to \dots \to c_{n+1})\circ Lh^c(c_0 \to c_1)(c\to c_0)+\\
& \sum_{i=1}^n(-1)^{i}f(c_0\to \dots\to \hat{c_{i}}\to \dots \to c_{n+1})(c\to c_0)+\\
& (-1)^{n+1}F(c_n\to c_{n+1})f(c_0\to \dots \to c_n)(c\to c_0).
\end{array}$$ We let $\tilde{f}(c\to c_0\to \dots \to c_n)=f(c_0\to \dots\to c_n)(c\to c_0)$. We check that the correspondence $f\to \tilde{f}$ is a morphism of complexes: $$\begin{array}{c}
d^n\tilde{f}(c\to c_0 \to \dots \to c_{n+1})=\tilde{f}(c\to \hat{c_{0}}\to c_1\to \dots \to c_{n+1}) \\
+(-1)^{i}\tilde{f}(c\to c_0\to \dots \to \hat{c_{i}}\to \dots \to c_{n+1})+\\
(-1)^{n+1}F(c_{n}\to c_{n+1})\tilde{f}(c\to c_0\to \dots \to c_n).
\end{array}$$ Thus, the map $f\to \tilde{f}$ is an isomorphism of complexes $K^*({\Bbb C},M)$ and $C^*(c/{\Bbb C},GQ_c)$. Consequently, the $n$-th cohomology groups of $K^*({\Bbb C}, M)$ are zeros for $n>0$. Hence $\lim\nolimits_{{\Bbb C}'}^n\{{{\rm{Ab}}}(Lh^c({{\rm{dom}}}\alpha),G({{\rm{cod}}}\alpha))\}=0$ for all $n>0$. The isomorphism (3) and preservation of products by $\lim\nolimits_{{\Bbb C}'}^n$ finish the proof.
Let ${\cal A}$ be an additive category with coproducts, ${\cal P}$ a nonempty Abelian structure. If the coproduct of every family of proper epimorphisms is proper, and $\cal A$ has enough $\cal P$-projectives, then for each small category $\Bbb C$ and functors $F,G:{\Bbb C}\to{\cal A}$ there exists a first quadrant spectral sequence with $$E_2^{p,q}=\lim\nolimits_{{\Bbb C}'}^p
\{{{\rm{Ext}}}_{\cal P}^q(F({{\rm{dom}}}\alpha),G({{\rm{cod}}}\alpha))\}
\Longrightarrow{{\rm{Ext}}}_{{\Bbb C}{\cal P}}^{p+q}(F,G).
$$
[Proof.]{} We will build an exact sequence $$0 \leftarrow F \leftarrow F_0 \leftarrow F_1 \leftarrow \dots$$ of proper morphisms with ${\Bbb C}{\cal P}$-projective $F_n$ for all $n\geq 0$. Recall that $O:{\cal A}^{\Bbb C}\to {\cal A}^{{{\rm{Ob}}}{\Bbb C}}$ is the restriction functor, and $\Lambda$ the left adjoint to $O$. The counit of adjunction $\varepsilon_F:\Lambda O F\to F$ is a retraction on each $c\in {{\rm{Ob}}}{\Bbb C}$. We choose a family $P=\{P(c)\}_{c\in {{\rm{Ob}}}{\Bbb C}}$ of $\cal P$-projectives and a family of proper epimorphisms $\psi=\{\psi_c:P(c)\to F(c)\}_{c\in{{\rm{Ob}}}{\Bbb C}}$ and apply the functor $\Lambda$. Then $\Lambda(\psi):\Lambda(P)\to \Lambda O F$ is the proper epimorphism as a coproduct of proper epimorphisms. We let $F_0=\Lambda(P)$. Let $K$ be a kernel of the morphism $\varepsilon\circ \Lambda (\psi):F_0\to F$. We apply above building to $K$ instead of $F$ and obtain some functor $K_0$. We let $F_1=K_0$. Then the inclusion $K\to F_0$ is a proper monomorphism in ${\cal A}^{\Bbb C}$. Consequently, the composition $F_1\to K \to F_0$ is a proper morphism. We denote it by $d_0$. By induction we build the members $F_2,F_3,\dots $ and morphisms $d_n:F_{n+1}\to F_n$. Now we consider the complex $$\{K^n(\alpha)\}=\{{\cal A}(F_n({{\rm{dom}}}\alpha),G({{\rm{cod}}}\alpha))\}$$ in the category ${{\rm{Ab}}}^{{\Bbb C}'}$ of natural systems on $\Bbb C$. By Grothendieck [@gro1957] there are two spectral sequences concerned with hyper cohomologies of the functor $\lim\nolimits_{{\Bbb C}'}$ with respect to the complex $K^*$ $$H^p(\lim\nolimits_{{\Bbb C}'}^q\{K^*(\alpha)\})\Longrightarrow H^n,\quad
\lim\nolimits_{{\Bbb C}'}^p \{H^qK^*(\alpha)\}\Longrightarrow H^n.
$$ For each $n\geq 0$ there exists a family $A = \{A_c\}_{c\in Ob{\Bbb C}}$ of objects of $\cal A$ such that $F_n=\Lambda A$. Applying Lemma \[l41\] we obtain $\lim\nolimits_{{\Bbb C}'}^q\{K^n(\alpha)\}=0$ for all $q>0$, and $n\geq 0$. Thus, the first spectral sequence degenerates and second gives the looking spectral sequence.
The [*preabelian category*]{} is the additive category with kernels and cokernels. A morphism $f: A \rightarrow B$ in the preabelian category is called [*strict*]{} if the associated morphism $Coim\, f \rightarrow Im\, f$ is the isomorphism. Kuz’minov and Cherevikin proved that any quasiabelian category has the Abelian structure in the sense of Heller where the class of proper morphisms consists of all strict morphisms. In particular the category of locally convex spaces and coninuous linear maps is quasiabelian. Palamadov proved that this category has enough injectives. The proper monomorphisms in this category are all the kernels. This category has the infinite products. The product of kernels is the kernel. Hence, the opposite to the conditions of Theorem 4.2 are satisfied.
Let ${\Bbb C}$ be a small category and $\cal A$ the category of locally convex spaces and continuous linear maps. Then for every diagrams $F,~G:~ {\Bbb C}\rightarrow {\cal A}$ there exists the spectral sequence with $$E_2^{p,q}=\lim\nolimits_{{\Bbb C}'}^p
\{{{\rm{Ext}}}_{\cal P}^q(F({{\rm{dom}}}\alpha),G({{\rm{cod}}}\alpha))\}
\Longrightarrow{{\rm{Ext}}}_{{\Bbb C}{\cal P}}^{p+q}(F,G).
$$ where $\cal P$ is the class of all strict morphisms.
[1]{}
H.-J. Baues, G. Wirshing, A. Grothendieck, A. Heller, M. A. Jibladze, T. I. Pirashvili, M. Jibladze, T. Pirashvili, A. A. Khusainov, V. I. Kuz’minov, A. Yu. Cherevikin, S. Mac Lane, S. Mac Lane, B. Mitchell, Oliver, B.
[^1]: In the preparation of this paper, we have been assisted by a grant from the TÜBİTAK and NATO. The first author is supported by Russian Federation Education Department
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abstract: 'A dual-channel AlN/GaN high electron mobility transistor (HEMT) architecture is demonstrated that leverages ultra-thin epitaxial layers to suppress surface-state related gate lag. Two high-density two-dimensional electron gas (2DEG) channels are utilized in an AlN/GaN/AlN/GaN heterostructure wherein the top 2DEG serves as a quasi-equipotential that screens potential fluctuations resulting from surface and interface trapped charge. The bottom channel serves as the transistor’s modulated channel. Dual-channel AlN/GaN heterostructures were grown by molecular beam epitaxy on free-standing HVPE GaN substrates where 300 nm long recessed and non-recessed gate HEMTs were fabricated. The recessed-gate HEMT demonstrated a gate lag ratio (GLR) of 0.88 with no collapse in drain current and supporting small signal metrics $f_t/f_{max}$ of 27/46 GHz. These performance results are contrasted with the non-recessed gate dual-channel HEMT with a GLR of 0.74 and 82 mA/mm current collapse with $f_t/f_{max}$ of 48/60 GHz.'
author:
- 'David A. Deen, *Member, IEEE*, David F. Storm, D. Scott Katzer, *Senior Member, IEEE*, R. Bass, David J. Meyer, *Senior Member, IEEE*'
title: 'Suppression of Surface-Originated Gate-Lag by a Dual-Channel AlN/GaN HEMT Architecture'
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degradation modes in GaN-based high electron mobility transistors (HEMTs) have received extensive scrutiny since the device’s inception. Such modes include dc current collapse, dc-RF frequency dispersion due to the virtual-gate effect, gate and drain lag, and power slump. Many of these performance impairments have been traced back to surface and bulk trapping [@Binari],[@Wang]. Post-growth passivation techniques have become the most popular method to address the deleterious effects of surface state traps and include conformal oxide and nitride depositions [@Medjdoub],[@Lee],[@Huang],[@Koehler]. Surface chemical treatments have been investigated to minimize the effects of virtual gating on frequency performance [@Meyer],[@Wang2],[@Liu]. Alternative approaches for epitaxial passivation have also been given some attention [@Shen]. In an attempt to ascertain the origin of surface states in GaN-based HEMTs, Higashiwaki et al. reported on the formation of surface states in AlGaN barriers due to the high temperature anneal process that alloys the metallic ohmic contacts [@Higashiwaki]. Based upon this premise, reports of regrown ohmic contacts have shown that by avoiding the high temperature anneal process, the surface state density is reduced [@Xing1]. Despite nearly two decades of ongoing work in these areas, HEMT performance still has yet to reach many of the theoretical limits. This suggests there are more performance gains to be obtained by the technology if the sources of these modes can be addressed, either by materials or device design.
![(a) Cross-sectional illustration of the recessed-gate HEMT with corresponding band diagrams (BD) taken vertically through the gated regions. BD$_1$ corresponds to the gated intrinsic region of the HEMT with only a single channel and BD$_2$ corresponds to the access region of the HEMT with two coincident channels. Band diagrams are shown including (dashed) and excluding (solid) 6$\times$10$^{13}$ cm$^{-2}$ trapped charge at the oxide/AlN interface. Interfacial trap states are depicted by red dashes in the device cross-section and band diagrams. (c) Surface morphology of the as-grown AlN surface with a RMS roughness of 0.64 nm is shown by a $2\times2$ $\mu m^2$ AFM scan.[]{data-label="fig:xsection"}](Xsectionv6){width="0.99\columnwidth"}
The binary-barrier AlN/GaN HEMT has set remarkable performance benchmarks due to the exceptionally high polarization-induced 2DEG density achievable (up to 6$\times$10$^{13}$ cm$^{-2}$) with high mobility (1800 cm$^2$/Vs) [@Medjdoub; @Zimmermann; @Shinohara; @Deen2]. Yet only a handful of HEMT designs have leveraged a few of the attributes that are inherent to this particular heterostructure [@Shinohara; @Cao3; @Deen3; @Ganguly]. In this letter we propose and demonstrate a novel alternative to post-growth surface passivation based on a dual-channel AlN/GaN/AlN/GaN heterostructure. The upper AlN/GaN heterojunction undergoes a recess etch, conformal oxidation, and gate metal deposition as illustrated in Fig. \[fig:xsection\]. The upper polarization-doped 2DEG serves solely to screen the potential fluctuations generated by surface trapped charge that would otherwise impose channel depletion leading to current collapse and gate lag. The trapped charge can also act as a source of remote ionized impurities that can scatter mobile channel electrons leading to mobility reduction in the current-carrying channel [@Deen1]. The bottom 2DEG serves as the gate-modulated channel. The HEMT access region includes both channels. Therefore, purely dual-channel AlN/GaN/AlN/GaN HEMTs have also been fabricated on the same wafer, serving as both a calibration structure for $CV$ and $IV$ characterization as well as a proxy to the recessed-gate HEMT access region. Several reports have been made on nitride-based dual-channel HEMTs with AlGaN or AlInN barriers with the intent to increase drain current density or assess HEMT noise characteristics and subsequently disregarded gate lag performance [@Chu],[@Jha],[@Zhang]. A notable attribute of using the AlN/GaN heterostructure for the HEMT design reported in this work is that the AlN barrier layers are inherently thin ($<$ 5 nm), which allows extremely shallow channels and therefore, multiple channel designs to maintain channels in close proximity to the surface. This is not the case for alloyed barriers that require a sufficient thickness in order to induce 2DEG formation.
AlN/GaN heterostructures were grown by RF-plasma assisted molecular beam epitaxy (MBE) on free-standing hydride vapor phase epitaxy (HVPE) GaN substrates. All heterostructures were grown at 650$^o$ C and were grown continuously without interrupts. Growth was initiated by a 60 second nitridation of the HVPE GaN substrate surface, immediately followed by growth of an ultra-thin, 1.5 nm AlN nucleation layer [@Cao3]. Next, a 1 $\mu$m thick beryllium-doped (10$^{19}$ cm$^{-3}$) GaN layer was deposited followed by a 0.3 $\mu$m thick lesser-doped region (2$\times$10$^{17}$ cm$^{-3}$) [@Storm],[@Deen2]. Next, a 200-nm unintentionally-doped GaN buffer layer was grown. We previously demonstrated that Be compensation-doped layers had minimal deleterious effects on electrical properties when separated from the 2DEG by at least 200 nm [@Storm2]. The active heterostructure layers were subsequently grown following an AlN/GaN/AlN sequence with thicknesses of 3/15/3 nm, respectively. These layer thicknesses were chosen in order to avoid lattice relaxation of the strained AlN layers while maintaining an optimal $\mu$-$n_s$ product. It has previously been shown that lattice relaxation begins to occur in the form of micro-cracks along crystalline planes in AlN layers greater than 4 nm when grown pseudomorpically on GaN [@Deen1],[@Jena1]. Post-growth characterization by atomic force microscopy showed (Fig. \[fig:xsection\](c)) a RMS surface roughness of 0.64 nm in a 2$\times$2 $\mu$m$^2$ scan with no indication of lattice relaxation of the AlN layers. An inductive-based contact-less sheet resistance measurement showed as-grown room-temperature sheet resistance across the wafer to be 210 $\Omega/\Box$ indicating 2DEG population in one or both of the channels in the as-grown structure.
![(a) Capacitance-voltage characteristics and (b) charge profile for the dual-channel test capacitor showing two distinct capacitance plateaus corresponding to the two parallel 2DEG distributions.[]{data-label="fig:cv"}](CVv3){width="\columnwidth"}
An ohmic-first processing schedule was employed to ensure the best conditions for forming low-resistance ohmic contacts to both parallel 2DEG channels. A pre-metallization Cl-based dry etch was employed to etch through the top AlN and GaN layers prior to contact metallization. The target etch depth was 18 nm below the terminal surface at the interface made between the GaN spacer and the bottom AlN layer. Electron beam evaporation was used to deposit a Ti/Al/Ni/Au metallic layer structure with corresponding thicknesses of 300/20/30/10 nm. An 860$^o$ C rapid thermal anneal was performed for 30 seconds following the metal contact deposition and a contact resistance of $\sim$2 $\Omega$-mm resulted. Mesa and inter-device isolation was facilitated by a Cl-based dry etch. Electron beam lithography was used to define the sub-micron gate-recess on half of the HEMTs on wafer. A Cl-based dry etch was utilized for the gate recess etch. The target depth was 17 nm from the terminal surface such that 1 nm of GaN spacer remained. The gate length was targeted at 300 nm by the recess etch with a 500 nm gate head length. Following the gate recess, a 7 nm thick atomic layer deposited Al$_2$O$_3$ film was used for gate insulation. Optical lithography was also used for the definition of 1 $\mu$m gates and other large-gated test structures following oxide deposition. A Ni/Au gate metal deposition and lift-off concluded the fabrication. Pertinent transistor geometries were source-drain separation ($L_{DS}$) of 3 $\mu$m and gate width ($W_G$) of 150 $\mu$m. Post-oxidation Hall effect measurements were performed on van der Pauw structures that included both channels and the room temperature sheet resistance and constituent parameters were measured to be $R_{sh}$ = 180 $\Omega/\square$, $n_s$ = 2.3 $\times$ 10$^{13}$ cm$^{-2}$, $\mu$ $\sim$ 1500 cm$^2$/Vs. The measured mobility represents an averaged mobility of both parallel channels since there was no convenient means to differentiate between the channels with the standard on-wafer Hall measurement. However, the individual charge densities of each channel were determined through capacitance-voltage ($CV$) profiling.
$CV$ measurements on a 100 $\mu$m diameter Al$_2$O$_3$/AlN/GaN/AlN/GaN test capacitor at 10 MHz showed two distinct plateaus (Fig. \[fig:cv\]) indicating two separate charge distributions in the heterostructure. The plot shows three curves with corresponding bias voltage sweep directions. Curve $(1)$ was the initial downward sweep, curve $(2)$ followed in the same direction, and curve $(3)$ was a final upward sweep. A clear deviation is seen between curve $(1)$ and subsequent curves that may amount to charging of interface and bulk traps in the hetero-system [@Zhang],[@Jena2]. The curve evolution from $(2)$ to $(3)$ shows some hysteresis, which may indicate oxide or confined mobile charge in the GaN spacer layer or buffer layer. It is assumed that curve $(1)$ gives the best evidence of the pristine heterostructure. Therefore, the integration of the lower capacitance plateau in (1) yields a charge density of 1.5 $\times$ 10$^{13}$ cm$^{-2}$ corresponding to the lower 2DEG. The integration of the total $CV$ profile yields a combined charge density of 2.6 $\times$ 10$^{13}$ cm$^{-2}$ which is in agreement with the Hall effect results. The difference gives the upper 2DEG density and was found to be 1.1 $\times$ 10$^{13}$ cm$^{-2}$. $CV$ curve $(1)$ was used to calculate the approximate charge density profile by $n(z) = (C^3/q\epsilon_s)(dC/dV)^{-1}$ and is show in Fig. \[fig:cv\](b). These charge densities were used to calibrate the electrostatic conditions used to calculate the band diagrams shown in Fig. 1. A self-consistent Poisson-Schrodinger solver [@Snider] was used to calculate band diagrams to show the effect with and without 6$\times$10$^{13}$ cm$^{-2}$ trapped charge at the Al$_2$O$_3$/AlN interface. The trap state density of 6$\times$10$^{13}$ cm$^{-2}$ was chosen for the band diagram simulations based off previous works that have extracted similar trap state densities of the oxide/AlN junction from high-frequency $CV$ methods and have correlated the trap density to spatially-fixed interfacial polarization states of the AlN barrier [@Jena2; @Deen4; @Ibbetson]. The work function ($\Phi_B = \chi_{Ni} - \chi_{ox}$, where $\chi$ is the electron affinity of the designated material layer) used for the Ni-Al$_2$O$_3$ gate was 2 eV and band offsets taken from Ref. [@Jena2] were used for the gated heterostructure simulations. Based off the simulation as well as prior work on single-channel AlN/GaN heterostructures [@Jena2], the presence of (ionized positive) donor-like trap states at the oxide/AlN interface cause downward band bending, which promotes an increase in the upper 2DEG density as well as the lower 2DEG density, though with a milder effect. However, it should be noted that though the 2DEG density increases under the influence of ionized surface/interface trap states, the ionized surface states trap electrons from the gate electrode and high-density 2DEG when under bias. Thus, reversing the downward band bending and promoting channel depletion near the gate electrode where the field is the highest. The effect has been referred to as virtual gate extension. The majority of the traps have slow (dis)charge times and cannot respond to gate modulation frequencies in the GHz range or sharp gate pulses. Therefore, in terms of frequency response, they serve to diminish frequency performance of the HEMT through unmodulated channel depletion (at GHz frequency) unless the trap states are either passivated or electrostatically screened. This point motivates the recessed-gate HEMT design reported in this work.
![Pulsed gate lag characteristics for the (a) dual-channel HEMT with bias conditions of $V_{DS}$ = $10 V$ and $-20V$ $<$ $V_{GS}$ $<$ 0$V$ and (b) the recessed-gate HEMT with bias conditions of $V_{DS}$ = $10 V$ and $-20V < V_{GS} < +3V$. Pulse width for both pulse sequences was 0.5 $\mu$s.[]{data-label="fig:GLR"}](GLRv3){width="\columnwidth"}
Gate lag refers to the time delay of a HEMT’s drain current recovery in response to a gate voltage pulse. Gate lag results from a slow recovery from depletion of the channel charge due to proximal trapped charge [@Binari; @Wang]. Interfacial trapped charge such as those at the oxide/AlN interface [@Deen4] can lead to gate lag [@Binari]. Therefore, a temporally-sequential pulsed gate voltage lag measurement has been used to quantify the gate lag response of the dual-channel and recessed-gate HEMTs. The measurement schedule began by measuring open-channel drain current where predetermined values of $V_{DS}$ and $V_{GS}$ where chosen based off the dc $IV$ characteristics. Those values were $V_{GS} = 0 V$ and $V_{GS} = +3 V$ with $V_{DS} = 10 V$ for the dual-channel and recessed-gate HEMTs, respectively. The measured drain current density in the dc open-channel condition ($I_{DS,o}$) is later used as the normalization value when calculated the gate lag ratio as defined by $GLR = I_{DS,pulse}/I_{DS,o}$. Normalization allows for the comparison of the gate lag response between dissimilar devices. The open-channel drain currents for the stated $V_{GS}$ values used in our measurements were $I_{DS} = 0.95$ A/mm and $I_{DS} = 0.38$ A/mm for the dual-channel and recessed-gate HEMTs, respectively (see Fig. \[fig:GLR\]). These values were chosen as the drain current resulting at knee voltage under a resistively-loaded drain (see Fig. \[fig:drain\]). Following the dc $I_{DS,o}$ measurement, $V_{GS}$ is brought to a value within the sub-threshold regime for a prescribed amount of time (0.8 $\mu$s in our schedule), which allows charging of trap states. Then $V_{GS}$ is abruptly pulsed back to the open-channel value previously listed for a specified pulse duration (0.5 $\mu$s in our schedule) and the drain current is monitored during the pulse cycle ($I_{DS,pulse}$) before $V_{GS}$ is brought back into sub-threshold. This concludes the primary gate lag measurement whereby a GLR value may be determined. In our measurement schedule shown in Fig. \[fig:GLR\], we have additionally made successive gate pulses 1 minute apart ($V_{GS}$ held in sub-threshold between pulses) in order to observe the effects of higher trapped charge density on GLR. Moreover, our measurement schedule included a restart where all bias voltages where brought to $0 V$ immediately before repeating the measurement schedule just described. This allows for the observation of how degraded the dc $I_{DS,o}$ value has become (grey region in Fig. \[fig:GLR\]) and serves as a proxy for current slump. The quantity, $\Delta I_{DS} = I_{DS,o} - I_{DS,1}$, where $I_{DS,1}$ is the dc value of $I_{DS}$ measured upon restarting the gate pulse schedule.
The results of pulsed gate lag measurements are shown in Fig. \[fig:GLR\] for the (a) dual channel AlN/GaN HEMT and the (b) charge-screening HEMT. The dual channel HEMT in (a) demonstrated a GLR of 0.74 which decreased while the HEMT was biased in sub-threshold to 0.67 for subsequent pulses. The hypothesis is that this reduction is mainly due to surface state charging and corresponding depletion of the upper channel (current collapse) since it was shown that the upper channel makes up a large fraction of the total drain current for the dual-channel HEMTs. Upon restarting the measurement the open channel drain current was found to have diminished by 82 mA/mm which is indicative of 2DEG channel depletion and possibly some buffer trapping. The recessed-gate HEMT in (b) demonstrated a GLR of 0.88-0.86, which indicates strong suppression of interface trap related gate lag degradation by its near unity value. Thus, the recessed-gate HEMT showed significant GLR performance improvement over the non-recessed dual-channel HEMT. The traps are assumed to be located at the oxide/AlN interface as is denoted in Fig. \[fig:xsection\] [@Zhang],[@Jena2]. The recessed-gate HEMT showed an emphasized (dis)charge curvature of the drain current pulse in Fig. \[fig:GLR\](b). This may be a manifestation of increased gate-to-channel capacitive charging time between the gate metal and upper 2DEG and would corroborate the slower frequency response of the recessed-gate HEMT compared to the dual-channel HEMT in Fig. \[fig:freq\](b). Further design enhancements are anticipated to alleviate some of the $RC$ charging in the HEMT design. A notable result of the recessed-gate HEMT is that after the GLR sequence was stopped and restarted, the initial drain current density had not diminished despite the absence of a passivation layer other than the thin Al$_2$O$_3$ gate insulation. Although other reports have been made on nitride-based dual-channel HEMTs with alloyed barrier layers [@Chu; @Jha; @Zhang], none have included gate lag measurements. To the best of our knowledge, this is the first report on dual-channel AlN/GaN HEMT gate lag as well as its innovation to the recessed-gate HEMT architecture. Although not shown, we have typically observed single channel AlN/GaN HEMTs grown on sapphire or SiC substrates with comparable oxide layer thicknesses to have GLRs of $<$ 0.5. Further refinements in the contacts, gate process, and layer structure are anticipated to advance the design to fully mitigate the detriment of surface traps observed in these ultra-shallow channel AlN/GaN HEMTs.
Dual-channel and recessed-gate HEMTs were fabricated on the same wafer and standard dc and small-signal rf HEMT characterizations were performed in order to qualify and contextualize the operational performance of the HEMTs through standard techniques. Both HEMT varieties’ dc drain characteristics are shown in Fig. \[fig:drain\] (a) and (b), respectively. The dual channel HEMT had a maximum drain current density of $\sim$2X that of the charge screening HEMT which indicates there was contact made to the etched sidewalls of the top channel. However, slight non-linearity in the drain characteristic at low voltage of the dual-channel HEMT indicates that the contact made was not purely ohmic. The recessed-gate HEMT showed peculiar transfer characteristics with an absence of a well defined maximum current density and non-linear characteristic as shown in Fig. \[fig:xfer\](a). Consequently, a multi-peaked $g_m$ was observed as shown in Fig. \[fig:xfer\](b) as a result of the non-linear transfer characteristic. The non-linearity in the recessed-gate HEMT transfer characteristic is a manifestation of the two channels in the access region immediately beneath the T-gate head where a small portion of the dual-channel access region (source-side and drain-side) is covered by gate metal (see Fig. \[fig:xsection\]a). The gate overlap in the access region introduces additional charge from the upper 2DEG that undergoes depletion simultaneously with the conduction channel, a form of capacitive coupling. While capacitive coupling between both 2DEG channels may occur, charge transfer between 2DEGs is not expected for the heterostructure investigated in this work due to the thick GaN spacer and high energy barrier of the buried AlN layer. Capacitive coupling can cause some reduction in frequency performance but can be eliminated by proper engineering of the gate electrode by a self-aligned process that does not have gate overlap in the access region.
![Drain characteristics for the (a) dual-channel HEMT and (b) the recessed-gate HEMT.[]{data-label="fig:drain"}](drainIVv4){width="\columnwidth"}
![(a) Transfer and (b) transconductance characteristics for the dual-channel and recessed-gate HEMTs.[]{data-label="fig:xfer"}](xferv3){width="\columnwidth"}
![Small signal frequency performance of the (a) dual-channel HEMT and (b) recessed-gate HEMT.[]{data-label="fig:freq"}](freqv3){width="\columnwidth"}
Small signal performance was measured up to 50 GHz for both the non-recessed and recessed-gate dual channel HEMTs, as shown in Fig. \[fig:freq\] (a) and (b), respectively. Unity current gain frequency, $f_t$, and maximum frequency of oscillation, $f_{max}$ were measured to be 27 GHz and 46 GHz, respectively, for the recessed-gate HEMT and 48 GHz and 60 GHz, respectively, for the dual-channel HEMT. The product of $f_t$ and gate length give a measure of average electron velocity in the channel. The resulting $f_t$-$L_g$ products were 8.1 GHz-$\mu$m and 14.4 GHz-$\mu$m for the recessed-gate and non-recessed HEMTs, respectively. The gate length used for the $f_t$-$L_g$ product was 300 nm for both the dual-channel and recessed-gate HEMTs since that length corresponds to the length of the gate stem closest to the channel (Fig. \[fig:xsection\](a)). It is noted that the non-recessed dual-channel HEMT demonstrated higher small signal frequency metrics. This is likely due to the recessed-gate HEMT having a larger $C_{GS}$ and $C_{GD}$ resulting in a higher gate charging time. The three charging times involved in setting $f_t$ are the parasitic charging time ($C_{GD}(R_S + R_D$), channel charging time ($(C_{GD}+C_{GS})\times g_{DS}/g_m$), and the drain delay ($C_{GD}/g_m$) [@Shinohara; @Deen2]. In the case of the recessed-gate HEMT, $C_{GD}$ and $C_{GS}$ increase due to the presence of the upper 2DEG while the lower 2DEG is the only modulated current-carrying channel, and therefore may slightly suffer in terms of frequency performance due to the increased (parasitic) gate capacitance. Nevertheless, the $f_t$-$L_g$ product of 8.1 GHz-$\mu$m is comparable to many reports of GaN-based HEMTs, which implies that if gate capacitance poses an issue, it is not severe. Additionally, a lower access resistance from the two parallel channels may also serve to improve frequency performance for the dual-channel HEMT.
In summary, we have demonstrated a novel recessed-gate dual-channel AlN/GaN/AlN/GaN HEMT architecture that suppresses the deleterious effect surface and interface-originated trapped charges have on drain current recovery under pulsed-gate conditions. This is achieved through the employment of the upper 2DEG as an equipotential that screens the potential fluctuations of the trapped charge. The primary indicator of the recessed-gate HEMT design’s efficacy for reducing the effect of surface/interface trapping effects is a GLR of 0.88 demonstrated with no observable decrease in drain current during subsequent gate voltage pulses over time. Small signal performance of 27/46 GHz was achieved for $f_t$/$f_{max}$ in the recessed-gate HEMT with gate lengths of 300 nm. The dual-channel recessed-gate AlN/GaN HEMT demonstrates the feasibility for alternative designs to enhance pulsed-gate performance in HEMTs.
The authors acknowledge N. Green for help with device processing and Professors D. Jena and H. Xing at Cornell University for constructive technical discussion. This work was funded by the Office of Naval Research (P. Maki).
[\[1\]]{} S. C. Binari, K. Ikossi, J. A. Roussos, W. Kruppa, D. Park, H. B. Dietrich, D. D. Koleske, A. E. Wickenden, R. L. Henry, “Trapping Effects and Microwave Power Performance in AlGaN/GaN HEMTs”, IEEE Trans. Elec. Dev., Vol. 48, No. 3, 2001. M. Wang, D. Yan, C. Zhang, B. Xie, C. P. Wen, J. Wang, Y. Hao, W. Wu, B. Shen, “Investigation of Surface- and Buffer-Induced Current Collapse in GaN High-Electron Mobility Transistors Using a Soft Switched Pulsed I-V Measurement”, IEEE Elec. Dev. Lett., Vol. 35, No. 11, 1094, 2014. F. Medjdoub, M. Zegaoui, D. Ducatteau, N. Rolland, P. A. Rolland, “High-Performance Low-Leakage-Current AlN/GaN HEMTs Grown on Silicon Substrate”, IEEE Elec. Dev. Lett. 32, 874, 2011. D. S. Lee, O. Laboutin, Y. Cao, W. Johnson, E. Beam, A. Ketterson, M. Schuette, P. Saunier, T. Palacios, “Impact of Al2O3 Passivation Thickness in Highly Scaled GaN HEMTs”, IEEE Elec. Dev. Lett., Vol. 33, No. 7, 976, 2012. S. Huang, Q. Jiang, S. Yang, C. Zhou, K. J. Chen, “Effective Passivation of AlGaN/GaN HEMTs by ALD-Grown AlN Thin Film”, IEEE Elec. Dev. Lett., Vol. 33, No. 4, 516, 2012. A. D. Koehler, N. Nepal, T. J. Anderson, M. J. Tadjer, K. D. Hobart, C. R. Eddy, F. J. Kub, “Atomic Layer Epitaxy AlN for Enhanced AlGaN/GaN HEMT Passivation”, IEEE Elec. Dev. Lett., Vol. 34, No. 9, 1115, 2013. D. J. Meyer, J. R. Flemish, J. M. Redwing, “Prepassivation surface treatment effects on pulsed and dc I-V performance of AlGaN/GaN high-electron-mobility transistors”, Appl. Phys. Lett. 92, 193505, 2008. R. Wang, G. Li, O. Laboutin, Y. Cao, W. Johnson, G. Snider, P. Fay, D. Jena, H. Xing, “210-GHz InAlN/GaN HEMTs With Dielectric-Free Passivation”, IEEE Elec. Dev. Lett. 32, 7, 2011. H.-Y. Liu, B.-Y. Chou, W.-C. Hsu, C.-S. Lee, C.-S. Ho, “Novel Oxide-Passivated AlGaN/GaN HEMT by Using Hydrogen Peroxide Treatment”, IEEE Trans. Elec. Dev., Vol. 58, No. 12, 4430, 2011. L. Shen, R. Coffie, D. Buttari, S. Heikman, A. Chakraborty, A. Chini, S. Keller, S. P. DenBaars, U. K. Mishra, “High-Power Polarization-Engineered GaN/AlGaN/GaN HEMTs Without Surface Passivation”, IEEE Elec. Dev. Lett., Vol. 25, No. 1, 7, 2004. M. Higashiwaki, S. Chowdhury, B. Swenson, U. Mishra, “Effects of oxidation on surface chemical states and barrier height of AlGaN/GaN Heterostructures”, Appl. Phys. Lett., Vol. 97, No. 22, 222105, 2010. Y. Yue, Z. Hu, J. Guo, B. Sensale-Rodriguez, G. Li, R. Wang, F. Faria, T. Fang, B. Song, X. Gao, S. Guo. T. Kosel, G. Snider, P. Fay, D. Jena, H. Xing, “InAlN/AlN/GaN HEMTs With Regrown Ohmic Contacts and $f_t$ of 370 GHz”, IEEE Elec. Dev. Lett., Vol. 33, No. 7, 2012. T. Zimmermann, D. A. Deen, Y. Cao, J. Simon, P. Fay, D. Jena, H. G. Xing, “AlN/GaN Insulated-Gate HEMTs With 2.3 A/mm Output Current and 480 mS/mm Transconductance”, IEEE Elec. Dev. Lett. 29, 661, 2008. K. Shinohara, D. Regan. I. Milosavljevic, A. L. Corrion, D. F. Brown, P. J. Willadsen, C. Butler, A. Schmitz, S. Kim, V. Lee, A. Ohoka, P. M. Asbeck, M. Micovic, “Electron Velocity Enhancement in Laterally Scaled GaN DH-HEMTs With $f_t$ of 260 GHz”, IEEE Elec. Dev. Lett., Vol. 32, No. 8, 2011. D. J. Meyer, D. A. Deen, D. F. Storm, M. G. Ancona, D. S. Katzer, R. Bass, J. A. Roussos, B. P. Downey, S. C. Binari, T. Gougousi et al., “High Electron Velocity Submicrometer AlN/GaN MOS-HEMTs on Freestanding GaN Substrates”, IEEE Elec. Dev. Lett. 34, 199, 2013. Y. Cao, T. Zimmermann, H. Xing, D. Jena, “Polarization-engineered removal of buffer leakage for GaN transistors”, Appl. Phys. Lett. 96, 042102, 2010. D. A. Deen, A. Osinsky, R. Miller, “Bimodal wireless sensing with dual-channel wide bandgap heterostructure varactors”, Appl. Phys. Lett. 104, 093506, 2014. G. Li, B. Song, S. Ganguly, M. Zhu, R. Wang, X. Yan, J. Verma, V. Protasenko, H. G. Xing, D. Jena, “Two-dimensional electron gases in strained quantum wells for AlN/GaN/AlN double heterostructure field-effect transistors on AlN”, Appl. Phys. Lett. 104, 193506, 2014. D. A. Deen, D. F. Storm, D. J. Meyer, R. Bass, S. C. Binari, T. Gougousi, K. R. Evans, “Impact of barrier thickness on transistor performance in AlN/GaN high electron mobility transistors grown on free-standing GaN substrates”, Appl. Phys. Lett. 105, 093503, 2014. R. Chu, Y. Zhou, J. Liu, D. Wang, K. J. Chen, K. M. Lau, “AlGaN-GaN Double-Channel HEMTs”, IEEE Trans. Elec. Dev. 52, 4, 2005. S. K. Jha, C. Surya, K. J. Chen, K. M. Lau, E. Jelencovic, “Low-frequency noise properties of double channel AlGaN/GaN HEMTs”, Sol.-Sta. Elec. 52, 606, 2008. K. Zhang, J. Xue, M. Cao, L. Yang, Y. Chen. J. Zhang, X. Ma, Y. Hao, “Trap states in InAlN/AlN/GaN-based double-channel high electron mobility transistors”, J. Appl. Phys. 113, 174503, 2013. D. F. Storm, D. A. Deen, D. S. Katzer, D. J. Meyer, S. C. Binari, T. Gougousi, T. Paskova, E. A. Preble, K. R. Evans, D. J. Smith, “Ultrathin-barrier AlN/GaN heterostructures grown by rf plasma-assisted molecular beam epitaxy on freestanding GaN substrates”, J. Crys. Growth 380, 14, 2013. D. F. Storm, D. S. Katzer, D. A. Deen, R. Bass, J. A. Roussos, S. C. Binari, T. Paskova, E. A. Preble, K. R. Evans, “Proximity effects of beryllium-doped GaN buffer layers on the electrical properties of epitaxial AlGaN/GaN heterostructures”, Sol. Stat. Elec. 54, 1470, 2010. Yu Cao and Debdeep Jena, “High-mobility window for two-dimensional electron gasses at ultrathin AlN/GaN heterojunctions”, Appl. Phys. Lett. 90, 182112, 2007. S. Ganguly, J. Verma, G. Li, T. Zimmermann, H. Xing, D. Jena, “Presence and origin of interface charges at atomic-layer deposited Al$_2$O$_3$/III-nitride heterojunctions”, Appl. Phys. Lett. 99, 193504, 2011. I. H. Tan, G. L. Snider, L. D. Chang, E. L. Hu, “A self-consistent solution of Schrodinger-Poisson equations using a nonuniform mesh”, J. Appl. Phys. 68, 4071, 1990. D. A. Deen, D. F. Storm, R. Bass, D. J. Meyer, D. S. Katzer, S. C. Binari, J. W. Lacis, T. Gougousi, “Atomic layer deposited Ta$_2$O$_5$ gate insulation for enhancing breakdown voltage of AlN/GaN high electron mobility transistors”, Appl. Phys. Lett. 98, 023506, 2011. J. P. Ibbetson, P. T. Fini, K. D. Ness, S. P. DenBaars, J. S. Speck, U. K. Mishra, “Polarization effects, surface states, and the source of electrons in AlGaN/GaN heterostructure field effect transistors”, Appl. Phys. Lett. 77, 250, 2000.
[^1]: D. A. Deen was with the Naval Research Laboratory, Electronic Science and Technology Division, SW Washington, DC 20375 USA. He is now with Seagate Technology, Read Head Operations, Bloomington, MN 55435, (e-mail: david.deen@alumni.nd.edu.)
[^2]: D. F. Storm, D. S. Katzer, and D. J. Meyer are with the Naval Research Laboratory, Electronic Science and Technology Division, SW Washington, DC 20375 USA.
[^3]: R. Bass is with Sotera Defense Solutions, Herndon VA 20171-5393.
|
LPENSL - 2014\
IMB - 2014
[Open spin chains with generic integrable boundaries:\
Baxter equation and Bethe ansatz completeness from SOV]{}
[**N. Kitanine**, **J.-M. Maillet**, **G. Niccoli** ]{}
**Abstract**
Introduction
============
The functional characterization of the complete transfer matrix spectrum associated to the most general spin-1/2 representations of the 6-vertex reflection algebra on general inhomogeneous chains is a longstanding open problem. It has attracted much attention in the framework of quantum integrability producing so far only partial results. The interest in the solution of this problem is at least twofold. On the one hand, the quantum integrable system associated to the limit of the homogeneous chain, i.e. the open spin-1/2 XXZ quantum chain with arbitrary boundary magnetic fields, is an interesting physical quantum model. It appears, in particular, in the context of out-of-equilibrium physics ranging from the relaxation behavior of some classical stochastic processes, as the asymmetric simple exclusion processes [@EssD05; @EssD06], to the transport properties of the quantum spin systems [@SirPA09; @Pro11]. Their solution can lead to non-perturbative physical results and a complete and manageable functional characterization of their spectrum represents the first fundamental steps in this direction. On the other hand, it is important to remark that the analysis of the spectral problem of these integrable quantum models turned out to be quite involved by standard Bethe ansatz [@Bet31; @FadST79] techniques. Therefore, these quantum models are natural laboratories where to define alternative non-perturbative approach to their exact solution. Indeed, the algebraic Bethe ansatz, introduced for open systems by Sklyanin [@Skl88] based on the Cherednik’s reflection equation [@Che84], in the case of open XXZ quantum spin chains can be applied directly only in the case of parallel z-oriented boundary magnetic fields. Under these special boundary conditions the spectrum is naturally described by a finite system of Bethe ansatz equations. Moreover the dynamics of such systems can be studied by exact computation of correlation functions [@KitKMNST07; @KitKMNST08], derived from a generalisation of the method introduced in [@KitMT99; @KitMT00; @MaiT00] for periodic spin chains.
Introducing a Baxter $T$-$Q$ equation, Nepomechie [@Nep02; @Nep04] first succeeded to describe the spectrum of the XXZ spin chain with non-diagonal boundary terms in the case of an anisotropy parameter associated to the roots of unity; furthermore, the result was obtained there only if the boundary terms satisfied a very particular constraint relating the magnetic fields on the two boundaries. This last constraint was also used in [@CaoLSW03] to introduce a generalized algebraic Bethe ansatz approach to this problem inspired by papers of Baxter [@Bax72; @Bax72a] and of Faddeev and Takhtadjan [@FadT79] on the XYZ spin chain. This method has led to the first construction of the eigenstates of the XXZ spin chain with non z-oriented boundary magnetic fields and this construction has been obtained for a general anisotropy parameter, i.e., not restricted to the roots of unity cases[^1]. In [@YanZ07] a different version of this technique based on the vertex-IRF transformation was proposed but in fact it required one additional constraint on the boundary parameters to work. It is worth mentioning that even if these constrained boundary conditions are satisfied and generalized Bethe ansatz method gives a possibility to go beyond the spectrum, as it was done for the diagonal boundary conditions, no representation for the scalar product of Bethe vectors[^2] and hence for the correlation functions were obtained.
This spectral problem in the most general setting has then been also addressed by other approaches. It is worth mentioning a new functional method leading to nested Bethe ansatz equations presented in [@Gal08] for the eigenvalue characterization and analogous to those previously introduced in [@MurN05] by a generalized $T$-$Q$ formalism. The eigenstate construction has been considered in these general settings in [BasK05a,Bas06]{} by developing the so-called $q$-Onsager algebra formalism. In this last case the characterization of the spectrum is given by classifying the roots of some characteristic polynomials. More recently, in [@CaoYSW13-4] an ansatz $T$-$Q$ functional equations for the spin chains with non-diagonal boundaries has been proposed[^3].
It is extremely important to remark that in general all methods based on Bethe ansatz (or generalized Bethe ansatz) are lacking proofs of the completeness of the spectrum and in most cases the only evidences of completeness are based on numerical checks for short length chains. This is the case for the XXZ chain with non-diagonal boundary matrices with the boundary constraint for which the completeness of the spectrum description by the associated system of Bethe ansatz equations has been studied numerically [@Nep-R-2003; @Nep-R-2003add]. In the case of the XXZ chain with completely general non-diagonal boundary matrices some numerical analysis is also presented in [@CaoYSW13-4]. Further numerical analysis have been developed in a much simpler case of the isotropic XXX spin chain where the most general boundary conditions can be always reduced by using the $SU(2)$ symmetry to one diagonal and one non-diagonal boundary matrices. For the XXX chains the ansatz introduced in [@CaoYSW13-2] was also applied and the completeness of the Bethe ansatz spectrum was checked numerically [@CaoJYW2013]. It is also important to mention a simplified ansatz proposed by Nepomechie based on a standard second order difference functional $T$-$Q$ equation with an additional inhomogeneous term. The completeness of the Bethe ansatz spectrum has been verified numerically for small XXX chains in [@Nep-2013] while in [@Nep-W-2013] the problem of the description of some thermodynamical properties has been addressed.
These interesting developments attracted our attention in connection to the quantum separation of variables (SOV) method pioneered by Sklyanin [@Skl85; @Skl92]. The first analysis of the spin chain in the classical limit from this point of view was performed in [@Skl89a; @Skl89b]. This alternative approach allows to obtain (mainly by construction) the complete set of eigenvalues and eigenvectors of quantum integrable systems. In particular, it was recently developed [NicT10,N10-1,N10-2,GroMN12,GroN12,Nic12b,Nic13a,N13-1,Nic13b,Nic13c,Fald-KN13,FaldN13,GroMN13]{} for a large variety of quantum models not solvable by algebraic Bethe ansatz. Moreover it has been shown first in [@GroMN12] that once the SOV spectrum characterization is achieved manageable and rather universal determinant formulae can be derived for matrix elements of local operators between transfer matrix eigenstates. In particular, this SOV method was first developed in [@Nic12b] for the spin-1/2 representations of the 6-vertex reflection algebra with quite general non-diagonal boundaries and then generalized to the most general boundaries in [@Fald-KN13]. There, it gives the complete spectrum (eigenvalues and eigenstates) and already allows to compute matrix elements of some local operators within this most general boundary framework. However, it is important to remark that this SOV characterization of the spectrum is somehow unusual in comparison to more traditional characterizations like those obtained from Bethe ansatz techniques. More precisely, the spectrum is described not in terms of the set of solutions to a standard system of Bethe ansatz equations but is given in terms of sets of solutions to a characteristic system of $\mathsf{N}$ quadratic equations in $\mathsf{N}$ unknowns, $\mathsf{N}$ being the number of sites of the chain. While the clear advantage of this SOV characterization is that it permits to characterize completely the spectrum without introducing any ansatz one has to stress that the classification of the sets of solutions of the SOV system of quadratic equations represents a new problem in quantum integrability which requires a deeper and systematic analysis.
The aim of the present article is to show that the SOV analysis of the transfer matrix spectrum associated to the most general spin-1/2 representations of the 6-vertex reflection algebra on general inhomogeneous chains is strictly equivalent to a system of *generalized* Bethe ansatz equations. This ensures that this system of Bethe equations characterizes automatically the entire spectrum of the transfer matrix. More in detail, we prove that the SOV characterization is equivalent to a second order finite difference functional equation of Baxter type: $$\tau (\lambda )Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}
(-\lambda )Q(\lambda +\eta )+F(\lambda ),$$ which contains an inhomogeneous term $F(\lambda )$ independent on the $\tau $ and $Q$-functions and entirely fixed by the boundary parameters. It vanishes only for some special but yet interesting non-diagonal boundary conditions (corresponding to the boundary constraints mentioned above). One central requirement in our construction of this functional characterization is the polynomial character of the $Q$-function. Indeed, it is this requirement that allows then to show that a finite system of equations of generalized Bethe ansatz type can be used to describe the complete transfer matrix spectrum. Note that similar results on the reformulation of the SOV spectrum characterization in terms of functional $T$-$Q$ equations with $Q$-function solutions in a well defined model dependent set of polynomials were previously derived [@N10-1; @N10-2; @GroN12] for the cases of transfer matrices associated to cyclic representations of the Yang-Baxter algebra. The analysis presented here is also interesting as it introduces the main tools to generalize this type of reformulation to other classes of integrable quantum models. The article is organized as follows. In Section 2 we set the main notations and we recall the main results of previous papers on SOV necessary for our purposes. Section 3 contains the main results of the paper with the reformulation of the SOV characterization of the transfer matrix spectrum in terms of the inhomogeneous Baxter functional equation and the associated finite system of generalized Bethe ansatz equations. In Section 4 we define the boundary conditions for which the inhomogeneity in the Baxter equation identically vanishes, in this way deriving the completeness of standard Bethe ansatz equations. There, we moreover derive the SOV spectrum functional reformulation for the remaining boundary conditions compatibles with homogeneous Baxter equations. Section 5 contains the description of a set of discrete transformations which leave unchanged the SOV characterization of the spectrum in this way proving the isospectrality of the transformed transfer matrices. These symmetries are used to find equivalent functional equation characterizations of the spectrum which allow to generalize the results described in Section 3 and 4. In Section 6 we present the SOV characterization of the spectrum for the rational 6-vertex representation of the reflection algebra and the reformulation of the spectrum by inhomogeneous Baxter equation. Finally, in Section 7, we present a comparison with the known numerical results in the literature for both the XXZ and XXX chains; the evidenced compatibility suggests that even in the homogenous chains our spectrum description is still complete.
Separation of variable for spin-1/2 representations of the reflection algebra
=============================================================================
Spin-1/2 representations of the reflection algebra and open XXZ quantum chain
-----------------------------------------------------------------------------
The representation theory of the reflection algebra can be studied in terms of the solutions $\mathcal{U}(\lambda )$ (monodromy matrices) of the following reflection equation:$$R_{12}(\lambda -\mu )\,\mathcal{U}_{1}(\lambda )\,R_{21}(\lambda +\mu -\eta
)\,\mathcal{U}_{2}(\mu )=\mathcal{U}_{2}(\mu )\,R_{12}(\lambda +\mu -\eta )\,\mathcal{U}_{1}(\lambda )\,R_{21}(\lambda -\mu ). \label{bYB}$$Here we consider the reflection equation associated to the 6-vertex trigonometric $R$ matrix $$R_{12}(\lambda )=\left(
\begin{array}{cccc}
\sinh (\lambda +\eta ) & 0 & 0 & 0 \\
0 & \sinh \lambda & \sinh \eta & 0 \\
0 & \sinh \eta & \sinh \lambda & 0 \\
0 & 0 & 0 & \sinh (\lambda +\eta )\end{array}\right) \in \text{End}(\mathcal{H}_{1}\otimes \mathcal{H}_{2}),$$where $\mathcal{H}_{a}\simeq \mathbb{C}^{2}$ is a 2-dimensional linear space. The 6-vertex trigonometric $R$-matrix is a solution of the Yang-Baxter equation:$$R_{12}(\lambda -\mu )R_{13}(\lambda )R_{23}(\mu )=R_{23}(\mu )R_{13}(\lambda
)R_{12}(\lambda -\mu ).$$The most general scalar solution ($2\times 2$ matrix) of the reflection equation reads$$K(\lambda ;\zeta ,\kappa ,\tau )=\frac{1}{\sinh \zeta }\left(
\begin{array}{cc}
\sinh (\lambda -\eta /2+\zeta ) & \kappa e^{\tau }\sinh (2\lambda -\eta ) \\
\kappa e^{-\tau }\sinh (2\lambda -\eta ) & \sinh (\zeta -\lambda +\eta /2)\end{array}\right) \in \text{End}(\mathcal{H}_{0}\simeq \mathbb{C}^{2}), \label{ADMFKK}$$where $\zeta ,$ $\kappa $ and $\tau $ are arbitrary complex parameters. Using it and following [@Skl88] we can construct two classes of solutions to the reflection equation (\[bYB\]) in the 2$^{\mathsf{N}}$-dimensional representation space:$$\mathcal{H}=\otimes _{n=1}^{\mathsf{N}}\mathcal{H}_{n}.$$Indeed, starting from$$K_{-}(\lambda )=K(\lambda ;\zeta _{-},\kappa _{-},\tau _{-}),\text{ \ \ \ \ }K_{+}(\lambda )=K(\lambda +\eta ;\zeta _{+},\kappa _{+},\tau _{+}),$$where $\zeta _{\pm },\kappa _{\pm },\tau _{\pm }$ are the boundary parameters, the following boundary monodromy matrices can be introduced $$\begin{aligned}
\mathcal{U}_{-}(\lambda ) &=&M_{0}(\lambda )K_{-}(\lambda )\widehat{M}_{0}(\lambda )=\left(
\begin{array}{cc}
\mathcal{A}_{-}(\lambda ) & \mathcal{B}_{-}(\lambda ) \\
\mathcal{C}_{-}(\lambda ) & \mathcal{D}_{-}(\lambda )\end{array}\right) \in \text{End}(\mathcal{H}_{0}\otimes \mathcal{H}), \\
\mathcal{U}_{+}^{t_{0}}(\lambda ) &=&M_{0}^{t_{0}}(\lambda
)K_{+}^{t_{0}}(\lambda )\widehat{M}_{0}^{t_{0}}(\lambda )=\left(
\begin{array}{cc}
\mathcal{A}_{+}(\lambda ) & \mathcal{C}_{+}(\lambda ) \\
\mathcal{B}_{+}(\lambda ) & \mathcal{D}_{+}(\lambda )\end{array}\right) \in \text{End}(\mathcal{H}_{0}\otimes \mathcal{H}).\end{aligned}$$ These matrices $\mathcal{U}_{-}(\lambda )$ and $\mathcal{V}_{+}(\lambda )=\mathcal{U}_{+}^{t_{0}}(-\lambda )$ define two classes of solutions of the reflection equation (\[bYB\]). Here, we have used the notations:$$M_{0}(\lambda )=R_{0\mathsf{N}}(\lambda -\xi _{\mathsf{N}}-\eta /2)\dots
R_{01}(\lambda -\xi _{1}-\eta /2)=\left(
\begin{array}{cc}
A(\lambda ) & B(\lambda ) \\
C(\lambda ) & D(\lambda )\end{array}\right) \label{T}$$and $$\widehat{M}(\lambda )=(-1)^{\mathsf{N}}\,\sigma _{0}^{y}\,M^{t_{0}}(-\lambda
)\,\sigma _{0}^{y}, \label{Mhat}$$where $M_{0}(\lambda )\in $ End$(\mathcal{H}_{0}\otimes \mathcal{H})$ is the bulk inhomogeneous monodromy matrix (the $\xi _{j}$ are the arbitrary inhomogeneity parameters) satisfing the Yang-Baxter relation:$$R_{12}(\lambda -\mu )M_{1}(\lambda )M_{2}(\mu )=M_{2}(\mu )M_{1}(\lambda
)R_{12}(\lambda -\mu ). \label{YB}$$The main interest of these boundary monodromy matrices is the property shown by Sklyanin [@Skl88] that the following family of transfer matrices:$$\mathcal{T}(\lambda )=\text{tr}_{0}\{K_{+}(\lambda )\,M(\lambda
)\,K_{-}(\lambda )\widehat{M}(\lambda )\}=\text{tr}_{0}\{K_{+}(\lambda )\mathcal{U}_{-}(\lambda )\}=\text{tr}_{0}\{K_{-}(\lambda )\mathcal{U}_{+}(\lambda
)\}\in \text{\thinspace End}(\mathcal{H}), \label{transfer}$$defines a one parameter family of commuting operators in End$(\mathcal{H})$. The Hamiltonian of the open XXZ quantum spin 1/2 chain with the most general integrable boundary terms can be obtained in the homogeneous limit ($\xi _{m}=0$ for $m=1,\ldots ,\mathsf{N}$) from the following derivative of the transfer matrix (\[transfer\]):$$H=\frac{2(\sinh \eta )^{1-2\mathsf{N}}}{\text{tr}\{K_{+}(\eta /2)\}\,\text{tr}\{K_{-}(\eta /2)\}}\frac{d}{d\lambda }\mathcal{T}(\lambda )_{\,\vrule height13ptdepth1pt\>{\lambda =\eta /2}\!}+\text{constant,} \label{Ht}$$and its explicit form reads: $$\begin{aligned}
H& =\sum_{i=1}^{\mathsf{N}-1}(\sigma _{i}^{x}\sigma _{i+1}^{x}+\sigma
_{i}^{y}\sigma _{i+1}^{y}+\cosh \eta \sigma _{i}^{z}\sigma _{i+1}^{z})
\notag \\
& +\frac{\sinh \eta }{\sinh \zeta _{-}}\left[ \sigma _{1}^{z}\cosh \zeta
_{-}+2\kappa _{-}(\sigma _{1}^{x}\cosh \tau _{-}+i\sigma _{1}^{y}\sinh \tau
_{-})\right] \notag \\
& +\frac{\sinh \eta }{\sinh \zeta _{+}}[(\sigma _{\mathsf{N}}^{z}\cosh \zeta
_{+}+2\kappa _{+}(\sigma _{\mathsf{N}}^{x}\cosh \tau _{+}+i\sigma _{\mathsf{N}}^{y}\sinh \tau _{+}). \label{H-XXZ-Non-D}\end{aligned}$$Here $\sigma _{i}^{a}$ are local spin $1/2$ operators (Pauli matrices), $\Delta =\cosh \eta $ is the anisotropy parameter and the six complex boundary parameters $\zeta _{\pm }$, $\kappa _{\pm }$ and $\tau _{\pm }$ define the most general integrable magnetic interactions at the boundaries.
Some relevant properties
------------------------
The following quadratic linear combination of the generators $\mathcal{A}_{-}(\lambda ),$ $\mathcal{B}_{-}(\lambda ),$ $\mathcal{C}_{-}(\lambda )$ and $\mathcal{D}_{-}(\lambda )$ of the reflection algebra: $$\begin{aligned}
\frac{\mathrm{det}_{q}\,\mathcal{U}_{-}(\lambda )}{\sinh (2\lambda -2\eta )}& =\mathcal{A}_{-}(\epsilon \lambda +\eta /2)\mathcal{A}_{-}(\eta /2-\epsilon
\lambda )+\mathcal{B}_{-}(\epsilon \lambda +\eta /2)\mathcal{C}_{-}(\eta
/2-\epsilon \lambda ) \label{q-detU_1} \\
& =\mathcal{D}_{-}(\epsilon \lambda +\eta /2)\mathcal{D}_{-}(\eta
/2-\epsilon \lambda )+\mathcal{C}_{-}(\epsilon \lambda +\eta /2)\mathcal{B}_{-}(\eta /2-\epsilon \lambda ), \label{q-detU_2}\end{aligned}$$where $\epsilon =\pm 1$, is the *quantum determinant* . It was shown by Sklyanin that it is a central element of the reflection algebra$$\lbrack \mathrm{det}_{q}\,\mathcal{U}_{-}(\lambda ),\mathcal{U}_{-}(\mu )]=0.$$The quantum determinant plays a fundamental role in the characterization of the transfer matrix spectrum and it admits the following explicit expressions:$$\begin{aligned}
\mathrm{det}_{q}\,\mathcal{U}_{-}(\lambda ) &=&\mathrm{det}_{q}K_{-}(\lambda )\mathrm{det}_{q}M_{0}(\lambda )\mathrm{det}_{q}M_{0}(-\lambda )
\label{q-detU_-exp} \\
&=&\sinh (2\lambda -2\eta )\mathsf{A}_{-}(\lambda +\eta /2)\mathsf{A}_{-}(-\lambda +\eta /2),\end{aligned}$$where: $$\mathrm{det}_{q}M(\lambda )=a(\lambda +\eta /2)d(\lambda -\eta /2),
\label{bulk-q-det}$$is the bulk quantum determinant and$$\mathrm{det}_{q}K_{\pm }(\lambda )=\mp \sinh (2\lambda \pm 2\eta )g_{\pm
}(\lambda +\eta /2)g_{\pm }(-\lambda +\eta /2).$$Here, we used the following notations:$$\mathsf{A}_{-}(\lambda )=g_{-}(\lambda )a(\lambda )d(-\lambda ),\text{ \ }d(\lambda )=a(\lambda -\eta ),\text{ \ \ }a(\lambda )=\prod_{n=1}^{\mathsf{N}}\sinh (\lambda -\xi _{n}+\eta /2), \label{eigenA}$$and$$g_{\pm }(\lambda )=\frac{\sinh (\lambda +\alpha _{\pm }-\eta /2)\cosh
(\lambda \mp \beta _{\pm }-\eta /2)}{\sinh \alpha _{\pm }\cosh \beta _{\pm }}, \label{g_PM}$$where $\alpha _{\pm }$ and $\beta _{\pm }$ are defined in terms of the boundary parameters by:$$\sinh \alpha _{\pm }\cosh \beta _{\pm }=\frac{\sinh \zeta _{\pm }}{2\kappa
_{\pm }},\text{ \ \ \ \ \ }\cosh \alpha _{\pm }\sinh \beta _{\pm }=\frac{\cosh \zeta _{\pm }}{2\kappa _{\pm }}. \label{alfa-beta}$$
\[normality\]The transfer matrix $\mathcal{T}(\lambda )$ is an even function of the spectral parameter $\lambda $:$$\mathcal{T}(-\lambda )=\mathcal{T}(\lambda ), \label{even-transfer}$$and it is central for the following special values of the spectral parameter: $$\begin{aligned}
\lim_{\lambda \rightarrow \pm \infty }e^{\mp 2\lambda (\mathsf{N}+2)}\mathcal{T}(\lambda ) &=&2^{-(2\mathsf{N}+1)}\frac{\kappa _{+}\kappa
_{-}\cosh (\tau _{+}-\tau _{-})}{\sinh \zeta _{+}\sinh \zeta _{-}},
\label{Central-asymp} \\
\mathcal{T}(\pm \eta /2) &=&(-1)^{\mathsf{N}}2\cosh \eta \mathrm{det}_{q}M(0),
\label{Central-1} \\
\mathcal{T}(\pm (\eta /2-i\pi /2)) &=&-2\cosh \eta \coth \zeta _{-}\coth
\zeta _{+}\mathrm{det}_{q}M(i\pi /2). \label{Central-2}\end{aligned}$$Moreover, the monodromy matrix $\mathcal{U}_{\pm }(\lambda )$ satisfy the following transformation properties under Hermitian conjugation:
- Under the condition $\eta \in i\mathbb{R}$ (massless regime), it holds: $$\mathcal{U}_{\pm }(\lambda )^{\dagger }=\left[ \mathcal{U}_{\pm }(-\lambda
^{\ast })\right] ^{t_{0}}, \label{ml-Hermitian_U}$$ for $\{i\tau _{\pm },i\kappa _{\pm },i\zeta _{\pm },\xi
_{1},...,\xi _{\mathsf{N}}\}\in \mathbb{R}^{\mathsf{N}+3}.$
- Under the condition $\eta \in \mathbb{R}$ (massive regime), it holds: $$\mathcal{U}_{\pm }(\lambda )^{\dagger }=\left[ \mathcal{U}_{\pm }(\lambda
^{\ast })\right] ^{t_{0}}, \label{m-Hermitian_U}$$for $\{\tau _{\pm },\kappa _{\pm },\zeta _{\pm },i\xi
_{1},...,i\xi _{\mathsf{N}}\}\in \mathbb{R}^{\mathsf{N}+3}.$
So under the same conditions on the parameters of the representation it holds: $$\mathcal{T}(\lambda )^{\dagger }=\mathcal{T}(\lambda ^{\ast }),
\label{I-Hermitian_T}$$i.e. $\mathcal{T}(\lambda )$ defines a one-parameter family of normal operators which are self-adjoint both for $\lambda $ real and purely imaginary.
SOV representations for $\mathcal{T}(\protect\lambda )$-spectral problem
------------------------------------------------------------------------
Let us recall here the characterization obtained in [@Nic12b; @Fald-KN13] by SOV method of the spectrum of the transfer matrix $\mathcal{T}(\lambda )$. First we introduce the following notations:$$X_{k,m}^{(i,r)}(\tau _{\pm },\alpha _{\pm },\beta _{\pm })\equiv \left(
-1\right) ^{i}\left( 1-r\right) \eta +\tau _{-}-\tau_{+}+(-1)^{k}(\alpha
_{-}+\beta _{-})-(-1)^{m}(\alpha _{+}-\beta _{+})+i\pi (k+m),
\label{SOV-cond-}$$and by using these linear combinations of the boundary parameters we introduce the set $N_{SOV}\subset\mathbb{C}^6$ of boundary parameters for which the separation of variables cannot be applied directly. More precisely $$(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha_{-},\beta _{-})\in N_{SOV},$$ if $\exists (k,h,m,n)\in \left\{ 0,1\right\} $ such that $$X_{k,m}^{(0,\mathsf{N})}(\tau _{\pm },\alpha _{\pm },\beta _{\pm })=0 \quad\text{and}\quad
X_{h,n}^{(1,\mathsf{N})}(\tau _{\pm },\alpha _{\pm },\beta _{\pm })=0.$$All the results in the following will be obtained for the generic values of the boundary parameters, not belonging to this set. The SOV method applicability can be further extended applying the discrete symmetries discussed in the Section \[sect-descretesym\].
Following [@Fald-KN13] we define the functions:$$\begin{aligned}
g_{a}(\lambda ) &=&\frac{\cosh ^{2}2\lambda -\cosh ^{2}\eta }{\cosh
^{2}2\zeta _{a}^{(0)}-\cosh ^{2}\eta }\,\prod_{\substack{ b=1 \\ b\neq a}}^{\mathsf{N}}\frac{\cosh 2\lambda -\cosh 2\zeta _{b}^{(0)}}{\cosh 2\zeta
_{a}^{(0)}-\cosh 2\zeta _{b}^{(0)}}\quad \text{ \ for }a\in \{1,...,\mathsf{N}\}, \\
\mathbf{A}(\lambda ) &=&(-1)^{\mathsf{N}}\frac{\sinh (2\lambda +\eta )}{\sinh 2\lambda }g_{+}(\lambda )g_{-}(\lambda )a(\lambda )d(-\lambda ),\end{aligned}$$and$$\begin{aligned}
f(\lambda )=& \frac{\cosh 2\lambda +\cosh \eta }{2\cosh \eta }\prod_{b=1}^{\mathsf{N}}\frac{\cosh 2\lambda -\cosh 2\zeta _{b}^{(0)}}{\cosh \eta -\cosh
2\zeta _{b}^{(0)}}\mathbf{A}(\eta /2) \notag \\
& -(-1)^{\mathsf{N}}\frac{\cosh 2\lambda -\cosh \eta }{2\cosh \eta }\prod_{b=1}^{\mathsf{N}}\frac{\cosh 2\lambda -\cosh 2\zeta _{b}^{(0)}}{\cosh
\eta +\cosh 2\zeta _{b}^{(0)}}\mathbf{A}(\eta /2+i\pi /2) \notag \\
& +2^{(1-\mathsf{N})}\frac{\kappa _{+}\kappa _{-}\cosh (\tau _{+}-\tau _{-})}{\sinh \zeta _{+}\sinh \zeta _{-}}(\cosh ^{2}2\lambda -\cosh ^{2}\eta
)\prod_{b=1}^{\mathsf{N}}(\cosh 2\lambda -\cosh 2\zeta _{b}^{(0)}),
\label{f-function}\end{aligned}$$where$$\zeta _{n}^{(h_{n})}=\xi _{n}+(h_{n}-\frac{1}{2})\eta \quad \forall n\in
\{1,...,\mathsf{N}\},\text{ }h_{n}\in \{0,1\}\text{.}$$We can now recall the main result on the characterization of the set $\Sigma
_{\mathcal{T}}$ formed by all the eigenvalue functions of the transfer matrix $\mathcal{T}(\lambda )$.
\[C:T-eigenstates-\]Let $(\tau _{+},\alpha _{+},\beta _{+},\tau
_{-},\alpha _{-},\beta _{-})\in \mathbb{C}^{6}\backslash N_{SOV}$ and let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ be generic:$$\xi _{a}\neq \pm\xi _{b}+r\eta\text{\ \ mod\,}2\pi \text{ \ }\forall a\neq b\in \{1,...,\mathsf{N}\}\,\,\text{and\thinspace \thinspace }r\in \{-1,0,1\}, \label{xi-conditions}$$then $\mathcal{T}(\lambda )$ has simple spectrum and the set of its eigenvalues $\Sigma _{\mathcal{T}}$ is characterized by:$$\Sigma _{\mathcal{T}}=\left\{ \tau (\lambda ):\tau (\lambda )=f(\lambda
)+\sum_{a=1}^{\mathsf{N}}g_{a}(\lambda )x_{a},\text{ \ \ }\forall
\{x_{1},...,x_{\mathsf{N}}\}\in \Sigma _{T}\right\} ,
\label{Interpolation-Form-T}$$where $\Sigma _{T}$ is the set of solutions to the following inhomogeneous system of $\mathsf{N}$ quadratic equations:$$x_{n}\sum_{a=1}^{\mathsf{N}}g_{a}(\zeta _{n}^{(1)})x_{a}+x_{n}f(\zeta
_{n}^{(1)})=q_{n},\text{ \ \ \ }q_{n}=\frac{\mathrm{det}_{q}K_{+}(\xi _{n})\mathrm{det}_{q}\,\mathcal{U}_{-}(\xi _{n})}{\sinh (\eta +2\xi _{n})\sinh (\eta
-2\xi _{n})},\text{ \ \ }\forall n\in \{1,...,\mathsf{N}\},
\label{Quadratic System}$$in $\mathsf{N}$ unknowns $\{x_{1},...,x_{\mathsf{N}}\}$.
Inhomogeneous Baxter equation
==============================
Here we show that the SOV characterization of the spectrum admits an equivalent formulation in terms of a second order functional difference equation of Baxter type:$$\tau (\lambda )Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )+F(\lambda ), \label{Inhom-Baxter-Eq}$$which contains a non-zero inhomogeneous term $F(\lambda )$ non-zero for generic integrable boundary conditions and the $Q$-functions are [*trigonometric polynomials*]{}. In this paper we will call $f(\lambda)$ a trigonometric polynomial of degree $\mathsf{M}$ if $e^{\mathsf{M}\lambda}\,f(\lambda)$ is a polynomial of $e^{2\lambda}$ of degree $\mathsf{M}$. Most trigonometric polynomials we will consider in the following sections will be even functions of $\lambda$ and will satisfy an additional condition $f({\lambda}+i\pi)=f({\lambda})$. It is easy to see in this situation that such functions can be written as polynomials of $\cosh 2\lambda$.
Main functions in the functional equation
-----------------------------------------
Let $Q(\lambda )$ be an even trigonometric polynomial of degree $2\mathsf{N}$. It can be written in the following form:$$\begin{aligned}
Q(\lambda )& =\sum_{a=1}^{\mathsf{N}}\prod_{\substack{ b=1 \\ b\neq a}}^{\mathsf{N}}\frac{\cosh 2\lambda -\cosh 2\zeta _{b}^{(0)}}{\cosh 2\zeta
_{a}^{(0)}-\cosh 2\zeta _{b}^{(0)}}Q(\zeta _{a}^{(0)})+2^{\mathsf{N}}\prod_{a=1}^{\mathsf{N}}\left( \cosh 2\lambda -\cosh 2\zeta
_{a}^{(0)}\right) \label{Q-form1} \\
& =2^{\mathsf{N}}\prod_{a=1}^{\mathsf{N}}\left( \cosh 2\lambda -\cosh
2\lambda _{a}\right) , \label{Q-form2}\end{aligned}$$where from now on the $Q(\zeta _{a}^{(0)})$ are arbitrary complex numbers or similarly the $\lambda _{a}$ are arbitrary complex numbers. Then, introducing the function:$$Z_{Q}(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda
)Q(\lambda +\eta )$$we can prove the following Lemma
Let $Q(\lambda )$ be any function of the form $\left( \ref{Q-form2}\right) $ then the associated function $Z_{Q}(\lambda )$ is an even trigonometric polynomial of degree $4\mathsf{N}+4$ of the following form:$$Z_{Q}(\lambda )=\sum_{a=0}^{2(\mathsf{N}+1)}z_{a}\cosh ^{a}2\lambda ,\text{
with }z_{2(\mathsf{N}+1)}=\frac{2\kappa _{+}\kappa _{-}\cosh (\alpha
_{+}+\alpha _{-}-\beta _{+}+\beta _{-}-(\mathsf{N}+1)\eta )}{\sinh \zeta
_{+}\sinh \zeta _{-}}.$$
The fact that the function $Z_{Q}(\lambda )$ is even in $\lambda $ is a trivial consequence of the fact that $Q(\lambda )$ is even; in fact, it holds:$$\begin{aligned}
Z_{Q}(-\lambda ) &=&\mathbf{A}(-\lambda )Q(-\lambda -\eta )+\mathbf{A}(\lambda )Q(-\lambda +\eta ) \notag \\
&=&\mathbf{A}(-\lambda )Q(\lambda +\eta )+\mathbf{A}(\lambda )Q(\lambda
-\eta )=Z_{Q}(\lambda ).\end{aligned}$$The fact that $Z_{Q}(\lambda )$ is indeed a trigonometric polynomial follows from its definition once we observe that $\lambda=0$ is not a singular point and the following identity holds:$$\lim_{\lambda \rightarrow 0}Z_{Q}(\lambda )=2g_{+}(0)g_{-}(0)a(0)a(-\eta
)Q(0)\cosh \eta .$$Now the functional form of $Z_{Q}(\lambda )$ is a consequence of the following identities:$$Z_{Q}(\lambda +i\pi )=Z_{Q}(\lambda ),\text{ \ }\lim_{\lambda \rightarrow
\pm \infty }\frac{Z_{Q}(\lambda )}{e^{\pm 4(\mathsf{N}+1)\lambda }}=\frac{\kappa _{+}\kappa _{-}\cosh (\alpha _{+}+\alpha _{-}-\beta _{+}+\beta _{-}-(\mathsf{N}+1)\eta )}{2^{(2\mathsf{N}+1)}\sinh \zeta _{+}\sinh \zeta _{-}},$$where the second identity follows from:$$\begin{aligned}
\lim_{\lambda \rightarrow \pm \infty }e^{\mp (2\mathsf{N}+4)\lambda }\mathbf{A}(\lambda )& =2^{-2(\mathsf{N}+1)}\frac{\kappa _{+}\kappa _{-}\exp \pm
(\alpha _{+}+\alpha _{-}-\beta _{+}+\beta _{-}+(\mathsf{N}-1)\eta )}{\sinh
\zeta _{+}\sinh \zeta _{-}}, \\
\lim_{\lambda \rightarrow \pm \infty }e^{\mp 2\mathsf{N}\lambda }Q(\lambda
)& =1.\end{aligned}$$
On the need of an inhomogeneous term in the functional equation
---------------------------------------------------------------
Here, we would like to point out that it is simple to define the boundary conditions for which one can prove that the homogeneous version of the Baxter equation $\left( \ref{Inhom-Baxter-Eq}\right) $ does not admit trigonometric polynomial solutions for $\tau (\lambda )\in \Sigma _{\mathcal{T}}$.
\[impossible\_hom\] Assume that the boundary parameters satisfy the following conditions:$$\kappa _{+}\neq 0,\kappa _{-}\neq 0,\text{ \ }Y^{(i,r)}(\tau _{\pm },\alpha
_{\pm },\beta _{\pm })\neq 0\text{ \ }\forall i\in \left\{ 0,1\right\} ,r\in
\mathbb{Z} \label{Inhomogeneous-boundary conditions}$$where we have defined:$$Y^{(i,r)}(\tau _{\pm },\alpha _{\pm },\beta _{\pm })\equiv \tau _{-}-\tau
_{+}+\left( -1\right) ^{i}\left[ \left( \mathsf{N}-1-r\right) \eta +(\alpha
_{-}+\alpha _{+}+\beta _{-}-\beta _{+})\right] ,$$then for any $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ the homogeneous Baxter equation:$$\tau (\lambda )Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta ),$$does not admit any (non identically zero) $Q(\lambda )$ of Laurent polynomial form in $e^{\lambda }$.
If we consider the following function:$$Q(\lambda )=\sum_{a=-s}^{r}y_{a}e^{a\lambda },\text{ \ with }r,s\in \mathbb{N}$$we can clearly always chose the coefficients $y_{a}$ such that the r.h.s. of the homogeneous Baxter equation has no poles as required. However, it is enough to consider now the asymptotics:$$\begin{aligned}
\lim_{\lambda \rightarrow +\infty }\frac{\left[ \mathbf{A}(\lambda
)Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )\right] }{e^{(2\mathsf{N}+4+r)\lambda }}& =\frac{y_{r}\kappa _{+}\kappa _{-}\cosh (\alpha
_{+}+\alpha _{-}-\beta _{+}+\beta _{-}+(\mathsf{N}-1-r)\eta )}{2^{2(\mathsf{N}+1)}\sinh \zeta _{+}\sinh \zeta _{-}} \\
\lim_{\lambda \rightarrow +\infty }e^{-(2\mathsf{N}+4+r)\lambda }\tau
(\lambda )Q(\lambda )& =\frac{y_{r}\kappa _{+}\kappa _{-}\cosh (\tau
_{+}-\tau _{-})}{2^{2(\mathsf{N}+1)}\sinh \zeta _{+}\sinh \zeta _{-}}\end{aligned}$$and use the conditions $\left( \ref{Inhomogeneous-boundary conditions}\right) $ to observe that for any $r\in \mathbb{Z}$ the asymptotic of the homogeneous Baxter equation cannot be satisfied which implies the validity of the lemma.
SOV spectrum in terms of the inhomogeneous Baxter equation
----------------------------------------------------------
We introduce now the following function of the boundary parameters:$$F_{0}=\frac{2\kappa _{+}\kappa _{-}\left( \cosh (\tau _{+}-\tau _{-})-\cosh
(\alpha _{+}+\alpha _{-}-\beta _{+}+\beta _{-}-(\mathsf{N}+1)\eta )\right) }{\sinh \zeta _{+}\sinh \zeta _{-}},$$and then the function:$$\begin{aligned}
F(\lambda ) &=&2^{\mathsf{N}}\,F_{0}\,(\cosh ^{2}2\lambda -\cosh ^{2}\eta
)a(\lambda )a(-\lambda )d(-\lambda )d(\lambda ) \\
&=&F_{0}\, (\cosh ^{2}2\lambda -\cosh ^{2}\eta )\prod_{b=1}^{\mathsf{N}}\prod_{i=0}^{1}(\cosh 2\lambda -\cosh 2\zeta _{b}^{(i)}).\end{aligned}$$We introduce also the set of functions $\Sigma _{\mathcal{Q}}$ such that $Q(\lambda)\in\Sigma _{\mathcal{Q}}$ if it has a form $\left( \ref{Q-form2}\right) $ and $$\tau(\lambda)=\frac{Z_{Q}(\lambda )+F(\lambda )}{Q(\lambda )}$$ is a trigonometric polynomial. We are now ready to prove the main theorem of this article:
\[T-eigenvalue-F-eq\]Let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ be generic [(\[xi-conditions\])]{} and let the boundary parameters $(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta
_{-})\in \mathbb{C}^{6}\backslash N_{SOV}$ satisfy the following conditions:$$\kappa _{+}\neq 0,\kappa _{-}\neq 0,\text{ \ }Y^{(i,2r)}(\tau _{\pm },\alpha
_{\pm },\beta _{\pm })\neq 0\text{ \ }\forall i\in \left\{ 0,1\right\} ,r\in
\left\{ 0,...,\mathsf{N}-1\right\} , \label{Inhom-cond-BaxEq}$$then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $\exists
!Q(\lambda )\in \Sigma _{\mathcal{Q}}$ such that $$\tau (\lambda
)Q(\lambda )=Z_{Q}(\lambda )+F(\lambda ).$$
First we prove that if $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ then there is a trigonometric polynomial $Q(\lambda )\in \Sigma _{\mathcal{Q}}$ satisfying the inhomogeneous functional Baxter equation: $$\tau (\lambda )Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )+F(\lambda ).$$To prove it we will show that there is the unique set of values $Q(\zeta _{b}^{(0)})$ such that $Q(\lambda)$ of the form (\[Q-form1\]) satisfies this equation. It is straightforward to verify that if $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ and $Q(\lambda )$ has the form $\left( \ref{Q-form2}\right) $ then the left and right hand sides of the above equation are both even trigonometric polynomials of $\lambda
$ and both can be written (using the asymptotic behavior) in the form:$$\frac{2\kappa _{+}\kappa _{-}\cosh (\tau _{+}-\tau _{-})\prod_{b=1}^{2\mathsf{N}+2}(\cosh 2\lambda -\cosh 2y_{b}^{\left( lhs/rhs\right) })}{\sinh
\zeta _{+}\sinh \zeta _{-}}.$$Then to prove that we can introduce a $Q(\lambda )$ of the form $\left( \ref{Q-form2}\right) $ which satisfies the inhomogeneous Baxter equation $\left( \ref{Inhom-Baxter-Eq}\right) $ with $\tau (\lambda )\in \Sigma _{\mathcal{T}} $, we have only to prove that $\left( \ref{Inhom-Baxter-Eq}\right) $ is satisfied in $4\mathsf{N}+4$ different values of $\lambda $. As the [*r.h.s*]{} and [*l.h.s*]{} of $\left( \ref{Inhom-Baxter-Eq}\right) $ are even functions we need to check this identity only for $2N+2$ non-zero points $\mu_j$ such that $\mu_j\neq \pm \mu_k$. It is a simple exercise verify that the equation $\left( \ref{Inhom-Baxter-Eq}\right) $ is satisfied automatically for any $Q(\lambda )$ of the form $\left( \ref{Q-form2}\right) $ in the following two points, $ \eta /2$ and $ \eta
/2+i\pi /2$:$$\tau (\eta /2)Q(\eta /2)=\mathbf{A}(\eta /2)Q(\eta /2-\eta )=\mathbf{A}(\eta /2)Q(\eta /2), \label{System-A}$$and:$$\tau (\eta /2+i\pi /2)Q(\eta /2+i\pi /2)=\mathbf{A}(\eta /2+i\pi /2)Q(i\pi /2-\eta /2) \\
=\mathbf{A}(\eta /2+i\pi /2)Q(\eta /2+i\pi /2) . \label{System-B}$$Indeed, these equations reduce to:$$\tau (\eta /2)=\mathbf{A}(\eta /2),\text{ \ \ \ }\tau (\eta /2+i\pi /2)=\mathbf{A}(\eta /2+i\pi /2)$$and so they are satisfied by definition for any $\tau (\lambda )\in \Sigma _{\mathcal{T}}$. Then we check the explicit form of the equation $\left( \ref{Inhom-Baxter-Eq}\right) $ in the $2\mathsf{N}$ points $ \zeta _{b}^{(0)}$ and $\zeta _{b}^{(1)}$:$$\tau (\zeta _{b}^{(0)})Q(\zeta _{b}^{(0)})=\mathbf{A}(-\zeta _{b}^{(0)})Q(\zeta _{b}^{(0)}+\eta )=\mathbf{A}(-\zeta
_{b}^{(0)})Q(\zeta _{b}^{(1)}),$$and:$$\tau (\zeta _{b}^{(1)})Q(\zeta _{b}^{(1)})=\mathbf{A}(\zeta _{b}^{(1)})Q(\zeta _{b}^{(1)}-\eta )=\mathbf{A}(\zeta
_{b}^{(1)})Q(\zeta _{b}^{(0)}).$$They are equivalent to the following system of equations:$$\begin{aligned}
\frac{\mathbf{A}(\zeta _{b}^{(1)})}{\tau (\zeta _{b}^{(1)})}& =\frac{\tau
(\zeta _{b}^{(0)})}{\mathbf{A}(-\zeta _{b}^{(0)})}\text{ \ \ \ \ \ }\forall
b\in \{1,...,\mathsf{N}\} \label{System1} \\
\frac{Q(\zeta _{b}^{(0)})\tau (\zeta _{b}^{(0)})}{\mathbf{A}(-\zeta
_{b}^{(0)})}& =\sum_{a=1}^{\mathsf{N}}\prod_{\substack{ c=1 \\ c\neq a}}^{\mathsf{N}}\frac{\cosh 2\zeta _{b}^{(1)}-\cosh 2\zeta _{c}^{(0)}}{\cosh
2\zeta _{a}^{(0)}-\cosh 2\zeta _{c}^{(0)}}Q(\zeta _{a}^{(0)})+2^{\mathsf{N}}\prod_{a=1}^{\mathsf{N}}\left( \cosh 2\zeta _{b}^{(1)}-\cosh 2\zeta
_{a}^{(0)}\right) \label{System2}\end{aligned}$$Now using the following quantum determinant identity $$\frac{\det_{q}K_{+}(\lambda-\eta/2)\det_{q}
\mathcal{U}_{-}(\lambda -\eta /2)}{\sinh (2\lambda +\eta )\sinh (2\lambda -\eta )}=\mathbf{A}(\lambda )\mathbf{A}(-\lambda +\eta ).\label{Tot-q-det-tt}$$ it is easy to see that the system of equations $\left( \ref{System1}\right) $ is certainly satisfied as $\tau (\lambda )\in \Sigma _{\mathcal{T}}$, once we recall the SOV characterization (\[Interpolation-Form-T\]) of $\Sigma _{\mathcal{T}}$. Indeed there is a set $\{x_1,\dots,x_n\}$ satisfying the equations (\[Quadratic System\]) and $\tau(\zeta _{b}^{(0)})=x_b$.
So we are left with $\left( \ref{System2}\right) $ a linear system of $\mathsf{N}$ inhomogeneous equations with $\mathsf{N}$ unknowns $Q(\zeta _{a}^{(0)})$. Here, we prove that the matrix of this linear system$$c_{a b}\equiv \prod_{\substack{ c=1 \\ c\neq a}}^{\mathsf{N}}\frac{\cosh
2\zeta _{b}^{(1)}-\cosh 2\zeta _{c}^{(0)}}{\cosh 2\zeta _{a}^{(0)}-\cosh
2\zeta _{c}^{(0)}}-\delta _{a b}\frac{\tau (\zeta _{b}^{(0)})}{\mathbf{A}(-\zeta _{b}^{(0)})}\text{ \ \ \ \ \ }\forall a,b\in \{1,...,\mathsf{N}\}$$ has nonzero determinant for the given $\tau (\lambda
)\in \Sigma _{\mathcal{T}}$. Indeed, let us suppose that for some $\tau (\lambda )\in \Sigma _{\mathcal{T}}$:$$\mathrm{det}_{\mathsf{N}}\left[ c_{a b}\right] =0. \label{det-coeff}$$Then there is at least one nontrivial solution $\{Q(\zeta
_{1}^{(0)}),...,Q(\zeta _{\mathsf{N}}^{(0)})\}\neq \{0,...,0\}$ to the homogeneous system of equations:$$\frac{Q(\zeta _{b}^{(0)})\tau (\zeta _{b}^{(0)})}{\mathbf{A}(-\zeta
_{b}^{(0)})}=\sum_{a=1}^{\mathsf{N}}\prod_{\substack{ c=1 \\ c\neq a}}^{\mathsf{N}}\frac{\cosh 2\zeta _{b}^{(1)}-\cosh 2\zeta _{c}^{(0)}}{\cosh
2\zeta _{a}^{(0)}-\cosh 2\zeta _{c}^{(0)}}Q(\zeta _{a}^{(0)})
\label{System2-homo}$$and hence we can define:$$Q_{\mathsf{M}}(\lambda )=\sum_{a=1}^{\mathsf{N}}\prod_{\substack{ b=1 \\ b\neq a}}^{\mathsf{N}}\frac{\cosh 2\lambda -\cosh 2\zeta _{b}^{(0)}}{\cosh
2\zeta _{a}^{(0)}-\cosh 2\zeta _{b}^{(0)}}Q(\zeta _{a}^{(0)})=\lambda _{\mathsf{M}+1}^{(\mathsf{M})}\prod_{b=1}^{\mathsf{M}}\left( \cosh 2\lambda
-\cosh 2\lambda _{b}^{(\mathsf{M})}\right) .$$It is an even trigonometric polynomial of degree $2\mathsf{M}$ such that $0\leq
\mathsf{M}\leq \mathsf{N}-1$ fixed by the solution $\{Q(\zeta
_{1}^{(0)}),...,Q(\zeta _{\mathsf{N}}^{(0)})\}$. Now using the $Q_{\mathsf{M}}(\lambda )$ and $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ we can define two functions:$$W_{1}(\lambda )=Q_{\mathsf{M}}(\lambda )\tau (\lambda )\text{ \, and \, }W_{2}(\lambda )=\mathbf{A}(\lambda )Q_{\mathsf{M}}(\lambda -\eta )+\mathbf{A}(-\lambda )Q_{\mathsf{M}}(\lambda +\eta )$$which are both even trigonometric polynomials of degree $2\mathsf{M}+2\mathsf{N}+4$. Then it is straightforward to observe that the systems of equations $\left( \ref{System1}\right) $ and $\left( \ref{System2-homo}\right) $ plus the conditions $\left( \ref{System-A}\right) $ and $\left( \ref{System-B}\right)
$, which are also satisfied with the function $Q_{\mathsf{M}}(\lambda )$, imply that $W_{1}(\lambda )$ and $W_{2}(\lambda )$ coincide in $4\mathsf{N}+4 $ different values of $\lambda $ ($\pm \eta /2$, $\pm (\eta /2+i\pi /2)$, $\pm \zeta _{b}^{(0)}$ and $\pm \zeta _{b}^{(1)}$). It means that $W_{1}(\lambda )\equiv W_{2}(\lambda )$, as these are two polynomials of maximal degree $4\mathsf{N}+2$. So, we have shown that from the assumption $\exists \tau (\lambda )\in \Sigma _{\mathcal{T}}$ such that $\left( \ref{det-coeff}\right) $ holds it follows that $\tau (\lambda )$ and $Q_{\mathsf{M}}(\lambda )$ have to satisfy the following homogeneous Baxter equations: $$\tau (\lambda )Q_{\mathsf{M}}(\lambda )=\mathbf{A}(\lambda )Q_{\mathsf{M}}(\lambda -\eta )+\mathbf{A}(-\lambda )Q_{\mathsf{M}}(\lambda +\eta ).
\label{Baxter-eq-homo}$$Now we can apply the Lemma \[impossible\_hom\] which implies that $Q_{\mathsf{M}}(\lambda )=0$ for any $\lambda$, which contradicts the hypothesis of the existence of a nontrivial solution to the homogeneous system [(\[System2-homo\])]{}. Hence, we have proven that $\mathrm{det}_{\mathsf{N}}\left[ c_{a b}\right] \neq 0.$ Therefore there is a unique solution $\{Q(\zeta
_{1}^{(0)}),...,Q(\zeta _{\mathsf{N}}^{(0)})\}$ of the inhomogeneous system $\left( \ref{System2}\right) $ which defines one and only one $Q(\lambda )$ of the form $\left( \ref{Q-form1}\right) $ satisfying the functional inhomogeneous Baxter’s equation $\left( \ref{Inhom-Baxter-Eq}\right) $.
We prove now that if $Q(\lambda )\in \Sigma _{\mathcal{Q}}$ then $\tau (\lambda )=\left( Z_{Q}(\lambda )+F(\lambda )\right) /Q(\lambda )\in
\Sigma _{\mathcal{T}}$. By definition of the functions $Z_{Q}(\lambda ),$ $F(\lambda )$ and $Q(\lambda )$ the function $\tau (\lambda )$ has the desired form:$$\tau (\lambda )=f(\lambda )+\sum_{a=1}^{\mathsf{N}}g_{a}(\lambda )\tau
(\zeta _{a}^{(0)}).$$To prove now that $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ we have to write the inhomogeneous Baxter equation $\left( \ref{Inhom-Baxter-Eq}\right) $ in the $2\mathsf{N}$ points $ \zeta _{b}^{(0)}$ and $ \zeta
_{b}^{(1)}$. Indeed, we have already proved that this reproduce the systems $\left( \ref{System1}\right) $ and $\left( \ref{System2}\right) $ and it is simple to observe that the system of equations $\left( \ref{System1}\right) $ just coincides with the inhomogeneous system of $\mathsf{N}$ quadratic equations:$$x_{n}\sum_{a=1}^{\mathsf{N}}g_{a}(\zeta _{n}^{(1)})x_{a}+x_{n}f(\zeta
_{n}^{(1)})=q_{n},\text{ \ \ \ }\forall n\in \{1,...,\mathsf{N}\},$$once we define $x_{a}=\tau (\zeta _{a}^{(0)})$ for any $a\in \{1,...,\mathsf{N}\}$ and we write $\tau (\zeta _{n}^{(1)})$ in terms of the $x_{a}$. Thus we show that $$\tau (\lambda )=\left( Z_{Q}(\lambda )+F(\lambda )\right)
/Q(\lambda )\in \Sigma _{\mathcal{T}},$$ completing the proof of the theorem.
Completeness of the Bethe ansatz equations
------------------------------------------
In the previous section we have shown that to solve the transfer matrix spectral problem associated to the most general representations of the trigonometric 6-vertex reflection algebra we have just to classify the set of functions $Q(\lambda )$ of the form $\left( \ref{Q-form2}\right) $ for which $\left( Z_{Q}(\lambda )+F(\lambda )\right) /Q(\lambda )$ is a trigonometric polynomial; i.e. the set of functions $\Sigma _{\mathcal{Q}}$ completely fixes the set $\Sigma _{\mathcal{T}}$. We can show now that the previous characterization of the transfer matrix spectrum allows to prove that $\Sigma _{InBAE}\subset \mathbb{C}^{\mathsf{N}}$ the set of all the solutions of inhomogeneous Bethe equations $$\{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \Sigma _{InBAE}$$ if $$\mathbf{A}(\lambda _{a})Q_{\mathbf{\lambda }}(\lambda _{a}-\eta )+\mathbf{A}(-\lambda _{a})Q_{\mathbf{\lambda }}(\lambda
_{a}+\eta )=-F(\lambda _{a}),\text{ \ }\forall a\in \{1,...,\mathsf{N}\} , \label{I-BAE}$$ defines the complete set of transfer matrix eigenvalues. In particular, the following corollary follows:
\[Theo-InBAE\] Let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ be generic $\left( \ref{xi-conditions}\right) $ and let the boundary parameters $(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha
_{-},\beta _{-})\in \mathbb{C}^{6}\backslash N_{SOV}$ satisfy $\left( \ref{Inhom-cond-BaxEq}\right) $ then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $\exists
!\{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \Sigma _{InBAE}$ such that:$$\tau (\lambda )=\frac{Z_{Q}(\lambda )+F(\lambda )}{Q(\lambda )}\text{ \ \
with \ \ }Q(\lambda )=2^{\mathsf{N}}\prod_{a=1}^{\mathsf{N}}\left( \cosh
2\lambda -\cosh 2\lambda _{a}\right) .$$Moreover, under the condition of normality defined in Proposition [normality]{}, the set $\Sigma _{InBAE}$ of all the solutions to the inhomogeneous system of Bethe equations $\left( \ref{I-BAE}\right) $ contains $2^{\mathsf{N}}$ elements.
Homogeneous Baxter equation
===========================
Boundary conditions annihilating the inhomogeneity of the Baxter equation
-------------------------------------------------------------------------
The description presented in the previous sections can be applied to completely general integrable boundary terms including as a particular case the boundary conditions for which the inhomogeneous term in the functional Baxter equation vanishes. As these are still quite general boundary conditions it is interesting to point out how the previous general results explicitly look like in these cases.
\[homogeneousBE\_N\] Let $(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta _{-})\in
\mathbb{C}^{6}\backslash N_{SOV}$ satisfying the condition:$$\kappa _{+}\neq 0,\kappa _{-}\neq 0,\text{ \ }\exists i\in \left\{
0,1\right\} \text{\ }:Y^{(i,2\mathsf{N})}(\tau _{\pm },\alpha _{\pm },\beta
_{\pm })=0 \label{ond-homo-boundary}$$and let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}
$ $^{\mathsf{N}}$ be generic [(\[xi-conditions\])]{}, then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $ \exists
!Q(\lambda )\in \Sigma _{\mathcal{Q}}$ such that $$\tau (\lambda
)Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda
)Q(\lambda +\eta ).$$Or equivalently, $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $\exists !\{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \Sigma _{BAE}$ such that:$$\tau (\lambda )=\frac{\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )}{Q(\lambda )}\text{ \ \ with \ \ }Q(\lambda
)=2^{\mathsf{N}}\prod_{a=1}^{\mathsf{N}}\left( \cosh 2\lambda -\cosh
2\lambda _{a}\right) .$$where:$$\Sigma _{BAE}=\left\{ \{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \mathbb{C}^{\mathsf{N}}:\mathbf{A}(\lambda _{a})Q_{\mathbf{\lambda }}(\lambda
_{a}-\eta )+\mathbf{A}(-\lambda _{a})Q_{\mathbf{\lambda }}(\lambda _{a}+\eta
)=0,\text{ \ }\forall a\in \{1,...,\mathsf{N}\}\right\} . \label{BAE}$$Moreover, under the condition of normality defined in Proposition [normality]{}, the set $\Sigma _{BAE}$ of the solutions to the homogeneous system of Bethe ansatz type equations $\left( \ref{BAE}\right) $ contains $2^{\mathsf{N}}$ elements.
This theorem is just a rewriting of the results presented in the Theorem \[T-eigenvalue-F-eq\] and Corollary \[Theo-InBAE\] for the case of vanishing inhomogeneous term. Indeed if the conditions $\left( \ref{BAE}\right) $ are satisfied then automatically the conditions of the main theorem $\left( \ref{Inhom-cond-BaxEq}\right) $ are satisfied too that implies that the map from the $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ to the $\{\lambda
_{1},...,\lambda _{\mathsf{N}}\}\in \Sigma _{BAE}$ is indeed an isomorphism.
More general boundary conditions compatibles with homogeneous Baxter equations
------------------------------------------------------------------------------
We address here the problem of describing the boundary conditions:$$\kappa _{+}\neq 0,\kappa _{-}\neq 0,\text{ \ }\exists i\in \left\{
0,1\right\} ,\mathsf{M}\in \left\{ 0,...,\mathsf{N}-1\right\} :Y^{(i,2\mathsf{M})}(\tau _{\pm },\alpha _{\pm },\beta _{\pm })=0,
\label{Cond-homo-M}$$for which the conditions $\left( \ref{Inhom-cond-BaxEq}\right) $ are not satisfied and then the Theorem \[T-eigenvalue-F-eq\] cannot be directly applied. In these $2\mathsf{N}$ hyperplanes in the space of the boundary parameters we have just to modify this theorem to take into account that the Baxter equation associated to the choice of coefficient $\mathbf{A}(\lambda
) $ is indeed compatible with the homogeneous Baxter equation for a special choice of the polynomial $Q(\lambda )$. First we define the following functions$$Q_{\mathsf{M}}(\lambda )=2^{\mathsf{M}}\prod_{b=1}^{\mathsf{M}}\left( \cosh
2\lambda -\cosh 2\lambda _{b}^{(\mathsf{M})}\right) . \label{Q-form-M}$$We introduce also the set of polynomials $\Sigma _{\mathcal{Q}}^{\mathsf{M}}$ such that $Q_{\mathsf{M}}(\lambda )\in \Sigma _{\mathcal{Q}}^{\mathsf{M}}$ if $Q_{\mathsf{M}}(\lambda )$ has a form $\left( \ref{Q-form-M}\right)$ and $$\tau (\lambda )=\frac{\mathbf{A}(\lambda )Q_{\mathsf{M}}(\lambda -\eta )+\mathbf{A}(-\lambda )Q_{\mathsf{M}}(\lambda +\eta )}{Q_{\mathsf{M}}(\lambda )}$$is a trigonometric polynomial. Then we can define the corresponding set $\Sigma _{\mathcal{T}}^{\mathsf{M}}$$$\Sigma _{\mathcal{T}}^{\mathsf{M}}=\left\{ \tau (\lambda ):\tau (\lambda
)\equiv \frac{\mathbf{A}(\lambda )Q_{\mathsf{M}}(\lambda -\eta )+\mathbf{A}(-\lambda )Q_{\mathsf{M}}(\lambda +\eta )}{Q_{\mathsf{M}}(\lambda )}\text{
\, if }Q_{\mathsf{M}}(\lambda )\in \Sigma _{\mathcal{Q}}^{\mathsf{M}}\right\}
.$$It is simple to prove the validity of the following:
\[mixed-condition\] Let the boundary conditions $\left( \ref{Cond-homo-M}\right) $ be satisfied, then $\Sigma _{\mathcal{T}}^{\mathsf{M}}\subset\Sigma _{\mathcal{T}}$ and moreover for any $\tau (\lambda )\in \Sigma _{\mathcal{T}}^{\mathsf{M}}$ there exists one and only one $Q_{\mathsf{M}}(\lambda )\in \Sigma _{\mathcal{Q}}^{\mathsf{M}}$ such that:$$\label{homogen-Bax-eq-M}
\tau (\lambda )Q_{\mathsf{M}}(\lambda )=\mathbf{A}(\lambda )Q_{\mathsf{M}}(\lambda -\eta )+\mathbf{A}(-\lambda )Q_{\mathsf{M}}(\lambda +\eta ),$$and for any $\tau (\lambda )\in \Sigma _{\mathcal{T}}\backslash \Sigma _{\mathcal{T}}^{\mathsf{M}}$ there exists one and only one $Q(\lambda )\in
\Sigma _{\mathcal{Q}}$ such that:$$\label{inhomogen-Bax-eq-M}
\tau (\lambda )Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )+F(\lambda ).$$
The proof follows the one given for the main Theorem \[T-eigenvalue-F-eq\] we have just to observe that thanks to the boundary conditions $\left( \ref{Cond-homo-M}\right) $ the set $\Sigma _{\mathcal{T}}^{\mathsf{M}}$ is formed by transfer matrix eigenvalues as the Baxter equation implies that for any $\tau (\lambda )\in \Sigma _{\mathcal{T}}^{\mathsf{M}}$ the systems of equations $\left( \ref{System-A}\right) ,$ $\left( \ref{System-B}\right) $ and $\left( \ref{System1}\right) $ are satisfied and moreover that the asymptotics of the $\tau (\lambda )\in \Sigma _{\mathcal{T}}^{\mathsf{M}}$ is exactly that of the transfer matrix eigenvalues.
Finally, it is interesting to remark that under the boundary conditions $\left( \ref{Inhom-cond-BaxEq}\right) $ the complete characterization of the spectrum of the transfer matrix is given in terms of the even polynomials $Q(\lambda )$ all of fixed degree $2\mathsf{N}$ and form $\left( \ref{Q-form2}\right) $ which are solutions of the inhomogeneous/homogeneous Baxter equation. However, in the cases when the boundary parameters satisfy the constraints $\left( \ref{Cond-homo-M}\right) $ for a given $\mathsf{M}\in \left\{ 0,...,\mathsf{N}-1\right\} $ a part of the transfer matrix spectrum can be defined by polynomials of smaller degree; i.e. the $Q_{\mathsf{M}}(\lambda
)\in \Sigma _{\mathcal{Q}}^{\mathsf{M}}$ for the fixed $\mathsf{M}\in
\left\{ 0,...,\mathsf{N}-1\right\} $.
Discrete symmetries and equivalent Baxter equations {#sect-descretesym}
===================================================
It is important to point out that we have some large amount of freedom in the choice of the functional reformulation of the SOV characterization of the transfer matrix spectrum. We have reduced it looking for trigonometric polynomial solutions $Q(\lambda )$ of the second order difference equations with coefficients $\mathbf{A}(\lambda )$ which are rational trigonometric functions. It makes the finite difference terms $\mathbf{A}(\lambda
)Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )$ in the functional equation a trigonometric polynomial. Indeed, this assumption reduces the possibility to use the following gauge transformations of the coefficients allowed instead by the SOV characterization:$$\mathbf{A}_{\alpha }(\lambda )=\alpha (\lambda )\mathbf{A}(\lambda ),\text{
\ }\mathbf{D}_{\alpha }(\lambda )=\frac{\mathbf{A}(-\lambda )}{\alpha
(\lambda +\eta )}.$$In the following we discuss simple transformations that do not modify the functional form of the coefficients allowing equivalent reformulations of the SOV spectrum by Baxter equations.
Discrete symmetries of the transfer matrix spectrum
---------------------------------------------------
It is not difficult to see that the spectrum (eigenvalues) of the transfer matrix presents the following invariance:
\[Lem-invariance\]We denote explicitly the dependence from the boundary parameters in the set of boundary parameters $\Sigma _{\mathcal{T}}^{(\tau _{+},\alpha
_{+},\beta _{+},\tau _{-},\alpha _{-},\beta _{-})}$ of the eigenvalue functions of the transfer matrix $\mathcal{T}(\lambda )$, then this set is invariant under the following $Z_{2}^{\otimes 3}$ transformations of the boundary parameters:$$\begin{aligned}
&\Sigma _{\mathcal{T}}^{(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha
_{-},\beta _{-})}\equiv \Sigma _{\mathcal{T}}^{(\epsilon _{\tau }\tau
_{+},\epsilon _{\alpha }\alpha _{+},\epsilon _{\beta }\beta _{+},\epsilon
_{\tau }\tau _{-},\epsilon _{\alpha }\alpha _{-},\epsilon _{\beta }\beta
_{-})}\ \\ &
\forall (\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})\in \{-1,1\}\times \{-1,1\}\times \{-1,1\}.\nonumber\end{aligned}$$
To prove this statement it is enough to look at the SOV characterization which defines completely the transfer matrix spectrum, i.e. the set $\Sigma _{\mathcal{T}}$, and to prove that it is invariant under the above considered $Z_{2}^{\otimes 3}$ transformations of the boundary parameters. We have first to remark that the central values $\left( \ref{Central-asymp}\right) $-$\left( \ref{Central-2}\right) $ of the transfer matrix $\mathcal{T}(\lambda )$ are invariant under these discrete transformations and then the function $f(\lambda )$, defined in $\left( \ref{f-function}\right) $, is invariant too and the same is true for the form $\left( \ref{Interpolation-Form-T}\right) $ of the interpolation polynomial describing the elements of $\Sigma _{\mathcal{T}}$. Then the invariance of the SOV characterization $\left( \ref{Quadratic System}\right) $ follows from the invariance of the quantum determinant$$\begin{aligned}
\mathrm{det}_{q}K_{+}(\lambda )\mathrm{det}_{q}\,\mathcal{U}_{-}(\lambda )
&=&\sinh (2\eta -2\lambda )\sinh (2\lambda +2\eta )g_{+}(\lambda +\eta
/2)g_{+}(-\lambda +\eta /2)g_{-}(\lambda +\eta /2) \notag \\
&&\times g_{-}(-\lambda +\eta /2)a(\lambda +\eta /2)d(\lambda -\eta
/2)a(-\lambda +\eta /2)d(-\lambda -\eta /2)\end{aligned}$$ under these discrete transformations.
It is important to underline that the above $Z_{2}^{\otimes 3}$ transformations of the boundary parameters do indeed change the transfer matrix $\mathcal{T}(\lambda )$ and the Hamiltonian and so this invariance is equivalent to the statement that these different transfer matrices are all isospectral. In particular, it is simple to find the similarity matrices implementing the following $Z_{2}$ transformations of the boundary parameters:$$\begin{aligned}
\mathcal{T}(\lambda |-\tau _{+},-\zeta _{+},\kappa _{+},-\tau _{-},-\zeta
_{-},\kappa _{-}) &=&\Gamma _{y}\mathcal{T}(\lambda |\tau _{+},\zeta
_{+},\kappa _{+},\tau _{-},\zeta _{-},\kappa _{-})\Gamma _{y},\text{ \ \ \ }\Gamma _{y}\equiv \otimes _{n=1}^{\mathsf{N}}\sigma _{n}^{y}, \\
\mathcal{T}(\lambda |\tau _{+},\zeta _{+},-\kappa _{+},\tau _{-},\zeta
_{-},-\kappa _{-}) &=&\Gamma _{z}\mathcal{T}(\lambda |\tau _{+},\zeta
_{+},\kappa _{+},\tau _{-},\zeta _{-},\kappa _{-})\Gamma _{z},\text{ \ \ \ }\Gamma _{z}\equiv \otimes _{n=1}^{\mathsf{N}}\sigma _{n}^{z}.\end{aligned}$$
Equivalent Baxter equations and the SOV spectrum
------------------------------------------------
The invariance of the spectrum $\Sigma _{\mathcal{T}}$ under these $Z_{2}^{\otimes 3}$ transformations of the boundary parameters can be used to define equivalent Baxter equation reformulation of $\Sigma _{\mathcal{T}}$. More precisely, let us introduce the following functions $\mathbf{A}_{(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})}(\lambda )$ and $F_{(\epsilon _{\tau },\epsilon _{\alpha },\epsilon
_{\beta })}(\lambda )$ obtained respectively by implementing the $Z_{2}^{\otimes 3}$ transformations:$$(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta
_{-})\rightarrow (\epsilon _{\tau }\tau _{+},\epsilon _{\alpha }\alpha
_{+},\epsilon _{\beta }\beta _{+},\epsilon _{\tau }\tau _{-},\epsilon
_{\alpha }\alpha _{-},\epsilon _{\beta }\beta _{-}),$$then the following characterizations hold for any fixed $(\epsilon _{\tau
},\epsilon _{\alpha },\epsilon _{\beta })\in \{-1,1\}\times \{-1,1\}\times
\{-1,1\}$:
\[T-eigenvalue-F-eq-gen\]Let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ be generic [(\[xi-conditions\])]{} and let the boundary parameters $(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta
_{-})\in \mathbb{C}^{6}\backslash N_{SOV}$ satisfy the following conditions:$$\kappa _{+}\neq 0,\kappa _{-}\neq 0,\text{ \ }Y^{(i,2r)}(\epsilon _{\tau
}\tau _{\pm },\epsilon _{\alpha }\alpha _{\pm },\epsilon _{\beta }\beta
_{\pm })\neq 0\text{ \ }\forall i\in \left\{ 0,1\right\} ,r\in \left\{ 0,...,\mathsf{N}-1\right\} , \label{Inhom-cond-BaxEq-gen}$$then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $\exists
!Q(\lambda )\in \Sigma _{\mathcal{Q}}$ such that $$\tau (\lambda
)Q(\lambda )=Z_{Q,(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})}(\lambda )+F_{(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})}(\lambda ),$$where:$$Z_{Q,(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta })}(\lambda )=\mathbf{A}_{(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})}(\lambda )(\lambda )Q(\lambda -\eta )+\mathbf{A}_{(\epsilon _{\tau
},\epsilon _{\alpha },\epsilon _{\beta })}(-\lambda )Q(\lambda +\eta ).$$
The proof follows step by step the one given for the main Theorem \[T-eigenvalue-F-eq\].
General validity of the inhomogeneous Baxter equations
------------------------------------------------------
The previous reformulations of the spectrum in terms of different inhomogeneous Baxter equations and the observation that the conditions under which the Theorem does not apply are related to the choice of the $(\epsilon
_{\tau },\epsilon _{\alpha },\epsilon _{\beta })\in \{-1,1\}\times
\{-1,1\}\times \{-1,1\}$ allow us to prove that unless the boundary parameters are lying on a finite lattice of step $\eta $ we can always use an inhomogeneous Baxter equations to completely characterize the spectrum of the transfer matrix. More precisely, let us introduce the following hyperplanes in the space of the boundary parameters:$$M\equiv \left\{
\begin{array}{l}
(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta _{-})\in
\mathbb{C}^{6}:\exists (r_{+,+},r_{-,+},r_{-,-})\in \{0,...,\mathsf{N}-1\}
\\
\text{ such that: \ \ \ }\left\{
\begin{array}{l}
r_{+,+}+r_{-,-}-r_{-,+}\in \{0,...,\mathsf{N}-1\} \\
\alpha _{+}+\alpha _{-}=(r_{-,+}-r_{+,+})\eta \\
\beta _{-}-\beta _{+}=(r_{-,-}-r_{-,+})\eta \\
\tau _{-}-\tau _{+}=(\mathsf{N}-1+r_{-,-}-3r_{+,+})\eta\end{array}\right.\end{array}\right\} \label{Def-M}$$then the following theorem holds:
Let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ satisfy the conditions [(\[xi-conditions\])]{} and let $(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta
_{-})\in \mathbb{C}^{6}\backslash \left( M\cup N_{SOV}\right) $ then we can always find a $(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta })\in
\{-1,1\}\times \{-1,1\}\times \{-1,1\}$ such that $\tau (\lambda )\in\Sigma _{\mathcal{T}}$ if and only if $\exists
!Q(\lambda )\in \Sigma _{\mathcal{Q}}$ such that $$\tau (\lambda
)Q(\lambda )=Z_{Q,(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})}(\lambda )+F_{(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})}(\lambda ).$$
The Theorem \[T-eigenvalue-F-eq-gen\] does not apply if $\exists i\in
\left\{ 0,1\right\} $ and $\exists r\in \left\{ 0,...,\mathsf{N}-1\right\} $ such that the following system of conditions on the boundary parameters are satisfied:$$Y^{(i,2r)}(\epsilon _{\tau }\tau _{\pm },\epsilon _{\alpha }\alpha _{\pm
},\epsilon _{\beta }\beta _{\pm })=0\text{ \ }\forall (\epsilon _{\tau
},\epsilon _{\alpha },\epsilon _{\beta })\in \{-1,1\}^{\otimes 3}
\label{Cond-general-homo}$$ then by simple computations it is possible to observe that the set $M$ defined in $\left( \ref{Def-M}\right) $ indeed coincides with the following set:$$\left\{ (\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta
_{-})\in \mathbb{C}^{6}:\text{ }\exists i\in \left\{ 0,1\right\} ,r\in
\left\{ 0,...,\mathsf{N}-1\right\} \text{ such that }\left( \ref{Cond-general-homo}\right) \text{ is satisfied}\right\} ,$$from which the theorem clearly follows.
Homogeneous Baxter equation
---------------------------
The discrete symmetries of the transfer matrix allow also to define the general conditions on the boundary parameters for which the spectrum can be characterized by a homogeneous Baxter equation. In particular the following corollary holds:
Let $(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta _{-})\in
\mathbb{C}^{6}\backslash N_{SOV}$ satisfy the condition:$$\begin{aligned}
&\kappa _{+}\neq 0,\kappa _{-}\neq 0,\nonumber\\
&\exists i\in \left\{
0,1\right\} ,\text{\ }\exists (\epsilon _{\tau },\epsilon _{\alpha
},\epsilon _{\beta })\in \{-1,1\}\times \{-1,1\}\times \{-1,1\}:Y^{(i,2\mathsf{N})}(\epsilon _{\tau }\tau _{\pm },,\epsilon _{\alpha }\alpha _{\pm
},\epsilon _{\beta }\beta _{\pm })=0\end{aligned}$$and let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}
$ $^{\mathsf{N}}$ be generic [(\[xi-conditions\])]{}, then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $\exists
!Q(\lambda )\in \Sigma _{\mathcal{Q}}$ such that $$\tau (\lambda
)Q(\lambda )=\mathbf{A}_{(\epsilon _{\tau },\epsilon _{\alpha },\epsilon
_{\beta })}(\lambda )(\lambda )Q(\lambda -\eta )+\mathbf{A}_{(\epsilon
_{\tau },\epsilon _{\alpha },\epsilon _{\beta })}(-\lambda )Q(\lambda +\eta
).$$Or equivalently we can define the set of all the solutions of the Bethe equations $$\Sigma _{BAE}=\left\{ \{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \mathbb{C}^{\mathsf{N}}:\mathbf{A}(\lambda _{a})Q_{\mathbf{\lambda }}(\lambda
_{a}-\eta )+\mathbf{A}(-\lambda _{a})Q_{\mathbf{\lambda }}(\lambda _{a}+\eta
)=0,\text{ \ }\forall a\in \{1,...,\mathsf{N}\}\right\} .$$Then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $\exists !\{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \Sigma _{BAE}$ such that:$$\tau (\lambda )=\frac{\mathbf{A}_{(\epsilon _{\tau },\epsilon _{\alpha
},\epsilon _{\beta })}(\lambda )Q(\lambda -\eta )+\mathbf{A}_{(\epsilon
_{\tau },\epsilon _{\alpha },\epsilon _{\beta })}(-\lambda )Q(\lambda +\eta )}{Q(\lambda )},$$with $$Q(\lambda )=2^{\mathsf{N}}\prod_{a=1}^{\mathsf{N}}\left( \cosh 2\lambda -\cosh 2\lambda _{a}\right).$$ Moreover, under the condition of normality defined in Proposition [normality]{}, the set $\Sigma _{BAE}$ of the solutions to the homogeneous system of Bethe ansatz type equations $\left( \ref{BAE}\right) $ contains $2^{\mathsf{N}}$ elements.
XXX chain by SOV and Baxter equation
====================================
The construction of the SOV characterization can be naturally applied in the case of the rational 6-vertex $R$-matrix, which in the homogeneous limit reproduces the XXX open quantum spin-1/2 chain with general integrable boundary conditions[^4]. Let us define:$$R_{12}(\lambda )=\left(
\begin{array}{cccc}
\lambda +\eta & 0 & 0 & 0 \\
0 & \lambda & \eta & 0 \\
0 & \eta & \lambda & 0 \\
0 & 0 & 0 & \lambda +\eta\end{array}\right) \in \text{End}(\mathcal{H}_{1}\otimes \mathcal{H}_{2}).$$Due to the $SU(2)$ invariance of the bulk monodromy matrix the boundary matrices defining the most general integrable boundary conditions can be always recasted in the following form:$$K_{-}(\lambda ;p)=\left(
\begin{array}{cc}
\lambda -\eta /2+p & 0 \\
0 & p-\lambda +\eta /2\end{array}\right) ,\text{ \ \ \ }K_{+}(\lambda ;q,\xi )=\left(
\begin{array}{cc}
\lambda +\eta /2+q & \xi (\lambda +\eta /2) \\
\xi (\lambda +\eta /2) & q-(\lambda +\eta /2)\end{array}\right) ,$$leaving only three arbitrary complex parameters here denoted with $\xi ,$ $p$ and $q$. Then the one parameter family of commuting transfer matrices:$$\mathcal{T}(\lambda )=\text{tr}_{0}\{K_{+}(\lambda )\,M(\lambda
)\,K_{-}(\lambda )\hat{M}(\lambda )\}\in \text{\thinspace End}(\mathcal{H}),$$in the homogeneous limit leads to the following Hamiltonian:$$H=\sum_{n=1}^{\mathsf{N}}\left( \sigma _{n}^{x}\sigma _{n+1}^{x}+\sigma
_{n}^{y}\sigma _{n+1}^{y}+\sigma _{n}^{z}\sigma _{n+1}^{z}\right) +\frac{\sigma _{\mathsf{N}}^{z}}{p}+\frac{\sigma _{1}^{z}+\xi \sigma _{1}^{x}}{q}.$$It is simple to show that the following identities hold:$$\mathrm{det}_{q}K_{+}(\lambda )\mathrm{det}_{q}\,\mathcal{U}_{-}(\lambda
)=4(\lambda ^{2}-\eta ^{2})(\lambda ^{2}-p^{2})((1+\xi ^{2})\lambda
^{2}-q^{2})\prod_{b=1}^{\mathsf{N}}(\lambda ^{2}-(\xi _{n}+\eta
)^{2})(\lambda ^{2}-(\xi _{n}-\eta )^{2}).$$We define:$$\mathbf{A}(\lambda )=(-1)^{\mathsf{N}}\frac{2\lambda +\eta }{2\lambda }(\lambda -\eta /2+p)(\sqrt{(1+\xi ^{2})}(\lambda -\eta /2)+q)\prod_{b=1}^{\mathsf{N}}(\lambda -\zeta _{b}^{(0)})(\lambda +\zeta _{b}^{(1)}),$$then it is easy to derive the following quantum determinant identity:$$\frac{\mathrm{det}_{q}K_{+}(\lambda )\mathrm{det}_{q}\,\mathcal{U}_{-}(\lambda )}{(4\lambda ^{2}-\eta ^{2})}=\mathbf{A}(\lambda +\eta /2)\mathbf{A}(-\lambda
+\eta /2).$$ From the form of the boundary matrices it is clear that for the rational 6-vertex case one can directly derive the SOV representations using the method developed in [@Nic12b] without any need to introduce Baxter’s gauge transformations. Some results in this case also appeared in [@FraSW08; @FraGSW11] based on a functional version of the separation of variables of Sklyanin, a method which allows to define the eigenvalues and wave-functions but which does not allow to construct in the original Hilbert space of the quantum chain the transfer matrix eigenstates.
The separation of variable description in this rational 6-vertex case reads:
\[C:T-eigenstates- copy(1)\]Let the inhomogeneities $\{\xi _{1},...,\xi
_{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ be generic: $$\xi _{a}\neq \pm\xi _{b}+r\eta \text{ \ }\forall a\neq b\in \{1,...,\mathsf{N}\}\,\,\text{and\thinspace \thinspace }r\in \{-1,0,1\}, \label{xi-conditions-xxx}$$ then $\mathcal{T}(\lambda )$ has simple spectrum and $\Sigma _{\mathcal{T}}$ is characterized by:$$\Sigma _{\mathcal{T}}=\left\{ \tau (\lambda ):\tau (\lambda )=f(\lambda
)+\sum_{a=1}^{\mathsf{N}}g_{a}(\lambda )x_{a},\text{ \ \ }\forall
\{x_{1},...,x_{\mathsf{N}}\}\in \Sigma _{T}\right\} ,$$where:$$g_{a}(\lambda )=\frac{4\lambda ^{2}-\eta ^{2}}{4{\zeta
_{a}^{(0)}} ^{2}-\eta ^{2}}\,\prod_{\substack{ b=1 \\ b\neq a}}^{\mathsf{N}}\frac{\lambda ^{2}-{ \zeta _{b}^{(0)}} ^{2}}{{
\zeta _{a}^{(0)}} ^{2}-{ \zeta _{b}^{(0)}} ^{2}}\quad \text{
\ for }a\in \{1,...,\mathsf{N}\},$$and$$f(\lambda )=\prod_{b=1}^{\mathsf{N}}\frac{\lambda ^{2}-{ \zeta
_{b}^{(0)}}^{2}}{{ \zeta _{a}^{(0)}} ^{2}-{\zeta
_{b}^{(0)}} ^{2}}\mathbf{A}(\eta /2)+2\left( 4\lambda ^{2}-\eta
^{2}\right) \,\prod_{b=1}^{\mathsf{N}}\lambda ^{2}-{ \zeta
_{b}^{(0)}} ^{2},$$$\Sigma _{T}$ is the set of solutions to the following inhomogeneous system of $\mathsf{N}$ quadratic equations:$$x_{n}\sum_{a=1}^{\mathsf{N}}g_{a}(\zeta _{n}^{(1)})x_{a}+x_{n}f(\zeta
_{n}^{(1)})=q_{n},\text{ \ \ \ }q_{n}=\frac{\mathrm{det}_{q}K_{+}(\xi _{n})\mathrm{det}_{q}\,\mathcal{U}_{-}(\xi _{n})}{\eta -4\xi _{n}^{2}},\text{ \ \ }\forall n\in \{1,...,\mathsf{N}\},$$in $\mathsf{N}$ unknowns $\{x_{1},...,x_{\mathsf{N}}\}$.
We are now ready to present the following equivalent characterization of the transfer matrix spectrum:
\[T-eigenvalue-F-eq copy(1)\]Let the inhomogeneities $\{\xi _{1},...,\xi
_{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ be generic [(\[xi-conditions-xxx\])]{}, then for $\xi \neq 0$ the set of transfer matrix eigenvalue functions $\Sigma _{\mathcal{T}}$ is characterized by:$$\tau (\lambda )\in \Sigma _{\mathcal{T}}\text{ \ if and only if }\exists
!Q(\lambda )=\prod_{b=1}^{\mathsf{N}}\left( \lambda ^{2}-\lambda
_{b}^{2}\right) \text{ such that }\tau (\lambda )Q(\lambda )=Z_{Q}(\lambda
)+F(\lambda ),$$with$$F(\lambda )=2(1-\sqrt{(1+\xi ^{2})})\left( 4\lambda ^{2}-\eta ^{2}\right)
\,\prod_{b=1}^{\mathsf{N}}\prod_{i=0}^{1}\left( \lambda ^{2}- {�\zeta
_{b}^{(i)}} ^{2}\right) .$$
The proof presented in Theorem \[T-eigenvalue-F-eq\] applies with small modifications also to present rational case.
The previous characterization of the transfer matrix spectrum allows to prove that the set $\Sigma _{InBAE}\subset\mathbb{C}^\mathsf{N}$ of all the solutions of the Bethe equations$$\{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in\Sigma _{InBAE}$$ if $$\mathbf{A}(\lambda _{a})Q_{\mathbf{\lambda }}(\lambda _{a}-\eta )+\mathbf{A}(-\lambda _{a})Q_{\mathbf{\lambda }}(\lambda
_{a}+\eta )=-F(\lambda _{a}),\text{ \ }\forall a\in \{1,...,\mathsf{N}\} , \label{I-BAE-XXX}$$ define the complete set of transfer matrix eigenvalues. In particular, the following corollary can be proved:
\[Theo-InBAE-XXX\]Let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ satisfy the following conditions [(\[xi-conditions\])]{}, then $\mathcal{T}(\lambda )$ has simple spectrum and for $\xi \neq 0$ then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $\exists
!\{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \Sigma _{InBAE}$ such that:$$\tau (\lambda )=\frac{Z_{Q}(\lambda )+F(\lambda )}{Q(\lambda )}\text{ \ \
with \ \ }Q(\lambda )=\prod_{b=1}^{\mathsf{N}}\left( \lambda ^{2}-\lambda
_{b}^{2}\right) .$$
Homogeneous chains and existing numerical analysis
==================================================
It is important to stress that the spectrum construction together with the corresponding statements of completeness presented in this paper strictly work for the most general spin 1/2 representations of the 6-vertex reflection algebra only for generic inhomogeneous chains. However, it is worth mentioning that the transfer matrix as well as the coefficients and the inhomogeneous term in our functional equation characterization of the SOV spectrum are analytic functions of the inhomogeneities $\{\xi _{j}\}$ so we can take without any problem the homogeneous limit ($\xi _{a}\rightarrow
0$ $\forall a\in \{1,...,\mathsf{N}\}$) in the functional equations. The main problem to be addressed then is the completeness of the description by this functional equations. Some first understanding of this central question can be derived looking at the numerical analysis [@Nep-R-2003; @Nep-R-2003add] of the completeness of Bethe Ansatz equations when the boundary constraints are satisfied and for the open XXX chain with general boundary terms [@Nep-2013].
Comparison with numerical results for the XXZ chain
---------------------------------------------------
The numerical checks of the completeness of Bethe Ansatz equations for the open XXZ quantum spin 1/2 chains were first done in [@Nep-R-2003] for the chains with non-diagonal boundaries satisfying boundary constraints: $$\kappa _{+}\neq 0,\kappa _{-}\neq 0,\text{ \ }\exists i\in \left\{
0,1\right\} ,\mathsf{M}\in \mathbb{N}\text{\ }:Y^{(i,2\mathsf{M})}(\tau _{\pm },\alpha _{\pm
},\beta _{\pm })=0.$$Indeed, under these conditions some generalizations of algebraic Bethe Ansatz can be used and so the corresponding Bethe equations can be defined.
In particular, the Nepomechie-Ravanini’s numerical results reported in [Nep-R-2003,Nep-R-2003add]{} suggest that the Bethe ansatz equations $\left( \ref{BAE}\right) $ in the homogeneous limit for the roots of the $Q$ function:$$Q(\lambda )=2^{\mathsf{M}}\prod_{a=1}^{\mathsf{M}}\left( \cosh 2\lambda -\cosh 2\lambda
_{a}\right) ,$$with the degree $\mathsf{M}$ obtained from the boundary constraint
- for $\mathsf{M}=\mathsf{N}$ they define the complete transfer matrix spectrum.
- for $\mathsf{M}<\mathsf{N}$ the complete spectrum of the transfer matrix contains two parts described by different Baxter equations. The first one has trigonometric polynomial solutions of degree $2\mathsf{M}$ the second one has a trigonometric polynomial solutions of degree $2\mathsf{N}-2-2\mathsf{M}$.
- for $\mathsf{M}>\mathsf{N}$ the complete spectrum of the transfer matrix spectrum plus $\tau (\lambda )$ functions which do not belong to the spectrum of the transfer matrix.
These results seem to be compatible with our characterization for the inhomogeneous chains. Indeed, the case $\mathsf{M}=\mathsf{N}$ coincides with the case in which our Baxter functional equation becomes homogeneous. Theorem \[homogeneousBE\_N\] states that in this case for generic inhomogeneities the Bethe ansatz is complete so we can expect (from the numerical analysis) that completeness will survive in the homogeneous limit. In the case $\mathsf{M}<$, our description of the spectrum by Lemma [mixed-condition]{} separates the spectrum in two parts. A first part of the spectrum is described by trigonometric polynomial solutions of degree $2\mathsf{M}$ to the homogeneous Baxter equation [(\[homogen-Bax-eq-M\])]{} and a second part is instead described by trigonometric polynomial solutions of degree 2 of the inhomogeneous Baxter equation [(\[inhomogen-Bax-eq-M\])]{}. However, by implementing the following discrete symmetry transformations $\alpha _{\pm
}\rightarrow -\alpha _{\pm }$, $\beta _{\pm }\rightarrow -\beta _{\pm }$, $\tau _{\pm }\rightarrow -\tau _{\pm }$ and applying the same Lemma [mixed-condition]{} w.r.t. the Baxter equations with coefficients $\mathbf{A}_{(-,-,-)}(\lambda )$ we get an equivalent description of the spectrum separated in two parts. One part of the spectrum is described in terms of the solutions of the transformed homogeneous Baxter equation which should be trigonometric polynomials of degree $2\mathsf{M}^{\prime }$, with $\mathsf{M}^{\prime }=\mathsf{N}-1-\mathsf{M}$ and the second part by the inhomogeneous Baxter equation. The comparison with the numerical results then suggests that, at least in the limit of homogeneous chains, the part of the spectrum generated by the trigonometric polynomial solutions of degree 2 of the inhomogeneous Baxter equation [(\[inhomogen-Bax-eq-M\])]{} coincides with the part generated by the trigonometric polynomial solutions of degree $2\mathsf{M}^{\prime }$ of the transformed homogeneous Baxter equation.
Finally, in the case $\mathsf{M}>\mathsf{N}$ we have a complete characterization of the spectrum given by an inhomogeneous Baxter functional equation however nothing prevent to consider solutions to the homogeneous Baxter equation once we take the appropriate $Q$-function with $\mathsf{M}>\mathsf{N}$ Bethe roots. The numerical results however seem to suggest that considering the homogeneous Baxter equations is not the proper thing to do in the homogeneous limit.
The previous analysis seems to support the idea that in the limit of homogeneous chain our complete characterization still describe the complete spectrum of the homogeneous transfer matrix.
Comparison with numerical results for the XXX chain
---------------------------------------------------
In the case of the open spin 1/2 XXX chain an ansatz based on two $Q$-functions and an inhomogeneous Baxter functional equation has been first introduced in [@CaoYSW13-2], the completeness of the spectrum obtained by that ansatz has been later verified numerically for small chains [@CaoJYW2013]. Using these results Nepomechie has introduced a simpler ansatz and developed some further numerical analysis in [@Nep-2013] confirming once again that the ansatz defines the complete spectrum for small chains. Here, we would like to point out that our complete description of the transfer matrix spectrum in terms of a inhomogeneous Baxter functional equation obtained for the inhomogeneous chains has the following well defined homogeneous limit: $$\tau (\lambda )Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )+F(\lambda )$$where:$$\begin{aligned}
F(\lambda ) &=&8(1-\sqrt{(1+\xi ^{2})})\left( \lambda ^{2}-\left( \eta
/2\right) ^{2}\right) ^{2\mathsf{N}+1}, \\
\mathbf{A}(\lambda ) &=&(-1)^{\mathsf{N}}\frac{2\lambda +\eta }{2\lambda }\left(\vphantom{\sqrt{(1+\xi ^{2})}}\lambda -\eta /2+p\right)\left(\sqrt{(1+\xi ^{2})}(\lambda -\eta /2)+q\right)\left(\lambda ^{2}-\left( \eta /2\right) ^{2}\right)^{\mathsf{N}}.\end{aligned}$$Taking into account the shift in our definition of the monodromy matrix which insures that the transfer matrix is an even function of the spectral parameter, the limit of our inhomogeneous Baxter functional equation coincides with the ansatz proposed by Nepomechie in [@Nep-2013]. Then the numerical evidences of completeness derived by Nepomechie in [@Nep-2013] suggest that the exact and complete characterization that we get for the inhomogeneous chain is still valid and complete in the homogeneous limit.
Conclusion and outlook {#conclusion-and-outlook .unnumbered}
======================
In this paper we have shown that the transfer matrix spectrum associated to the most general spin-1/2 representations of the 6-vertex reflection algebras (rational and trigonometric), on general inhomogeneous chains is completely characterized in terms of a second order difference functional equations of Baxter $T$-$Q$ type with an inhomogeneous term depending only on the inhomogeneities of the chain and the boundary parameters. This functional $T$-$Q$ equation has been shown to be equivalent to the SOV complete characterization of the spectrum when the $Q$-functions belong to a well defined set of polynomials. The polynomial character of the $Q$-function is a central feature of our characterization which allows to introduce an equivalent finite system of generalized Bethe ansatz equations. Moreover, we have explicitly proven that our functional characterization holds for all the values of the boundary parameters for which SOV works, clearly identifying the only 3-dimensional hyperplanes in the 6-dimensional space of the boundary parameters where our description cannot be applied. We have also clearly identified the 5-dimensional hyperplanes in the space of the boundary parameters where the spectrum (or a part of the spectrum) can be characterized in terms of a homogeneous $T$-$Q$ equation and the polynomial character of the $Q$-functions is then equivalent to a standard system of Bethe equations. Completeness of this description is a built in feature due to the equivalence to the SOV characterization. The equivalence between our functional $T$-$Q$ equation and the SOV characterization holds for generic values of the $\xi _{a}$ in the $\mathsf{N}$-dimensional space of the inhomogeneity parameters however there exist hyperplanes for which the conditions [(\[xi-conditions\])]{} are not satisfied and so a direct application of the SOV approach is not possible (at least for the separate variables described in [@Fald-KN13]) and the limit of homogeneous chains ($\xi _{a}\rightarrow 0$ $\forall a\in \{1,...,\mathsf{N}\}$) clearly belong to these hyperplanes. From the analyticity of the transfer matrix eigenvalues, of the coefficients of the functional $T$-$Q$ equation and of the inhomogeneous term in it w.r.t. the inhomogeneity parameters it is possible to argue that these functional equations still describes transfer matrix eigenvalues on the hyperplanes where SOV method cannot be applied and, in particular, in the homogeneous limit. However, in all these cases the statements about the simplicity of the transfer matrix spectrum and the completeness of the description by our functional $T$-$Q$ equation are not anymore granted and they require independent proofs. These fundamental issues will be addressed in a future publication. Here we want just to recall that the comparison with the few existing numerical results on the subject seems to suggests that the statement of completeness should be satisfied even in the homogeneous limit of special interest as it allows to reproduce the spectrum of the Hamiltonian of the spin-1/2 open XXZ quantum chains under the most general integrable boundary conditions.
Finally, it is important to note that the form of the Baxter functional equation for the most general spin-1/2 representations of the 6-vertex reflection algebras and in particular the necessity of an inhomogeneous term are mainly imposed by the requirement that the set of solutions is restricted to polynomials. Then the problem to get homogeneous Baxter equations relaxing this last requirement remains an interesting open problem.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank E. Sklyanin and V. Terras for discussions. J.M.M. and G. N. are supported by CNRS. N.K and J.M.M. are supported by ANR grant “DIADEMS”. N. K. would like to thank LPTHE, University Paris VI and Laboratoire de Physique, ENS-Lyon for hospitality.
[99]{} P. Baseilhac. The $q$-deformed analogue of the Onsager algebra: Beyond the Bethe ansatz approach 754 (2006) 309.
P. Baseilhac, K. Koizumi. A deformed analogue of Onsager’s symmetry in the XXZ open spin chain (2005) P10005.
R. Baxter. Partition function of the eight-vertex lattice model. , 70:193–228, 1972.
R. J. Baxter. One-dimensional anisotropic [H]{}eisenberg chain. , 70:323–37, 1972.
H. Bethe. Zür [T]{}heorie der [M]{}etalle [I]{}. [E]{}igenwerte und [E]{}igenfunktionen [A]{}tomkete. , 71:205–226, 1931.
J. Cao, H.-Q. Lin, K.-J. Shi, and Y. Wang. Exact solution of [$XXZ$]{} spin chain with unparallel boundary fields. , 663(3):487–519, 2003.
J. Cao, W. Yang, K. Shi, and Y. Wang. Off-diagonal Bethe ansatz solutions of the anisotropic spin-1/2 chains with arbitrary boundary fields. , 887:152-175, 2013.
J. Cao, W. Yang, K. Shi, and Y. Wang. Off-diagonal Bethe ansatz and exact solution a topological spin ring. 111, 137201, 2013.
J. Cao, W. Yang, K. Shi, and Y. Wang. Off-diagonal bethe ansatz solution of the XXX spin-chain with arbitrary boundary conditions. , 875:152–165, 2013.
J. Cao, W. Yang, K. Shi, and Y. Wang. Spin-1/2 XYZ model revisit: general solutions via off-diagonal Bethe ansatz. arXiv preprint arXiv:1307.0280, 2013.
I. V. Cherednik. Factorizing particles on a half-line and root systems. , 61:977–983, 1984.
N. Crampé, E. Ragoucy, D. Simon. Eigenvectors of open XXZ and ASEP models for a class of non-diagonal boundary conditions P11038 (2010).
N. Crampé, E. Ragoucy Generalized coordinate Bethe ansatz for non-diagonal boundaries 858 (2012) 502.
J. de Gier and F. H. L. Essler. Bethe ansatz solution of the asymmetric exclusion process with open boundaries. , 95(24):240601, 4, 2005.
J. de Gier and F. H. L. Essler. Exact spectral gaps of the asymmetric exclusion process with open boundaries. , 2006(12):P12011, 2006.
W. Galleas. Functional relations from the Yang-Baxter algebra: Eigenvalues of the XXZ model with non-diagonal twisted and open boundary conditions 790 (2008) 524.
J. de Gier and P. Pyatov. Bethe ansatz for the Temperley-Lieb loop model with open boundaries (2004).
N. Grosjean, J. M. Maillet, and G. Niccoli. On the form factors of local operators in the lattice sine-Gordon model. , P10006, 2012.
N. Grosjean, J.-M. Maillet, G. Niccoli, On the form factors of local operators in the Bazhanov-Stroganov and chiral Potts models. arXiv:1309.4701.
N. Grosjean, G. Niccoli, The $\tau_2$-model and the chiral Potts model revisited: completeness of Bethe equations from Sklyanin’s SOV method. P11005 (2012).
L. D. Faddeev, E. K. Sklyanin, and L. A. Takhtajan. Quantum inverse problem method [I]{}. , 40:688–706, 1979.
S. Faldella, N. Kitanine, G. Niccoli. Complete spectrum and scalar products for the open spin-1/2 XXZ quantum chains with non-diagonal boundary terms. Accepted for publication on *J. Stat. Mech.: Theory Exp.* arXiv:1307.3960.
S. Faldella, G. Niccoli. SOV approach for integrable quantum models associated to the most general representations on spin-1/2 chains of the 8-vertex reflection algebra. arXiv:1307.5531.
G. Filali and N. Kitanine. Spin chains with non-diagonal boundaries and trigonometric [SOS]{} model with reflecting end. , 7:Paper 012, 22, 2011.
H. Frahm, A. Seel, T. Wirth. Separation of variables in the open XXX chain. (2008) 351.
H. Frahm, J. H. Grelik, A. Seel, T. Wirth. Functional Bethe ansatz methods for the open XXX chain. (2011) 015001.
Y. Jiang, S. Cui, J. Cao, Wen-Li Yang and Y. Wang, Completeness and Bethe root distribution of the spin- 1/2 Heisenberg chain with arbitrary boundary fields. arXiv:1309.6456v1.
N. Kitanine, K. Kozlowski, J. Maillet, G. Niccoli, N. Slavnov, and V. Terras. On correlation functions of the open [$XXZ$]{} chain [I]{}. , pages P10009, 37 pp. (electronic), 2007.
N. Kitanine, K. Kozlowski, J. Maillet, G. Niccoli, N. Slavnov, and V. Terras. On correlation functions of the open [$XXZ$]{} chain [II]{}. , page P07010, 2008. arXiv:0803.3305.
N. Kitanine, J. M. Maillet, and V. Terras. Form factors of the [$XXZ$]{} [H]{}eisenberg spin-1/2 finite chain. , 554 \[FS\]:647–678, 1999.
N. Kitanine, J. M. Maillet, and V. Terras. Correlation functions of the [$XXZ$]{} heisenberg spin-1/2 chain in a magnetic field. , 567 \[FS\]:554–582, 2000.
J. M. Maillet and V. Terras. On the quantum inverse scattering problem. , 575:627, 2000. Preprint LPENSL-TH-19/99, hep-th/9911030.
R. Murgan and R. I. Nepomechie. Bethe ansatz derived from the functional relations of the open XXZ chain for new special cases. , (2005) P08002.
R. I. Nepomechie. Solving the open [XXZ]{} spin chain with nondiagonal boundary terms at roots of unity. , 622(3):615 – 632, 2002.
R. I. Nepomechie. Bethe ansatz solution of the open [$XXZ$]{} chain with nondiagonal boundary terms. , 37(2):433–440, 2004. Special issue on recent advances in the theory of quantum integrable systems.
R. I. Nepomechie, Inhomogeneous T-Q equation for the open XXX chain with general boundary terms: completeness and arbitrary spin. arXiv:1307.5049.
R. I. Nepomechie and F. Ravanini, Completeness of the Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms , 11391-11402, 2003.
R. I. Nepomechie and F. Ravanini, Addendum to ‘Completeness of the Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms’ , 1945-1946, 2004.
R. I. Nepomechie and C. Wang, Boundary energy of the open XXX chain with a non-diagonal boundary term. arXiv:1310.6305.
G. Niccoli. Reconstruction of Baxter Q-operator from Sklyanin SOV for cyclic representations of integrable quantum models. , 835: 263-283, 2010.
G. Niccoli. Completeness of Bethe Ansatz by Sklyanin SOV for Cyclic Representations of Integrable Quantum Models. , 1103:123, 2011.
G. Niccoli. Non-diagonal open spin 1/2 [X]{}[X]{}[Z]{} quantum chains by separation of variables: complete spectrum and matrix elements of some quasi-local operators. , 2012(10):P10025, 2012.
G. Niccoli, On the form factors of local operators in the Bazhanov-Stroganov and chiral Potts models. ICMP12 Proceedings by World Scientific. arXiv:1301.4924.
G. Niccoli. Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and form factors. , 870: 397 – 420, 2013.
G. Niccoli. Form factors and complete spectrum of XXX antiperiodic higher spin chains by quantum separation of variables. 54, 053516 (2013).
G. Niccoli. Antiperiodic dynamical 6-vertex model I: Complete spectrum by SOV, matrix elements of the identity on separate states and connections to the periodic 8-vertex model. 46 075003, 2013.
G. Niccoli and J. Teschner. The sine-Gordon model revisited: I. , P09014 (2010).
A. Nichols, V. Rittenberg and J. de Gier. One-boundary Temperley-Lieb algebras in the XXZ and loop models. P03003 (2005).
T. Prosen. Open [X]{}[X]{}[Z]{} spin chain: Nonequilibrium steady state and a strict bound on ballistic transport. , 106:217206, May 2011.
J. Sirker, R. G. Pereira, and I. Affleck. Diffusion and ballistic transport in one-dimensional quantum systems. , 103:216602, Nov 2009.
E. K. Sklyanin. The quantum [T]{}oda chain. In *Nonlinear equations in classical and quantum field theory ([M]{}eudon/[P]{}aris, 1983/1984)*, volume 226 of *Lecture Notes in Phys.*, pages 196–233. Springer, Berlin, 1985.
E. Sklyanin. Boundary conditions for integrable quantum systems. , 21:2375–2389, 1988.
E. K. Sklyanin. Poisson structure of a periodic classical XYZ chain. [*J. Soviet Math.*]{}, 1989, v.46, n.1., p.1664-1683.
E. K. Sklyanin. Poisson structure of classical XXZ chain. [*J. Soviet Math.*]{}, 1989, v.46, n.5, p.2104-2111.
E. K. Sklyanin. Quantum inverse scattering method. selected topics. In M.-L. Ge, editor, *Quantum group and Quantum Integrable Systems*, pages 63–97. Nankai Lectures in Mathematical Physics, World Scientific, 1992.
L. A. Takhtajan and L. D. Faddeev. The quantum method of the inverse problem and the Heisenberg [XYZ]{} model. , 34(5):11–68, 1979.
W.-L. Yang and Y.-Z. Zhang. On the second reference state and complete eigenstates of the open [$XXZ$]{} chain. , pages 044, 11 pp. (electronic), 2007.
[^1]: Different methods leading to Bethe ansatz equations have been also proposed under the same boundary conditions by using the framework of the Temperley-Lieb algebra in [@deGP04; @NicRd05] and by making a combined use of coordinate Bethe ansatz and matrix ansatz in [@CraRS10; @CraR12].
[^2]: Some partial results in this direction were achieved in [@FilK11] but only in the special case of double boundary constrains introduced in [YanZ07]{}.
[^3]: See also the papers [@CaoYSW13-1; @CaoYSW13-2; @CaoYSW13-3] for the application of the same method to different models.
[^4]: Here we use notations similar to those introduced in the papers [@CaoJYW2013] and [@Nep-2013] where some inhomogeneous Baxter equation ansatzs appear with the aim to make simpler for the reader a comparison when the limit of homogeneous chain is implemented.
|
---
abstract: 'We develop an event-driven Receding Horizon Control (RHC) scheme for a Ride Sharing System (RSS) in a transportation network where vehicles are shared to pick up and drop off passengers so as to minimize a weighted sum of passenger waiting and traveling times. The RSS is modeled as a discrete event system and the event-driven nature of the controller significantly reduces the complexity of the vehicle assignment problem, thus enabling its real-time implementation. Simulation results using actual city maps and real taxi traffic data illustrate the effectiveness of the RH controller in terms of real-time implementation and performance relative to known greedy heuristics.'
author:
- 'Rui Chen and Christos G. Cassandras [^1] [^2]'
bibliography:
- 'CSS1206.bib'
title: '[**Optimization of Ride Sharing Systems Using Event-driven Receding Horizon Control$^{\star}$**]{}'
---
Introduction
============
It has been abundantly documented that the state of transportation systems worldwide is at a critical level. Based on the $2011$ Urban Mobility Report, the cost of commuter delays has risen by $260$% over the past $25$ years and $28$% of U.S. primary energy is now used in transportation [@schrank2011urban]. Traffic congestion also leads to an increase in vehicle emissions; in large cities, as much as $90$% of CO emissions are due to mobile sources. Disruptive technologies that aim at dramatically altering the transportation landscape include vehicle connectivity and automation as well as shared personalized transportation through emerging mobility-on-demand systems. Focusing on the latter, the main idea of a Ride Sharing System (RSS) is to assign vehicles in a given fleet so as to serve multiple passengers, thus effectively reducing the total number of vehicles on a road network, hence also congestion, energy consumption, and adverse environmental effects.
The main objectives of a RSS are to minimize the total Vehicle-Miles-Traveled (VMT) over a given time period (equivalently, minimize total travel costs), to minimize the average waiting and traveling times experienced by passengers, and to maximize the number of satisfied RSS participants (both drivers and passengers) [@agatz2012optimization]. When efficiently managed, a RSS has the potential to reduce the total number of private vehicles in a transportation network, hence also decreasing overall energy consumption and traffic congestion, especially during peak hours of a day. From a passenger standpoint, a RSS is able to offer door-to-door transportation with minimal delays which makes traveling more convenient. From an operator’s standpoint a RSS provides a considerable revenue stream. A RSS also provides an alternative to public transportation or can work in conjunction with it to reduce possible low uitization of vehicles and long passenger delays.
In this paper, we concentrate on designing dynamic vehicle assignment strategies in a RSS aiming to minimize the system-wide waiting and traveling times of passengers. The main challenge in obtaining optimal vehicle assignments is the complexity of the optimization problem involved in conjunction with uncertainties such as random passenger service request times, origins, and destinations, as well as unpredictable traffic conditions which determine the times to pick up and drop off passengers. Algorithms used in RSS are limited by the NP-complete nature of the underlying traveling salesman problem [@chen2017hierarchical] which is a special case of the much more complex problems encountered in RSS optimization. Therefore, a global optimal solution for such problems is generally intractable, even in the absence of the aforementioned uncertainties. Moreover, a critical requirement in such algorithms is a guarantee that they can be implemented in a real-time context.
Several methods have been proposed to solve the RSS problem addressing the waiting and traveling times of passengers. In [@agatz2011dynamic], a greedy approach is used to match vehicles to passenger requests which can on one hand guarantee real-time assignments but, on the other, lacks performance guarantees. The optimization algorithm in [@santi2014quantifying] improves the average traveling time performance but limits the seat capacity of each vehicle to $2$ (otherwise, the problem becomes intractable for $4$ or more seats) and allows no dynamic allocation of new passengers after a solution is determined. Although vehicles can be dynamically allocated to passengers in [@berbeglia2010dynamic], all pickup and drop-off events are constrained to take place within a specified time window. The RTV-graph algorithm [@alonso2017demand] can also dynamically allocate passengers, but its complexity increases dramatically with the number of agents (passengers and vehicles) and the seat capacity of vehicles. To address the issue of increasing complexity with the size of a RSS, a hierarchical approach is proposed in [@chen2017hierarchical] such that the system is decomposed into smaller regions. Within a region, a mixed-integer linear programs is formulated so as to obtain an optimal vehicle assignment over a sequence of fixed time horizons. Although this method addresses the complexity issue, it involves a large number of unnecessary calculations since there is no need to always re-evaluate an optimal solution over every such horizon. Another approach to reducing complexity, is to abstract a RSS model through passenger and vehicle flows as in [@calafiore2017flow],[@tsao2018stochastic] and [@salazar2018interaction]. In [@salazar2018interaction], for example, the interaction between autonomous mobility-on-demand and public transportation systems is considered so as to maximize the overall social welfare.
In order to deal with the well-known curse of dimensionality [@bertsekas2005dynamic] that characterizes optimization problem formulations for a RSS, we adopt an *event-driven* *Receding Horizon Control* (RHC) approach. This is in the same spirit as Model Predictive Control (MPC) techniques [@camacho2013model] with the added feature of exploiting the event-driven nature of the control process in which the RHC algorithm is invoked only when certain events occur. Therefore, compared with conventional time-driven MPC this approach can avoid unnecessary calculations and can significantly improve the efficiency of the RH controller by reacting to random events as they occur in real time. The basic idea of event-driven RHC introduced in [@li2006cooperative] and extended in [@khazaeni2016event] is to solve an optimization problem over a given *planning horizon* when an event is observed in a way which allows vehicles to cooperate; the resulting control is then executed over a generally shorter *action horizon* defined by the occurrence of the next event of interest to the controller. Compared to methods such as [@santi2014quantifying]-[@alonso2017demand], the RHC scheme is not constrained by vehicle seating capacities and is specifically designed to dynamically re-allocate passengers to vehicles at any time. Moreover, compared to the time-driven strategy in [@chen2017hierarchical], the event-driven RHC scheme refrains from unnecessary calculations when no event in the RSS occurs. Finally, in contrast to models used in [@tsao2018stochastic] and [@salazar2018interaction], we maintain control of every vehicle and passenger in a RSS at a microscopic level while ensuring that real-time optimal (over each receding horizon) vehicle assignments can be made.
The paper is organized as follows. We first present in Section II a discrete event system model of a RSS and formulate an optimization problem aimed at minimizing a weighted sum of passenger waiting and traveling times. Section III first reviews the basic RHC scheme previously used and then identifies how it is limited in the context of a RSS. This motivates the new RHC approach described in Section IV, specifically designed for a RSS. Extensive simulation results are given in Section V for actual maps in Ann Arbor, MI and New York City, where, in the latter case, real taxi traffic data are used to drive the simulation model. We conclude the paper in Section VI.
Problem Formulation
===================
We consider a Ride Sharing System (RSS) in a traffic network consisting of $N$ nodes $\mathcal{N}=\{1,...,N\}$ where each node corresponds to an intersection. Nodes are connected by arcs (i.e., road segments). Thus, we view the traffic network as a directed graph $\mathbb{G}$ which is embedded in a two-dimensional Euclidean space and includes all points contained in every arc, i.e., $\mathbb{G}\subset\mathbb{R}^{2}$. In this model, a node $n\in\mathcal{N}$ is associated with a point $\nu_{n}\in\mathbb{G}$, the actual location of this intersection in the underlying two-dimensional space. The set of vehicles present in the RSS at time $t$ is $\mathcal{A}(t)$, where the index $j\in\mathcal{A}(t)$ will be used to uniquely denote a vehicle, and let $A(t)=|\mathcal{A}(t)|$. The set of passengers is $\mathcal{P}(t)$, where the index $i$ will be used to uniquely denote a passenger, and let $P(t)=|\mathcal{P}(t)|$. Note that $\mathcal{A}(t)$ is time-varying since vehicles may enter or leave the RSS at any time and the same is true for $\mathcal{P}(t)$.
There are two points in $\mathbb{G}$ associated with each passenger $i$, denoted by $o_{i},r_{i}\in\mathbb{G}$: $o_{i}$ is the origin where the passenger issues a service request (pickup point) and $r_{i}$ is the passenger’s destination (drop-off point). Let ${O}(t)=\{o_{1},...,o_{P}\}$ be the set of all passenger origins and ${R}(t)=\{r_{1},...,r_{P}\}$ the corresponding destination set. Vehicles pick up passengers and deliver them to their destinations according to some policy. We assume that the times when vehicles join the RSS are not known in advance, but they become known as a vehicle joins the system. Similarly, the times when passenger service requests occur are random and their destinations become known only upon being picked up.
**State Space:** In addition to $\mathcal{A}(t)$ and $\mathcal{P}(t)$ describing the state of the RSS, we define the states associated with each vehicle and passenger as follows. Let $x_{j}(t)\in\mathbb{G}$ be the position of vehicle $j$ at time $t$ and let $N_{j}(t)\in\{0,1,...,C_{j}\}$ be the number of passengers in vehicle $j$ at time $t$, where $C_{j}$ is the capacity of vehicle $j$. The state of passenger $i$ is denoted by $s_{i}(t)$ where $s_{i}(t)=0$ if passenger $i$ is waiting to be picked up and $s_{i}(t)=j\in\mathcal{A}(t)$, where $j>0$, when the passenger is in vehicle $j$ after being picked up. Finally, we associate with passenger $i$ a left-continuous clock value $z_{i}(t)\in$ $\mathbb{R}$ whose dynamics are defined as follows: when the passenger joins the system and is added to $\mathcal{P}(t)$, the initial value of $z_{i}(t)$ is $0$ and we set $\dot
{z}_{i}(t)=1$, as illustrated in Fig.\[z\_dynamic\] where the passenger service request time is $\varphi_{i}$. Thus, $z_{i}(t)$ may be used to measure the waiting time of passenger $i$. When $i$ is picked up by some vehicle $j$ at time $\rho_{i,j}$ (see Fig.\[z\_dynamic\]), $z_{i}(t)$ is reset to zero and thereafter measures the traveling time until the passenger’s destination is reached at time $\sigma_{i,j}$. In summary, the state of the RSS is $\mathbf{X}(t)=\{\mathcal{A}(t),x_{1}(t),\dots,x_{A}(t),N_{1}(t),\dots
,N_{A}(t),\mathcal{P}(t),s_{1}(t),\dots,s_{P}(t),\newline z_{1}(t),\dots
,z_{P}(t)\}$.
![A typical sample path of passenger $i$’s clock state $z_{i}(t)$.[]{data-label="z_dynamic"}](z_dynamic.jpg)
**Events:** All state transitions in the RSS are event-driven with the exception of the passenger clock states $z_{i}(t)$, $i\in\mathcal{P}(t)$, in which case it is the reset conditions (see Fig.\[z\_dynamic\]) that are event-driven. As we will see, all control actions (to be defined) affecting the state $\mathbf{X}(t)$ are taken only when an event takes place. Therefore, regarding a vehicle location $x_{j}(t)$, $j\in\mathcal{A}(t)$, for control purposes we are interested in its value only when events occur, even though we assume that $x_{j}(t)$ is available to the RSS for all $t$ based on an underlying localization system.
We define next the set $E$ of all events whose occurrence causes a state transition. We set $E=E_{U}\cup E_{C}$ to differentiate between uncontrollable events contained in $E_{U}$ and controllable events contained in $E_{C}$. There are six possible event types, defined as follows:
\(1) $\alpha_{i}\in E_{U}$: a service request is issued by passenger $i$.
\(2) $\beta_{j}\in E_{U}$: vehicle $j$ joins the RSS.
\(3) $\gamma_{j}\in E_{U}$: vehicle $j$ leaves the RSS.
\(4) $\pi_{i,j}\in E_{C}$: vehicle $j$ picks up passenger $i$ (at $o_{i}\in\mathbb{G}$).
\(5) $\delta_{i,j}\in E_{C}$: vehicle $j$ drops off passenger $i$ (at $r_{i}\in\mathbb{G}$).
\(6) $\zeta_{m,j}\in E_{C}$: vehicle $j$ arrives at intersection (node) $m\in\mathcal{N}$.
Note that events $\alpha_{i}$, $\beta_{j}$ are uncontrollable exogenous events. Event $\gamma_{j}$ is also uncontrollable, however it may not occur unless the guard condition $N_{j}(t)=0$ is satisfied, that is, the number of passengers in vehicle $j$ must be zero when it leaves the system. On the other hand, the remaining three events are controllable. First, $\pi_{i,j}$ depends on the control policy (to be defined) through which a vehicle is assigned to a passenger and is feasible only when $s_{i}(t)=0$ and $N_{j}(t)<C_{j}$. Second, $\delta_{i,j}$ is feasible only when $s_{i}(t)=j\in\mathcal{A}(t)$. Finally, $\zeta_{m,j}$ depends on the policy (to be defined) and occurs when the route taken by vehicle $j$ involves intersection $m\in\mathcal{N}$.
**State Dynamics:** The events defined above determine the state dynamics as follows.
\(1) Event $\alpha_{i}$ adds an element to the passenger set $\mathcal{P}(t)$ and increases its cardinality, i.e., $P(t^{+})=P(t)+1$ where $t$ is the occurrence time of this event. In addition, it initializes the passenger state and associated clock:$$s_{i}(t^{+})=0,\text{ \ \ }\dot{z}_{i}(t^{+})=1\text{ with }z_{i}(t)=0
\label{alpha_dynamics}$$ and generates the origin information of this passenger $o_{i}\in\mathbb{G}$.
\(2) Event $\beta_{j}$ adds an element to the vehicle set $\mathcal{A}(t)$ and increases its cardinality, i.e., $A(t^{+})=A(t)+1$. It also initializes $x_{j}(t)$ to the location of vehicle $j$ at time $t$.
\(3) Event $\gamma_{j}$ removes vehicle $j$ from $\mathcal{A}(t)$ and decreases its cardinality, i.e., $A(t^{+})=A(t)-1$.
\(4) Event $\pi_{i,j}$ occurs when $x_{j}(t)=o_{i}$ and it generates the destination information of this passenger $r_{i}\in\mathbb{G}$. This event affects the states of both vehicle $j$ and passenger $i$:$$N_{j}(t^{+})=N_{j}(t)+1,\text{ \ \ }s_{i}(t^{+})=j$$ and, since the passenger was just picked up, the associated clock is reset to $0$ and starts measuring traveling time towards the destination $r_{i}$:$$z_{i}(t^{+})=0,\text{ \ }\dot{z}_{i}(t^{+})=1 \label{pi _dynamics}$$
\(5) Event $\delta_{i,j}$ occurs when $x_{j}(t)=r_{i}$ and it causes a removal of passenger $i$ from $\mathcal{P}(t)$ and decreases its cardinality, i.e., $P(t^{+})=P(t)-1$. In addition, it affects the state of vehicle $j$:$$N_{j}(t^{+})=N_{j}(t)-1$$
\(6) Event $\zeta_{m,j}$ occurs when $x_{j}(t)=\nu_{m}$. This event triggers a potential change in the control associated with vehicle $j$ as described next.
**Control:** The control we exert is denoted by $u_{j}(t)\in\mathbb{G}$ and sets the destination of vehicle $j$ in the RSS. We note that the destination $u_{j}(t)$ may change while vehicle $j$ is en route to it based on new information received as various events may take place. The control is initialized when event $\beta_{j}$ occurs at some point $x_{j}(t)$ by setting $u_{j}(t)=\nu_{m}$ where $m\in\mathcal{N}$ is the intersection closest to $x_{j}(t)$ in the direction vehicle $j$ is headed. Subsequently, the vector $\mathbf{u}(t)=\{u_{1}(t),\dots,u_{A}(t)\}$ is updated according to a given policy whenever an event from the set $E$ occurs (we assume that all events are observable by the RSS controller). Our control policy is designed to optimize the objective function described next.
**Objective Function:** Our objective is to minimize the combined *waiting* and *traveling* times of passengers in the RSS over a given finite time interval $[0,T]$. In order to incorporate all passengers who have received service over $[0,T]$, we define the set $$\mathcal{P}_{T}=\cup_{t\in\lbrack0,T]}\mathcal{P}(t)$$ to include all passengers $i\in\mathcal{P}(t)$ for any $t\in\lbrack0,T]$. In simple terms, $\mathcal{P}_{T}$ is used to record all passengers who are either currently active in the RSS at $t=T$ or were active and departed at some time $t<T$ when the associated $\delta_{i,j}$ event occurred for some $j\in\mathcal{A}(t)$.
We define $w_{i}$ to be the waiting time of passenger $i$ and note that, according to (\[alpha\_dynamics\]), $w_{i}=z_{i}(t)$ where $t$ is the time when event $\pi_{i,j}$ occurs. Similarly, letting $y_{i}$ be the total traveling time of passenger $i$, according to (\[pi \_dynamics\]) we have $y_{i}=z_{i}(t)$ where $t$ is the time when event $\delta_{i,j}$ occurs. We then formulate the following problem, given an initial state $\mathbf{X}_{0}$ of the RSS: $$\min_{\mathbf{u}(t)}E\left[ \sum_{i\in\mathcal{P}_{T}}[\mu_{w}w_{i}+\mu
_{y}y_{i}]\right] \label{cost_func1}$$ where $\mu_{w},\mu_{y}$ are weight coefficients defined so that $\mu_{w}=\frac{\omega}{W_{\max}}$ and $\mu_{y}=\frac{1-\omega}{Y_{\max}}$, $\omega
\in\lbrack0,1]$, and $W_{\max}$ and $Y_{\max}$ are upper bounds of the waiting and traveling time of passengers respectively. The values of $W_{\max}$ and $Y_{\max}$ are selected based on user experience to capture the worst case tolerated for waiting and traveling times respectively. This construction ensures that $w_{i}$ and $y_{i}$ are properly normalized so that (\[cost\_func1\]) is well-defined.
The expectation in (\[cost\_func1\]) is taken over all random event times in the RSS defined in an appropriate underlying probability space. Clearly, modeling the random event processes so as to analytically evaluate this expectation is a difficult task. This motivates viewing the RSS as unfolding over time and adopting a control policy based on observed actual events and on estimated future events that affect the RSS state.
Assuming for the moment that the system is deterministic, let $t_{k}$ denote the occurrence time of the $k$th event over $[0,T]$. A control action $\mathbf{u}(t_{k})$ may be taken at $t_{k}$ and, for simplicity, is henceforth denoted by $\mathbf{u}_{k}$. Along the same lines, we denote the state $\mathbf{X}(t_{k})$ by $\mathbf{X}_{k}$. Letting $K_{T}$ be the number of events observed over $[0,T]$, the optimal value of the objective function when the initial state is $\mathbf{X}_{0}$ is given by $$J(\mathbf{X}_{0})=\min_{u_{0},\cdots u_{K_{T}}}\left[ \sum_{i\in
\mathcal{P}_{T}}[\mu_{w}w_{i}+\mu_{y}y_{i}]\right]$$ We convert this into a maximization problem by considering $[-\mu_{w}w_{i}-\mu_{y}y_{i}]$ for each $i\in\mathcal{P}_{T}$. Moreover, observing that both $w_{i}$ and $y_{i}$ are upper-bounded by $T$, we consider the non-negative rewards $T-w_{i}$ and $T-w_{i}$ and rewrite the problem above as $$J(\mathbf{X}_{0})=\max_{u_{0},\cdots u_{K_{T}}}\left[ \sum_{i\in
\mathcal{P}_{T}}[\mu_{w}(T-w_{i})+\mu_{y}(T-y_{i})]\right]
\label{Reward_Function}$$ Then, determining an optimal policy amounts to solving the following Dynamic Programming (DP) equation [@bertsekas2005dynamic]:$$J(\mathbf{X}_{k})=\max_{\mathbf{u}_{k}\in\mathbb{G}}[C(\mathbf{X}_{k},\mathbf{u}_{k})+J_{{k+1}}(\mathbf{X}_{k+1})],\text{ \ }k=0,1,\ldots,K_{T}$$ where $C(\mathbf{X}_{k},\mathbf{u}_{k})$ is the immediate reward at state $\mathbf{X}_{k}$ when control $\mathbf{u}_{k}$ is applied and $J_{{k+1}}(\mathbf{X}_{k+1})$ is the future reward at the next state $\mathbf{X}_{k+1}$. Our ability to solve this equation is limited by the well-known curse of dimensionality [@bertsekas2005dynamic] even if our assumption that the RSS is fully deterministic were to be valid. This further motivates adopting a *Receding Horizon Control* (RHC) approach as in similar problems encountered in [@li2006cooperative] and [@khazaeni2016event]. This is in the same spirit as Model Predictive Control (MPC) techniques [@camacho2013model] with the added feature of exploiting the event-driven nature of the control process. In particular, in the event-driven RHC approach, a control action taken when the $k$th event is observed is selected to maximize an immediate reward defined over a *planning horizon* $H_{k}$, denoted by $C(\mathbf{X}_{k},\mathbf{u}_{k},H_{k})$, followed by an estimated future reward $\hat{J}_{{k+1}}(\mathbf{X(}t_{k}+H_{k}))$ when the state is $\mathbf{X(}t_{k}+H_{k})$. The optimal control action $\mathbf{u}_{k}^{\ast}$ is, therefore,$$\mathbf{u}_{k}^{\ast}=\arg\max_{\mathbf{u}_{k}\in\mathbb{G}}[C(\mathbf{X}_{k},\mathbf{u}_{k},H_{k})+\hat{J}_{{k+1}}(\mathbf{X(}t_{k}+H_{k}))]
\label{RHC_algorithm}$$ The control action $\mathbf{u}_{k}^{\ast}$ is subsequently executed only over a generally shorter *action horizon* $h_{k}\leq H_{k}$ so that $t_{k+1}=t_{k}+h_{k}$ (see Fig.\[RHC\]). The selection of $H_{k}$ and $h_{k}$ will be discussed in the next section.
![Event-Driven receding horizon control.[]{data-label="RHC"}](rhc.jpg)
Receding Horizon Control (RHC)
==============================
In this section, we first review the basic RHC scheme as introduced in [@li2006cooperative], and a modified version in [@khazaeni2016event] intended to overcome some of the original scheme’s limitations. We refer to the RHC in [@li2006cooperative] as *RHC1* and the RHC in [@khazaeni2016event] as *RHC2*.
The basic RHC scheme in [@li2006cooperative] considers a set of cooperating agents and a set of targets in a Euclidean space. The purpose of agents is to visit targets and collect a certain time-varying reward associated with each target. The key steps of the scheme are as follows: (1) Determine a planning horizon $H_{k}$ at the current time $t_{k}$. (2) Solve an optimization problem to minimize an objective function defined over the time interval $[t_{k},t_{k}+H_{k}]$. (3) Determine an action horizon $h_{k}$ and execute the optimal solution over $[t_{k},t_{k}+h_{k}]$. (4) Set $t_{k+1}=t_{k}+h_{k}$ and return to step (1).
Letting $\mathcal{A}(t)$ be the agent set and $\mathcal{P}(t)$ the target set, we define $d_{i,j}(t)$ for any $i\in\mathcal{P}(t)$, $j\in\mathcal{A}(t)$ to be the distance between target $i$ and agent $j$ at time $t$. In [@li2006cooperative], the planning horizon $H_{k}$ is defined as the earliest time that any agent can visit any target in the system: $$H_{k}=\min_{i\in\mathcal{P}(t),j\in\mathcal{A}(t)}\left\{ \dfrac
{d_{i,j}(t_{k})}{v}\right\} \label{planninghorizon}$$ where $v$ is the fixed speed of agents. The action horizon $h_{k}$ is defined to be the earliest time in $[t_{k},t_{k}+H_{k}]$ when an event in the system occurs (e.g., a new target appears). In some cases, $h_{k}$ is alternatively defined through $h_{k}=\epsilon H_{k}$ for some $\epsilon\in(0,1]$ so as to ensure that $h_{k}\leq H_{k}$.
In order to formulate the optimization problem to be solved at every control action point $t_{k}$, the concept of *neighborhood* for a target is defined in [@li2006cooperative] as follows. The $k$th nearest agent neighbor to target $l$ is$$\beta^{k}(l,t)=\underset{i\in\mathcal{A}(t),i\neq\beta_{l}^{1}(t),\dots
,i\neq\beta_{l}^{k-1}(t)}{\operatorname{arg}\,\operatorname{min}}\;{d_{l,i}(t)}$$ where $k=1,2,\ldots$, and the $b$-neighborhood of the target is given by the set of the $b$ closest neighbors to it: $$B_{l}^{b}(t)=\{\beta^{1}(l,t),\dots,\beta^{b}(l,t)\} \label{neighborset}$$ Based on , for any given $b\geq1$ the *relative distance* between agent $i$ and target $l$ is defined as $$\bar{d}_{l,i}(t)=\left\{
\begin{array}
[c]{l}\dfrac{d_{l.i}(t)}{\sum_{q\in B_{l}^{b}(t)}d_{l,q}(t)}\\
1
\end{array}
\right.
\begin{array}
[c]{l}\text{if }i\in B_{l}^{b}(t)\\
\text{otherwise}\end{array}$$ Then, the *relative responsibility* function of agent $i$ for target $l$ is defined as: $$p(\bar{d}_{l,i}(t))=\left\{
\begin{array}
[c]{l}1\\
\dfrac{1-\Gamma-\bar{d}_{l,i}}{1-2\Gamma}\\
0
\end{array}
\right.
\begin{array}
[c]{l}\text{if }\bar{d}_{l,i}\leq\Gamma\\
\text{if }\Gamma\leq\bar{d}_{l,i}\leq1-\Gamma\\
\text{otherwise}\end{array}
\label{RelativeResponsibility}$$ where $p(\bar{d}_{l,i}(t))$ can be viewed as the probability that agent $i$ is the one to visit target $l$. In particular, when the relative distance is small, then $i$ is committed to visit $l$, whereas if the relative distance is large, then $i$ takes no responsibility for $l$. All other cases define a cooperative region where agent $i$ visits $l$ with some probability dependent on the parameter $\Gamma$ which is selected so that $\Gamma\in\lbrack0,\frac{1}{2})$ and reflects a desired level of cooperation among agents; this cooperation level increases as $\Gamma$ decreases.
The use of $p(\bar{d}_{l,i}(t))$ allows the RHC to avoid early commitments of agents to target visits, since changes in the system state may provide a better opportunity for an agent to improve the overall system performance. A typical example arises when agent $i$ is committed to target $l$ and a new target, say $l^{\prime}$, appears which is in close proximity to $i$; in such a case, it may be beneficial for $i$ to visit $l^{\prime}$ and let $l$ become the responsibility of another agent that may be relatively close to $l$ and uncommitted. This is possible if $p(\bar{d}_{l,i}(t))<1$. In what follows, we will generalize the definition of distance $d_{i,j}(t)$ between target $i$ and agent $j$ to the distance between any two points $x,y\in\mathbb{R}^{2}$ expressed as ${d(x,y)}$.
Using the relative responsibility function, the optimization problem solved by the RHC at each control action point assigns an agent to a point which minimizes a given objective function and which is not necessarily a target point. Details of how this problem is set up and solved and the properties of the *RHC1* scheme may be found in [@li2006cooperative].
**Limitations of** *RHC1***:** There are three main limitations of the original RHC scheme:
*(1) Agent trajectory instabilities:* A key benefit of *RHC1* is the fact that early commitments of agents to targets are avoided. As already described above, if a new target appears in the system, an agent en route to a different target may change its trajectory to visit the new one if this is deemed beneficial to the cooperative system as a whole. This benefit, however, is also a cause of potential instabilities when agents frequently modify their trajectories, thus potentially wasting time. It is also possible that an agent may oscillate between two targets and never visit either one. In [@li2006cooperative], necessary and sufficient conditions were provided for some simple cases to quantify such instabilities, but these conditions may not always be satisfied.
*(2) Future cost estimation inaccuracies:* The effectiveness of *RHC1* rests on the accuracy of the future cost estimation term $\hat
{J}_{{k+1}}(\mathbf{X(}t_{k}+H_{k})$ in (\[RHC\_algorithm\]). In [@li2006cooperative], this future cost is estimated through its lower bound, thus resulting in an overly optimistic outlook.
*(3) Algorithm complexity:* In [@li2006cooperative], the optimization problem at each algorithm iteration involves the selection of each agent’s heading over $[0,2\pi]$. This is because the planning horizon $H_{k}$ defines a set of feasible reachable points $F_{j}(t_{k},H_{k})=\{w:d(w,x_{j}(t_{k})=vH_{k}\}$ which is a disk of radius $H_{k}/v$ (where $v$ is each agent’s speed) around the agent’s position at time $t_{k}$. This problem must be solved over all agents and incurs considerable computational complexity: if $[0,2\pi]$ is discretized with discretization level $G$, then the complexity of this algorithm at each iteration is $O(G^{A(t)})$.
The modified RCH scheme *RHC2* in [@khazaeni2016event] was developed to address these limitations. To deal with issues *(1)* and *(3)* above, a set of *active targets* $S_{j}(t_{k},H_{k})$ is defined for agent $j$ at each iteration time $t_{k}$. Its purpose is to limit the feasible reachable set $F_{j}(t_{k},H_{k})$ defined by all agent headings over $[0,2\pi]$ so that it is reduced to a finite set of points. Let $x\in
F_{j}(t_{k},H_{k})$ be a reachable point and define a *travel cost* function $\eta_{i}(x,t)$ associated with every target $i\in\mathcal{P}(t)$ measuring the cost of traveling from a point $x$ at time $t$ to a target $i\in\mathcal{P}(t)$. The active target set is defined in [@khazaeni2016event] as $$\begin{aligned} S_{j}(t_{k},H_{k})=\{l:l=\arg\min_{i\in\mathcal{P}(t)}\;\eta_{i}(x,t_{k}+H_{k})\\ \text{ for some }x\in F_{j}(t_{k},H_{k})\} \end{aligned} \label{ActiveTragets_RHC2}$$ Clearly, $S_{j}(t_{k},H_{k})\subseteq\mathcal{P}(t)$ is a finite set of targets defined by the following property: an active target is closer to some reachable point $x$ than any other target in the sense of minimizing the metric $\eta_{i}(x,t_{k}+H_{k})$. Therefore, if there is some target $l^{\prime}\notin S_{j}(t_{k},H_{k})$, then there is no incentive in considering it as a candidate for agent $j$ to head towards. Restricting the feasible headings of an agent to its active target set not only reduces the complexity of optimally selecting a heading at $t_{k}$, but it also limits oscillatory trajectory behavior, since by (\[planninghorizon\]) there is always an active target on the set $F_{j}(t_{k},H_{k})$ so that eventually all targets are guaranteed to be visited.
Let $\mathbf{u}_{k}$ be the control applied at time $t_{k}$ under planning horizon $H_{k}$. The $j$th component of $\mathbf{u}_{k}$ is the control $u_{j}(t_{k})$ applied to agent $j$, where $u_{j}(t_{k})\in S_{j}(t_{k},H_{k})$ as defined in (\[ActiveTragets\_RHC2\]). The estimated time for agent $j$ to reach a target $u_{j}(t_{k})$ is denoted by $\hat{\tau}_{u,j}(\mathbf{u}_{k},t_{k},H_{k})$ where (for notational simplicity) we set $u_{j}(t_{k})=u$. This time is given by$$\hat{\tau}_{u,j}(\mathbf{u}_{k},t_{k},H_{k})=t_{k}+H_{k}+\frac{1}{v}d(x_{j}(t_{k}),x_{u}),\text{ \ \ }u\in S_{j}(t_{k},H_{k}) \label{tauhat_1}$$ where $x_{u}$ is the location of target $u=u_{j}(t_{k})$.
To address issue *(2)* regarding future cost estimation inaccuracies, a new estimation framework is introduced in [@khazaeni2016event] by defining a set of targets $\mathcal{T}_{k,j}\subseteq\mathcal{P}(t)-\{u\}$ that agent $j$ would visit in the future, i.e., at $t>t_{k}+H_{k}$, as follows: $$\mathcal{T}_{k,j}=\{l:p(\bar{d}_{l,j}(t_{k}))>p(\bar{d}_{l,q}(t_{k})),\text{
}\forall q\in\mathcal{A}(t)\} \label{targetset}$$ This set limits the targets considered by agent $j$ to those with a current relative responsibility value in (\[RelativeResponsibility\]) which exceeds that of any other agent. The estimated time to reach a target $l\in
\mathcal{T}_{k,j}$ under control $\mathbf{u}_{k}$ and planning horizon $H_{k}$ is denoted by $\hat{\tau}_{l,j}(\mathbf{u}_{k},t_{k},H_{k})$. The first target to be visited in $\mathcal{T}_{k,j}$, denoted by $l^{1}$, is the one with the minimal travel cost from target $u\in S_{j}(t_{k},H_{k})$, i.e., $l^{1}=\arg\min_{l\in\mathcal{T}_{k,j}}\{\eta_{l}(x_{u},\hat{\tau}_{u,j}(\mathbf{u}_{k},t_{k},H_{k}))\}$. Then, all subsequent targets in $\mathcal{T}_{k,j}-\{l^{1}\}$ are similarly ordered as $\{l^{2},l^{3},\ldots\}$. Therefore, setting $\mathcal{T}_{k,j}^{n}=\mathcal{T}_{k,j}-\{l^{1},\ldots,l^{n-1}\}$, $n=2,\ldots,|\mathcal{T}_{k,j}|$, we have $$l^{n+1}=\arg\min_{l\in\mathcal{T}_{k,j}^{n}}\{\eta_{l}(x_{l^{n}},\hat{\tau
}_{l^{n},j}(\mathbf{u}_{k},t_{k},H_{k}))\},\text{ \ }n=1,\ldots,|\mathcal{T}_{k,j}|$$ and $$\hat{\tau}_{l^{n+1},j}(\mathbf{u}_{k},t_{k},H_{k})=\hat{\tau}_{l^{n},j}(\mathbf{u}_{k},t_{k},H_{k})+\frac{1}{v}d(x_{l^{n}},x_{l^{n+1}})$$
**Limitations of the** *RHC2* **with respect to a RSS**:
*(1) Euclidean vs. Graph topology:* Both *RHC1* and *RHC2* are based on an underlying Euclidean space topology. In a RSS, however, we are interested in a graph-based topology which requires the adoption of a different distance metric.
*(2) Future cost estimation inaccuracies*: The travel cost metric $\eta_{i}(x,t)$ used in *RHC2* assumes that all future targets to be visited at $t>t_{k}+H_{k}$ are independent of each other and that an agent can visit any target. However, in a RSS, each agent $j$ has a capacity limit $C_{j}$. This has two implications: $(i)$ If a vehicle is full, it must first be assigned to a drop-off point before it can visit a new pickup point, and $(ii)$ The number of future pickup points is limited by $C_{j}-N_{j}(t)$, the residual capacity of vehicle $j$.
The fact that there are two types of targets in a RSS (pickup points and drop-off points), also induces an interdependence in the rewards associated with target visits. Whereas in [@khazaeni2016event] a reward is associated with each target visit, in a RSS the rewards are $w_{i}$ and $y_{i}$ where $y_{i}$ can only be collected after $w_{i}$. This necessitates a new definition of the set $\mathcal{T}_{k,j}$ in (\[targetset\]). For example, if $i\in\mathcal{T}_{k,j}$ and vehicle $j$ is full and must drop off a passenger at a remote location, then using (\[targetset\]) would cause vehicle $j$ to first go to the drop-off location and then return to pick up $i$; however, there may be a free vehicle $k$ in the vicinity of $j$’s current location which is obviously a better choice to assign to passenger $i$.
*(3) Agent trajectory instabilities*: *RHC2* does not resolve the possibility of agent trajectory instabilities. Moreover, the nature of such instabilities is different due to the graph topology used in a RSS.
In view of this discussion, we will present in the next section a new RHC scheme specifically designed for a RSS and addressing the issues identified above. We will keep using the term target to refer to points $o_{i}$ and $r_{i}$ for all $i\in\mathcal{P}(t)$.
The New RHC Scheme
==================
We begin by introducing some variables used in the new RHC scheme as follows.
\(1) $d(u,v)$ is defined as the *Manhattan distance* [@farris1972estimating] between two points $u,v\in\mathbb{G}$. This measures the shortest path distance between two points on a directed graph that includes points on an arc of this graph which belong to $\mathbb{G}\subset\mathbb{R}^{2}$.
\(2) $\mathcal{R}_{i,j}(t)$ is the set of the $n$ closest pickup locations in the sense of the Manhattan distance defined above, where $n=C_{j}-N_{j}(t)-1$ if $j$ picks up $i$ at $o_{i}$ at time $t$, and $n=C_{j}-N_{j}(t)+1$ if $j$ drops off $i$ at $r_{i}$ at time $t$. Clearly, the set may contain fewer than $n$ elements if there are insufficient pickup locations in the RSS at time $t$.
\(3) $\hat{\mathcal{R}}_{i,j}(t)$ is the set of $n$ drop-off locations for $j$, where $n=N_{j}(t)+1$ if $j$ picks up $i$ at $o_{i}$, and $n=N_{j}(t)-1$ if $j$ drops off $i$ at $r_{i}$.
\(4) $\varphi_{i}$ and $\rho_{{i,j}}$ denote the occurrence time of events $\alpha_{i}$ (passenger $i$ joins the RSS) and $\pi_{i,j}$ (pickup of passenger $i$ by vehicle $j$) respectively.
In the rest of this section we present the new RHC scheme which overcomes the issues previously discussed through four modifications: $(i)$ We define the *travel value* of a passenger for each vehicle considering the distance between vehicles and passengers, as well as the vehicle’s residual capacity. $(ii)$ Based on the new travel value and the graph topology of the map, we introduce a new *active target set* for each vehicle during $[t_{k},t_{k}+H_{k})$. This allows us to reduce the feasible solution set of the optimization problem (\[RHC\_algorithm\]) at each iteration. $(iii)$ We develop an improved future reward estimation mechanism to better predict the time that a passenger is served in the future. $(iv)$ To address the potential instability problem, a method to restrain oscillations is introduced in the optimization algorithm at each iteration.
Each of these modifications is described below, leading to the new RHC scheme. We begin by defining the planning horizon $H_{k}$ at the $k$th control update consistent with (\[planninghorizon\]) as $$H_{k}=\min_{i\in\mathcal{P}(t_{k}),j\in\mathcal{A}(t_{k})}\left\{
\frac{d(x_{j}(t_{k}),c_{i})}{v_{j}(t_{k})}\right\} \label{Horizon}$$ where $$c_{i}=\left\{
\begin{array}
[c]{l}o_{i}\\
r_{i}\\
\end{array}
\right.
\begin{array}
[c]{l}\text{if }s_{i}(t)=0\text{ and }N_{j}(t_{k})<C_{j}\\
\text{if }s_{i}(t)=j\\
\end{array}
\label{ci_def}$$ and $v_{j}(t_{k})$ is the maximal speed of vehicle $j$ at time $t_{k}$, assumed to be maintained over $[t_{k},t_{k}+H_{k}]$. Thus, $H_{k}$ is the shortest Manhattan distance from any vehicle location to any target (either $o_{i}$ or $r_{i}$) at time $t_{k}$. Note that $c_{i}$ is undefined if $s_{i}(t)=0$ and $N_{j}(t_{k})=C_{j}$. Formally, to ensure consistency, we set $d(x_{j}(t_{k}),c_{i})=\infty$ if $s_{i}(t)=0$ and $N_{j}(t_{k})=C_{j}$ since $o_{i}$ is not a valid pickup point for $j$ in this case.
The action horizon $h_{k}\leq H_{k}$ is defined by the occurrence of the next event in $E$, i.e., $h_{k}=\tau_{k+1}-t_{k}$ where $\tau_{k+1}$ is the time of the next event to occur after $t_{k}$. If no such event occurs over $[t_{k},t_{k}+H_{k}]$, we set $h_{k}=H_{k}$.
Vehicle Travel Value Function
-----------------------------
Recall that in *RHC2* a travel cost function $\eta_{i}(x,t)$ was defined for any agent measuring the cost of traveling from a point $x$ at time $t$ to a target $i\in\mathcal{P}(t)$. In our case, we define instead a *travel value* measuring the reward (rather than cost) associated with a vehicle $j$ when it considers any passenger $i\in\mathcal{P}(t)$. There are three cases to consider depending on the state $s_{i}(t)$ for any $i\in\mathcal{P}(t)$ as follows:
*Case 1:* If $s_{i}(t)=0$, then passenger $i$ is waiting to be picked up. From a vehicle $j$’s point of view, there are two components to the value of picking up this passenger at point $o_{i}$: $(i)$ The accumulated waiting time $t-\varphi_{i}$ of passenger $i$; the larger this waiting time, the higher the value of this passenger is. $(ii)$ The distance of $j$ from $o_{i}$; the shorter the distance, the higher the value of this passenger is. To ensure this value component is non-negative, we define $D$ to be the largest possible travel time between any two points in the RSS (often referred to as the diameter of the underlying graph) and consider $D-d(x_{j}(t),o_{i})$ as this value component.
In order to properly normalize each component and ensure its associated value is restricted to the interval $[0,1]$, we use the waiting time upper bound $W_{\max}$ introduced in (\[cost\_func1\]) and the distance upper bound $D$ to define the total travel value function as $$V_{i,j}(x_{j}(t),t)=(1-\mu)\cdot\dfrac{t-\varphi_{i}}{W_{\max}}+\mu\cdot
\frac{D-d(x_{j}(t),o_{i})}{D} \label{vaa}$$ where $\mu\in\lbrack0,1]$ is a weight coefficient depending on the relative importance the RSS places on passenger satisfaction (measured by waiting time) and vehicle distance traveled. In the latter case, a large value of $d(x_{j}(t),o_{i})$ implies that vehicle $j$ wastes time either traveling empty (if $N_{j}(t)=0$) or adding to the traveling time of passengers already on board (if $N_{j}(t)>0$).
*Case 2:* If $s_{i}(t)=j\in\mathcal{A}(t)$, then passenger $i$ is already on board with destination $r_{i}$. From vehicle $j$’s point of view, there are again two components to the value of delivering this passenger to point $r_{i}$: $(i)$ The accumulated travel time $t-\rho_{i,j}$ of passenger $i$. $(ii)$ The distance of $j$ from $r_{i}$. Similar to (\[vaa\]), we define $$V_{i,j}(x_{j}(t),t)=(1-\mu)\cdot\dfrac{t-\rho_{i,j}}{Y_{\max}}+\mu\cdot
\frac{D-d(x_{j}(t),r_{i})}{D} \label{vbb}$$ where $Y_{\max}$ is the travel time upper bound introduced in (\[cost\_func1\]).
*Case 3:* If $s_{i}(t)=k\neq j$, $k\in\mathcal{A}(t)$, then passenger $i$ is already on board some other vehicle $k\neq j$. Therefore, from vehicle $j$’s point of view, the value of this passenger is $V_{i,j}(x_{j}(t),t)=0$.
We summarize the definition of the travel value function as follows:$$V_{i,j}(x_{j}(t),t)=\left\{
\begin{array}
[c]{cc}(1-\mu)\cdot\frac{t-\varphi_{i}}{W_{\max}}+\mu\cdot\frac{D-d(x_{j}(t),o_{i})}{D} & \text{if }s_{i}(t)=0\\
(1-\mu)\cdot\frac{t-\rho_{i,j}}{Y_{\max}}+\mu\cdot\frac{D-d(x_{j}(t),r_{i})}{D} & \text{if }s_{i}(t)=j\\
0 & \text{otherwise}\end{array}
\right. \label{TravelValueFunction}$$ In addition to this immediate value associated with passenger $i$, there is a future value for vehicle $j$ to consider depending on the sets $\mathcal{R}_{i,j}(t)$ and $\hat{\mathcal{R}}_{i,j}(t)$ defined earlier. In particular, if $s_{i}(t)=0$ and vehicle $j$ proceeds to the pickup location $o_{i}$, then the value associated with $\mathcal{R}_{i,j}(t)$ is defined as $$V_{i,j}^{\mathcal{R}}(x_{j}(t),t)=\max_{n\in\mathcal{R}_{i,j}(t)}V_{n,j}(o_{i},t)$$ which is the maximal travel value among all passengers in $\mathcal{R}_{i,j}(t)$ to be collected if vehicle $j$ selects $o_{i}$ as its destination at time $t$. On the other hand, if $s_{i}(t)=j$ and vehicle $j$ proceeds to the drop-off location $r_{i}$, then $V_{n,j}(o_{i},t)$ above is replaced by $V_{n,j}(r_{i},t)$. Since the value of $s_{i}(t)$ is known to $j$, we will use $c_{i}$ as defined in (\[ci\_def\]) and write$$V_{i,j}^{\mathcal{R}}(x_{j}(t),t)=\max_{n\in\mathcal{R}_{i,j}(t)}V_{n,j}(c_{i},t)$$ Similarly, the value of $\hat{\mathcal{R}}_{i,j}(t)$ is defined as $$V_{i,j}^{\hat{\mathcal{R}}}(x_{j}(t),t)=\max_{n\in\hat{\mathcal{R}}_{i,j}(t)}V_{n,j}(c_{i},t)$$ We then define the total travel value associated with a vehicle $j$ when it considers any passenger $i\in\mathcal{P}(t)$ as $$\bar{V}_{i,j}(x_{j}(t),t)=V_{i,j}(x_{j}(t),t)+\max\{V_{i,j}^{\mathcal{R}}(x_{j}(t),t),V_{i,j}^{\hat{\mathcal{R}}}(x_{j}(t),t)\} \label{V_definition}$$ Figure \[estimatefuturevalue\] shows an example of how $\bar{V}_{i,j}(x_{j}(t),t)$ is evaluated by vehicle $j$ in the case where $c_{i}=o_{i}$ (i.e., $s_{i}(t)=0$). In this case, $\mathcal{R}_{i,j}(t)=\{k,l,p\}$ and $\hat{\mathcal{R}}_{i,j}(t)=\{m,n\}$.
![Travel value of passenger $i$ evaluated by vehicle $j$ when $s_{i}(t)=0$.[]{data-label="estimatefuturevalue"}](estimatefuturevalue.jpg)
Active Target Sets
------------------
The concept of an active target set was introduced in [@khazaeni2016event]. Clearly, this cannot be used in a RSS since the topology is no longer Euclidean and the travel cost function $\eta_{i}(x,t)$ has been replaced by the travel value function (\[V\_definition\]).
We begin by defining the reachability (or feasible) set $F_{j}(t_{k},H_{k})$ for vehicle $j$ in the RSS topology specified by $\mathbb{G}\subset
\mathbb{R}^{2}$. This is now a finite set consisting of *horizon points* in $\mathbb{G}$ reachable through some path starting from $x_{j}(t_{k})$ and assuming a fixed speed $v_{j}(t_{k})$ as defined in (\[Horizon\]). This is illustrated in Fig. \[horizon\_example\] where $F_{j}(t_{k},H_{k})$ consists of 10 horizon points (one-way streets have been taken into account as directed arcs in the underlying graph). Observe that $H_{k}$ in this example is defined by $o_{2}$, the pickup location of passenger 2 (horizon point $5$) in accordance with (\[Horizon\]). Note that since the actual speed of the vehicle may be lower than $v_{j}(t_{k})$, it is possible that no horizon point is reached at time $t_{k}+h_{k}$ even if $h_{k}=H_{k}$. This simply implies that a new planning horizon $H_{k+1}$ is evaluated at $t_{k}+H_{k}$ (which might still be defined by $o_{2}$).
![Example of the reachability set of vehicle $j$.[]{data-label="horizon_example"}](gridmap.jpg){height="5.5cm" width="8cm"}
We can now define the active target set of vehicle $j$ to consist of any target (pickup or drop-off locations of passengers) which has the largest travel value to $j$ for at least one horizon point $x\in F_{j}(t_{k},H_{k})$.
**Definition:** The set of *Active Targets* of vehicle $j$ is defined as ** $$\begin{aligned} S_{j}(t_{k},H_{k})=\{l:l=\arg\max_{i\in\mathcal{P}(t)}\;\bar{V}_{i,j}(x,t_{k}+H_{k})\\ \text{ for some }x\in F_{j}(t_{k},H_{k})\} \end{aligned} \label{active_targets_RSS}$$ Observe that $S_{j}(t_{k},H_{k})\subseteq\mathcal{P}(t_{k})$ and may reduce the number of passengers to consider as potential destinations assigned to $j$ when $S_{j}(t_{k},H_{k})\subset\mathcal{P}(t_{k})$ since $$u_{j}(t_{k})\in S_{j}(t_{k},H_{k})$$ In the example of Fig. \[horizon\_example\], $\mathcal{P}(t_{k})$ contains 6 passengers where $s_{1}(t_{k})=s_{2}(t_{k})=s_{4}(t_{k})=0$ and $s_{3}(t_{k})=s_{5}(t_{k})=s_{6}(t_{k})=j$. Thus, we can immediately see that $P(t_{k}) =6<$ $\left\vert F_{j}(t_{k},H_{k})\right\vert =10$. Further, observe that the drop-off points $r_{5}$ and $r_{6}$ are such that $r_{5},r_{6}\notin S_{j}(t_{k},H_{k})$ since both points are farther away from $x_{j}(t_{k})$ than $r_{3}$ and $o_{2}$ respectively. Therefore, the optimal control selection to be considered at $t_{k}$ is reduced to $u_{j}(t_{k})\in
S_{j}(t_{k},H_{k})=\{o_{1},o_{2},r_{3},o_{4}\}$. In addition, if the capacity $C_{j}$ happens to be such that $C_{j}=3$, then the only feasible control would be $u_{j}(t_{k})=r_{3}$.
Future Reward Estimation
------------------------
In order to solve the optimization problem (\[RHC\_algorithm\]) at each RHC iteration time $t_{k}$, we need to estimate the time that a future target is visited when $t>t_{k}+H_{k}$ so as to evaluate the term $\hat{J}_{{k+1}}(\mathbf{X(}t_{k}+H_{k}))$. Let us start by specifying the immediate reward term $C(\mathbf{X}_{k},\mathbf{u}_{k},H_{k})$ in (\[RHC\_algorithm\]). In view of (\[Reward\_Function\]), there are three cases: $(i)$ As a result of $\mathbf{u}_{k}$, an event $\pi_{i,j}$ (where $s_{i}(t)=j$) occurs at time $t_{k+1}$ with an associated reward $C(\mathbf{X}_{k},\mathbf{u}_{k},H_{k})=\mu_{w}(T-w_{i})$ where $w_{i}=t_{k+1}-\varphi_{i}$, $(ii)$ As a result of $\mathbf{u}_{k}$, an event $\delta_{i,j}$ occurs at time $t_{k+1}$ with an associated reward $C(\mathbf{X}_{k},\mathbf{u}_{k},H_{k})=\mu
_{y}(T-y_{i})$ where $y_{i}=t_{k+1}-\rho_{{i,j}}$, and $(iii)$ Any other event results in no immediate reward. In summary, adopting the notation $C(\mathbf{u}_{k},t_{k+1})$ for the immediate reward resulting from control $\mathbf{u}_{k}$, we have $$C(\mathbf{u}_{k},t_{k+1})=\left\{
\begin{array}
[c]{cc}\mu_{w}(T-w_{i}) & \text{if event }\pi_{i,j}\text{ occurs at }t_{k+1}\\
\mu_{y}(T-y_{i}) & \text{if event }\delta_{i,j}\text{ occurs at }t_{k+1}\\
0 & \text{otherwise}\end{array}
\right. \label{Immediate_Reward}$$ In order to estimate future rewards at times $t>t_{k+1}$, recall that $\mathcal{T}_{k,j}\subseteq\mathcal{P}(t)-\{u_{j}(t_{k})\}$ is a set of targets that vehicle $j$ would visit in the future, after reaching $u_{j}(t_{k})$. This set was defined in [@khazaeni2016event] through (\[targetset\]) and a new definition suitable for the RSS will be given below. Then, for each target $n\in\mathcal{T}_{k,j}$ the associated reward is $C(\mathbf{u}_{k},\hat{\tau}_{n,j})$ where $\hat{\tau}_{n,j}$ is the estimated time that vehicle $j$ reaches target $n$. If $n=o_{i}$ for some passenger $i$, then, from (\[Immediate\_Reward\]), $C(\mathbf{u}_{k},\hat{\tau}_{n,j})=\mu_{w}(T-\hat{w}_{i})$ where $\hat{w}_{i}=\hat{\tau}_{n,j}-\varphi_{i}$, whereas if $n=r_{i}$ for some passenger $i$, then $C(\mathbf{u}_{k},\hat{\tau
}_{n,j})=\mu_{y}(T-\hat{y}_{i})$ where $\hat{y}_{i}=\hat{\tau}_{n,j}-\rho_{{i,j}}$. Further, we include a *discount factor* $\lambda_{n}(\hat{\tau}_{n,j})$ to account for the fact that the accuracy of our estimate $\hat{\tau}_{n,j}$ is monotonically decreasing with time, hence $\lambda
_{n}(\hat{\tau}_{n,j})\in(0,1]$. Therefore, for each vehicle $j$ the associated term for $\hat{J}_{{k+1}}(\mathbf{X(}t_{k}+H_{k}))$ is $$\hat{J}_{j}(\mathbf{X}(t_{k}+H_{k}))=\sum_{n=1}^{\left\vert \mathcal{T}_{k,j}\right\vert }\lambda_{n}(\hat{\tau}_{n,j})C(u_{k,j},\hat{\tau}_{n,j})
\label{future_reward}$$ and$$\hat{J}(\mathbf{X}(t_{k}+H_{k}))=\sum_{j\in\mathcal{A}(t_{k})}\hat{J}_{j}(\mathbf{X}(t_{k}+H_{k})) \label{Total_future_reward}$$
We now need to derive estimates $\hat{\tau}_{n,j}$ for each $n\in
\mathcal{T}_{k,j}$. These estimates clearly depend on the order imposed on the elements of $\mathcal{T}_{k,j}$, i.e., the expected order that vehicle $j$ follows in reaching the targets (after it reaches $u_{j}(t_{k})$) contained in this set. As already explained under *(2)* at the end of the last section, this order depends on the passenger states and the residual capacity of the vehicle. Suppose that the order is specified through $\theta_{n}^{j}$ defined as the $n$th target label in $\mathcal{T}_{k,j}$ (e.g., $\theta
_{1}^{j} = 4$ indicates that target $4$ is the first to be visited). Then, (\[future\_reward\]) is rewritten as$$\hat{J}_{j}(\mathbf{X}(t_{k}+H_{k}))=\sum_{n=1}^{\left\vert \mathcal{T}_{k,j}\right\vert }\lambda_{\theta_{n}^{j}}(\hat{\tau}_{\theta_{n}^{j},j})C(u_{k,j},\hat{\tau}_{\theta_{n}^{j},j}) \label{future_reward_ordered}$$
It now remains to $(i)$ define the set $\mathcal{T}_{k,j}$, suitably modified from (\[targetset\]) to apply to a RSS, so as to address the inaccuracy limitation *(2)* described at the end of the last section, and $(ii)$ Specify the ordering $\{\theta_{1}^{j},\ldots,\theta_{\left\vert
\mathcal{T}_{k,j}\right\vert }^{j}\}$ imposed on the elements of $\mathcal{T}_{k,j}$.
We proceed by defining target subsets of $\mathcal{T}_{k,j}$ ordered in terms of the priority of vehicle $j$ to visit these targets compared to other vehicles. This is done using the relative responsibility function in (\[RelativeResponsibility\]) with the Manhattan distance used in evaluating $\bar{d}_{l,i}(t)$. Thus, let $\mathcal{T}_{k,j}=\mathcal{T}_{k,j}^{1}\cup\dots\cup\mathcal{T}_{k,j}^{M}$ where $\mathcal{T}_{k,j}^{m}$ has the $m$th highest priority among all subsets and $M\leq P(t)$ is the number of subsets. When $m=1$, we have $$\mathcal{T}_{k,j}^{1}=\{l:p(\bar{d}_{l,j}(t_{k}))>p(\bar{d}_{l,q}(t_{k})),\text{ }\forall q\in\mathcal{A}(t),\text{ \ }\forall l\in
\mathcal{P}(t)\}$$ which is the same as (\[targetset\]): this is the passenger responsibility set of vehicle $j$ in the sense that this vehicle has a higher responsibility value in (\[RelativeResponsibility\]) for each passenger in $\mathcal{T}_{k,j}^{1}$ than that of any other vehicle. Note that if $s_{l}(t_{k})=j$, then by default we have $l\in T_{k,j}^{1}$ since the drop-off location $r_{i}$ is the exclusive responsibility of vehicle $j$. For passengers with $s_{l}(t_{k})=0$, they are included in $T_{k,j}^{1}$ as long as there is no other vehicle $q\neq
j$ with a higher relative responsibility for $l$ than that of $j$.
Next, let $\mathcal{A}_{l,m}(t)$ be a subset of vehicles defined as$$\mathcal{A}_{l,m}(t_{k})=\{j:l\notin\mathcal{T}_{k,j}^{n},\text{ }n<m,\text{
\ }j\in\mathcal{A}(t_{k})\}$$ This subset contains all vehicles which do not have target $l$ included in any of their top $m-1$ priority subsets. We then define $\mathcal{T}_{k,j}^{m}$ when $m>1$ as follows:$$\begin{aligned} \mathcal{T}_{k,j}^{m}=\{l:p(\bar{d}_{l,j}(t_{k}))>p(\bar{d}_{l,q}(t_{k})),\text{ }\forall q\in\mathcal{A}_{l,m}(t_{k}),\\ \text{ }\forall l\notin\mathcal{T}_{k,j}^{n},n<m\text{\ }\} \end{aligned} \label{Tsets}$$ This set contains all targets for which $j$ has a higher relative responsibility than any other vehicle and which have not been included in any higher priority set $\mathcal{T}_{k,j}^{n},$ $n<m$. As an example, suppose passenger $i$ is waiting to be picked up and belongs to $T_{k,j_{1}}^{1}$, $T_{k,j_{2}}^{2}$ and $T_{k,j_{3}}^{3}$, where $j_{1}$ is the closest vehicle to $i$. Suppose vehicle $j_{1}$ is full and needs to drop off a passenger first whose destination is far away. Because vehicle $j_{2}$ has the $2$nd highest priority, then $j_{2}$ may serve $i$ provided it has available seating capacity. If $j_{2}$ cannot serve $i$, then vehicle $j_{3}$ with a lower priority is the next to consider serving $i$. In this manner, we overcome the limitation of (\[targetset\]) where no agent capacity is taken into account.
The last step is to specify the ordering $\{\theta_{1}^{j},\ldots
,\theta_{\left\vert T_{k,j}^{m}\right\vert }^{j}\}$ imposed on each set $T_{k,j}^{m},$ $j\in\mathcal{A}(t)$, $m=1,\ldots,M$. This is accomplished by using the travel value function $\bar{V}_{i,j}(x_{j}(t),t)$ in (\[V\_definition\]) as follows: $$\begin{aligned}
\bar{V}_{\theta_{n+1}^{j},j}(c_{\theta_{n}^{j}},\hat{\tau}_{\theta_{n}^{j},j}) & \leq\bar{V}_{i,j}(c_{\theta_{n}^{j}},\hat{\tau}_{\theta_{n}^{j},j})\label{v_last}\\
\text{for all }i & \in T_{k,j}^{m}-\{\theta_{1}^{j},\dots,\theta_{n}^{j}\}\nonumber\end{aligned}$$ where we have used the definition of $c_{i}$ in (\[ci\_def\]). Setting $u=u_{j}(t_{k})$, the estimated times are given by$$\begin{aligned}
\hat{\tau}_{\theta_{1}^{j},j} & =t_{k}+\frac{1}{v}d(x_{j}(t_{k}),x_{u})+\frac{1}{v}d(u,c_{\theta_{1}^{j},j})\label{tau1}\\
\hat{\tau}_{\theta_{n}^{j},j} & =\hat{\tau}_{\theta_{n-1}^{j},j}+\frac{1}{v}d(c_{\theta_{n-1}^{j},j},c_{\theta_{n}^{j},j}),\text{ \ }n>1 \label{taun}$$ where $\hat{\tau}_{\theta_{1}^{j},j}$ is the estimated time of reaching the target with the highest travel value beyond the one selected as $u_{j}(t_{k})$ among all targets in $T_{k,j}^{m}$ and $\hat{\tau}_{\theta_{n}^{j},j}$ for $n>1$ is the estimated time of reaching the $n$th target in the order established through (\[v\_last\]). Note that this approach takes into account the state of vehicle $j$; in particular, if $N_{j}(t)=C_{j}$, then the ordering of targets in $T_{k,j}^{m}$ is limited to those such that $s_{i}(t_{k})=j$.
This completes the evaluation of the estimated future reward in (\[Total\_future\_reward\]) based on (\[Immediate\_Reward\]) and (\[future\_reward\]), along with the ordering of future targets specified through (\[v\_last\]).
Preventing Vehicle Trajectory Instabilities
-------------------------------------------
Our final concern is the issue of instabilities discussed under *(3)* at the end of the last section. This problem arises when a new passenger joins the system and introduces a new target for one or more vehicles in its vicinity which may have higher travel value in the sense of (\[V\_definition\]) than current ones. As a result, a vehicle may switch its current destination $u_{j}(t_{k})$ and this process may repeat itself with additional future new passengers. In order to avoid frequent such switches, we introduce a threshold parameter denoted by $\Theta$ and react to any event $\alpha_{i}$ (a service request issued by a new passenger $i$) that occurs at time $t_{k}$ as follows:$$u_{j}(t_{k})=\left\{
\begin{array}
[c]{cc}o_{i} &
\begin{array}
[c]{c}\text{if }\bar{V}_{i,j}(x_{j}(t_{k}),o_{i})-\bar{V}_{u,j}(x_{j}(t_{k}),x_{u})>\Theta\text{,}\\
N_{j}(t)<C_{j}\text{, }j=1,\ldots,A(t_{k})
\end{array}
\\
u & \text{otherwise}\end{array}
\right. \label{Threshold_Control}$$ where $u=u_{j}(t_{k-1})$ is the current destination of $j$. In simple terms, the current control remains unaffected unless the new passenger provides an incremental value relative to this control which exceeds a given threshold. Since (\[Threshold\_Control\]) is applied to all vehicles in the current vehicle set $\mathcal{A}(t)$, the vehicle with the largest incremental travel value ends up with $o_{i}$ as its control as long as it exceeds $\Theta$. Note that the new passenger may not be assigned to $j$ unless this vehicle has a positive residual capacity.
RHC optimization scheme
-----------------------
The RHC scheme consists of a sequence of optimization problems solved at each event time $t_{k}$, $k=1,2,\ldots$with each problem of the form$$\begin{aligned} \mathbf{u}_{k}^{\ast}=&\arg\max_{\substack{u_{k,j}\in S_{j}(t_{k},H_{k} )\\j\in\mathcal{A}(t_{k})}}[C(\mathbf{u}_{k},t_{k+1})\\ &+\sum_{j\in \mathcal{A}(t_{k})}\sum_{n=1}^{\left\vert T_{k,j}^{m}\right\vert }\lambda_{\theta_{n}^{j}}(\hat{\tau}_{\theta_{n}^{j},j})C(u_{k,j},\hat{\tau }_{\theta_{n}^{j},j})]\text{, \ }m=1,\ldots,M\label{RHCalgo}\end{aligned}$$ where $S_{j}(t_{k},H_{k})$ is the active target of vehicle $j$ at time $t_{k}$ obtained through (\[active\_targets\_RSS\]), $C(\mathbf{u}_{k},t_{k+1})$ is given by (\[Immediate\_Reward\]), and $\hat{\tau}_{\theta_{n}^{j},j}$ is evaluated through (\[tau1\])-(\[taun\]) with the ordering $\{\theta
_{1}^{j},\ldots,\theta_{\left\vert T_{k,j}^{m}\right\vert }^{j}\}$ given by (\[v\_last\]) and the sets $\mathcal{T}_{k,j}^{m}$, $m=1,\ldots,M$, defined through (\[Tsets\]). Note that (\[RHCalgo\]) must be augmented to include (\[Threshold\_Control\]) when the event occurring at $t_{k}$ is of type $\alpha_{i}$.
An algorithmic description of the RHC scheme is given in **Algorithm $1$**
[ 1) Determine $H_k$ through (\[Horizon\])]{} [ 2) Determine the active target set $S_j(t_k, H_k)$ through (\[active\_targets\_RSS\]) for all $j\in \mathcal{A}(t)$]{} [ 3) Evaluate the estimated future reward through (\[tau1\]) and (\[taun\]) for all candidate optimal controls]{}
[4) Determine the optimal control $\mathbf{u^*_k}$ in (\[RHCalgo\])]{}
[5) Execute $\mathbf{u^*_k}$ until an event occurs ]{}
**Complexity of Algorithm $1$**: The complexity of the original RHC in [@li2006cooperative] was discussed in Section III. For the new RHC we have developed, the optimal control for vehicle $j$ at any iteration is selected from the finite set $S_{j}(t_{k},H_{k})$ defined by active targets. Thus, the complexity is $O(\Omega^{A(t)})$ where $\Omega\leq P(t)$ (the number of targets) is the maximum number of active targets. Observe that $\Omega$ decreases as targets are visited if new ones are not generated.
Simulation Results
==================
We use the SUMO (Simulation of Urban Mobility) [@sumo] transportation system simulator to evaluate our RHC for a RSS applied to two traffic networks (in Ann Arbor, MI and in New York City, NY). Among other convenient features, SUMO may be employed to simulate large-scale traffic networks and to use traffic data and maps from other sources, such as OpenStreetMap and VISUM. Vehicle speeds are set by the simulation and they include random factors like different road speed limits, turns, traffic lights, etc.
![A RSS in the Ann Arbor map.[]{data-label="SUMO"}](sumo.jpg){height="6.5cm" width="8.5cm"}
RHC for a RSS in the Ann Arbor map
----------------------------------
A RSS for part of the Ann Arbor map is shown in Fig.\[SUMO\]. Green colored vehicles are idle while red colored ones contain passengers to be served. A triangle along a road indicates a waiting passenger. We pre-load in SUMO a fixed number of vehicles, while passengers request service at random points in time as the simulation runs. Passenger arrivals are modeled as a Poisson process with a rate of $3$ passengers/min. The remaining RSS system parameters are selected as follows: $C_{j}=4$, $T=300$ min, $W_{max}=47$ min, $Y_{max}=47$ min, $D=3000$ m and the threshold in (\[Threshold\_Control\]) is set at $\Theta=0.3$.
![Waiting and traveling time histograms under different weights $\omega$ for the Ann Arbor RSS.[]{data-label="AA_730_weight"}](AA_730Wweighs.jpg){height="12cm" width="8cm"}
In Table \[tab:AA\_730\_weight\], the average waiting and traveling times under RHC are shown for different weights $\omega$ in the Ann Arbor RSS. The results are averaged over three independent simulation runs. In this example, the number of pre-loaded vehicles is $7$ and simulations end after $30$ passengers are delivered to associated destinations (which is within T=300 min set above). In order to evaluate the performance of the RSS at steady state, we allow a simulation to warm up before starting to measure the $30$ passengers served over the course of a simulation run.
The first column of Table \[tab:AA\_730\_weight\] shows different values of the weights $\omega$ as defined in specifying the relative importance assigned to passenger waiting and traveling respectively. As expected, emphasizing waiting results in larger vehicle occupancy and longer average travel times. In Fig. \[AA\_730\_weight\] we provide the waiting and traveling time histograms for all cases in Table \[tab:AA\_730\_weight\].
In Table \[tab:AA\_comparison\], we compare our RHC method with a greedy heuristic (GH) algorithm (similar to [@agatz2011dynamic]) which operates as follows. When passenger $i$ joins the RSS and generates the pickup point $o_{i}$, we evaluate the incremental cost this point incurs to vehicle $j\in\mathcal{A}(t)$ when placed in every possible position in this vehicle’s current destination sequence, as long as the capacity constraint $N_{j}(t)<C_{j}$ is never violated. The optimal position is the one that minimizes this incremental cost. Once this is done for all vehicles $j\in\mathcal{A}(t)$, we select the minimal incremental cost incurred among all vehicles. Then, passenger $i$ is assigned to the associated vehicle. As seen in Table \[tab:AA\_comparison\] with $\omega=0.5$, the RHC algorithm achieves a substantially better weighted sum performance (approximately by a factor of $2$) which are averaged over three independent simulation runs. In Fig. \[AA\_730\_Comparison\] we compare the associated waiting and traveling time histograms showing in greater detail the substantially better performance of RHC relative to GH. Table \[tab:AA\_comparison\_vehicle\_num\] compares different vehicle numbers when the delivered passenger number is $30$ showing waiting and traveling times, vehicle occupancy and the objective in . The larger the vehicle number, the better the performance can be achieved.
![Comparison of waiting and traveling time histograms under RHC and GH for the Ann Arbor RSS ($\omega=0.5$).[]{data-label="AA_730_Comparison"}](AA_730WComparisons.jpg){height="10cm" width="8cm"}
![A RSS covering an area of $10\times10$ blocks in New York City.[]{data-label="RSS_NY"}](NY_map.jpg){height="6.5cm" width="8cm"}
RHC for a RSS in the New York City map
--------------------------------------
A RSS covering an area of $10\times10$ blocks in New York City is shown in Fig.\[RSS\_NY\]. In this case, we generate passenger arrivals based on actual data from the NYC Taxi and Limousine Commission which provides exact timing of arrivals and the associated origins and destinations. We pre-loaded $8$ vehicles and run the simulations until $50$ passengers are served based on actual data from a weekday of January, 2016 (the approximate passenger rate is $16$ passengers/min). All other RSS settings are the same as before.
![Waiting and traveling time histograms under different weights $\omega$ for the New York City RSS with $8$ vehicles.[]{data-label="NYC_850_weight"}](NYC_850Wweights.jpg){height="11cm" width="8cm"}
![Comparison of waiting and traveling time histograms under RHC and GH in the New York City RSS with $8$ vehicles.[]{data-label="NYC_850_Comparison"}](NYC_850WComparisons.jpg){height="11cm" width="8cm"}
![Waiting and traveling time histograms under different weights $\omega$ for the New York City RSS with $28$ vehicles.[]{data-label="NYC_28160_weight"}](NYC_28160Wweights.jpg){height="11cm" width="8cm"}
![Comparisons of waiting and traveling time histograms between the RHC and GH methods in the New York City RSS when the vehicle number is $28$.[]{data-label="NYC_28160_Comparison"}](NYC_28160WComparisons.jpg){height="11cm" width="8cm"}
In Table \[tab:NYC\_850\_weight\], the average waiting and traveling times under RHC are shown for different weights $\omega$ in the New York City RSS. The results are averaged over three independent simulation runs. The first column of Table \[tab:NYC\_850\_weight\] shows different values of the weights $\omega$ as defined in specifying the relative importance assigned to passenger waiting and traveling resepctively. As in the case of the Ann Arbor RSS, emphasizing waiting results in larger vehicle occupancy with longer average travel times. In Fig. \[NYC\_850\_weight\] we provide the waiting and traveling time histograms for all cases in Table \[tab:NYC\_850\_weight\].
In Table \[tab:NYC\_comparison\], we compare RHC with $\omega=0.5$ with the aforementioned greedy heuristic algorithm GH in terms of the average waiting and traveling times. We can see once again that the RHC algorithm achieves a substantially better performance. In Fig.\[NYC\_850\_Comparison\] we compare the associated waiting and traveling time histograms for RHC relative to GH.
We have also tested a relatively long RSS operation based on actual passenger data from a weekday of January 2016 which is the same as before for the shorter time intervals. We pre-loaded $28$ vehicles and run simulations until $160$ passengers are served. All other settings are the same as before.
Table \[tab:NYC\_28160\_weight2\] shows the associated waiting and traveling times under different weights with similar results as before. Figure \[NYC\_28160\_weight\] shows the associated waiting and traveling time histograms for all cases in Table \[tab:NYC\_28160\_weight2\].
In Table \[tab:NYC\_comparison2\], we compare RHC to the GH algorithm in terms of the average waiting and traveling times with results consistent with those of Table \[tab:NYC\_comparison\].
Table \[tab:NYC\_comparison\_vehicle\_num\] compares different vehicle numbers when the delivered passenger number is $160$ showing waiting and traveling times, vehicle occupancy and the objective in whose performance is consistent with that of Table \[tab:AA\_comparison\_vehicle\_num\].
Table \[tab:RHC\_assignment\_time\] shows real execution times for our RHC regarding different vehicle and passenger numbers.
Finally, we tested a relatively longer RSS operation with $38$ vehicles based on the same actual passenger data as before which generates $1000$ passengers over approximately $1.2$ ’real’ operation hours. Simulations will not end until $900$ passengers are delivered. In Table \[tab:NYC\_comparison3\], we compare RHC to the GH algorithm in terms of the average waiting and traveling times with results consistent with those of Table \[tab:NYC\_comparison\].
Conclusions and Future Work
===========================
An event-driven RHC scheme is developed for a RSS where vehicles are shared to pick up ad drop off passengers so as to minimize a weighted sum of passenger waiting and traveling times. The RSS is modeled as a discrete event system whose event-driven nature significantly reduces the complexity of the vehicle assignment problem, thus enabling its implementation in a real-time context. Simulation results adopting actual city maps and real taxi traffic data show the effectiveness of the RHC controller in terms of real-time implementation and performance relative to known greedy heuristics. In our ongoing work, an important problem we are considering is where to optimally position idle vehicles so that they are best used upon receiving future calls. Moreover, depending on real execution times of our RHC algorithm (see Table \[tab:RHC\_assignment\_time\]), we will use this information as a rational measure for decomposing a map into regions such that within each region the RHC vehicle assignment response times remain manageable.
[^1]: $^{\star}$Supported in part by NSF under grants ECCS-1509084, CNS-1645681, and IIP-1430145, by AFOSR under grant FA9550-15-1-0471, by the DOE under grant DE-AR0000796, by the MathWorks and by Bosch.
[^2]: The authors are with the Division of Systems Engineering and Center for Information and Systems Engineering, Boston University, Brookline, MA 02446, USA `{ruic,cgc}@bu.edu`
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---
abstract: 'Multi-cavity photonic systems, known as photonic molecules (PMs), are ideal multi-well potential building blocks for advanced quantum and nonlinear optics[@Abbarchi:2013fk; @Dousse:2010ys; @Gerace:2009vn; @Liew:2010uq]. A key phenomenon arising in double well potentials is the spontaneous breaking of the inversion symmetry, i.e. a transition from a delocalized to two localized states in the wells, which are mirror images of each other. Although few theoretical studies have addressed mirror-symmetry breaking in micro and nanophotonic systems [@rodrigues2013symmetry; @Maes:05; @Maes:06; @Bulgakov:12; @bulgakov2013light], no experimental evidence has been reported to date. Thanks to the potential barrier engineering implemented here, we demonstrate spontaneous mirror-symmetry breaking through a pitchfork bifurcation in a PM composed of two coupled photonic crystal nanolasers. Coexistence of localized states is shown by switching them with short pulses. This offers exciting prospects for the realization of ultra-compact, integrated, scalable optical flip-flops based on spontaneous symmetry breaking. Furthermore, we predict such transitions with few intracavity photons for future devices with strong quantum correlations.'
author:
- Philippe Hamel
- Samir Haddadi
- Fabrice Raineri
- Paul Monnier
- Gregoire Beaudoin
- Isabelle Sagnes
- Ariel Levenson
- 'Alejandro M. Yacomotti'
title: 'Spontaneous mirror-symmetry breaking in a photonic molecule'
---
Spontaneous symmetry breaking (SSB) unifies diverse physical mechanisms through which a given symmetric system ends up in an asymmetric state[@golubitsky1988singularities]. It explains many central questions from particle and atomic physics to nonlinear optics (the Goldstone boson and the Higgs mechanism[@strocchi2005symmetry; @Endres:2012nx], phase transitions in Bose-Einstein condensates –BECs–[@sadler2006spontaneous; @Zibold:2010kl], metamaterials[@Liu:2014kx], bifurcations in lasers[@Green:1990kx; @Ackemann:2013ys], photorrefractive media[@Kevrekidis:2005bh], to mention just a few). A paradigmatic symmetry in this context is given by reflection in a double-well potential (DWP), as it is the case of pyramidal molecules (e.g. ammonia)[@jona2002interaction]: SSB dictates whether the state of a system will be delocalized or, in turn, confined within either well. In photonics, such a mechanism is possible provided the third order nonlinearities overcome photon tunneling[@Malomed:2013fk]. In this work we experimentally show SSB in a photonic molecule (PM) given by two evanescently coupled photonic crystal (PhC) nanolasers. Switchable localized modes with broken mirror-symmetry will be demonstrated herein. This can be prospected as a nanoscale version of a laser flip-flop[@Liu:2010fk]; the memory is pumped incoherently, set and reset can be induced with positive pulses and there is no coherent driving beam to bias the device, as in conventional bistable cavities. This paves the way for the realization of ultra-small flip-flop optical memories based on SSB.
![[**Photonic molecule**]{}. [**a**]{}, 3D sketch of the photonic molecule composed of two coupled L3 PhC nanocavites. PhC lattice period is $a=425$ nm, hole radius $r=0.266a$. Embedded QWs are represented by a dark blue line. Blue holes ($r_{blue}=r-0.06a$), shifted in the $\Gamma$K direction by $\Delta r_{blue}=0.15a$, increase the cavity Q-factor. Red holes ($r_{red}=r+0.05a$) improve beaming of the radiated photons. Green holes ($r_{green}=r-20\%$) control the coupling strength. Orange holes ($r_{orange}=r_{red}-20\%$) combine both effects. [**b**]{}, SEM image of the fabricated sample; dashed circles highlight the modified holes. [**c**]{}, Splitted energy levels of the PhC molecule (note that the ground state is the anti-bonding mode). Insets: far-field emission profiles of bonding (B, top) and anti bonding (AB, bottom) modes.[]{data-label="coupled_cavities"}](Fig1a_1.eps "fig:") ![[**Photonic molecule**]{}. [**a**]{}, 3D sketch of the photonic molecule composed of two coupled L3 PhC nanocavites. PhC lattice period is $a=425$ nm, hole radius $r=0.266a$. Embedded QWs are represented by a dark blue line. Blue holes ($r_{blue}=r-0.06a$), shifted in the $\Gamma$K direction by $\Delta r_{blue}=0.15a$, increase the cavity Q-factor. Red holes ($r_{red}=r+0.05a$) improve beaming of the radiated photons. Green holes ($r_{green}=r-20\%$) control the coupling strength. Orange holes ($r_{orange}=r_{red}-20\%$) combine both effects. [**b**]{}, SEM image of the fabricated sample; dashed circles highlight the modified holes. [**c**]{}, Splitted energy levels of the PhC molecule (note that the ground state is the anti-bonding mode). Insets: far-field emission profiles of bonding (B, top) and anti bonding (AB, bottom) modes.[]{data-label="coupled_cavities"}](Fig1b.eps "fig:") ![[**Photonic molecule**]{}. [**a**]{}, 3D sketch of the photonic molecule composed of two coupled L3 PhC nanocavites. PhC lattice period is $a=425$ nm, hole radius $r=0.266a$. Embedded QWs are represented by a dark blue line. Blue holes ($r_{blue}=r-0.06a$), shifted in the $\Gamma$K direction by $\Delta r_{blue}=0.15a$, increase the cavity Q-factor. Red holes ($r_{red}=r+0.05a$) improve beaming of the radiated photons. Green holes ($r_{green}=r-20\%$) control the coupling strength. Orange holes ($r_{orange}=r_{red}-20\%$) combine both effects. [**b**]{}, SEM image of the fabricated sample; dashed circles highlight the modified holes. [**c**]{}, Splitted energy levels of the PhC molecule (note that the ground state is the anti-bonding mode). Insets: far-field emission profiles of bonding (B, top) and anti bonding (AB, bottom) modes.[]{data-label="coupled_cavities"}](Fig1c.eps "fig:")
We represent the PM as a DWP, symmetric with respect to the inversion plane. We describe the dynamics in terms of the complex amplitudes of the photonic field at the left ($\psi_L$) and right ($\psi_R$) sites, $|\psi|^2$ being photon number. A finite potential barrier leads to a tunneling rate $K$. We further consider a local (nonlinear) interaction $U |\psi_{L,R}|^2$, and a lifetime $\tau$ due to losses. SSB instabilities occur as long as $K$ is lower than a critical value $K_c$ ($|K|<|K_c|$), with $|K_c\tau|\sim |U| \cdot |\psi|^2$[@PhysRevE.64.025202]. In the case of our PM laser, $|\psi|^2$ will linearly increase with the pump power $P_p$ after the threshold power $P_{th}^-$ corresponding to the anti-bonding mode, which is the hybrid mode of the PM minimizing the optical losses. Hence $|\psi|^2 \rightarrow |\psi_-|^2=\Delta P_-=P_p-P_{th}^-$. In a semiconductor medium, $U$ can be related to the phase-amplitude coupling factor $\alpha$ (the Henri factor); $K_c$ then yields $K_c \tau \sim \alpha \Delta P_-$, which can also be recast as $$\Delta P_c\sim K\tau/\alpha.
\label{critical}$$ In absence of cavity detuning, $K$ is related to mode splitting ($\Delta \lambda _{split}$) as $K\tau=\Delta \lambda _{split}/\delta \lambda$, where $\delta \lambda$ is the cavity linewidth. On the other hand, $\alpha$ lies between $\sim 5$ and $10$. Hence a good DWP-candidate to demonstrate SSB with low pump powers has to fulfill the condition that mode splitting be of the order of the resonance width, i.e. $K\tau \sim1$. Finely controlling coupling strength in PMs is thus a key ingredient to achieve SBB transitions. We implement such a control –together with both efficient laser emission and free-space photon collection– by means of an original PhC cavity design, as explained hereafter.
![[**Laser emission of coupled nanocavities**]{}. [**a**]{}, Spectral measurements as a function of pump power when pumping at the center of the system; both modes (B, short wavelength, and AB, higher wavelength) are observed, but only AB undergoes laser emission. [**b**]{}, AB laser mode output vs input power. Green filled squares: experimental measurements obtained from spectral peaks in ([**a**]{}) for increasing input power. Black line: Numerical solution of coupled lasers rate equations. Black open square: pitchfork bifurcation. Colored open squares: first broken parity state. Colored crosses: second broken parity state. Open circles: bifurcations leading to oscillations. Insets: illustrations of the double-well potential with the unique stable solution before bifurcation (left inset) and the two co-existing solutions after bifurcation (right inset). []{data-label="pitchfork"}](Fig2a_1.eps "fig:") ![[**Laser emission of coupled nanocavities**]{}. [**a**]{}, Spectral measurements as a function of pump power when pumping at the center of the system; both modes (B, short wavelength, and AB, higher wavelength) are observed, but only AB undergoes laser emission. [**b**]{}, AB laser mode output vs input power. Green filled squares: experimental measurements obtained from spectral peaks in ([**a**]{}) for increasing input power. Black line: Numerical solution of coupled lasers rate equations. Black open square: pitchfork bifurcation. Colored open squares: first broken parity state. Colored crosses: second broken parity state. Open circles: bifurcations leading to oscillations. Insets: illustrations of the double-well potential with the unique stable solution before bifurcation (left inset) and the two co-existing solutions after bifurcation (right inset). []{data-label="pitchfork"}](Fig2b_2.eps "fig:")
Our DWP landscape is obtained with the specific PhC molecule depicted in Fig. \[coupled\_cavities\]a. This is formed by two evanescently coupled PhC L3 cavities (three holes missing in the $\Gamma$K direction of a triangular lattice) in a semiconductor free standing membrane (Fig. \[coupled\_cavities\]a,b). With the aim of realizing high-Q cavities with improved beaming and controlled coupling strength, three conception tools are used: i) end-holes of each L3-cavity are shifted and shrunk in order to increase theoretical Q-factors up to $\sim \, 10^5$ (blue holes in Fig. \[coupled\_cavities\]a,b)[@Akahane:2003zr; @Portalupi:2010]; ii) the radius of neighbor holes are modified in order to confine radiated photons within a $\sim \, 30^\circ$ emission cone [@Haddadi2012; @Haddadi2013] (red holes in Fig. \[coupled\_cavities\]a,b); iii) the hole size of the central row (green holes in Fig. \[coupled\_cavities\]a,b) is modified in order to control the coupling strength towards $K\tau\sim1$[@Haddadi:14]. Both single and coupled cavities have been etched on InP membranes containing InGaAs/InGaAsP quantum wells (see Methods). Resonant wavelengths are about $\lambda\sim 1540 \,\mathrm{nm}$, and measured Q-factors of bare cavities (i.e. at QW transparency) are $Q=4970$ ($\tau \sim 8 \, \mathrm{ps}$) for single, and $Q=4300$ ($\tau\sim 7 \, \mathrm{ps}$) for coupled cavities.
Mode splitting has been measured through room-temperature micro-photoluminescence spectroscopy. A cw pump beam is focussed down to a $\sim 1.5 \, \mu \mathrm{m}$-diameter spot at the center of the coupled cavity system (see Methods). Two modes of the PM can be observed: the anti-bonding, “AB” (ground state), and bonding, “B” (excited state)[@ground]. Far-field patterns showing intensity maxima (B) and minima (AB) at $k=0$ are shown in Fig. \[coupled\_cavities\]c. From a mode splitting of $\Delta \lambda_{split}\approx1.4\, \mathrm{nm}$ (Fig. \[coupled\_cavities\]c), the normalized coupling constant is $K \tau=3.3$ . A solitary nano-cavity laser exhibits a S-shaped, output vs. input power curve, where its sharpness is related to the spontaneous emission $\beta$ factor. Now, what is the expected behavior for two evanescently coupled nanolasers when pumped at the center of the PM? Out of the two hybrid modes, the lasing mode is the one with lower optical losses, i.e. the AB mode; the B mode is strongly attenuated (Fig. \[pitchfork\]a). In Fig. \[pitchfork\]b we depict the AB maxima (green symbols) superimposed to a numerical solution of coupled lasers rate equations, with $\beta=0.017$ given by a fit of the experimental points (black line). A S-shaped curve for the AB mode is observed up to $P_p=1.33 \, P_{th}$. Within this range the solution is delocalized in the DWP (Fig. \[pitchfork\]b, left inset). Above this value two branches of steady state solutions come up (plus a third one being the destabilized AB mode), corresponding to two co-existing solutions: the “Left cavity on” together with the “Right cavity off” (“L1R0” from now on), and the “Left cavity off” together with the “Right cavity on” solution (“L0R1”), see Fig. \[pitchfork\]b, right inset. Unlike the AB mode, these new solutions have no defined parity: a SSB instability takes place at $P_p=1.33 P_{th}$ in the form of a pitchfork bifurcation (Fig. \[pitchfork\], black square). The two new branches (upper and lower) remain stable up to $P_{p}=1.37 \, P_{th}$ where the system undergoes secondary instabilities (Hopf bifurcations, Fig. \[pitchfork\], circles) leading to ultrafast oscillations (predicted frequencies $\sim \, 150-180 \, \mathrm{GHz}$ depending on the pump power), larger but close to the beating note $\nu_{beat}=K/\pi\sim \, 148 \, \mathrm{GHz}$. These can be related to ac Josephson oscillations[@Abbarchi:2013fk]. For a lasing AB mode in presence of self-focussing nonlinearities (positive nonlinear refractive index above QW transparency), such SSB scenario is only possible for a specific sign of the optical coupling parameter $K$ –positive with our sign convention– which imposes a lower energy for the AB mode. The L3 cavity-based PM implemented here fulfills this requirement.
![[**Time domain measurements and pitchfork bifurcation**]{}. [**a**]{}, Pulse sequence of a $600\, \mu$s-duration output signal from both nanocavities (blue: L, red: R). The time series shows alternation of “High red-Low blue” and “Low red-High blue” states (dead time windows between pulses are omitted in a sequence). The zoom shows this alternation over two periods. (b) Superimposed time traces after identification of two different states, L1R0 and L0R1; averages are shown in thick line. (c) Same data as (b), plotted as a function of the instantaneous pump power, showing the SSB bifurcation (full squares: L1R0, open diamonds: L0R1). The black line shows a numerical (Runge-Kutta) integration of the coupled laser rate equations (x-axis is rescaled $\times2$ for a qualitative comparison with the experiment). Grey areas correspond to fast (filtered by the photodetector) oscillations in the time domain. The inset presents a theoretical bifurcation diagram with imperfect symmetry when the pump position is slightly shifted from the center of the molecule by $\delta p=2\times10^{-4}$ (pump beam slightly shifted to the R cavity).[]{data-label="time_series"}](Fig3a_1.eps "fig:") ![[**Time domain measurements and pitchfork bifurcation**]{}. [**a**]{}, Pulse sequence of a $600\, \mu$s-duration output signal from both nanocavities (blue: L, red: R). The time series shows alternation of “High red-Low blue” and “Low red-High blue” states (dead time windows between pulses are omitted in a sequence). The zoom shows this alternation over two periods. (b) Superimposed time traces after identification of two different states, L1R0 and L0R1; averages are shown in thick line. (c) Same data as (b), plotted as a function of the instantaneous pump power, showing the SSB bifurcation (full squares: L1R0, open diamonds: L0R1). The black line shows a numerical (Runge-Kutta) integration of the coupled laser rate equations (x-axis is rescaled $\times2$ for a qualitative comparison with the experiment). Grey areas correspond to fast (filtered by the photodetector) oscillations in the time domain. The inset presents a theoretical bifurcation diagram with imperfect symmetry when the pump position is slightly shifted from the center of the molecule by $\delta p=2\times10^{-4}$ (pump beam slightly shifted to the R cavity).[]{data-label="time_series"}](Fig3b_1.eps "fig:") ![[**Time domain measurements and pitchfork bifurcation**]{}. [**a**]{}, Pulse sequence of a $600\, \mu$s-duration output signal from both nanocavities (blue: L, red: R). The time series shows alternation of “High red-Low blue” and “Low red-High blue” states (dead time windows between pulses are omitted in a sequence). The zoom shows this alternation over two periods. (b) Superimposed time traces after identification of two different states, L1R0 and L0R1; averages are shown in thick line. (c) Same data as (b), plotted as a function of the instantaneous pump power, showing the SSB bifurcation (full squares: L1R0, open diamonds: L0R1). The black line shows a numerical (Runge-Kutta) integration of the coupled laser rate equations (x-axis is rescaled $\times2$ for a qualitative comparison with the experiment). Grey areas correspond to fast (filtered by the photodetector) oscillations in the time domain. The inset presents a theoretical bifurcation diagram with imperfect symmetry when the pump position is slightly shifted from the center of the molecule by $\delta p=2\times10^{-4}$ (pump beam slightly shifted to the R cavity).[]{data-label="time_series"}](Fig3c_1.eps "fig:")
The spectral measurements presented in Fig. \[pitchfork\] show a saturation of the integrated output laser power for $P_p>1.4 \, P_{th}$. This is consistent with the predicted bifurcation, since mean power for broken symmetry states is lower compared to the AB mode. In order to investigate SSB experimentally, fast time detection with spatial filtering of individual cavity outputs has been set up. Two identical APD photodiodes coupled to single-mode optical fibers are used to collect L and R cavity signals simultaneously. The diffraction limited collection area is smaller than the inter-cavity distance ($d=1.47 \, \mu \mathrm{m}$) such that less than 10% cross-talk is observed (see Methods). The modulated pump beam (50 KHz, 30 ns-rise time), impinging the sample with a peak power of $\sim 6 \, \mathrm{mW}$, is aligned at the center of the PM.
Fig. \[time\_series\]a shows a sequence of simultaneous outputs from both cavities. Segments of alternating “High blue-Low red”, and “Low blue-High red” peaks can be observed. We average out these time traces by superimposing events using a peak detection algorithm (see Methods). The result is shown in Fig. \[time\_series\]b. Two types of events are clearly identified: L1R0 and L0R1. These are plotted in Fig. \[time\_series\]b as function of time, and in Fig. \[time\_series\]c as a function of the instantaneous pump power, together with a numerical integration of the coupled lasers rate equations. The AB mode builds up from noise, with a laser threshold of ${\cal P}_{th}\approx 2.7 \, \mathrm{mW}$, and evolves in the usual way up to $({\cal P}_p-{\cal P}_{th})/{\cal P}_{th}\sim 0.7$ (${\cal P}_p\approx 4.5 \, \mathrm{mW}$) where the two distinct branches of output states come up. It is important to point out that the lower power branch, instead of monotonically decreasing as ${\cal P}_p$ is increased, raises again for $({\cal P}_p-{\cal P}_{th})/{\cal P}_{th}>1$. This is in good agreement with the the model, being a consequence of the fast oscillations. Experimentally, $\sim100$ GHz oscillations are filtered out by the APD bandwidth and only the DC intensity component is measured, which is higher than the steady state intensity of the lower branch (Fig. \[time\_series\]c, inset).
In absence of external noise terms, a small shift of the pump beam from the center of the PM ($\delta p=2\times10^{-4}$ in the model) may be responsible for triggering one state or another within each pump pulse[@detuning] . Experimentally, the fact that fast alternation between stable states is observed may be due to i) mechanical vibrations; and ii) spontaneous emission fluctuation noise making the system to spontaneously choose L1R0 or L0R1 states. In either case, a way to experimentally prove that our scenario indeed corresponds to SSB, i.e. that L1R0 and L0R1 states [*do coexist*]{}, is to be able to switch from one to the other within the same pump pulse, as it will be done in the following.
![[**Coexistence of parity broken states**]{}. [**a**]{}, Switching from one state to the other is triggered by a short ($100$ ps) pulse on each cavity. Top: non-perturbed spontaneous switching. Center: the pulse is applied to the L cavity (sketched as a blue peak). Bottom: the perturbation is applied to the R cavity (sketched as a red peak). Continuous and dashed lines: states “1” and “2” respectively, defined as the states before the arrival of the pulse (“1”: L1R0 in the top and center frames, L0R1 in the bottom frame; “2”: L0R1 in the top and center frames, L1R0 in the bottom frame). [**b**]{}, Image of the intensity profile from an InGaAs camera after stabilizing states using an early perturbation. Insets: Time traces corresponding to early induced L0R1 (top) and L1R0 (bottom) states.[]{data-label="switch_exp"}](Fig4a.eps "fig:") ![[**Coexistence of parity broken states**]{}. [**a**]{}, Switching from one state to the other is triggered by a short ($100$ ps) pulse on each cavity. Top: non-perturbed spontaneous switching. Center: the pulse is applied to the L cavity (sketched as a blue peak). Bottom: the perturbation is applied to the R cavity (sketched as a red peak). Continuous and dashed lines: states “1” and “2” respectively, defined as the states before the arrival of the pulse (“1”: L1R0 in the top and center frames, L0R1 in the bottom frame; “2”: L0R1 in the top and center frames, L1R0 in the bottom frame). [**b**]{}, Image of the intensity profile from an InGaAs camera after stabilizing states using an early perturbation. Insets: Time traces corresponding to early induced L0R1 (top) and L1R0 (bottom) states.[]{data-label="switch_exp"}](Fig4b.eps "fig:")
Coexistence of L1R0 and L0R1 is experimentally investigated by means of an additional short ($\sim100 \, \mathrm{ps}$) pulse laser synchronously superimposed to the modulated cw pump beam. Pulses can be spatially aligned to either cavity, while the position of the cw pump beam is kept fixed at the center of the PM. Fig. \[switch\_exp\]a shows the initial (non-perturbed, top) situation. As the short pulse is aligned at the left cavity, two events are observed: either L cavity was off before the arrival of the short pulse, i.e. the L0R1 state anticipated the pulse perturbation, in such a case L0R1 survives (Fig. \[switch\_exp\]a, center, dashed lines); or L1R0 state anticipated the pulse perturbation, in such a case a switch to L0R1 is observed (Fig. \[switch\_exp\]a, center, continuous lines). When the perturbation laser is shifted to the right cavity the situation is reversed: L0R1 switches to L1R0, while L1R0 remains unchanged. Note that a cavity in the on-state can be switched off with an extra pulse on it, which simultaneously switches on the adjacent cavity. This pretty much resembles a light rocker switch in a house: “rocking” the lever by pushing it on the raised half makes the mechanism switching. Pulsed external control is well reproduced by our theoretical model.
The short pulse laser can also be used to stabilize spontaneous switching such that photon trapping in one of the two cavities can be measured with a slower 2D detector. Fig. \[switch\_exp\]b shows intensity images captured on an InGaAs camera: the short pulse laser is used to stabilize either the L0R1 state with a pulse on the L cavity, or the L1R0 state with a pulse on the R cavity (see Fig. \[switch\_exp\]b, insets). These results further illustrate photon confinement states around the cavity regions. Switching asymmetric states is a clear advantage in the context of applications, for instance all-optical flip-flops in photonic integrated circuits[@Liu:2010fk]. This demonstration of coexistence of broken parity states through optical switching constitute an experimental proof of spontaneous mirror-symmetry breaking in a PM nanolaser. A major interest of our system is that the photonic barrier amplitude and sign can be controlled by design[@Haddadi:14]. It can be shown that i) changing the sign of interactions is equivalent to reversing the coupling constant, an alternative to the control of the local nonlinearity through an external magnetic field by the Feshbach-resonance effect in BECs; ii) the symmetry of the bifurcated ground state can be exchanged without modifying the nature of interactions. In our case, the latter would be observable as far as the bonding state becomes the lasing mode.
In our experiments, the inferred photon number in each cavity is ${\cal S}\sim 100$ (the normalization photon number is ${\cal S}_{norm}\approx 135$) at the onset of SSB, compatible with the signal level measured by the ADP detectors (see Methods). Shrinking the middle row hole radius $r_{green}$ by just a few percent would result in a reduced mode splitting, from $K\tau= 3.3$ to, e.g., $K\tau \sim 0.7$. The predicted bifurcation point would then decrease to $\Delta P_c=K\tau/\alpha \sim 0.1$, which is 3% far from laser threshold. This means that the pitchfork bifurcation point would be further shifted towards the laser threshold, eventually occurring on the steep portion of the S-curve. In such conditions, photon number in each cavity, ${\cal S} \approx {\cal S}_{norm} \Delta P_c$ becomes ${\cal S} \sim 10$ at the SSB instability. Quantum interference in PMs in presence of –even modest– nolinearities are expected to leave its fingerprints on the quantum correlations of the laser photons[@Liew:2010uq]. This PhC molecule, combined with quantum dot technology[@faraon2008coherent], might then constitute a building block for a news class of light emitting nano-sources with strong photonic correlations.
**Methods**
[**Sample fabrication**]{}
The active membrane is grown by metalorganic chemical vapor deposition on an InP substrate with an intermediate InGaAs etch-stop layer and a SiO$_2$ sacrificial layer on top. It includes four InGaAs/InGaAsP quantum wells (photoluminescence centered at $\sim1510$ nm, FWHM =63 nm). This structure is bonded upside-down to a Si substrate coated with benzocyclobuten (BCB). The InP and etch-top layer are removed chemically leaving a structure composed of, from bottom to top: a $\sim 280\,\mu$m-thick Si substrate, a $\sim400$ nm-thick BCB layer, a $\sim1\, \mu$m-thick SiO2 layer and the 265 nm-thick membrane. Sample fabrication is achieved by deposition of a $\sim 200$ nm SiN layer (hard mask), e-beam lithography (2 nm-resolution) to write the PhC on a poly-methyl-methacrylate resist, and inductively-coupled-plasma reactive ion etching to etch the mask and the membrane. SiO$_2$ layer is removed chemically by AF acid penetrating the holes.
[**Setup description**]{}
The nanolasers are pumped at $\lambda=808$ nm with a cw single-mode fibered laser diode (Lumics L808M100), modulated using a 120 MHz (AGILENT 81150A, minimum rise time $\sim2$ ns) waveform generator (pulses of few tens of ns, repetition rates from 10 to 200 kHz). A $\times100$ magnification, 0.95 numerical aperture (N.A) and IR antireflection coated microscope objective (OLYMPUS MPLAN 100xIR) is used to focus the pump on the sample down to a $1.5\,\mu$m spot diameter. Its relative position with respect to the cavities is adjusted (5 nm resolution) thanks to a nano-positioning sample holder (Melles Griot APT 600 6-axis stage) with piezoactuators and feedback loops. The emitted signal, collected through the same objective, is separated into three paths: i) temporal analysis (see details below), ii) spectral analysis with a spectrometer (Princeton Instruments, Acton SP2500i) coupled to a Ni cooled InGaAs 1D array detector (Princeton Instruments, OMA V, $\sim0.1$ nm resolution) and iii) IR imaging with an InGaAs camera (Sensors Unlimited SU 320) measuring both intensity and far-field emission profiles. The latter is obtained using a Fourier imaging technique (an additional lens on a kinematic base images the back focal plane of the objective).
[**Detection of laser emission**]{}
The emission of each laser is measured separately using diffraction-limited spatial filtering. The beam in the temporal detection path is separated via a 50/50 beam splitter. Each beam is coupled into a single mode, $8\,\mu$m-diameter core, fiber through a $\times20$ (N.A. = 0.3, NACHET, IR coated) microscope objective. The fiber acts as a pinhole to select a specific region of the PhC. The ends of the fibers are connected to two identical 660 MHz-bandwidth, low noise avalanche photodiodes (APD, Princeton Lightwave PLA-841-FIB, equivalent noise power $\sim200\,\mathrm{fW}/\sqrt{Hz}$) to perform time domain measurements. The detection area is selected by superimposing on the camera a 1550 nm laser spot, injected from the detection fiber, to a given nanolaser emission area. The cross-talk between the two detection channels (Ch1 and Ch2) has been quantified as follows: i) detection of the coupled nanolasers is optimized in Ch1 and Ch2; ii) sample is translated in the plane such that a single nanolaser of identical parameters is brought into one detection area and maximized in Ch1; iii) a residual signal in Ch2 reveals a cross-talk of less than 10% . The APD responsivity ($\sim60$ kV/W) and the signal level at bifurcation (7 mV) yield an optical power of 115 nW. Taking into account the transmission of optical elements and the coupling efficiency into the fiber, the output power emitted by one cavity (to a half space) is $P_{out}\approx1\,\mu$W. The equivalent total photon number inside the cavity is $S = 2\times P_{out}\tau/h\nu\approx110$ at the pitchfork bifurcation.
[**Peak detection algorithm**]{}
Time traces in each channel are simultaneously recorded using a 13 GHz-bandwidth oscilloscope (Lecroy Wavemaster 813Zi) in the form of a sequence of 100 consecutive, 50 ns-duration time windows (one output pulse per window); 100 pulses are then recorded for each cavity in one shot. Within SSB conditions, two types of events are observed: a high pulse in cavity L (Ch 1) together with a low pulse in cavity R (Ch 2), called L1R0, and vice versa (L0R1). Such states appear either in long clusters (up to hundreds of pulses), or in segments of rapidly alternating L1R0-L0R1 events (few tens of pulses). We attribute long clusters to a small long-lived drift (typically due to mechanical vibrations), and segments with alternating events to spontaneous switching. The 600 $\mu$s-duration segment picked up in Fig. \[time\_series\]a is a typical example of spontaneous switching, containing 31 pulses (50% L1R0 and 50% L0R1). A peak detection algorithm with threshold (75% of the peak amplitude) is implemented in Ch2 to discriminate two cases: peak in Ch2 is larger than that in Ch1 (case L0R1), or smaller (case L1R0). Averages are performed over each type of events (Fig. \[time\_series\]b). When applying a short ($\sim\,100$ ps) perturbation pulse for demonstration of coexistence, peak detection is restricted to a time window starting with the pump pulse and ending at the occurrence of the perturbation pulse. Averages are subsequently performed (Fig. \[switch\_exp\]).
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**ACKNOWLEDGEMENTS**
We thank A. Amo, J. Bloch, S. Barbay, J. Dudley and A. Aspect for enlightening comments. This work was supported by the CNRS, ANR (ANR-12-BS04-0011) and the RENATECH network.
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author:
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Eliot Bolduc$^{1,*}$, Genevieve Gariepy$^{1}$ and Jonathan Leach$^{1}$\
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title: 'Direct measurement of large-scale quantum states'
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**In quantum mechanics, predictions are made by way of calculating expectation values of observables, which take the form of Hermitian operators. It is far less common to exploit non-Hermitian operators to perform measurements. Here, we show that the expectation values of a particular set of non-Hermitian matrices, which we call column operators, directly yield the complex coefficients of a quantum state vector. We provide a definition of the state vector in terms of measurable quantities by decomposing the column operators into observables. The technique we propose renders very-large-scale quantum states significantly more accessible in the laboratory, as we demonstrate by experimentally characterising a 100 000-dimensional entangled state. This represents an improvement of two orders of magnitude with respect to previous characterisations of discrete entangled states. Furthermore, in numerical studies, we consider mixed quantum states and show that for purities greater that 0.81, we can reliably extract the most significant eigenvector of the density matrix with a probability greater than 99%. We anticipate that our method will prove to be a useful asset in the quest for understanding and manipulating large-scale quantum systems.**
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One of the current challenges in the field of computing is harnessing the potential processing power provided by quantum devices that exploit entanglement. Experimental research aimed at overcoming this challenge is driven by the production, control and detection of larger and larger entangled quantum states [@Monz:2011; @Wong:2012; @Yokoyama:2013; @Krenn:2014]. However, the task of characterising these entangled states quickly becomes intractable as the number of parameters that define a many-body system scales exponentially with the system size. To keep up with the ever-growing quantum state dimensionality, much effort is put into developing efficient characterisation methods [@Smith:2005; @Banaszek:2013; @Flammia:2005; @Cramer:2010; @Bogdanov:2010; @Toth:2010; @Gross:2010; @Mahler:2013; @Schwemmer:2014; @Shabani:2011; @Teo:2013; @Tonolini:2014; @Lloyd:2014; @Ferrie:2014; @Lundeen:2011].
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Quantum state tomography is the process of retrieving the values that define a quantum system. The process typically involves two steps: i) gathering an informationally complete set of data and ii) finding the quantum state most consistent with the data set using post-measurement processing such as the maximum likelihood estimation algorithm [@Banaszek:1999]. Many efficient tomographic methods capitalize on the first step by making simplifying assumptions about the state[@Flammia:2005; @Cramer:2010; @Toth:2010; @Schwemmer:2014; @Gross:2010; @Shabani:2011; @Tonolini:2014; @Lloyd:2014; @Lundeen:2011], thus reducing the number of measurements required to uniquely identify it. In particular, tomography via compressed sensing allows one to efficiently reconstruct quantum states based on the fact that low-rank density matrices, i.e. quasi-pure states, are sparse in a particular basis [@Gross:2010; @Liu:2012; @Schwemmer:2014; @Tonolini:2014]. Compared to assumption-free tomography, compressive sensing provides a square-root improvement on the required number of measurements [@Banaszek:2013]. This improvement enabled the reconstruction of the density matrices of a 6-qubit state [@Schwemmer:2014] and a (17$\times$17)-dimensional state [@Tonolini:2014], the largest phase-and-amplitude measurement of an entangled state reported to date. Although compressed sensing does not make use of maximum likelihood estimation, it does require non-trivial post-measurement processing.
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Recently, Lundeen [*et al.*]{} reported on the direct measurement of a wavefunction using a method that, for the first time, required no involved post-measurement processing [@Lundeen:2011]. Their method is based on weak measurements, whereby one weakly couples a quantum system to a pointer state and subsequently performs a few standard strong measurements on the pointer state. The outcome of a weak measurement is known as the “weak value", and in the conditions exposed in Ref. [@Lundeen:2011] the weak value is proportional to a given state vector coefficient. The method of Lundeed [*et al.*]{} can be used in combination with the assumption that the quantum state at hand is pure, providing the same square-root improvement as compressed sensing. Variations on the original scheme allow measurements of mixed states and increased detection efficiency [@Bamber:2014; @Salvail:2013; @Wu:2013gb].
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An important contribution of the work by Lundeen [*et al.*]{} was to link the state vector elements to the expectation value of weak measurements. We take a different approach, and point out that the enabling feature that allows access to the complex state vector is not weak measurement but the use of particular non-Hermitian operators. Although weak measurements provide a way to decompose these non-Hermitian operators, it is not the only suitable approach. Moreover, the introduction of weak values in the measurement procedure adds complexity to the experiment and the formalism that links weak values to measurement outcomes involves an approximation that breaks down in a variety of circumstances [@Duck:1989; @Salvail:2013; @Malik:2013].
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In this paper, we propose an alternative approach to the direct measurement of quantum states that is exact in the case of pure states, proves to be reliable in the presence of noise, and is consistent with results obtained with well-established tomographic techniques. The key principle of our formalism is to decompose the particular non-Hermitian matrices that yield the complex state vector coefficients using only observables. Our method therefore only requires strong measurements, as in standard tomography, while maintaining the directness of weak-value-assisted tomography. The simplicity in both the experimental procedure and post-measurement processing renders our method ideally suited for the characterisation of large-scale systems, which can be high-dimensional, many-body or both. We begin by developing the theory on which our method is based and then demonstrate the potential of this scheme by experimentally retrieving the complex coefficients of a (341$\times$341)-dimensional entangled state.
####
Consider a quantum system in a $d$-dimensional Hilbert space, whose state vector $$\label{eq:psi}
{|\Psi\rangle}=\sum_{j=0}^{d-1} c_j {|j\rangle}$$ is expanded in the basis $\{ {|j\rangle}\}$ and where $c_j$ are unknown complex expansion coefficients. In order to retrieve these coefficients, we introduce the column operators $\widehat{C}_j={|a\rangle}{\langle j|}$, where ${|a\rangle}$ is an arbitrary reference vector. Each column operator has an expectation value $$\label{eq:column}
\langle{\widehat{C}_j}\rangle={\langle\Psi|a\rangle}{c_j},$$ which is proportional to a complex state vector expansion coefficient. Since the value of ${{\langle\Psi|a\rangle}}$ is independent of $j$, we can express the state vector in terms of the column operators up to a phase factor: $$\label{eq:master}
{|\Psi\rangle}=\frac{\text{e}^{i\phi}}{\nu}\sum_{j=0}^{d-1} \langle\widehat{C}_j\rangle{|j\rangle},$$ where $\nu = |{{\langle\Psi|a\rangle}}|$ is a normalization constant. We can ignore the phase factor $\text{e}^{i\phi}$ since it bears no physical significance.
####
Most column operators $\widehat{C}_j$ are not Hermitian matrices and are thus not observables. To overcome this apparent constraint, we recognize that any non-Hermitian matrix can be constructed from a complex-weighted sum of Hermitian matrices. Hence, the crucial step to our method is to construct the column operators in terms of measurable quantities: $\widehat{C}_j= \sum_q {w}_{jq} {\widehat{\mathcal O}_{jq}}$, where ${w}_{jq}$ are complex weights and ${\widehat{\mathcal O}_{jq}}$ are observables. As a result, this allows us to retrieve any state vector element with a complex-weighted sum of measurement outcomes: $$\label{eq:coef}
c_j=\frac{1}{\nu}\sum_{q} {w}_{jq} \langle\widehat{\mathcal{O}}_{jq}\rangle.$$ Equation \[eq:coef\] is an exact definition of the pure state vector that is provided in terms of measurable quantities. The above formalism readily applies to a general class of quantum states, including high-dimensional and many-body systems.
####
As an example, consider the case of a qubit ${|\Psi\rangle}=c_0{|0\rangle}+c_1{|1\rangle}$ with ${|a\rangle}={|0\rangle}$ as the reference vector. The first column operator $\widehat{C}_0$ is Hermitian and given by the projector ${|0\rangle}{\langle 0|}$. The second column operator $\widehat{C}_1={|0\rangle}{\langle 1|}$ is not Hermitian but can be constructed a number of ways. The first construction – which, as pointed out earlier, is a key part of the weak value formalism – is the complex-weighted sum of Pauli matrices: $\widehat{C}_1=(\hat\sigma_x+i\hat\sigma_y)/2$, a decomposition that requires two observables, each of which is made of two projectors or eigenvectors. A second decomposition requiring only three projectors is given by $$\label{eq:tetrarec}
\widehat{C}_1= \sum_{q=0}^2 \frac{2}{3}\text{e}^{i 2\pi q /3} {|s_{q}\rangle}{\langle s_{q}|},$$ where ${|s_{q}\rangle}=({|0\rangle}+\text{e}^{i 4 \pi q/3 } {|1\rangle})/\sqrt{2}$ are the states onto which the observables $\widehat{\mathcal{O}}_{1q}$ project. In both cases, the qubit state vector is exactly given by ${|\Psi\rangle}=(\langle\widehat{C}_0\rangle {|0\rangle} + \langle\widehat{C}_1\rangle{|1\rangle})/\langle\widehat{C}_0\rangle^\frac{1}{2}$.
####
To demonstrate the power and scalability of our scheme, we apply it to the measurement of a state entangled in greater than 100 000 dimensions. We provide a complete characterisation of the spatially entangled two-photon field produced through spontaneous parametric downconversion (SPDC). In general, SPDC can give rise to spatial and frequency correlations between two photons [@Miatto:2011; @Dada:2011; @Agnew:2011; @Leach:2012; @Salakhutdinov:2012; @Tasca:2012; @Geelen:2013; @Krenn:2014; @Osorio:2008; @Mosley:2008; @Osorio:2013]. The purity of the spatial part of the full state can only be guaranteed if the two types of correlations are completely decoupled, which can be achieved in the collinear regime [@Osorio:2008] – see Supplementary Information section A for a theoretical estimation of our system purity. The consequences of applying our scheme to a quantum state with non-unit purity, which is always the case in the presence of noise, will be discussed below.
####
We express the spatial part of the entangled state in a discrete cylindrical basis of transverse spatial modes. The azimuthal part of the modes is given by $\text{e}^{i\ell\phi}$, where $\ell$ is an integer between $-\infty$ and $\infty$ and $\phi$ is the azimuthal angle. This type of phase profile is known to carry $\ell$ units of orbital angular momentum (OAM). We decompose the radial part of the field with the recently introduced Walsh modes, labelled by the integer $k$ ranging from 0 to $\infty$ [@Geelen:2013]. The Walsh modes all have the same Gaussian amplitude envelope, but different $\pi$-steps radial phase profiles. Combining the OAM modes with the Walsh modes yields a complete basis for coherent two-dimensional images. To perform the characterisation of the two-photon spatial field, we consider 31 OAM modes and 11 Walsh modes for each photon. The state vector thus takes the form $$\label{eq:c}
{|\Phi\rangle} = \sum_{\ell_1=-15}^{15}\sum_{k_1=0}^{10} \sum_{\ell_2=15}^{-15}\sum_{k_2=0}^{10} c_{\ell_1,k_1}^{\ell_2,k_2} {|\ell_1,k_1\rangle}{|\ell_2,k_2\rangle}.$$ Using the column-operator decomposition described in the Methods section, we sequentially measure all 116 281 coefficients $c_{\ell_1,k_1}^{\ell_2,k_2}$, which are shown in figure \[fig:Walsh\]a and \[fig:Walsh\]b. The total Hilbert space dimensionality of this measured state is more than two orders of magnitude larger than any previously reported amplitude-and-phase-characterised discrete entangled state [@Tonolini:2014]. As a simple verification of the accuracy of our method, we calculate the probabilities associated with each joint mode via the Born rule, $|c_{\ell_1,k_1}^{\ell_2,k_2}|^2$, as shown in figure \[fig:Walsh\]c. This result is consistent with the directly measured correlation matrix shown in figure \[fig:Walsh\]e, showing that we retrieve the correct magnitude of the amplitudes.
####
To rigorously assess the validity of the directly measured complex quantum state ${|\psi\rangle}$, i.e. both the amplitudes and the phases, we compare it to the results obtained through full tomography (i.e. assumption-free tomography). As full tomography cannot be performed on a (341$\times$341)-dimensional entangled state in a reasonable time, we characterise a ($5\times 5$)-dimensional subset of the SPDC two-photon state. We perform the comparison in a basis of various OAM modes ($\ell_1 \in \{1,-1,2,-2,3 \}$, $\ell_2 \in \{1,-1,2,-2,-3 \}$ ) and a fixed radial Walsh mode ($k_1=k_2=0$). The total number of unknown parameters in the corresponding density matrix is equal to $624$. After performing the direct measurement procedure in this basis, we record 8000 random projective measurements that we break into 8 sets of 1000. For each set, we recover a density matrix $\rho_{\textrm{exp}}$ and calculate its purity and the fidelity with the directly measured state ${|\psi\rangle}$; fidelity is defined as $\sqrt{{\langle \psi|}\rho_{\textrm{exp}}{|\psi\rangle}}$. On average, the purity calculation yields ($0.96\pm0.02$), and the fidelity gives ($0.985\pm0.004$), where the uncertainties correspond to one standard deviation. After reconstruction of a density matrix, we find that the average error between the measured count rates and the count rates predicted by the density matrix is 5.5%. This can be explained by shot noise, the pixelated nature of the SLM, and the finite aperture of the optical elements. While we expect unit purity, the 5% noise level accounts for the discrepancy with the measured value.
####
The extremely high fidelity between the tomography results $\rho_{\textrm{exp}}$ and the directly measured state ${|\psi\rangle}$ indicates the validity of our approach for quantum state measurements applied to near pure states. To evaluate our method in the context of mixed states, we perform a series of numerical simulations where we vary the rank, purity, and dimension of an unknown state $\rho_{\textrm{sim}}$, where no sources of noise are added to the simulated measurement outcomes. We apply our direct measurement procedure to these states and calculate the fidelity $|{\langle\psi|\psi_{\textrm{sim}}\rangle}|$, where ${|\psi_{\textrm{sim}}\rangle}$ is the eigenvector of $\rho_{\textrm{sim}}$ with the largest eigenvalue. For initial states $\rho_{\textrm{sim}}$ with purity greater than $0.81$, we measure a fidelity greater than $0.99$ in at least 99% of the cases. The dependency of this result on the dimensionality of the state is negligible. This result indicates that our direct method is able to extract the primary eigenvector of a density matrix, even for a partially mixed state. Full details of this analysis and the density matrix reconstruction are presented in the Supplementary Information.
####
Knowledge of the amplitude and phase of the state vector elements allows us to perform otherwise inaccessible calculations. As an example, we perform a calculation of the Schmidt decomposition [@Ekert:1995]. This is equivalent to the singular value decomposition for the case of optical transfer matrices. The Schmidt decomposition yields a new joint basis in which the photons are perfectly correlated and where the joint modes have equal phases, as shown in figure \[fig:Walsh\]d. When the Schmidt decomposition is applied to the entire state, we calculate a number of Schmidt modes equal to 142; this represents the effective number of independent joint modes contained within the state (the maximum for a (341$\times$341)-dimensional state being 341). The Schmidt decomposed two-photon field is a good candidate for the violation of very-high-dimensional Bell inequalities [@Dada:2011]. Further details on the Schmidt decomposition can be found in the Supplementary Information.
####
There are a number of approaches to reducing the necessary cost and effort for measuring large-scale quantum states. These include, but are not limited to, developing technologies for mode sorting [@Berkhout:2010] and arbitrary unitary transformations [@Morizur:2010; @Miller:2013], reducing the required number of measurement settings, and circumventing the requirement for reconstruction procedures. It is clear that there is significant interplay between each of these approaches. The theoretical implementation of an approach that combines the principles of our work with generalised measurements, such as POVMs (positive operator value measures), is considered in the Supplementary Information. The ability to use POVMs in the laboratory relies on the aforementioned technologies. Access to these types of technologies would reduce the overall number of measurement settings to uniquely recover a quantum state. However, such a system requires arbitrary unitary transformations for spatial states, which is in itself an active area of research [@Berkhout:2010; @Morizur:2010; @Miller:2013]. Given the limitations of mode sorters for very large dimensions, and the practical nature of projective measurements, our scheme provides a simple and elegant method for the characterisation of large-scale quantum states.
####
Our scheme allows direct access to the complex coefficients that define large-scale quantum states. The main result of our work is a novel method for retrieving a state vector coefficient with a complex-weighted sum of strong measurement outcomes. One challenge in reconstructing a quantum state from measurement outcomes lies in data processing; our scheme trades the difficulty of data processing for theoretical analysis prior to the experiment, that is, finding the measurements one has to perform. We anticipate that our work will have an impact on a number of disciplines, for example, quantum parameter estimation, measurement in quantum computing, quantum information and metrology.\
**Methods**\
**Experiment** The two-photon field is generated via SPDC with a 405-nm laser diode pumping a 1-mm-long periodically-poled KTP (PPKTP) crystal with 50 mW of power. The experimental setup is shown in figure \[fig:setup\]. We separate the two photons with a right angle prism and image the plane of the crystal to a Holoeye spatial light modulator (SLM) with a magnification of -10. We simultaneously display two holograms, one on each side of the SLM, to control the amplitude and phase profiles of the two photons independently. In order to make projective measurements of superposition modes, we make use of intensity masking [@Bolduc:2013]. We image the plane of the SLM with a magnification of $-1/2500$ to two single mode fibers. The combination of the SLM and singles mode fibers allows us to make arbitrary projective measurements. All measurements are performed in coincidence with two single photon avalanche detectors, with a timing window of 25 ns, an integration time of 1 s for modes outside the diagonal and 20 s for the diagonal elements ($\ell_1=-\ell_2$ and $k_1=k_2$). We start an automatic alignment procedure with the SLM every four hours to compensate for drift. Including the time it takes to calculate and display a hologram (about one second), the entire experiment takes two weeks; assumption-free tomography would take more than four centuries at the same acquisition rate. We perform no background subtraction and use the fundamental mode ($\ell_1=-\ell_2=k_1=k_2=0$) as the reference vector ${|a\rangle}$. The count rate of the fundamental mode is approximately 900 coincidences per second and varies by 10% over 24-hour periods. To correct for long term drift, we normalise each outcome to the count rate of the fundamental mode, which we measure before the measurement of each column operator. In standard tomography, the calculation of error bounds on the measured state is not a straightforward task [@Christandl:2012]. Here, we can calculate the error bound on a given coefficient with a weighted sum of the detector counts used to retrieve it. For a given state vector coefficient, the errors on the amplitude $|c_j|$ and phase arg$(c_j)$ are both inversely proportional to the overlap $\nu$ of the reference vector with the quantum state. In order to minimize the errors, it is important to choose a reference vector that has a high probability of occurrence within the state – the fundamental mode is the most probable one in our case.\
**Two-body column-operator decomposition** In order to decompose a given state vector coefficient $c_{\ell_1,k_1}^{\ell_2,k_2}$ into a set of measurement outcomes, we need to find a projector decomposition of the corresponding column operator $\widehat{C}_{\ell_1,k_1}^{\ell_2,k_2}={|0,0\rangle}{\langle \ell_1,k_1|}\otimes{|0,0\rangle}{\langle \ell_2,k_2|}$, as in equation \[eq:coef\]. We numerically find this column-operator decomposition, i.e. the complex weights ${w}_q$ and the observables $\widehat{\mathcal{O}}_q$, using the differential evolution algorithm (see Supplementary Information part D). By inspection, we find that the corresponding analytical form of the state vector coefficients is given by $$\label{eq:coef5}
\widehat{C}_{\ell_1,k_1}^{\ell_2,k_2}=\frac{1}{\nu} \sum_{q=0}^{4} \frac{4}{5} \text{e}^{i 2\pi q /5}{|s_{1,q}\rangle}{\langle s_{1,q}|}\otimes{|s_{2,q}\rangle}{\langle s_{2,q}|},$$ where $\sqrt{2}{|s_{m,q}\rangle}={|0,0\rangle}+\text{e}^{i 4 \pi q/5} {|\ell_m,k_m\rangle} $ with $m=\{1,2 \}$, and $\nu=|{\langle\Psi|0,0\rangle}|$ is a normalisation constant. This decomposition is only valid when the state of any photon is different from the reference vector, i.e. ${|\ell_m,k_m\rangle}\neq{|0,0\rangle}$. Each coefficient measured with the above column-operator decomposition requires five projective measurements, thus explaining the $5D^2$ scaling, where $D$ is the Hilbert space dimensionality of a single particle. The protocol scales much more favorably than assumption-free tomography, which requires $D^4$ projections.
####
Here, we briefly explain our protocol for measuring the entire SPDC state vector. We measure more than 99% of the coefficients using the decomposition of equation \[eq:coef5\]. The remaining column operators are the special cases ${|0,0\rangle}{\langle \ell_1,k_1|}\otimes{|0,0\rangle}{\langle 0,0|}$ and ${|0,0\rangle}{\langle 0,0|}\otimes{|0,0\rangle}{\langle \ell_2,k_2|}$, which respectively correspond to a row and a column of the result shown in figure \[fig:Walsh\]a. These column operators can be decomposed into only three joint local measurements using the projector ${|0,0\rangle}{\langle 0,0|}$ on one system and a column-operator decomposition similar to that of equation \[eq:tetrarec\] on the other system. Finally, the column operator ${|0,0\rangle}{\langle 0,0|}\otimes{|0,0\rangle}{\langle 0,0|}$ is a projector, and its expectation value can be measured in a single experimental configuration.\
**Full quantum tomography** We perform full tomography with high count rates in order to achieve high accuracy. We set the magnification between the plane of the SLM and that of the single mode fibers to 1/400. In this condition, we obtain a count rate of approximately 18,000 counts per second for the fundamental mode and integrate over 1 second for each individual projective measurement. The increase in the count rate of the fundamental mode comes at the price of lower count rates for high order modes. Regarding the full tomography measurements, we take an overcomplete set of 1000 random projective measurements in a $(5\times5)$-dimensional space. To minimize high-frequency components on the SLM, we limit the random superpositions to two-dimensional subsets of the state space.
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![[]{data-label="fig:Walsh"}](figure1.pdf)
![**Generation and characterisation of a two-photon field.** The entangled state is produced via SPDC in a PPKTP crystal and spatially separated by a prism. For the state determination stage, the crystal plane is imaged onto a spatial light modulator (SLM), which is in turn imaged to the input facet of two single mode fibers. In order to make a given projective measurement, we display the corresponding joint mode on the SLM and measure the coincidence rate between the two single photon avalanche diode detectors. The inset shows the five joint holograms that correspond to the column-operator decomposition of $\widehat{C}_{2,0}^{\hspace{1pt}\text{-}2,0}$. The state vector coefficient $c_{2,0}^{\text{-}2,0}$ is given by $\frac{4}{5\nu}\sum_q \langle{\mathcal{\widehat{O}}_q}\rangle \text{e}^{i 4\pi q/5}$, where the expectation value of a given observable is proportional to the measured count rate when displaying the corresponding hologram.[]{data-label="fig:setup"}](figure3.pdf)
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abstract: 'Let $({\varepsilon}_j)_{j\geq 0}$ be a sequence of independent $p-$dimensional random vectors and $\tau\geq1$ a given integer. From a sample ${\varepsilon}_1,\cdots,{\varepsilon}_{T+\tau-1},{\varepsilon}_{T+\tau}$ of the sequence, the so-called lag$-\tau$ auto-covariance matrix is $C_{\tau}=T^{-1}\sum_{j=1}^T{\varepsilon}_{\tau+j}{\varepsilon}_{j}^t$. When the dimension $p$ is large compared to the sample size $T$, this paper establishes the limit of the singular value distribution of $C_\tau$ assuming that $p$ and $T$ grow to infinity proportionally and the sequence satisfies a Lindeberg condition. Compared to existing asymptotic results on sample covariance matrices developed in random matrix theory, the case of an auto-covariance matrix is much more involved due to the fact that the summands are dependent and the matrix $C_\tau$ is not symmetric. Several new techniques are introduced for the derivation of the main theorem.'
address:
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Department of Statistics and Actuarial Science\
The University of Hong Kong\
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School of Physical & Mathematical Sciences\
Nanyang Technological University\
author:
-
-
-
title: 'On singular value distribution of large-dimensional autocovariance matrices'
---
,
Introduction
============
Let ${\varepsilon}_1,\ldots,{\varepsilon}_{T+\tau}$ be a sample from a stationary process with values in $\mathbb{R}^p$. The $p\times p$ matrix $$\label{Ceps}
C_\tau := \frac1T \sum_{j=1}^T {\varepsilon}_{\tau+j} {\varepsilon}_{j}^t,$$ is the so-called lag$-\tau$ [*sample auto-covariance matrix*]{} of the process (here $u^t$ denotes the transpose of a vector or matrix $u$). In a classical low-dimensional situation where the dimension $p$ is assumed much smaller than the sample size $T$, $C_\tau$ is very close to ${\mathop{\mathbb{E}}}C_\tau={\mathop{\mathbb{E}}}{\varepsilon}_{1+\tau} {\varepsilon}_{1}^t$ so that its asymptotic behavior when $T\to\infty$ (so $p$ is considered as fixed) is well known. In the high-dimensional context where typically the dimension $p$ is of same order as $T$, $C_\tau$ will not converge to ${\mathop{\mathbb{E}}}C_\tau$ and its asymptotic properties have not been well investigated. In this paper, we study the empirical spectral distribution (ESD) of $C_\tau$, namely, the distribution generated by its $p$ singular values. The main result of the paper is the establishment of the limit of this ESD when $({\varepsilon}_j)$ is an independent sequence with elements having a finite fourth moments while $p$ and $T$ grow to infinity proportionally.
In order to understand the importance of limiting spectral distribution (LSD) of singular values of the auto-covariance matrix $C_\tau$, we describe a statistical problem where these distributions are of central interest. In a recent stimulating paper, @LamYao12 considers the following dynamic factor model $$\label{model}
{x}_i=\Lambda {f}_i + {\varepsilon}_i +\mu,$$ where $\{{x}_i; ~ 0\le i\le T\}$ is an observed $p$-dimensional sequence, $\{f_i\}$ a sequence of $m$-dimensional “latent factor" ($m\ll p$) uncorrelated with the error process $\{{\varepsilon}_i\}$ and $\mu\in \mathbb{R}^p$ is the general mean. A particularly important question here is the determination of the number $m$ of factors. For any stationary process $\{ w_i\}$, let ${\Sigma}_{ w} ={\mathop{\text{cov}}}({ w}_i,{ w}_{i-1})$ be its (population) lag-$1$ auto-covariance matrix, we have $${\Sigma}_{x} = \Lambda {\Sigma}_{f} \Lambda^t.$$ It turns out that ${\Sigma}_{x}$ has exactly $m$ non-null singular values so that based on a sample ${x}_0,{x}_1,\ldots,{x}_T$ it seems natural to infer $m$ from the singular values of the sample lag-1 auto-covariance matrix $$\begin{aligned}
\Gamma_x &=& \frac1T \sum_{j=1}^T (\Lambda{f}_j+{\varepsilon}_j)(\Lambda{f}_{j-1}+{\varepsilon}_{j-1})^t\\
&=& \Lambda \left(\frac1T\sum_{j=1}^T {f}_j {f}^t_{j-1}\right)\Lambda^t
+ \Lambda \left(\frac1T\sum_{j=1}^T {f}_j {\varepsilon}^t_{j-1}\right)
+ \left(\frac1T\sum_{j=1}^T {\varepsilon}_j {f}^t_{j-1} \right)\Lambda^t
+ C_{1}~.\end{aligned}$$ Because $\Lambda$ has rank $m$, the first three terms all have rank bounded by $m$ and $\Gamma_{x}$ appears as a finite-rank perturbation of the lag-1 auto-covariance matrix $C_1$ which in general has rank $p\gg m$. Therefore, understanding the properties of the singular values of $C_1$ will be of primary importance for the understanding of the $m$ largest singular values of the matrix of $\Gamma_{x}$ which are, as said above, fundamental for the determination of the number of factors $m$. Notice however that this statistical problem is given here to describe a potential application of the theory established in this paper, but this theory on singular value distribution is general and can be applied to fields other than statistics.
If we take $\tau=0$ in , the matrix $S=\frac 1T\sum_{j=1}^T{\varepsilon}_j{\varepsilon}_j^t$ is the sample covariance matrix from the observations. The theory for eigenvalue distributions of $S$ has been extensively studied in the random matrix literature dating back to the seminal paper [@MP1967]. In this paper, the famous Mar$\check{c}$enko-Pastur law as limit of eigenvalue distributions has been found for a wide class of sample covariance matrices. Further development includes the almost sure convergence of these distributions ([@Silv95]) and conditions for convergence of the largest and the smallest eigenvalues, see [@BY93]. Meanwhile book-length analysis of sample covariance matrices can be found in [@BS10], [@AGZ10], [@PasShc10]. The situation of an auto-covariance matrix $C_\tau$ is completely different. To author’s best knowledge, none of the existing literature in random matrix theory treats the sample auto-covariance matrix and the limit for its eigenvalue distribution found in this paper is new.
There are basically two major differences between $C_\tau$ and $S$. First, while $S$ is a non-negative symmetric random matrix, $C_\tau$ is even not symmetric and we must rely on singular value distributions which are in general much more involved than eigenvalue distributions. Secondly, because of the positive lag $\tau$, the summands in $C_\tau$ are no more independent as it is the case for the sample covariance matrix $S$. This again makes the analysis of $C_\tau$ more difficult. As a consequence of these major differences, several new techniques are introduced in the paper in order to complete the proofs, although the general strategy is common in the random matrix theory (see @BS10 [@PasShc10]). For example, the characterization of the Stieltjes transform of the limiting distribution is obtained via a system of equations due to the time delay $\tau$ where for the case of sample covariance matrix, the characterization is given by a single equation([@MP1967], [@Silv95]).
The rest of the paper is organized as follows. The main theorem of the paper is introduced in Section \[results\]. Section \[proofs\] details the proof of the main theorem when time lag $\tau=1$. Section \[extension\] generalizes the proof from time lag $\tau=1$ to any given positive number. Meanwhile, in contrast to other aspects discussed above, the preliminary steps of truncation, centralization and standardization of the matrix entries are similar to the case of a sample covariance matrix. They are given in Appendix \[app\]. To ease the reading of the proof, technical lemmas are grouped in Section \[lemmas\].
Main Results {#results}
============
In this paper, we intend to derive the limiting singular value distribution of the lag$-\tau$ auto-covariance matrix defined in . It will be done in two steps. We derive the main result first for the lag-1$(\tau=1)$ sample auto-covariance matrix $C_1=\frac 1T\sum_{t=1}^T{\varepsilon}_j{\varepsilon}_{j-1}^t$. It turns out that the general case $\tau\geq 1$ is essentially the same and the extension is easily obtained. The details of the extension are given in Section \[extension\].
Therefore, we consider the lag-1 sample auto-covariance matrix $C_1= \frac1T \sum_{j=1}^T {\varepsilon}_j {\varepsilon}_{j-1}^t$. By definition, it is equivalent to study the limiting spectral distribution(LSD) of the matrix $$A=C_{1}C_{1}^t=\frac{1}{T^2}(\sum_{j=1}^T\varepsilon_j\varepsilon^t_{j-1})(\sum_{j=1}^T\varepsilon_{j-1}\varepsilon^t_j).$$ Alternatively, $$A=\frac{1}{T^{2}}XY^{t}YX^{t},$$ where $X=\left(\varepsilon_{1},\cdots,\varepsilon_{T}\right)_{p\times
T}$, $Y=\left(\varepsilon_{0},\cdots,\varepsilon_{T-1}\right)_{p\times
T}$. Here we define a modified version of the A matrix, $$B=\frac{1}{T^{2}}Y^{t}YX^{t}X=\sum_{j=1}^{p}s_{j}s_{j}^{t}\sum_{j=1}^{p}r_{j}r_{j}^{t},$$ where $s_{j}=\frac{1}{\sqrt{T}}\left(\varepsilon_{j0},\varepsilon_{j1},\cdots,\varepsilon_{j,T-1}\right)^{t}$ is the j-th row of $Y$, and $r_{j}=\frac{1}{\sqrt{T}}\left(\varepsilon_{j1},\varepsilon_{j2},\cdots,\varepsilon_{j,T}\right)^{t}$ the j-th row of $X$. As $A$ and $B$ have same nonzero eigenvalues, the LSD of $A$ can be derived from the LSD of $B$.
The main result of the paper is
\[th1\] Let the following assumptions hold:
- $\varepsilon_{i}=\left(\varepsilon_{1i},\cdots\varepsilon_{pi}\right)^{t},i=0,1,2,\cdots,T$ are independent p-dimensional real-valued random vectors with independent entries satisfying condition: $$\mathbb{E}(\varepsilon_{it})=0,~\mathbb{E}(\varepsilon^2_{it})=1,~\sup_{1\leq i\leq p,0\leq t\leq T}\mathbb{E}\left(|\varepsilon_{it}|^{4}\right)<M,$$ for some constant $M$ and for any $\eta>0$, $$\frac{1}{\eta^{4}pT}\sum_{i=1}^{p}\sum_{t=0}^{T}\mathbb{E}\left(|\varepsilon_{it}|^{4}I_{(|\varepsilon_{it}|\geq\eta T^{1/4})}\right)=o\left(1\right);$$
- As $p\rightarrow \infty$, the sample size $T=T(p)\rightarrow \infty$ and $p/T{\rightarrow}c>0$.
Then,
- as $p,T\rightarrow \infty$, almost surely, the empirical spectral distribution $F^B$ of $B$, converges to a non-random probability distribution $\b{F}$ whose Stieltjes transform $x=x(\alpha)$, $\alpha\in \mathbb{C}\setminus \mathbb{R}$, satisfies the equation $$\label{eq1}
\alpha^{2}x^{3}-2\alpha\left(c-1\right)x^{2}+\left(c-1\right)^{2}x-\alpha x-1=0.$$
- Moreover, for $\alpha\in \mathbb{C}^+=\{z:\mathfrak{Im}z>0\}$, equation admits an unique solution $\alpha\mapsto x(\alpha)$ with positive imaginary part and the density function of the LSD $\b{F}$ is:
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[$$\begin{aligned}
f(u)&=\frac{1}{\pi
u}\left\{-u-\frac{5(c-1)^2}{3}+\frac{2^{4/3}(3u+(c-1)^2)(c-1)}{3d(u)^{1/3}}
+\frac{2^{2/3}(c-1)d(u)^{1/3}}{3}\right.\\
&\quad + \left.\frac{1}{48}\left[-8(c-1)+\frac{2\times
2^{1/3}(3u+(c-1)^2)}{d(u)^{1/3}}+2^{2/3}d(u)^{1/3}\right]^2\right\}^{1/2},
\end{aligned}$$]{}
where $d(u)=-2(c-1)^3+9(1+2c)u+3\sqrt{3}\sqrt{u(-4u^2+(-1+4c(5+2c))u-4c(c-1)^3)}$.
--------------------------------------------------------------------------------------------
Moreover, the support of f(u) is $(0,b]$ for $0<c\leq 1$, and $[a,b]$ for $c>1$, where $$a=\frac{1}{8}(-1+20c+8c^2-(1+8c)^{3/2}),
\quad b=\frac{1}{8}(-1+20c+8c^2+(1+8c)^{3/2}).$$
It’s easy to check that when $c<1$, the LSD of $B$ has a point mass $1-c$ at the origin since $\mathrm{rank}(B)=p<T$ for large $p$ and $T$, and at the same time we have $$\left\{
\begin{array}{l}\displaystyle
\int_0^{b}f(u)du=c,\quad 0<c<1,\\[1.5mm]\displaystyle
\int_{a}^{b}f(u)du=1,\quad c\geq 1.
\end{array}
\right.$$
Since the matrix $A$ we are interested in has the same non-zero eigenvalues with $B$, the following proposition holds.
\[pro2\] Under the conditions of Theorem \[th1\], the ESD of $A$ converges a.s. to a non-random limit distribution $$F=\frac{1}{c}\b{F}+(1-\frac{1}{c})\delta_0,$$ whose Stieltjes transform $y=y(\alpha)$, $\alpha\in \mathbb{C}\setminus \mathbb{R}$, satisfies the equation $$\alpha^2 c^2 y^3+\alpha c(c-1)y^2-\alpha y-1=0.$$ In particular, $F$ has the density function $$\left\{
\begin{array}{l}\displaystyle
\frac{1}{c}f(u),~u\in (0,b],\mbox{ for }0<c<1,\\[1.5mm]\displaystyle
\frac{1}{c}f(u),~u\in [a,b],\mbox{ for }c\geq 1.
\end{array}
\right.$$ where in the later case $c\geq 1$, $F$ has an additional mass $(1-\frac{1}{c})$ at the origin.
The following details the density function of LSD of $A$ for different values of c.
- When $c=1$, the support is $0\leq u\leq
\frac{27}{4}$ and the density function is $$\frac{1}{c}f(u)=\frac{1}{\pi
u}\left[-u+3\left(\frac{u}{2^{2/3}d(u)^{1/3}}+\frac{d(u)^{1/3}}{6\times2^{1/3}}\right)^2\right]^{1/2},$$ where $d(u)=27u+3\sqrt{3}\times\sqrt{u(-4u^2+27u)}$. It’s easy to see that as $u\rightarrow 0_{+}$, $f(u)\rightarrow\infty$.
- If $c<1$, it can be seen from the explicit form of $f(u)$ that when $u{\rightarrow}0_{+}$, $\frac{1}{c}f(u)\rightarrow\infty$ because the $u$ in the denominator of the density function cannot be completely canceled out.
- If $c>1$, the shape of the density function turns out to be a little different from the case $c\leq 1$. Nevertheless it’s quite intuitive because the lower bound of the support is positive and the density function is bounded.
The density functions of LSD of A for $c=0.5,1,2,3$ are displayed on Figure \[fig1\].
![Density plots of the LSD of B.Top to bottom and left to right: c=0.5,1,2 and 3, respectively[]{data-label="fig1"}](pictures.jpg "fig:"){width="100.00000%"}\
Proofs
======
Proof of Theorem \[th1\]
------------------------
The proof of the theorem follows the general strategy based on the Stieltjes transform as presented in @Silv95, @BS10 and @PasShc10. However, the random matrix B here is no more a covariance matrix as considered in these references. Almost all the steps of the proof need new arguments and ideas compared to the case of sample covariance matrices considered so far in the literature. Following this method, the first step is to truncate the entries $\{\varepsilon_{jt}\}$ at a convenient rate using Assumption (a). After truncation and the follow-up steps of centralization and standardization, we may assume that $$|\varepsilon_{ij}|\leq\eta T^{1/4},\quad
\mathbb{E}\left(\varepsilon_{ij}\right)=0,\quad
Var\left(\varepsilon_{ij}\right)=1,\quad \sup_{1\leq i\leq p,0\leq
j\leq T}\mathbb{E}\left(|\varepsilon_{ij}|^{4}\right)<M.$$ The details of these technical steps are given in Appendix A.
By the rank inequality(Theorem A.44 of [@BS10]), it is enough to consider $$B=\sum_{j=1}^{p}s_{j}s_{j}^{t}\sum_{j=1}^{p}r_{j}r_{j}^{t}=P_1\tilde{C}P_1^t\tilde{C},$$ where $$s_{j}=P_{1}r_{j}=\frac{1}{\sqrt{T}}(0,\varepsilon_{j1},\cdots,\varepsilon_{j,T-1})^t,
\quad C=\sum_{j=1}^ps_js_j^t, \quad \tilde{C}=\sum_{j=1}^{p}r_{j}r_{j}^{t},\quad
P_{1}=\left(\begin{array}{cc}
\textbf{0} & 0\\
\textbf{I}_{T-1} & \textbf{0}
\end{array}\right).$$ At this stage, the important observation is that here we have replaced $s_j=\frac{1}{\sqrt{T}}(\varepsilon_{j0},\varepsilon_{j1},\cdots,\varepsilon_{j,T-1})^t$ by $\tilde{s}_j=\frac{1}{\sqrt{T}}(0,\varepsilon_{j1},\cdots,\varepsilon_{j,T-1})^t$ without altering the LSD of B since when $T{\rightarrow}\infty$, the effect of this substitution will vanish. For the sake of convenience, we still use $s_j$ to denote $\tilde{s}_j$.
For $\alpha\in \mathbb{C}\setminus\mathbb{R}$, define $$B\left(\alpha\right)=\sum_{j=1}^{p}s_{j}s_{j}^{t}\sum_{j=1}^{p}r_{j}r_{j}^{t}-\alpha I_{T}.$$ Let $$x_0=\frac{1}{T}tr(B^{-1}(\alpha)),\quad y_0=\frac{1}{T}tr(\tilde{C}B^{-1}(\alpha)),\quad z_0=\frac{1}{T}tr(B^{-1}(\alpha)C).$$ The method consists in finding a system of two asymptotic equations satisfied by $x_0$ and $y_0$. Solving the system yields an asymptotic equivalent for $x_0$ and finally leads to the equation satisfied by the limit of $x_0$. Nonetheless, $x_0$ is the Stieltjes transform of the matrix B which can be recovered from the inversion formula.
Let $$B_{j}\left(\alpha\right)=\sum_{k\neq j}s_{k}s_{k}^{t}\sum_{i\neq
j}r_{i}r_{i}^{t}-\alpha I_{T},\quad C_j=C-s_js_j^t,\quad \tilde{C}_j=\tilde{C}-r_jr_j^t,\quad 1\leq j\leq
p,$$ then $$\begin{aligned}
B\left(\alpha\right)&=B_{j}\left(\alpha\right)+\sum_{i\neq
j}s_{j}s_{j}^{t}r_{i}r_{i}^{t}+\sum_{k\neq
j}s_{k}s_{k}^{t}r_{j}r_{j}^{t}+s_{j}s_{j}^{t}r_{j}r_{j}^{t}\\
&=B_j\left(\alpha\right)+s_js_j^t\tilde{C}_j+C_jr_jr_j^t+s_js_j^tr_jr_j^t.\end{aligned}$$
We have $$I_{T}=B(\alpha)B^{-1}(\alpha)=\left(\sum_{j=1}^{p}s_{j}s_{j}^{t}\right)\left(\sum_{j=1}^{p}r_{j}r_{j}^{t}\right)B^{-1}\left(\alpha\right)-\alpha
B^{-1}\left(\alpha\right).$$ Taking trace and dividing both sides by $T$, we get
$$\label{eq:1}
1=\frac{1}{T}\sum_{j=1}^{p}s_{j}^{t}\tilde{C}B^{-1}\left(\alpha\right)s_{j}-\alpha\frac{1}{T}tr\left(B^{-1}\left(\alpha\right)\right).$$
Note that $x_0=\frac{1}{T}tr(B^{-1}(\alpha))$ is the Stieltjes transform of the ESD of the matrix B, and its limit will be found by letting $p,T\rightarrow\infty$ on both sides of the equation.
Consider $s_{j}^{t}\tilde{C} B^{-1}\left(\alpha\right)s_{j}$, using the identities $$\left(B+\sum_{j=1}^{m}ab_{j}^{t}\right)^{-1}a=\frac{B^{-1}a}{1+\sum_{j=1}^{m}b_{j}^{t}B^{-1}a},$$ and $$B^{-1}-D^{-1}=B^{-1}\left(D-B\right)D^{-1},$$ we have $$\begin{aligned}
s_{j}^{t}\tilde{C}B^{-1}\left(\alpha\right)s_{j}&=\frac{s_{j}^{t}\tilde{C}\left(B_{j}\left(\alpha\right)+C_{j}r_jr_j^t\right)^{-1}s_{j}}{1+s_{j}^{t}\tilde{C}\left(B_{j}\left(\alpha\right)+C_{j}r_jr_j^t\right)^{-1}s_{j}}\\
&=1-\frac{1}{1+s_j^t\tilde{C}_j\left(B_{j}\left(\alpha\right)+C_{j}r_jr_j^t\right)^{-1}s_j+s_j^tr_jr_j^t\left(B_{j}\left(\alpha\right)+C_{j}r_jr_j^t\right)^{-1}s_j}\\
& := 1- \frac{1}{1+L_1+L_2},\end{aligned}$$ where $L_1$ and $L_2$ are explicitly defined.
For $L_1$, by Lemma \[lem3\], or equivalently by Lemma 2.7 of [@BS98], we have $$\begin{aligned}
L_1&=&s_{j}^{t}\tilde{C}_j\left(B_{j}\left(\alpha\right)+C_{j}r_jr_j^t\right)^{-1}s_{j} \\
& = & s_{j}^{t}\tilde{C}_j B_{j}^{-1}\left(\alpha\right)s_{j}- s_{j}^{t}\tilde{C}_j B_{j}\left(\alpha\right)^{-1}C_{j}r_jr_j^t\left(B_{j}\left(\alpha\right)+C_{j}r_jr_j^t\right)^{-1}s_{j}\\
& = & s_{j}^{t}\tilde{C}_j B_{j}^{-1}\left(\alpha\right)s_{j}-\frac{s_{j}^{t}\tilde{C}_j B_{j}^{-1}\left(\alpha\right)C_{j}r_jr_j^tB_{j}\left(\alpha\right)^{-1}s_{j}}{1+r_j^tB_{j}^{-1}\left(\alpha\right)C_{j}r_j}\\
& = & \frac{1}{T}tr\left(\tilde{C}_j B_{j}^{-1}\left(\alpha\right)\right)-\frac{\frac{1}{T}tr\left(\tilde{C}_j B_{j}^{-1}\left(\alpha\right)C_{j}P_{1}^{t}\right)\cdot\frac{1}{T}tr\left(B_{j}^{-1}\left(\alpha\right)P_{1}\right)}{1+\frac{1}{T}tr\left(B_{j}\left(\alpha\right)^{-1}C_{j}\right)}+o_{a.s.}(1).\end{aligned}$$
For $L_2$, we have $$\begin{aligned}
L_2&=s_{j}^{t}r_{j}r_{j}^{t}\left(B_{j}\left(\alpha\right)+C_{j}r_jr_j^t\right)^{-1}s_{j}=s_{j}^{t}r_{j}r_{j}^{t}B_{j}^{-1}\left(\alpha\right)s_{j}-\frac{s_{j}^{t}r_{j}r_{j}^{t}B_{j}^{-1}\left(\alpha\right)C_jr_jr_j^tB^{-1}_j(\alpha)s_{j}}{1+r_j^tB_{j}^{-1}\left(\alpha\right)C_{j}r_{j}}\\
&=\left(s_{j}^{t}P_{1}^{t}s_{j}\right)\cdot\frac{1}{T}tr\left(B_{j}^{-1}\left(\alpha\right)P_{1}\right)-\frac{\left(s_{j}^{t}P_{1}^{t}s_{j}\right)\cdot\frac{1}{T}tr\left(B_{j}^{-1}(\alpha)C_j\right)\cdot\frac{1}{T}tr\left(B_{j}^{-1}\left(\alpha\right)P_{1}\right)}{1+\frac{1}{T}tr\left(B_{j}^{-1}\left(\alpha\right)C_{j}\right)}+o_{a.s.}(1)=o_{a.s.}(1).\end{aligned}$$
Therefore, by equation $\eqref{eq:1}$, we have $$\begin{aligned}
&~1+\alpha\frac{1}{T}tr(B^{-1}(\alpha))=o_{a.s.}(1)+ \label{yt1}\\
&\frac{p}{T}\left(1-\dfrac{1+\dfrac{1}{T}tr(B^{-1}(\alpha)C)}{\left(1+\dfrac{1}{T}tr\left(B^{-1}(\alpha)C\right)\right)\left(1+\dfrac{1}{T}tr(\tilde{C}B^{-1}(\alpha))\right)-\dfrac{1}{T}tr\left(\tilde{C} B^{-1}\left(\alpha\right)CP_{1}^{t}\right)\cdot\dfrac{1}{T}tr\left(B^{-1}\left(\alpha\right)P_{1}\right)}\right)\nonumber\end{aligned}$$
Here, we have used the following equivalents, uniformly in $j$, as $p,T\rightarrow\infty$, $$\frac{1}{T}tr\left(B_{j}^{-1}\left(\alpha\right)C_{j}\right)=z_{0}+o_{a.s.}(1),$$ $$\frac{1}{T}tr\left(B_{j}^{-1}\left(\alpha\right)\right)=x_0+o_{a.s.}(1),$$ $$\frac{1}{T}tr\left(\tilde{C}_jB_{j}^{-1}\left(\alpha\right)\right)=y_{0}+o_{a.s.}(1).$$
Similar to equation , we have
$$1=\frac{1}{T}\sum_{j=1}^{p}r_{j}^{t}B^{-1}\left(\alpha\right)Cr_{j}-\alpha\frac{1}{T}tr\left(B^{-1}\left(\alpha\right)\right).\label{eq:2}$$
Considering $r_{j}^{t} B^{-1}\left(\alpha\right)C r_{j}$, we have $$\begin{aligned}
r_{j}^{t}B^{-1}\left(\alpha\right)Cr_{j}&=\frac{r_{j}^{t}\left(B_{j}\left(\alpha\right)+s_j s_j^t \tilde{C}_{j}\right)^{-1}Cr_{j}}{1+r_{j}^{t}\left(B_{j}\left(\alpha\right)+s_j s_j^t \tilde{C}_{j}\right)^{-1}Cr_{j}}\\
&=1-\frac{1}{1+r_j^t\left(B_{j}\left(\alpha\right)+s_j s_j^t\tilde{C}_j\right)^{-1}C_jr_j+r_j^t\left(B_{j}\left(\alpha\right)+s_js_j^t\tilde{C}_j\right)^{-1}s_j s_j^tr_j}\\
& := 1- \frac{1}{1+W_1+W_2},\end{aligned}$$ where $W_1$ and $W_2$ are explicitly defined.
For $W_1$, we have $$\begin{aligned}
W_1&=&r_{j}^{t}\left(B_{j}\left(\alpha\right)+s_j s_j^t\tilde{C}_j\right)^{-1}C_{j}r_{j} \\
& = & r_{j}^{t}B_{j}^{-1}\left(\alpha\right)C_jr_{j}- r_{j}^{t} B_{j}^{-1}\left(\alpha\right)s_j s_j^t\tilde{C}_j\left(B_{j}\left(\alpha\right)+s_j s_j^t\tilde{C}_j\right)^{-1}C_{j}r_{j}\\
& = & r_{j}^{t}B_{j}^{-1}\left(\alpha\right)C_jr_{j}-\dfrac{r_j^tB_{j}^{-1}\left(\alpha\right)s_{j}s_{j}^{t}\tilde{C}_j B_{j}^{-1}\left(\alpha\right)C_{j}r_j}{1+s_j^t \tilde{C}_{j}B_{j}^{-1}\left(\alpha\right)s_j}\\
& = & \frac{1}{T}tr\left(B_{j}^{-1}\left(\alpha\right)C_j\right)-\frac{\frac{1}{T}tr\left(\tilde{C}_j B_{j}^{-1}\left(\alpha\right)C_{j}P_{1}^{t}\right)\cdot\frac{1}{T}tr\left(B_{j}^{-1}\left(\alpha\right)P_{1}\right)}{1+\frac{1}{T}tr\left(\tilde{C}_{j}B_{j}\left(\alpha\right)^{-1}\right)}+o_{a.s.}(1).\end{aligned}$$
For $W_2$, we have $$\begin{aligned}
W_2&=r_{j}^{t}\left(B_{j}\left(\alpha\right)+s_js_j^t\tilde{C}_{j}\right)^{-1}s_{j}s_{j}^{t}r_{j}=r_{j}^{t}B_{j}^{-1}\left(\alpha\right)s_{j}s_{j}^{t}r_{j}-\frac{r_{j}^{t}B_{j}^{-1}\left(\alpha\right)s_js_j^t\tilde{C}_jB^{-1}_j(\alpha)s_{j}s_{j}^{t}r_{j}}{1+s_j^t\tilde{C}_jB^{-1}_j(\alpha)s_{j}}\\
&=\left(s_{j}^{t}P_{1}^{t}s_{j}\right)\cdot\frac{1}{T}tr\left(B_{j}^{-1}\left(\alpha\right)P_{1}\right)-\frac{\left(s_{j}^{t}P_{1}^{t}s_{j}\right)\cdot\frac{1}{T}tr\left(\tilde{C}_j B_{j}^{-1}(\alpha)\right)\cdot\frac{1}{T}tr\left(B_{j}^{-1}\left(\alpha\right)P_{1}\right)}{1+\frac{1}{T}tr\left(\tilde{C}_{j}B_{j}^{-1}\left(\alpha\right)\right)}+o_{a.s.}(1)=o_{a.s.}(1).\end{aligned}$$
Therefore, by equation $\eqref{eq:2}$, we have $$\begin{aligned}
&~1+\alpha\frac{1}{T}tr(B^{-1}(\alpha))=o_{a.s.}(1)+ \label{yt2}\\
&\frac{p}{T}\left(1-\dfrac{1+\dfrac{1}{T}tr(B^{-1}(\alpha)\tilde{C})}{\left(1+\dfrac{1}{T}tr\left(B^{-1}(\alpha)C\right)\right)\left(1+\dfrac{1}{T}tr(\tilde{C}B^{-1}(\alpha))\right)-\dfrac{1}{T}tr\left(\tilde{C} B^{-1}\left(\alpha\right)CP_{1}^{t}\right)\cdot\dfrac{1}{T}tr\left(B^{-1}\left(\alpha\right)P_{1}\right)}\right)\nonumber\end{aligned}$$
Thus, according to equation and , we have $$\dfrac{1}{T}tr(B^{-1}(\alpha)\tilde{C})=\dfrac{1}{T}tr(B^{-1}(\alpha)C)+o_{a.s.}(1).$$
By Lemma \[lem1\], the second term is $o_{a.s.}(1)$ since both $\frac{1}{T}tr\left(P_{1}^{t}\tilde{C}_j B_{j}\left(\alpha\right)^{-1}C_{j}\right)$ and $\frac{1}{T}tr\left(B_{j}\left(\alpha\right)^{-1}C_{j}\right)$ are non-negative and bounded as $p,T\rightarrow\infty$. $$L_1 =\frac{1}{T} tr\left(\tilde{C}_jB_j^{-1}(\alpha)\right)+o_{a.s.}(1)=y_0+o_{a.s.}(1).$$
Finally, by equation $\eqref{eq:2}$, we find $$\label{eq3}
1+\alpha x_0=\frac{p}{T}\left(1-\frac{1}{1+y_0}\right)+o_{a.s.}(1).$$
To find a second equation satisfied by $x_0$ and $y_0$, using Lemma \[lem3\] and Lemma \[lem1\],
$$\begin{aligned}
\frac{1}{T}tr(\tilde{C}B^{-1}(\alpha))&=\frac{1}{T}tr(\sum_{j=1}^pr_jr_j^tB^{-1}(\alpha))=\frac{1}{T}\sum_{j=1}^pr_j^tB^{-1}(\alpha)r_j\\
&=\frac{1}{T}\sum_{j=1}^p\dfrac{r_j^t\left(B_j(\alpha)+s_js_j^t\tilde{C}_j\right)^{-1}r_j}{1+r_j^t\left(B_j(\alpha)+s_js_j^t\tilde{C}_j\right)^{-1}C_jr_j+r_j^t\left(B_j(\alpha)+s_js_j^t\tilde{C}_j\right)^{-1}s_js_j^tr_j}\\
&=\frac{1}{T}\sum_{j=1}^p\dfrac{r_j^tB_j^{-1}(\alpha)r_j-\dfrac{r_j^tB_j^{-1}(\alpha)s_js_j^t\tilde{C}_jB_j^{-1}(\alpha)r_j}{1+s_j^t\tilde{C}_jB_j^{-1}(\alpha)s_j}}{1+r_j^tB^{-1}_j(\alpha)C_jr_j-\dfrac{r_j^tB^{-1}_j(\alpha)s_js_j^t\tilde{C}_jB^{-1}_j(\alpha)C_jr_j}{1+s_j^t\tilde{C}_jB^{-1}_j(\alpha)s_j}}+o_{a.s.}(1)\\
&=\frac{p}{T}\cdot\dfrac{\dfrac{1}{T}tr(B^{-1}(\alpha))}{1+\dfrac{1}{T}tr(B^{-1}(\alpha)C)}+o_{a.s.}(1).\end{aligned}$$
This leads to $$\label{eq4}
y_0=\frac{p}{T}\cdot\frac{x_0}{1+y_0}+o_{a.s.}(1).$$
In conclusion, $(x_0,y_0)$ satisfy the system
$$\begin{cases}\displaystyle
1+\alpha x_0=\frac{cy_0}{1+y_0}+o_{a.s.}(1),\\[1.5mm]\displaystyle
y_0=\frac{cx_0}{1+y_0}+o_{a.s.}(1).
\end{cases}$$ Notice that for any $T,~|x_0|\leq \frac{1}{|\mathfrak{Im}(\alpha)|}$ is bounded, and by equation , $|y_0|$ is also bounded as $T\rightarrow \infty$, otherwise may not hold. Therefore, both $\{x_0\}$ and $\{y_0\}$ are bounded sequences. Let be two subsequences $\{x_{t_n}\},\{y_{t_n}\}$ so that $x_{t_n}\rightarrow x$ and $y_{t_n}\rightarrow y$ as $n\rightarrow\infty$. It can be concluded that the limiting functions $(x,y)$ satisfy the system of equations:
$$\begin{cases}\displaystyle
1+\alpha x=\frac{cy}{1+y}&(1)\\[1.5mm]\displaystyle
y=\frac{cx}{1+y}&(2)
\end{cases}$$
By eliminating $y$, we finally find the equation satisfied by the limiting function $x$. Denote by $\mathcal{F}$ all the analytical functions $\{f:~\mathbb{C}^+\mapsto \mathbb{C}^+\}$. Because according to the following proof we have one unique solution on $\mathcal{F}$ that satisfies equation , the whole bounded sequence $\{x_0\}$ has one unique limit $x$ in $\mathcal{F}$.
As for the second statement of Theorem \[th1\], in order to find the density function of the LSD $\b{F}$ of $B$, we use the inversion formula:
$$f\left(u\right)=\lim_{\varepsilon{\rightarrow}0_{+}}\frac{1}{\pi}{\mathfrak{Im}
}x\left(u+i\varepsilon\right)$$ where $x\left(\cdot\right)$ is the Stieltjes transform of $\b{F}$. Write $$\lim_{\varepsilon{\rightarrow}0_{+}}x(u+i\varepsilon)=\psi(u)+i\phi(u),$$ both $\psi$ and $\phi$ are real-valued functions of $u$. By substituting $\alpha=u+i\varepsilon$, $x=\psi+i\phi$ into equation and letting $\varepsilon{\rightarrow}0_{+}$, both the real part and the imaginary part of the LHS of equation should equal to 0, i.e. $$\begin{cases}\displaystyle
u^{2}{\psi}^{3}-3u^{2}\psi\cdot {\phi}^{2}-2u\left(c-1\right)\left({\psi}^{2}-{\phi}^{2}\right)-\left(u-\left(c-1\right)^{2}\right)\psi-1=0 & \left(3\right)\\[1.5mm]\displaystyle
-u^{2}{\phi}^{2}+3u^{2}{\psi}^{2}-4u\left(c-1\right)\psi-\left(u-\left(c-1\right)^{2}\right)=0
& \left(4\right)
\end{cases}$$ By plugging in (4) into (3), we get $$-8u^2\psi^3+16u(c-1)\psi^2+(2u-10(c-1)^2)\psi+\frac{2(c-1)^3}{u}-2c+1=0.$$ Solving this equation and substituting for $\psi$ in (4), we get $$\begin{aligned}
{\phi}_1^2(u)&=\frac{1}{u^2}\left\{-u-\frac{5(c-1)^2}{3}+\frac{2^{4/3}(3u+(c-1)^2)(c-1)}{3d(u)^{1/3}}+\frac{2^{2/3}(c-1)d(u)^{1/3}}{3}\right.\\
&\quad +\left.\frac{1}{48}\left[-8(c-1)+\frac{2\times
2^{1/3}(3u+(c-1)^2)}{d(u)^{1/3}}+2^{2/3}d(u)^{1/3}\right]^2\right\},\end{aligned}$$ $$\begin{aligned}
{\phi}_2^2(u)&=\frac{1}{u^2}\left\{-u-\frac{5(c-1)^2}{3}+\frac{1+i\sqrt{3}}{2}\cdot\frac{2^{4/3}(3u+(c-1)^2)(c-1)}{3d(u)^{1/3}}+\frac{1-i\sqrt{3}}{2}\cdot\frac{2^{2/3}(c-1)d(u)^{1/3}}{3}\right.\\
&\quad +\left.\frac{1}{48}\left[-8(c-1)+\frac{1+i\sqrt{3}}{2}\cdot\frac{2\times
2^{1/3}(3u+(c-1)^2)}{d(u)^{1/3}}+\frac{1-i\sqrt{3}}{2}\cdot2^{2/3}d(u)^{1/3}\right]^2\right\},\end{aligned}$$ $$\begin{aligned}
{\phi}_3^2(u)&=\frac{1}{u^2}\left\{-u-\frac{5(c-1)^2}{3}+\frac{1-i\sqrt{3}}{2}\cdot\frac{2^{4/3}(3u+(c-1)^2)(c-1)}{3d(u)^{1/3}}+\frac{1+i\sqrt{3}}{2}\cdot\frac{2^{2/3}(c-1)d(u)^{1/3}}{3}\right.\\
&\quad +\left.\frac{1}{48}\left[-8(c-1)+\frac{1-i\sqrt{3}}{2}\cdot\frac{2\times
2^{1/3}(3u+(c-1)^2)}{d(u)^{1/3}}+\frac{1+i\sqrt{3}}{2}\cdot2^{2/3}d(u)^{1/3}\right]^2\right\},\end{aligned}$$ where $$\label{eq5}
d(u)=-2(c-1)^3+9(1+2c)u+3\sqrt{3}\sqrt{u(-4u^2+(-1+4c(5+2c))u-4c(c-1)^3)}.$$ It can be checked that only the first solution is compatible with the fact that both $\psi$ and $\phi$ are real-valued functions of $u$, i.e. $$\begin{aligned}
{\phi}^2(u)&=\frac{1}{u^2}\left\{-u-\frac{5(c-1)^2}{3}+\frac{2^{4/3}(3u+(c-1)^2)(c-1)}{3d(u)^{1/3}}+\frac{2^{2/3}(c-1)d(u)^{1/3}}{3}\right.\\
&\quad +\left.\frac{1}{48}\left[-8(c-1)+\frac{2\times
2^{1/3}(3u+(c-1)^2)}{d(u)^{1/3}}+2^{2/3}d(u)^{1/3}\right]^2\right\}.\end{aligned}$$
From the explicit form of ${\phi}^2(u)$ we see that, necessarily, $$u(-4u^2+(-1+4c(5+2c))u-4c(c-1)^3)\geq 0,$$ since $u\geq 0$. Solving this quadratic inequality, we get two roots, $$\label{eq6}
a=\frac{1}{8}(-1+20c+8c^2-(1+8c)^{3/2}),\quad b=\frac{1}{8}(-1+20c+8c^2+(1+8c)^{3/2}).$$ It’s very easy to see that $a$ is an increasing function of $c$ and $a=0$ when $c=1$.
In other words, if $0<c<1$, $-\frac{1}{4}<a<0$, then the support of the density function should be $(0,b)$. If $c\geq 1$, $a\geq 0$, then the support of the density function is $(a,b)$.
Then the density function of the limiting spectral distribution of the $T\times T$ dimensional multiplied lag-1 sample auto-covariance matrix $B$ is
------------------------------------------------------------------------------------------
[$$\begin{aligned}
f(u)&=\frac{1}{\pi
u}\left\{-u-\frac{5(c-1)^2}{3}+\frac{2^{4/3}(3u+(c-1)^2)(c-1)}{3d(u)^{1/3}}
+\frac{2^{2/3}(c-1)d(u)^{1/3}}{3}\right.\\
&\quad + \left.\frac{1}{48}\left[-8(c-1)+\frac{2\times
2^{1/3}(3u+(c-1)^2)}{d(u)^{1/3}}+2^{2/3}d(u)^{1/3}\right]^2\right\}^{1/2},
\end{aligned}$$]{}
------------------------------------------------------------------------------------------
where $0<u\leq b$, for $0<c\leq 1$ and $a \leq u \leq b$, for $c>1$, with $(a,b)$ given in equation and $d(u)$ given in equation . Therefore, equation admits at least one solution $\alpha\mapsto x(\alpha)$ that corresponds to this density function of the LSD $\b{F}$. As for the uniqueness, suppose there exists another solution $x_1(\alpha)$ that satisfies equation , then there should be another density $f_1(u)$ that corresponds to $x_1(\alpha)$ while $f_1(u)\neq f(u)$. However, it can be seen from the previous deductions that the density function is unique. Therefore, $f_1(u)=f(u)$, $x_1(\alpha)=x(\alpha)$. Equation admits one unique solution.
Proof of Proposition \[pro2\]
-----------------------------
Under the same conditions in **Theorem \[th1\]**, the ESD of $A$ converges to a non-random limit distribution $F$ with Stieltjes transform $y=y(\alpha)$. On the other hand, the ESD of $B$ converges to $\b{F}$ with Stieltjes transform $x=x(\alpha)$ satisfying $$\alpha^2x^3-2\alpha(c-1)x^2+(c-1)^2x-\alpha x-1=0.$$
Since it’s known that $$F=\frac{1}{c}\b{F}+(1-\frac{1}{c})\delta_0,$$ conclusively we have $$(1-c)(-\frac{1}{\alpha})+cy(\alpha)=x(\alpha).$$ Substituting into the equation of $x$ we can get the equation of $y$, which is $$\alpha^2 c^2 y^3+\alpha c(c-1)y^2-\alpha y-1=0.$$
Extension to lag-$\tau$ sample auto-covariance matrix {#extension}
=====================================================
So far in previous sections, we have focused on the singular value distribution of the lag-1 sample auto-covariance matrix $C_1=T^{-1}\sum_{j=1}^T{\varepsilon}_j{\varepsilon}_{j-1}^t$, while in this section, for any given positive integer $\tau$, we discuss the singular value distribution of the lag-$\tau$ sample auto-covariance matrix $C_{\tau}=T^{-1}\sum_{j=1}^T{\varepsilon}_j{\varepsilon}_{j-\tau}^t$.
Here we follow exactly the same strategy used in the derivation of the LSD of the lag-1 sample auto-covariance matrix. It’s easy to see that the difference between $C_1$ and $C_\tau$ lies in that we have now for $C_\tau$, $$s_j=P_1^{\tau}r_j=\frac{1}{\sqrt{T}}(\underbrace{0,\cdots,0,}_{\tau~ 0's}{\varepsilon}_{j1},\cdots,{\varepsilon}_{j,T-\tau}),~B=\sum_{j=1}^ps_js_j^t\sum_{j=1}^pr_jr_j^t=P_1^{\tau}\tilde{C}(P_1^{\tau})^t\tilde{C}.$$ Meanwhile, the other matrices and notations remain the same using however the new definition of the $s_j's$ above. Consequently, equation becomes $$\begin{aligned}
&~1+\alpha\frac{1}{T}tr(B^{-1}(\alpha))=o_{a.s.}(1)+ \label{yt3}\\
&\frac{p}{T}\left(1-\dfrac{1+\dfrac{1}{T}tr(B^{-1}(\alpha)C)}{\left(1+\dfrac{1}{T}tr\left(B^{-1}(\alpha)C\right)\right)\left(1+\dfrac{1}{T}tr(\tilde{C}B^{-1}(\alpha))\right)-\dfrac{1}{T}tr\left(\tilde{C} B^{-1}\left(\alpha\right)C\left(P_{1}^{\tau}\right)^{t}\right)\cdot\dfrac{1}{T}tr\left(B^{-1}\left(\alpha\right)P_{1}^{\tau}\right)}\right)\nonumber\end{aligned}$$ Equation becomes
$$\begin{aligned}
&~1+\alpha\frac{1}{T}tr(B^{-1}(\alpha))=o_{a.s.}(1)+ \label{yt4}\\
&\frac{p}{T}\left(1-\dfrac{1+\dfrac{1}{T}tr(B^{-1}(\alpha)\tilde{C})}{\left(1+\dfrac{1}{T}tr\left(B^{-1}(\alpha)C\right)\right)\left(1+\dfrac{1}{T}tr(\tilde{C}B^{-1}(\alpha))\right)-\dfrac{1}{T}tr\left(\tilde{C} B^{-1}\left(\alpha\right)C\left(P_{1}^{\tau}\right)^{t}\right)\cdot\dfrac{1}{T}tr\left(B^{-1}\left(\alpha\right)P_{1}^{\tau}\right)}\right)\nonumber\end{aligned}$$
Thus, according to equation and , we still have $$\dfrac{1}{T}tr(B^{-1}(\alpha)\tilde{C})=\dfrac{1}{T}tr(B^{-1}(\alpha)C)+o_{a.s.}(1).$$ Meanwhile, by Lemma \[lem2\], we still have $$\label{lemeq}
\frac{1}{T}tr\left(B^{-1}(\alpha)P_1^{\tau}\right)=o_{a.s.}(1),$$ then by equation , we have $$\label{teq3}
1+\alpha x_0=\frac{p}{T}\left(1-\frac{1}{1+y_0}\right)+o_{a.s.}(1).$$ Similarly, as for the second equation satisfied by $x_0$ and $y_0$, equation persists. $$\label{teq4}
y_0=\frac{p}{T}\cdot\frac{x_0}{1+y_0}+o_{a.s.}(1).$$ Therefore, the system of equations satisfied by $x_0$ and $y_0$ remains the same when the time lag changes from 1 to $\tau$. In other words, for a given positive time lag $\tau$, the singular value distribution of $C_{\tau}$ is the same with that of $C_1$ established in Theorem \[th1\].
TECHNICAL LEMMAS {#lemmas}
================
\[lem3\] Under the same assumptions in **Theorem \[th1\]**, we have, $\forall 1\leq j\leq p$, almost surely, $$\label{eq7}
s_j^tB_j^{-1}(\alpha)s_j=\frac{1}{T}tr(B_j^{-1}(\alpha))+o_{a.s.}(1),$$ $$\label{eq8}
r_j^tB_j^{-1}(\alpha)P_1^kr_j=\frac{1}{T}tr(B_j^{-1}(\alpha)P_1^k)+o_{a.s.}(1),$$ $$\label{eq10}
r_j^t\tilde{C}_jB_j^{-1}(\alpha)P_1^kr_j=\frac{1}{T}tr(\tilde{C}_jB_j^{-1}(\alpha)P_1^k)+o_{a.s.}(1),$$ $$\label{eq9}
s_j^tB_j^{-1}(\alpha)C_js_j=\frac{1}{T}tr(B_j^{-1}(\alpha)C_j)+o_{a.s.}(1),$$ where the $o_{a.s.}(1)$ terms are uniform in $1\leq j\leq p$.
We detail the proof of and the proofs of , and are very similar, thus omitted.
Denote $B_j^{-1}(\alpha)$ by $(y_{kl})=Y$, $s_j=\frac{1}{\sqrt{T}}(\varepsilon_{j0},\cdots,\varepsilon_{j,T-1})^t$, then we have $$|y_{kl}|<\frac{1}{\nu},\quad |\varepsilon_{it}|<\eta
T^{\frac{1}{4}},\quad \sup_{1\leq i\leq p,0\leq t\leq
T}\mathbb{E}|\varepsilon_{it}|^4<M,$$ where $\nu$ is the image part of $\alpha$.
Following the scheme of **Lemma 9.1** of [@BS10] it’s easy to see that $$\begin{aligned}
\mathbb{E}\left\lvert s_j^t Y s_j-\frac{1}{T} tr(Y)\right\rvert^{2r}&=\mathbb{E}\left\lvert \frac{1}{T}\sum_{k,l=1}^T \varepsilon_{j,k-1}y_{kl}\varepsilon_{j,l-1}-\frac{1}{T}\sum_{k=1}^Ty_{kk}\right\rvert^{2r}\\
&=\mathbb{E}\left\lvert \frac{1}{T}\sum_{k=1}^T(\varepsilon_{j,k-1}^2-1)y_{kk}+\frac{1}{T}\sum_{k\neq l} \varepsilon_{j,k-1}y_{kl}\varepsilon_{j,l-1}\right\rvert^{2r}\\
&=\mathbb{E}\left\lvert S_1+S_2 \right\rvert^{2r}\leq 2^r
\frac{\mathbb{E}|S_1|^{2r}+\mathbb{E}|S_2|^{2r}}{2},
\end{aligned}$$ where $$S_1=\frac{1}{T}\sum_{k=1}^T(\varepsilon^2_{j,k-1}-1)y_{kk},\quad
S_2=\frac{1}{T}\sum_{1\leq k\neq l\leq
T}y_{kl}\varepsilon_{j,k-1}\varepsilon_{j,l-1},$$ What’s more, $$\begin{aligned}
\mathbb{E}|S_1|^{2r}&=\mathbb{E}\left\lvert \frac{1}{T}\sum_{k=1}^T(\varepsilon^2_{j,k-1}-1)y_{kk}\right\rvert^{2r}\\
&\leq \frac{1}{T^{2r}}\sum_{t=1}^r\sum_{1\leq k_1<\cdots<k_t\leq T}\sum_{{i_1+\cdots+i_t=2r}\atop{i_1\geq2,\cdots,i_t\geq 2}}(2r)!\prod_{l=1}^t\frac{\mathbb{E}(\varepsilon_{j,k_l-1}^2-1)^{i_l}y_{k_lk_l}^{i_l}}{i_l!}\\
&\leq \frac{1}{T^{2r}}\cdot\frac{1}{v^{2r}}\sum_{t=1}^r T^{t}\sum_{{i_1+\cdots+i_t=2r}\atop{i_1\geq2,\cdots,i_t\geq 2}}\frac{(2r)!}{\prod_{l=1}^ti_l!}\cdot M^t\frac{(\eta T^{\frac{1}{4}})^{4r}}{(\eta T^{\frac{1}{4}})^{4t}}\\
&\leq \frac{1}{T^{2r}}\cdot\frac{1}{v^{2r}}\sum_{t=1}^r T^{t}
t^{2r}M^t\frac{(\eta T^{\frac{1}{4}})^{4r}}{(\eta
T^{\frac{1}{4}})^{4t}}=O(\frac{1}{T^r}),
\end{aligned}$$ $$\mathbb{E}|S_2|^{2r}=\frac{1}{T^{2r}}\sum
y_{i_1j_1}y_{t_1l_1}\cdots
y_{i_rj_r}y_{t_rl_r}\mathbb{E}(\varepsilon_{j,i_1}\varepsilon_{j,j_1}\varepsilon_{j,t_1}\varepsilon_{j,l_1}\cdots\varepsilon_{j,i_r}\varepsilon_{j,j_r}\varepsilon_{j,t_r}\varepsilon_{j,l_r}).$$ Consider a graph G with $2r$ edges that link $i_t$ to $j_t$ and $l_t$ to $k_t$, $t=1,\cdots,r$. It’s easy to see that for any nonzero term, the vertex degrees of the graph are not less than 2. Write the non-coincident vertices as $v_1,\cdots,v_m$ with degrees $p_1,\cdots,p_m$ greater than 1, then, similarly in **Lemma 9.1** of @BS10, we have, $$\left\lvert
\mathbb{E}(\varepsilon_{j,i_1}\varepsilon_{j,j_1}\varepsilon_{j,t_1}\varepsilon_{j,l_1}\cdots\varepsilon_{j,i_r}\varepsilon_{j,j_r}\varepsilon_{j,t_r}\varepsilon_{j,l_r})
\right\rvert\leq (\eta T^{\frac{1}{4}})^{2(2r-m)},$$ $$\mathbb{E}|S_2|^{2r}\leq \frac{1}{T^{2r}\nu^{2r}}\sum_{m=2}^r
T^{m/2}(\eta T^{\frac{1}{4}})^{2(2r-m)}m^{4r}=O(\frac{1}{T^r}).$$ Therefore, by the Borel-Cantelli lemma, we have, $\forall 1\leq j\leq
p$, $$s_j^tB_j(\alpha)^{-1}s_j=\frac{1}{T}tr(B_j(\alpha)^{-1})+o_{a.s.}(1),$$ where the $o_{a.s.}(1)$ terms are uniform in $1\leq j\leq p$.
\[lem1\] Under the same assumptions in **Theorem \[th1\]**, we have, $\forall 1\leq j\leq p$, $1\leq k\leq T-1$, almost surely, $$r_j^{t}B_j^{-1}(\alpha)P_1^k r_j=\frac{1}{T}tr\left(B^{-1}\left(\alpha\right)P_{1}^k\right)+o_{a.s.}(1)=o_{a.s.}(1),$$ $$r_j^t\tilde{C}_jB_j^{-1}(\alpha)P_1^k r_j=\frac{1}{T}tr\left(\tilde{C}B^{-1}\left(\alpha\right)P_{1}^k\right)+o_{a.s.}(1)=o_{a.s.}(1),$$ where the $o_{a.s.}(1)$ terms are uniform in $1\leq j\leq p$.
Notice that, for $1\leq k\leq T-1$, $$P_{1}=\left(\begin{array}{cc}
\textbf{0} & 0\\
\textbf{I}_{T-1} & \textbf{0}
\end{array}\right),
\quad P_1^k=\left(\begin{array}{cc}
\textbf{0} & \bf{0}\\
\textbf{I}_{T-k} & \textbf{0}
\end{array}\right),\quad
P_1^T={\bf0},\quad s_j=P_1r_j.$$ Here $P_1^T$ represents the power $T$ of the $T\times T$ matrix $P_1$, we use $P_1^t$ to denote the transpose of matrix $P_1$. Denote, for $1\leq k\leq T$, $$\frac{1}{T}tr{\left( }B^{-1}(\alpha){\right) }:=x_0,\quad \frac{1}{T}tr{\left( }B^{-1}(\alpha)C {\right) }= \frac{1}{T}tr{\left( }\tilde{C}B^{-1}(\alpha){\right) }:=y_0,$$ $$\frac{1}{T}tr{\left( }B^{-1}(\alpha)P_1^k {\right) }:=x_k,\quad \frac{1}{T}tr{\left( }\tilde{C}B^{-1}(\alpha)P_1^k {\right) }:=y_k.$$ It’s easy to see that $$x_T=y_T=0.$$ In addition, for any $1\leq j\leq p$, $$\begin{aligned}
s_j^t\tilde{C}_jB_j^{-1}(\alpha)C_jr_j&=s_j^t\tilde{C}_j{\left( }C_j\tilde{C}_j-\alpha \bf{I}_T{\right) }^{-1} C_jr_j\\
&=s_j^t{\left( }{\bf I} -\alpha C_j^{-1}\tilde{C}_j^{-1} {\right) }^{-1} r_j=s_j^t\tilde{C}_jC_j{\left( }\tilde{C}_jC_j-\alpha \bf{I}{\right) }^{-1} r_j\\
&=\alpha \cdot s_j^t {\left( }\tilde{C}_jC_j-\alpha \bf{I}{\right) }^{-1} r_j+o_{a.s.}(1)\\
&=\alpha \cdot r_j^t {\left( }C_j\tilde{C}_j-\alpha \bf{I}{\right) }^{-1} s_j+o_{a.s.}(1)\\
&=\alpha\frac{1}{T}tr(B^{-1}(\alpha)P_1)+o_{a.s.}(1)=\alpha x_1+o_{a.s.}(1).\end{aligned}$$ Now we can derive the recursion equations between $x_k$ and $y_k$.
Firstly, for $x_k$, $1\leq k\leq T-1$, since $$P_1^k=\left(\sum_{j=1}^ps_js_j^t\sum_{j=1}^pr_jr_j^t\right)B^{-1}(\alpha)P_1^k-\alpha B^{-1}(\alpha)P_1^k,$$ taking trace and dividing $T$ on both sides of the equation, we get $$\begin{aligned}
& ~\alpha \cdot \frac{1}{T}tr{\left( }B^{-1}(\alpha)P_1^k{\right) }\\
=&\frac{1}{T}\sum_{j=1}^p s_j^t\tilde{C}B^{-1}(\alpha) P_1^ks_j\\
=&\frac{1}{T}\sum_{j=1}^p\dfrac{s_j^t\tilde{C}_j{\left( }B_j(\alpha)+C_jr_jr_j^t {\right) }^{-1}P_1^ks_j}{1+s_j^t\tilde{C}_j{\left( }B_j(\alpha)+C_jr_jr_j^t {\right) }^{-1}s_j}+o_{a.s.}(1)\\
=&\frac{1}{T}\sum_{j=1}^p\dfrac{1+y_0}{(1+y_0)^2-\alpha x_1^2}\left[s_j^t\tilde{C}_jB_j^{-1}(\alpha)P_1^ks_j-\dfrac{s_j^t\tilde{C}_jB_j^{-1}(\alpha)C_jr_jr_j^tB_j^{-1}(\alpha)P_1^ks_j}{1+r_j^tB^{-1}_j(\alpha)C_jr_j}\right]+o_{a.s.}(1)\\
=&\frac{p}{T}\dfrac{1+y_0}{(1+y_0)^2-\alpha x_1^2}\left[\frac{1}{T}tr(\tilde{C}B^{-1}(\alpha)P_1^k)-\frac{\alpha x_1}{1+y_0}\cdot\frac{1}{T}tr{\left( }B^{-1}(\alpha)P_1^{k+1}{\right) }\right]+o_{a.s.}(1),\end{aligned}$$ i.e. $$\label{rec1}
\alpha x_k=\frac{p}{T}\cdot\dfrac{1+y_0}{(1+y_0)^2-\alpha x_1^2}\cdot y_k-\frac{p}{T}\cdot\frac{\alpha x_1}{(1+y_0)^2-\alpha x_1^2}\cdot x_{k+1}+o_{a.s.}(1),~1\leq k\leq T-1.$$ Particularly, for $k=T-1$, we have $$\label{rec2}
\alpha x_{T-1}=\frac{p}{T}\cdot\dfrac{1+y_0}{(1+y_0)^2-\alpha x_1^2}\cdot y_{T-1}+o_{a.s.}(1).$$ Similarly, for $y_k$, $1\leq k\leq T$, $$\begin{aligned}
y_k&=\frac{1}{T}tr{\left( }\tilde{C}B^{-1}(\alpha)P_1^k{\right) }\\
&=\frac{1}{T}tr{\left( }\sum_{j=1}^p r_jr_j^t B^{-1}(\alpha)P_1^k{\right) }=\frac{1}{T}\sum_{j=1}^pr_j^t B^{-1}(\alpha)P_1^k r_j\\
&=\frac{1}{T}\sum_{j=1}^p \dfrac{r_j^t {\left( }B_j(\alpha)+s_js_j^t \tilde{C}_j {\right) }^{-1}P_1^kr_j}{1+r_j^t{\left( }B_j(\alpha)+s_js_j^t \tilde{C}_j {\right) }^{-1}C_jr_j}+o_{a.s.}(1)\\
&=\frac{1}{T}\sum_{j=1}^p\dfrac{1+y_0}{(1+y_0)^2-\alpha x_1^2}\cdot\left[r_j^tB_j^{-1}(\alpha)P_1^kr_j-\dfrac{r_j^tB_j^{-1}(\alpha)s_js_j^t\tilde{C}_jB_j^{-1}(\alpha)P_1^kr_j}{1+s_j^t\tilde{C}_jB^{-1}_j(\alpha)s_j}\right]+o_{a.s.}(1)\\
&=\frac{p}{T}\cdot \dfrac{1+y_0}{(1+y_0)^2-\alpha x_1^2}\cdot\left[\frac{1}{T}tr(B^{-1}(\alpha)P_1^k)-\frac{x_1}{1+y_0}\cdot\frac{1}{T}tr{\left( }\tilde{C}B^{-1}(\alpha)P_1^{k-1}{\right) }\right]+o_{a.s.}(1),\end{aligned}$$ i.e. $$\label{rec3}
y_k=\frac{p}{T}\cdot \dfrac{1+y_0}{(1+y_0)^2-\alpha x_1^2}\cdot x_k-\frac{p}{T}\cdot \dfrac{x_1}{(1+y_0)^2-\alpha x_1^2}\cdot y_{k-1}+o_{a.s.}(1), ~1\leq k\leq T-1.$$
Particularly, for $k=T$, we have $$\label{rec4}
y_T=\frac{p}{T}\cdot \dfrac{1+y_0}{(1+y_0)^2-\alpha x_1^2}\cdot x_{T}-\frac{p}{T}\cdot \dfrac{x_1}{(1+y_0)^2-\alpha x_1^2}\cdot y_{T-1}+o_{a.s.}(1).$$ Note that $$x_T=y_T=0,$$ then we have either $x_1=o_{a.s.}(1)$ or $y_{T-1}=o_{a.s.}(1)$.
If $x_1=o_{a.s.}(1)$, according to equation , we have $y_1=o_{a.s.}(1)$, then according to equation , we have $x_2=y_2=o_{a.s.}(1)$, recursively, we have for all $1\leq k\leq T-1$, $$x_k=y_k=o_{a.s.}(1).$$ Otherwise, if $y_{T-1}=o_{a.s.}(1)$, according to equation , we have $x_{T-1}=o_{a.s.}(1)$, then according to equation , we have $y_{T-2}=o_{a.s.}(1)$, then according to equation , we have $x_{T-2}=o_{a.s.}(1)$, recursively, we still have for all $1\leq k\leq T-1$, $$x_k=y_k=o_{a.s.}(1).$$ Therefore we have, $\forall 1\leq j\leq p$, $1\leq k\leq T-1$, almost surely, $$r_j^{t}B_j^{-1}(\alpha)P_1^k r_j=\frac{1}{T}tr\left(B^{-1}\left(\alpha\right)P_{1}^k\right)+o_{a.s.}(1)=o_{a.s.}(1),$$ $$r_j^t\tilde{C}_jB_j^{-1}(\alpha)P_1^k r_j=\frac{1}{T}tr\left(\tilde{C}B^{-1}\left(\alpha\right)P_{1}^k\right)+o_{a.s.}(1)=o_{a.s.}(1),$$ where the $o_{a.s.}(1)$ terms are uniform in $1\leq j\leq p$.
\[lem2\] Extension of Lemma \[lem1\] to time lag $\tau$:
we have, $\forall 1\leq j\leq p$, $1\leq k\leq [\frac{T}{\tau}]$, almost surely, $$r_j^{t}B_j^{-1}(\alpha)(P_1^{\tau})^k r_j=\frac{1}{T}tr\left(B^{-1}\left(\alpha\right)(P_1^{\tau})^k\right)+o_{a.s.}(1)=o_{a.s.}(1),$$ $$r_j^t\tilde{C}_jB_j^{-1}(\alpha)(P_1^{\tau})^k r_j=\frac{1}{T}tr\left(\tilde{C}B^{-1}\left(\alpha\right)(P_1^{\tau})^k\right)+o_{a.s.}(1)=o_{a.s.}(1),$$ where the $o_{a.s.}(1)$ terms are uniform in $1\leq j\leq p$.
Denote, for $1\leq k\leq \left[\frac{T}{\tau}\right]$, $$\frac{1}{T}tr{\left( }B^{-1}(\alpha){\right) }:=x_0,\quad \frac{1}{T}tr{\left( }B^{-1}(\alpha)C {\right) }= \frac{1}{T}tr{\left( }\tilde{C}B^{-1}(\alpha){\right) }:=y_0,$$ $$\frac{1}{T}tr{\left( }B^{-1}(\alpha)(P_1^{\tau})^k {\right) }:=x_k,\quad \frac{1}{T}tr{\left( }\tilde{C}B^{-1}(\alpha)(P_1^{\tau})^k {\right) }:=y_k.$$ It’s easy to see that $$x_{\left[\frac{T}{\tau}\right]+1}=y_{\left[\frac{T}{\tau}\right]+1}=0.$$ In addition, for any $1\leq j\leq p$, $$s_j^t\tilde{C}_jB_j^{-1}(\alpha)C_jr_j=\alpha\frac{1}{T}tr(B^{-1}(\alpha)P_1^{\tau})+o_{a.s.}(1)=\alpha x_1+o_{a.s.}(1).$$ Now we can derive the recursion equations between $x_k$ and $y_k$.
Firstly, for $x_k$, $1\leq k\leq \left[\frac{T}{\tau}\right]$, $$\begin{aligned}
\alpha &\cdot \frac{1}{T}tr{\left( }B^{-1}(\alpha)(P_1^{\tau})^k{\right) }=o_{a.s.}(1)+\\
&~\frac{p}{T}\dfrac{1+y_0}{(1+y_0)^2-\alpha x_1^2}\left[\frac{1}{T}tr(\tilde{C}B^{-1}(\alpha)(P_1^{\tau})^k)-\frac{\alpha x_1}{1+y_0}\cdot\frac{1}{T}tr{\left( }B^{-1}(\alpha)(P_1^{\tau})^{k+1}{\right) }\right],\end{aligned}$$ i.e. $$\label{rec5}
\alpha x_k=\frac{p}{T}\cdot\dfrac{1+y_0}{(1+y_0)^2-\alpha x_1^2}\cdot y_k-\frac{p}{T}\cdot\frac{\alpha x_1}{(1+y_0)^2-\alpha x_1^2}\cdot x_{k+1}+o_{a.s.}(1),~1\leq k\leq \left[\frac{T}{\tau}\right].$$
Similarly, for $y_k$, $1\leq k\leq \left[\frac{T}{\tau}\right]+1$, $$\begin{aligned}
y_k&=\frac{1}{T}tr{\left( }\tilde{C}B^{-1}(\alpha)(P_1^{\tau})^k {\right) }\\
&=\frac{p}{T}\cdot \dfrac{1+y_0}{(1+y_0)^2-\alpha x_1^2}\cdot\left[\frac{1}{T}tr(B^{-1}(\alpha)(P_1^{\tau})^k)-\frac{x_1}{1+y_0}\cdot\frac{1}{T}tr{\left( }\tilde{C}B^{-1}(\alpha)(P_1^{\tau})^{k-1}{\right) }\right]+o_{a.s.}(1),\end{aligned}$$ i.e. $$\label{rec6}
y_k=\frac{p}{T}\cdot \dfrac{1+y_0}{(1+y_0)^2-\alpha x_1^2}\cdot x_k-\frac{p}{T}\cdot \dfrac{x_1}{(1+y_0)^2-\alpha x_1^2}\cdot y_{k-1}+o_{a.s.}(1), ~1\leq k\leq \left[\frac{T}{\tau}\right]+1.$$
Particularly, for $k=\left[\frac{T}{\tau}\right]+1$, we have $$\label{rec4}
y_{\left[\frac{T}{\tau}\right]+1}=\frac{p}{T}\cdot \dfrac{1+y_0}{(1+y_0)^2-\alpha x_1^2}\cdot x_{\left[\frac{T}{\tau}\right]+1}-\frac{p}{T}\cdot \dfrac{x_1}{(1+y_0)^2-\alpha x_1^2}\cdot y_{\left[\frac{T}{\tau}\right]}+o_{a.s.}(1).$$ Note that $$x_{\left[\frac{T}{\tau}\right]+1}=y_{\left[\frac{T}{\tau}\right]+1}=0,$$ following the same arguments in Lemma \[lem1\], we have, $\forall 1\leq j\leq p$, $1\leq k\leq \left[\frac{T}{\tau}\right]$, almost surely, $$r_j^{t}B_j^{-1}(\alpha)(P_1^{\tau})^k r_j=\frac{1}{T}tr\left(B^{-1}\left(\alpha\right)(P_1^{\tau})^k\right)+o_{a.s.}(1)=o_{a.s.}(1),$$ $$r_j^t\tilde{C}_jB_j^{-1}(\alpha)(P_1^{\tau})^k r_j=\frac{1}{T}tr\left(\tilde{C}B^{-1}\left(\alpha\right)(P_1^{\tau})^k\right)+o_{a.s.}(1)=o_{a.s.}(1),$$ where the $o_{a.s.}(1)$ terms are uniform in $1\leq j\leq p$.
Justification of truncation, centralization and standardization {#app}
===============================================================
Recall that $\varepsilon_{t}=\left(\varepsilon_{1t},\cdots,\varepsilon_{pt}\right)^{t}$, $\varepsilon_{it}$ are independent real-valued random variables with $\mathbb{E}\left(\varepsilon_{it}\right)=0,\mathbb{E}\left(|\varepsilon_{it}|^{2}\right)=1$, and we are interested in is the LSD of time-lagged covariance matrix
$$A=\frac{1}{T^{2}}\left(\sum_{i=1}^{T}\varepsilon_{i}\varepsilon_{i-1}^{t}\right)\left(\sum_{j=1}^{T}\varepsilon_{j-1}\varepsilon_{j}^{t}\right).$$
The assumed moment conditions are: for some constant $M$,
$$\sup_{1\leq i\leq p,0\leq t\leq
T}\mathbb{E}\left(|\varepsilon_{it}|^{4}\right)<M,$$
and for any $\eta>0$,
$$\frac{1}{\eta^{4}pT}\sum_{i=1}^{p}\sum_{t=0}^{T}\mathbb{E}\left(|\varepsilon_{it}|^{4}I_{(|\varepsilon_{it}|\geq\eta
T^{1/4})}\right)=o\left(1\right).$$
The aim of the truncation, centralization and standardization procedure is that after these treatment, we may assume that
$$|\varepsilon_{ij}|\leq\eta T^{1/4},\quad
\mathbb{E}\left(\varepsilon_{ij}\right)=0,\quad
Var\left(\varepsilon_{ij}\right)=1,\quad
\mathbb{E}\left(|\varepsilon_{ij}|^{4}\right)<M.$$
Since the whole procedure is the same with respect to different time lag $\tau$, we focus on the case of lag-1 sample auto-covariance matrix.
Truncation
----------
Let $\tilde{\varepsilon}_{jt}=\varepsilon_{jt}I_{(|\varepsilon_{jt}|<\eta
T^{1/4})}$, $\tilde{\varepsilon}_{t}=\left(\tilde{\varepsilon}_{1t},\cdots,\tilde{\varepsilon}_{pt}\right)^{t}$, $\eta$ can be seen as a constant.
Define $$\tilde{A}=\frac{1}{T^{2}}\left(\sum_{i=1}^{T}\tilde{\varepsilon}_{i}\tilde{\varepsilon}_{i-1}^{t}\right)\left(\sum_{j=1}^{T}\tilde{\varepsilon}_{j-1}\tilde{\varepsilon}_{j}^{t}\right),$$ then according to Theorem A.44 of [@BS10] which states that
$$\lVert F^{AA^{*}}-F^{BB^{*}}\rVert \leq\frac{1}{p}
\mathrm{rank}\left(A-B\right),$$
we have $$\begin{aligned}
\lVert F^{A}-F^{\tilde{A}}\rVert & \leq \frac{1}{p} \mathrm{rank}\left(\frac{1}{T}\sum_{i=1}^{T}\tilde{\varepsilon}_{i}\tilde{\varepsilon}_{i-1}^{t}-\frac{1}{T}\sum_{i=1}^{T}\varepsilon_{i}\varepsilon_{i-1}^{t}\right)\\
&\leq \frac{1}{p}
\mathrm{rank}\left(\frac{1}{T}\sum_{i=1}^{T}\tilde{\varepsilon}_{i}(\tilde{\varepsilon}_{i-1}^{t}-\varepsilon_{i-1}^{t})\right)
+\frac{1}{p} \mathrm{rank}\left(\frac{1}{T}\sum_{i=1}^{T}(\tilde{\varepsilon}_{i}-\varepsilon_{i})\varepsilon_{i-1}^{t}\right)\\
&\leq \frac{1}{p}\sum_{i=1}^{T}
\mathrm{rank}\left(\frac{1}{T}\tilde{\varepsilon}_{i}(\tilde{\varepsilon}_{i-1}^{t}-\varepsilon_{i-1}^{t})\right)
+\frac{1}{p}\sum_{i=1}^{T} \mathrm{rank}\left(\frac{1}{T}(\tilde{\varepsilon}_{i}-\varepsilon_{i})\varepsilon_{i-1}^{t}\right)\\
& \leq
\frac{2}{p}\sum_{t=0}^{T}\sum_{i=1}^{p}I_{(|\varepsilon_{it}|\geq\eta
T^{1/4})},\end{aligned}$$
$$\begin{aligned}
\mathbb{E}\left(\frac{1}{p}\sum_{t=0}^{T}\sum_{i=1}^{p}I_{(|\varepsilon_{it}|\geq\eta
T^{1/4})}\right)
&\leq \frac{1}{p}\sum_{t=0}^{T}\sum_{i=1}^{p}\mathbb{E}\left(\frac{|\varepsilon_{it}|^{4}}{\eta^{4}\cdot T}I_{(|\varepsilon_{it}|\geq\eta T^{1/4})}\right)\\
& =
\frac{1}{\eta^{4}pT}\sum_{i=1}^{p}\sum_{t=0}^{T}\mathbb{E}\left(|\varepsilon_{it}|^{4}I_{(|\varepsilon_{it}|\geq\eta
T^{1/4})}\right) =o\left(1\right),\end{aligned}$$
$$\begin{aligned}
Var\left(\frac{1}{p}\sum_{t=0}^{T}\sum_{i=1}^{p}I_{(|\varepsilon_{it}|\geq\eta
T^{1/4})}\right)
&= \frac{1}{p^{2}}\sum_{t=0}^{T}\sum_{i=1}^{p}Var\left(I_{(|\varepsilon_{it}|\geq\eta T^{1/4})}\right)\\
&\leq
\frac{1}{p^{2}}\sum_{t=0}^{T}\sum_{i=1}^{p}\mathbb{E}\left(I_{(|\varepsilon_{it}|\geq\eta
T^{1/4})}\right) =o\left(\frac{1}{T}\right).\end{aligned}$$
Applying Bernstein’s inequality
$$\mathbb{P}\left(|S_{n}|\geq\varepsilon\right)\leq2\exp\left(-\frac{\varepsilon^{2}}{2\left(B_{n}^{2}+b\varepsilon\right)}\right),$$
where $S_{n}=\sum_{i=1}^{n}X_{i}$, $B_{n}^2=\mathbb{E}S_{n}^{2}$, $X_{i}$ are i.i.d bounded by b, we can get that, for any small $\varepsilon>0$, $$\mathbb{P}\left(\frac{1}{p}\sum_{t=0}^{T}\sum_{i=1}^{p}I_{(|\varepsilon_{it}|\geq\eta
T^{1/4})}\geq\varepsilon\right)\leq2\exp\left(-\frac{\varepsilon^{2}}{2\left(\frac{\varepsilon}{p}+o\left(\frac{1}{T}\right)\right)}\right)=2\exp\left(-K_{\varepsilon}p\right),$$ which is summable, then by Borel-Cantelli lemma,
$$a.s.\lVert F^{A}-F^{\tilde{A}}\rVert \rightarrow0, \mbox{as
}T\rightarrow\infty.$$
Centralization
--------------
Let $\hat{\varepsilon}_{it}=\tilde{\varepsilon}_{it}-\mathbb{E}\left(\tilde{\varepsilon}_{it}\right)$, $~\hat{\varepsilon}_{t}=\left(\hat{\varepsilon}_{1t},\cdots,\hat{\varepsilon}_{pt}\right)$, $~\hat{A}=\frac{1}{T^{2}}\left(\sum_{i=1}^{T}\hat{\varepsilon}_{i}\hat{\varepsilon}_{i-1}^{t}\right)\left(\sum_{j=1}^{T}\hat{\varepsilon}_{j-1}\hat{\varepsilon}_{j}^{t}\right)$.
With Theorem A.46 of [@BS10], $$L^{4}\left(F^{AA^{*}},F^{BB^{*}}\right)\leq\frac{2}{p^{2}}tr\left(AA^{*}+BB^{*}\right)tr\left(\left(A-B\right)\left(A-B\right)^{*}\right),$$
we have
$$\begin{aligned}
L^{4}\left(F^{\hat{A}},F^{\tilde{A}}\right) & \leq \frac{2}{p^{2}}tr\left(\frac{1}{T^{2}}\left(\sum_{i=1}^{T}\hat{\varepsilon}_{i}\hat{\varepsilon}_{i-1}^{t}\right)\left(\sum_{j=1}^{T}\hat{\varepsilon}_{j-1}\hat{\varepsilon}_{j}^{t}\right)+\frac{1}{T^{2}}\left(\sum_{i=1}^{T}\tilde{\varepsilon}_{i}\tilde{\varepsilon}_{i-1}^{t}\right)\left(\sum_{j=1}^{T}\tilde{\varepsilon}_{j-1}\tilde{\varepsilon}_{j}^{t}\right)\right)\\
& \quad \cdot tr\left(\frac{1}{T^{2}}\left(\sum_{i=1}^{T}\hat{\varepsilon}_{i}\hat{\varepsilon}_{i-1}^{t}-\sum_{i=1}^{T}\tilde{\varepsilon}_{i}\tilde{\varepsilon}_{i-1}^{t}\right)\left(\sum_{j=1}^{T}\hat{\varepsilon}_{j-1}\hat{\varepsilon}_{j}^{t}-\sum_{j=1}^{T}\tilde{\varepsilon}_{j-1}\tilde{\varepsilon}_{j}^{t}\right)\right)\\
& := N_1\cdot N_2.\end{aligned}$$
For $N_2$, $$\begin{aligned}
\label{eq12}
N_2 & = tr\left(\frac{1}{T^{2}}\left(\sum_{i=1}^{T}\hat{\varepsilon}_{i}\hat{\varepsilon}_{i-1}^{t}-\sum_{i=1}^{T}\tilde{\varepsilon}_{i}\tilde{\varepsilon}_{i-1}^{t}\right)\left(\sum_{j=1}^{T}\hat{\varepsilon}_{j-1}\hat{\varepsilon}_{j}^{t}-\sum_{j=1}^{T}\tilde{\varepsilon}_{j-1}\tilde{\varepsilon}_{j}^{t}\right)\right)\nonumber\\
&=
tr\left(\frac{1}{T^{2}}\sum_{i=1}^{T}\left(\mathbb{E}\left(\tilde{\varepsilon}_{i}\right)\mathbb{E}\left(\tilde{\varepsilon}_{i-1}^{t}\right)-\mathbb{E}\left(\tilde{\varepsilon}_{i}\right)\tilde{\varepsilon}_{i-1}^{t}-\tilde{\varepsilon}_{i}\mathbb{E}\left(\tilde{\varepsilon}_{i-1}^{t}\right)\right)\right.\nonumber
\\
&\quad \cdot \left.\sum_{i=1}^{T}\left(\mathbb{E}\left(\tilde{\varepsilon}_{i}\right)\mathbb{E}\left(\tilde{\varepsilon}_{i-1}^{t}\right)-\mathbb{E}\left(\tilde{\varepsilon}_{i}\right)\tilde{\varepsilon}_{i-1}^{t}-\tilde{\varepsilon}_{i}\mathbb{E}\left(\tilde{\varepsilon}_{i-1}^{t}\right)\right)\right)^{t}\nonumber\\
&= \left\lVert \frac{1}{T}\sum_{i=1}^{T}\left(\mathbb{E}\left(\tilde{\varepsilon}_{i}\right)\mathbb{E}\left(\tilde{\varepsilon}_{i-1}^{t}\right)-\mathbb{E}\left(\tilde{\varepsilon}_{i}\right)\tilde{\varepsilon}_{i-1}^{t}-\tilde{\varepsilon}_{i}\mathbb{E}\left(\tilde{\varepsilon}_{i-1}^{t}\right)\right)\right\rVert ^{2}\nonumber\\
&\leq 2\left\lVert
\frac{1}{T}\sum_{i=1}^{T}\mathbb{E}\left(\tilde{\varepsilon}_{i}\right)\mathbb{E}\left(\tilde{\varepsilon}_{i-1}^{t}\right)\right\rVert^{2}+2\left\lVert
\frac{1}{T}\sum_{i=1}^{T}\mathbb{E}\left(\tilde{\varepsilon}_{i}\right)\tilde{\varepsilon}_{i-1}^{t}\right\rVert^{2}+2\left\lVert
\frac{1}{T}\sum_{i=1}^{T}\tilde{\varepsilon}_{i}\mathbb{E}\left(\tilde{\varepsilon}_{i-1}^{t}\right)\right\rVert^{2}.\end{aligned}$$ Consider the second term, we have $$\begin{aligned}
& \left\lVert \frac{1}{T}\sum_{i=1}^{T}\mathbb{E}\left(\tilde{\varepsilon}_{i}\right)\tilde{\varepsilon}_{i-1}^{t}\right\rVert^{2}=\frac{1}{T^{2}}\sum_{i,j=1}^{p}\left(\sum_{t=1}^{T}\tilde{\varepsilon}_{j,t-1}\mathbb{E}\left(\tilde{\varepsilon}_{it}\right)\right)^{2}\\
=& \frac{1}{T^{2}}\sum_{i,j=1}^{p}\sum_{t=1}^{T}\tilde{\varepsilon}_{j,t-1}^{2}\left(\mathbb{E}\left(\tilde{\varepsilon}_{it}\right)\right)^{2}+\frac{1}{T^{2}}\sum_{i,j=1}^{p}\sum_{t_{1}\neq t_{2}}\tilde{\varepsilon}_{j,t_{1}-1}\tilde{\varepsilon}_{j,t_{2}-1}\mathbb{E}\left(\tilde{\varepsilon}_{it_{1}}\right)\mathbb{E}\left(\tilde{\varepsilon}_{it_{2}}\right)\\
{=:} & M_{1}+M_{2}.\end{aligned}$$
Notice that $\sup_{1\leq i\leq p,1\leq t\leq
T}\mathbb{E}\left(\varepsilon_{it}^{4}\right)<M$, we have
$$\begin{aligned}
\mathbb{E}\left(M_{1}\right) & = \frac{1}{T^{2}}\sum_{i,j=1}^{p}\sum_{t=1}^{T}\mathbb{E}\left(\tilde{\varepsilon}_{j,t-1}^{2}\right)\left(\mathbb{E}\left(\tilde{\varepsilon}_{it}\right)\right)^{2}\\
& \leq \frac{C_{1}}{T^{2}}\sum_{i,j=1}^{p}\sum_{t=1}^{T}\left(\mathbb{E}\left(|\varepsilon_{it}|I_{(|\varepsilon_{it}|\geq\eta T^{1/4})}\right)\right)^{2}\\
& \leq \frac{C_{1}}{T^{2}}\sum_{i,j=1}^{p}\sum_{t=1}^{T}\frac{1}{\eta^{6}\cdot T^{3/2}}\left(\mathbb{E}\left(|\varepsilon_{it}|^{4}I_{(|\varepsilon_{it}|\geq\eta T^{1/4})}\right)\right)^{2}\\
& = O\left(T^{-\frac{1}{2}}\right),\end{aligned}$$
Moreover, $$\begin{aligned}
Var\left(M_{1}\right) & = \frac{1}{T^{4}}\sum_{j=1}^{p}\sum_{t=1}^{T}\mathbb{E}\left(\tilde{\varepsilon}_{j,t-1}^{2}-\mathbb{E}\left(\tilde{\varepsilon}_{j,t-1}^{2}\right)\right)^{2}\left(\sum_{i=1}^{p}\left(\mathbb{E}\left(\tilde{\varepsilon}_{it}\right)\right)^{2}\right)^{2}\\
& \leq \frac{1}{T^{4}}\sum_{j=1}^{p}\sum_{t=1}^{T}\mathbb{E}\left(\tilde{\varepsilon}_{j,t-1}^{2}\right)^{4}\left(\sum_{i=1}^{p}\left(\mathbb{E}\left(|\varepsilon_{it}|I_{(|\varepsilon_{it}|\geq\eta\cdot T^{1/4})}\right)\right)^{2}\right)^{2}\\
& \leq
\frac{C_{2}}{T^{4}}\sum_{j=1}^{p}\sum_{t=1}^{T}\frac{1}{T^{3}}\left(\sum_{i=1}^{p}\left(\mathbb{E}\left(|\varepsilon_{it}|^{4}I_{(|\varepsilon_{it}|\geq\eta\cdot
T^{1/4})}\right)\right)^{2}\right)^{2}=O\left(T^{-3}\right).\end{aligned}$$ Therefore, $a.s. \quad M_{1}\rightarrow0, \mbox{as }
T\rightarrow\infty$.
For the term $M_2$, we have $$\begin{aligned}
\mathbb{E}\left(M_{2}\right) & = \frac{1}{T^{2}}\sum_{i,j=1}^{p}\sum_{t_{1}\neq t_{2}}\mathbb{E}\left(\tilde{\varepsilon}_{j,t_{1}-1}\tilde{\varepsilon}_{j,t_{2}-1}\right)\mathbb{E}\left(\tilde{\varepsilon}_{it_{1}}\right)\mathbb{E}\left(\tilde{\varepsilon}_{it_{2}}\right)\\
& = \frac{1}{T^{2}}\sum_{i,j=1}^{p}\sum_{t_{1}\neq t_{2}}\mathbb{E}\left(\tilde{\varepsilon}_{j,t_{1}-1}\right)\mathbb{E}\left(\tilde{\varepsilon}_{j,t_{2}-1}\right)\mathbb{E}\left(\tilde{\varepsilon}_{it_{1}}\right)\mathbb{E}\left(\tilde{\varepsilon}_{it_{2}}\right)\\
& \leq \frac{1}{T^{2}}\sum_{i,j=1}^{p}\sum_{t_{1}\neq
t_{2}}\frac{1}{\eta^{12}\cdot T^{3}}\left(\sup_{1\leq i\leq
p,0\leq t\leq
T}\mathbb{E}\left(|\varepsilon_{it}|^{4}I_{(|\varepsilon_{it}|\geq\eta\cdot
T^{1/4})}\right)\right)^{4}=O\left(T^{-1}\right),\end{aligned}$$
$$\begin{aligned}
Var\left(M_{2}\right) & = \frac{1}{T^{4}}\sum_{j=1}^{p}\sum_{t_{1}\neq t_{2}}Var\left(\tilde{\varepsilon}_{j,t_{1}-1}\tilde{\varepsilon}_{j,t_{2}-1}\right)\left(\sum_{i=1}^{p}\mathbb{E}\left(\tilde{\varepsilon}_{it_{1}}\right)\mathbb{E}\left(\tilde{\varepsilon}_{it_{2}}\right)\right)^{2}\\
& \leq \frac{1}{T^{4}}\sum_{j=1}^{p}\sum_{t_{1}\neq t_{2}}\mathbb{E}\left(\tilde{\varepsilon}_{j,t_{1}-1}^{2}\right)\mathbb{E}\left(\tilde{\varepsilon}_{j,t_{2}-1}^{2}\right)\left(\sum_{i=1}^{p}\left(\sup_{1\leq i\leq p,0\leq t\leq T}\mathbb{E}\left(\tilde{\varepsilon}_{it}\right)\right)^{2}\right)^{2}\\
& \leq \frac{C_{3}}{T^{4}}\sum_{j=1}^{p}\sum_{t_{1}\neq
t_{2}}\frac{1}{T^{3}}\left(\sum_{i=1}^{p}\left(\sup_{1\leq i\leq
p,0\leq t\leq
T}\mathbb{E}\left(|\varepsilon_{it}|^{4}I_{(|\varepsilon_{it}|\geq\eta\cdot
T^{1/4})}\right)\right)^{2}\right)^{2}=O\left(T^{-2}\right).\end{aligned}$$
Therefore, a.s. $M_{2}\rightarrow0$, as $T\rightarrow\infty$.
Consequently, $\left\lVert
\frac{1}{T}\sum_{i=1}^{T}\mathbb{E}\left(\tilde{\varepsilon}_{i}\right)\tilde{\varepsilon}_{i-1}^{t}\right\rVert^{2}\rightarrow
0, a.s.$ Similarly, we can prove that the last term in equation tends to zero almost surely. As for the first term, we have $$\begin{aligned}
\left\lVert\frac{1}{T}\sum_{i=1}^{T}\mathbb{E}\left(\tilde{\varepsilon}_{i}\right)\mathbb{E}\left(\tilde{\varepsilon}_{i-1}^{t}\right)\right\rVert^{2} & = \sum_{i,j=1}^{p}\left(\frac{1}{T}\sum_{t=1}^{T}\left(\mathbb{E}\left(\tilde{\varepsilon}_{it}\right)\mathbb{E}\left(\tilde{\varepsilon}_{j,t-1}\right)\right)\right)^{2}\\
& = \frac{1}{T^{2}}\sum_{i,j=1}^{p}\sum_{t_{1}=1}^{T}\sum_{t_{2}=1}^{T}\mathbb{E}\left(\tilde{\varepsilon}_{it_{1}}\right)\mathbb{E}\left(\tilde{\varepsilon}_{j,t_{1}-1}\right)\mathbb{E}\left(\tilde{\varepsilon}_{it_{2}}\right)\mathbb{E}\left(\tilde{\varepsilon}_{j,t_{2}-1}\right)\\
& \leq
\frac{C_{4}}{T^{2}}\sum_{i,j=1}^{p}\sum_{t_{1}=1}^{T}\sum_{t_{2}=1}^{T}\frac{1}{T^{3}}\left(\sup_{1\leq
i\leq p,0\leq t\leq
T}\mathbb{E}\left(|\varepsilon_{it}|^{4}I_{(|\varepsilon_{it}|\geq\eta\cdot
T^{1/4})}\right)\right)^{4}=O\left(T^{-1}\right).\end{aligned}$$ Therefore $$N_1=tr\left(\frac{1}{T^{2}}\left(\sum_{i=1}^{T}\hat{\varepsilon}_{i}\hat{\varepsilon}_{i-1}^{t}-\sum_{i=1}^{T}\tilde{\varepsilon}_{i}\tilde{\varepsilon}_{i-1}^{t}\right)\left(\sum_{j=1}^{T}\hat{\varepsilon}_{j-1}\hat{\varepsilon}_{j}^{t}-\sum_{j=1}^{T}\tilde{\varepsilon}_{j-1}\tilde{\varepsilon}_{j}^{t}\right)\right)\rightarrow0,
a.s.$$ Now, we consider $N_1$, $$\frac{1}{p^{2}}tr\left(\frac{1}{T^{2}}\left(\sum_{i=1}^{T}\hat{\varepsilon}_{i}\hat{\varepsilon}_{i-1}^{t}\right)\left(\sum_{j=1}^{T}\hat{\varepsilon}_{j-1}\hat{\varepsilon}_{j}^{t}\right)+\frac{1}{T^{2}}\left(\sum_{i=1}^{T}\tilde{\varepsilon}_{i}\tilde{\varepsilon}_{i-1}^{t}\right)\left(\sum_{j=1}^{T}\tilde{\varepsilon}_{j-1}\tilde{\varepsilon}_{j}^{t}\right)\right){=:}M_{3}+M_{4},$$ Firstly, for $M_{3}$, since $\mathbb{E}\left(\hat{\varepsilon}_{it}\right)=0$, $$\begin{aligned}
\mathbb{E}\left(M_{3}\right) & = \mathbb{E}\left(\frac{1}{p^{2}T^{2}}\sum_{i,j=1}^{p}\left(\sum_{t=1}^{T}\hat{\varepsilon}_{it}\hat{\varepsilon}_{j,t-1}\right)^{2}\right)\\
& =
\frac{1}{p^{2}T^{2}}\sum_{i,j=1}^{p}\sum_{t=1}^{T}\mathbb{E}\left(\hat{\varepsilon}_{it}^{2}\right)\mathbb{E}\left(\hat{\varepsilon}_{j,t-1}^{2}\right)=O\left(\frac{1}{T}\right).\end{aligned}$$ Moreover, $$\begin{aligned}
Var\left(M_{3}\right) & = \mathbb{E}\left(\frac{1}{p^{2}T^{2}}\sum_{i,j=1}^{p}\left(\sum_{t=1}^{T}\hat{\varepsilon}_{it}\hat{\varepsilon}_{j,t-1}\right)^{2}\right)^{2}-\left(\mathbb{E}\left(M_{3}\right)\right)^{2}\\
& = \frac{1}{p^{4}T^{4}}\mathbb{E}\left(\sum_{i,j=1}^{p}\sum_{t=1}^{T}\hat{\varepsilon}_{it}^{2}\hat{\varepsilon}_{j,t-1}^{2}\right)^{2}+\frac{1}{p^{4}T^{4}}\mathbb{E}\left(\sum_{i,j=1}^{p}\sum_{t_{1}\neq t_{2}}\hat{\varepsilon}_{it_{1}}\hat{\varepsilon}_{j,t_{1}-1}\hat{\epsilon}_{it_{2}}\hat{\varepsilon}_{j,t_{2}-1}\right)^{2}+O\left(\frac{1}{T^{2}}\right)\\
& \leq
O\left(\frac{1}{T^{2}}\right)+O\left(\frac{1}{T^{3}}\right)+O\left(\frac{1}{T^{2}}\right)=O\left(\frac{1}{T^{2}}\right).\end{aligned}$$ Therefore $M_3{\rightarrow}0$, a.s. Next for $M_4$,
$$\begin{aligned}
\mathbb{E}\left(M_{4}\right) & = \mathbb{E}\left(\frac{1}{p^{2}T^{2}}\sum_{i,j=1}^{p}\left(\sum_{t=1}^{T}\tilde{\varepsilon}_{it}\tilde{\varepsilon}_{j,t-1}\right)^{2}\right)\\
& = \frac{1}{p^{2}T^{2}}\sum_{i,j=1}^{p}\sum_{t=1}^{T}\mathbb{E}\tilde{\varepsilon}_{it}^{2}\mathbb{E}\tilde{\varepsilon}_{j,t-1}^{2}+\frac{1}{p^{2}T^{2}}\sum_{i,j=1}^{p}\sum_{t_{1}\neq t_{2}}\mathbb{E}\left(\tilde{\varepsilon}_{it_{1}}\right)\mathbb{E}\left(\tilde{\varepsilon}_{j,t_{1}-1}\right)\mathbb{E}\left(\tilde{\varepsilon}_{it_{2}}\right)\mathbb{E}\left(\tilde{\varepsilon}_{j,t_{2}-1}\right)\\
& \leq
O\left(\frac{1}{T}\right)+\frac{1}{p^{2}T^{2}}\sum_{i,j=1}^{p}\sum_{t_{1}\neq
t_{2}}\frac{1}{\eta^{12}T^{3}}\left(\sup_{1\leq i\leq p,0\leq
t\leq
T}\mathbb{E}\left(|\varepsilon_{it}|^{4}I_{(|\varepsilon_{it}|\geq\eta\cdot
T^{1/4})}\right)\right)^{4}=O\left(\frac{1}{T}\right).\end{aligned}$$
$$\begin{aligned}
Var\left(M_{4}\right) & = \frac{1}{p^{4}T^{4}}Var\left(\sum_{i,j=1}^{p}\left(\sum_{t=1}^{T}\tilde{\varepsilon}_{it}\tilde{\varepsilon}_{j,t-1}\right)^{2}\right)\\
& \leq \frac{1}{p^{4}T^{4}}\mathbb{E}\left(\sum_{i,j=1}^{p}\left(\sum_{t=1}^{T}\tilde{\varepsilon}_{it}\tilde{\varepsilon}_{j,t-1}\right)^{2}\right)^{2}\\
& = \frac{1}{p^{4}T^{4}}\mathbb{E}\left(\sum_{i,j=1}^{p}\sum_{t=1}^{T}\tilde{\varepsilon}_{it}^{2}\tilde{\varepsilon}_{j,t-1}^{2}\right)^{2}+\frac{1}{p^{4}T^{4}}\mathbb{E}\left(\sum_{i,j=1}^{p}\sum_{t_{1}\neq t_{2}}\tilde{\varepsilon}_{it_{1}}\tilde{\varepsilon}_{j,t_{1}-1}\tilde{\varepsilon}_{it_{2}}\tilde{\varepsilon}_{j,t_{2}-1}\right)^{2}\\
& \leq
O\left(\frac{1}{T^{2}}\right)+O\left(\frac{1}{T^{6}}\right)=O\left(\frac{1}{T^{2}}\right).\end{aligned}$$
Therefore, $M_{4}\rightarrow0, a.s. $. All in all, $$L^{4}\left(F^{\hat{A}},F^{\tilde{A}}\right)\leq N_1\cdot N_2
\leq4\left(M_{3}+M_{4}\right)\left(M_{1}+M_{2}\right)\rightarrow 0,
a.s. T\rightarrow\infty.$$
Rescaling
---------
Define $\hat{\sigma}_{ij}^{2}=\mathbb{E}|\hat{\varepsilon}_{ij}|^{2}=\mathbb{E}|\tilde{\varepsilon}_{ij}-\mathbb{E}\tilde{\varepsilon}_{ij}|^{2}$, we can see that as $T\rightarrow\infty$, $\hat{\sigma}_{ij}^{2}\rightarrow1$ since $\mathbb{E}(\varepsilon_{ij})=0$, $Var\left(\varepsilon_{ij}\right)=1$.
According to Theorem A.46 of [@BS10], we have $$\begin{aligned}
L^{4}\left(F^{\hat{A}},F^{\hat{\sigma}_{ij}^{-4}\hat{A}}\right) & \leq \frac{2}{p^{2}}\left[\frac{1+\hat{\sigma}_{ij}^{-4}}{T^{2}}tr\left(\left(\sum_{i=1}^{T}\hat{\varepsilon}_{i}\hat{\varepsilon}_{i-1}^{t}\right)\left(\sum_{j=1}^{T}\hat{\varepsilon}_{j-1}\hat{\varepsilon}_{j}^{t}\right)\right)\right]\\
&\cdot\left[\frac{1-\hat{\sigma}_{ij}^{-4}}{T^{2}}tr\left(\left(\sum_{i=1}^{T}\hat{\varepsilon}_{i}\hat{\varepsilon}_{i-1}^{t}\right)\left(\sum_{j=1}^{T}\hat{\varepsilon}_{j-1}\hat{\varepsilon}_{j}^{t}\right)\right)\right]\\
& =
2\left(1-\hat{\sigma}_{ij}^{-8}\right)\left[\frac{1}{pT^{2}}tr\left(\left(\sum_{i=1}^{T}\hat{\varepsilon}_{i}\hat{\varepsilon}_{i-1}^{t}\right)\left(\sum_{j=1}^{T}\hat{\varepsilon}_{j-1}\hat{\varepsilon}_{j}^{t}\right)\right)\right]^{2}.\end{aligned}$$
Consider $M_{5}:=\frac{1}{pT^{2}}tr\left(\left(\sum_{i=1}^{T}\hat{\varepsilon}_{i}\hat{\varepsilon}_{i-1}^{t}\right)\left(\sum_{j=1}^{T}\hat{\varepsilon}_{j-1}\hat{\varepsilon}_{j}^{t}\right)\right)$,
$$\begin{aligned}
\mathbb{E}\left(M_{5}\right) & = \frac{1}{pT^{2}}\sum_{i,j=1}^{p}\mathbb{E}\left(\sum_{t=1}^{T}\hat{\varepsilon}_{it}\hat{\varepsilon}_{j,t-1}\right)^{2}\\
& =
\frac{1}{pT^{2}}\sum_{i,j=1}^{p}\sum_{t=1}^{T}\mathbb{E}\left(\hat{\varepsilon}_{it}^{2}\right)\mathbb{E}\left(\hat{\varepsilon}_{j,t-1}^{2}\right)=c\hat{\sigma}_{ij}^{4}.\end{aligned}$$
Moreover, $$\begin{aligned}
Var\left(M_{5}\right) & \leq \mathbb{E}\left(\frac{1}{pT^{2}}\sum_{i,j=1}^{p}\left(\sum_{t=1}^{T}\hat{\varepsilon}_{it}\hat{\varepsilon}_{j,t-1}\right)^{2}\right)^{2}\\
& = \frac{1}{p^{2}T^{4}}\mathbb{E}\left(\sum_{i,j=1}^{p}\sum_{t=1}^{T}\hat{\varepsilon}_{it}^{2}\hat{\varepsilon}_{j,t-1}^{2}\right)^{2}+\frac{1}{p^{2}T^{4}}\mathbb{E}\left(\sum_{i,j=1}^{p}\sum_{t_{1}\neq t_{2}}\hat{\varepsilon}_{it_{1}}\hat{\varepsilon}_{j,t_{1}-1}\hat{\varepsilon}_{it_{2}}\hat{\varepsilon}_{j,t_{2}-1}\right)^{2}\\
& = O\left(1\right)+O\left(\frac{1}{T^{2}}\right)=O\left(1\right).\end{aligned}$$ Therefore $L^{4}\left(F^{\hat{A}},F^{\hat{\sigma}_{ij}^{-4}\hat{A}}\right)\rightarrow0,
a.s.$
[99]{}
Anderson, G. W., Guionnet, A., and Zeitouni, O. (2010). [*An introduction to random matrices*]{}(No. 118). Cambridge University Press.
Bai, Z.D. and Silverstein, J.W. (1998). No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. [*The Annals of Probability*]{}, Vol. 26, NO. 1, 316-345.
Bai, Z. and Silverstein, J. W. (2010). [*Spectral Analysis of Large Dimensional Random Matrices*]{} (2nd ed.) Springer.
Bai, Z.D. and Yin, Y.Q.(1993). Limit of the smallest eigenvalues of large dimensional covariance matrix. [*Ann. Probab.*]{} [**21**]{}(3), 1275-1294. Lam, C., Yao, Q. and Bathia, N. (2011). Estimation of latent factors for high-dimensional time series. [*Biometrika*]{} [**98(4)**]{}, 901-918. Lam, C. and Yao, Q. (2012). Factor modeling for high-dimensional time series: inference for the number of factors. [*Ann. Statist.* ]{} [**40(2)**]{}, 694-726.
Mar$\check{c}$enko, V.A. and Pastur, L.A.(1967), Distribution of eigenvalues in certain sets of random matrices. [*Math. USSR-Sb.*]{} [**1**]{}, 457-483.
Pastur, L. A. and Shcherbina, M. (2010). [*Eigenvalue distribution of large random matrices.*]{} American Mathematical Society, Providence, Rhode Island
Silverstein, J.W. (1995) Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices. [*J. Multivariate Anal.*]{} [**5**]{}, 331-339
|
[**Conservation laws of the system of equations\
of one-dimensional shallow water over uneven bottom in Lagrange’s variables**]{}
Alexander V. Aksenov$^{a,b}$ and Konstantin P. Druzhkov$^{a,b}$
$^{a}$ Lomonosov Moscow State University, 1 Leninskiye Gory,\
Main Building, 119991 Moscow, Russia\
$^{b}$ Keldysh Institute of Applied Mathematics RAS,\
4 Miusskaya Square, 125047 Moscow, Russia
E-mail: aksenov.av@gmail.com and Konstantin.Druzhkov@gmail.com
**Abstract**
The system of equations of one-dimensional shallow water over uneven bottom in Euler’s and Lagrange’s variables is considered. Intermediate system of equations is introduced. Hydrodynamic conservation laws of intermediate system of equations is used to find all first order conservation laws of shallow water equations in Lagrange’s variable for all bottom profiles. The obtained conservation laws are compared with the hydrodynamic conservation laws of the system of equations of one-dimensional shallow water over uneven bottom in Euler’s variables. Bottom profiles are given for which there are additional conservation laws.
Keywords: shallow water, conservation laws, Lagrange’s variable, Noether’s theorem.
Introduction
============
There are various approaches to finding conservation laws of equations of mathematical physics [@Noether; @Olver; @VinKr; @BlumanChAnco; @Ibr; @BlChA]. The most widely known method of constructing of conservation laws is based on Noether’s theorem [@Noether]. This method uses symmetries.
Many works are devoted to the construction of conservation laws of equations in hydro- and gas-dynamics [@Shmygl; @MelSK; @MelS; @PolZ; @PolZZh].
The hydrodynamic conservation laws of the one-dimensional shallow water equations over uneven bottom in Euler’s variables were obtained in [@AksDr].
In the present work, the first-order conservation laws of the shallow water equations in Lagrangian’s variables for all bottom profiles are obtained without using of symmetries.
Basic Equations
===============
In dimensionless variables, the system of one-dimensional shallow-water equations over an uneven bottom has the following form [@Stoker]: $$\begin{aligned}
&u_t + uu_x + \eta_x = 0,\\
&\eta_t + ((\eta + h(x))u)_x = 0.
\label{SW}
\end{aligned}$$ Here $h(x)$ is the thickness of the unperturbed layer of the liquid, $u=u(x,t)$ is the depth-average horizontal velocity, $\eta=\eta(x,t)$ is the deviation of the free surface $(\eta(x, t) + h(x)\ge 0)$. The bottom profile is given by the relation $z=-h(x)$ ($z$ is the vertical coordinate).
**Remark 1.** The system of equations is similar to the system of equations of one-dimensional gas dynamics [@Chernyi; @CourFriedr].
Using the second equation of the system of equations , we introduce a new variable $m=m(x,t)$ and consider the following system of equations $$\begin{aligned}
&u_t + uu_x + \rho_x = h'(x),\\
&m_x = \rho,\\
&m_t = -u\rho,
\end{aligned}
\label{SWLF}$$ where $\rho=\eta+h(x)$.
From the second and third equations it follows that the variable $m$ is Lagrangian’s variable due to the relation $$\frac{dm}{dt}=m_t + u m_x = 0.$$ One can get the equation of one-dimensional shallow water in Lagrange’s variables by choosing $m$ and $t$ as independent variables [@Chernyi] $$x_{tt} - \dfrac{x_{mm}}{x_{m}^3} = h'(x).
\label{LE}$$
One-to-one correspondence between the system of equations and the equation is given by the relations $$\begin{aligned}
u = x_t,\qquad \rho = \dfrac{1}{x_m}.
\end{aligned}
\label{Map}$$
Note that the system of equations is intermediate system between system of equations and the equation . The system of equations is a covering system [@VinKr] for the system of equations .
Conservation Laws of the equation in\
Lagrange’s variables {#ConsL}
=====================================
Under the conservation laws of the system of equations we understand divergent forms for which the solutions of the system equations satisfy the relation $$D_x(P) + D_t(Q) = 0.
\label{Div}$$ Here $P$, $Q$ are functions of independent and dependent variables and their derivatives; $$D_x=\frac{\partial}{\partial x}+m_x\frac{\partial}{\partial m}+
u_x\frac{\partial}{\partial u}+
\rho_x\frac{\partial}{\partial \rho}+\dots, \quad
D_t=\frac{\partial}{\partial t}+m_t\frac{\partial}{\partial m}+
u_t\frac{\partial}{\partial u}+
\rho_t\frac{\partial}{\partial \rho}+\dots$$ are total derivatives in variables $x$ and $t$. Conservation laws for which the equality is satisfied everywhere, we will call trivial conservation laws. The maximum order of derivatives included in the functions $P$ and $Q$ will be called the order of the conservation law. The conservation laws of the zero order will be called hydrodynamic. The conservation laws of the equation are defined similarly.
We recall that conservation laws in divergent form are equivalent to differential 1-forms [@VinKr] $$Q\, dx - P\, dt,$$ which are closed on solutions of the system .
**\[prop1\]Proposition 1.** According to the relations , hydrodynamic conservation law of the system of equations with functions $P$, $Q$ defines the first-order conservation law of the equation with functions $\widetilde{P}=P - x_t\, Q$, $\widetilde{Q}=x_mQ$. The opposite is true.
**Proof.** Denote the total derivatives in the variables $m$, $t$ as $\widetilde{D}_m$ and $\widetilde{D}_t$, and their restrictions on the equation as $\overline{D}_x$ and $\overline{D}_t$. By the relations , these derivatives are related in the following way $$\overline{D}_x = \dfrac{1}{x_m}\widehat{D}_m, \qquad
\overline{D}_t = \widehat{D}_t - \dfrac{x_t}{x_m}\widehat{D}_m.$$ Then $$\begin{aligned}
\overline{D}_x(P) + \overline{D}_t(Q) &= \dfrac{1}{x_m}\widehat{D}_m(P) +
\widehat{D}_t(Q) - \dfrac{x_t}{x_m}\widehat{D}_m(Q) ={}\\
&= \dfrac{1}{x_m}\Bigl(\widehat{D}_m(P - x_tQ) + \widehat{D}_t(x_mQ)\Bigr).
\end{aligned}$$ This implies the validity of the proposition being proved.
Also true the proposition
**Proposition 2.** If the functions $P$ and $Q$ in a conservation law of the system of equations are independent of $m$, then they determine the conservation law of the system of equations . All conservation laws of the system of equations , except the conservation law $$D_t(\eta) + D_x((\eta+h(x))u),$$ are obtained from the conservation laws of the system of equations .
Note that finding of the hydrodynamic conservation laws of the system of equations is easier than finding of the first-order conservation laws of the equation .
Relation on solutions of the system of equations takes the form $$P_x + u_xP_u + \rho_xP_{\rho} + \rho P_m+Q_t +
(h'(x) - \rho_x - uu_x)Q_u - (u\rho_x + u_x\rho)Q_{\rho} - u\rho Q_m \equiv 0.$$ Equating to zero the coefficients of the derivatives $u_x$ and $\rho_x$, we obtain the following overdetermined system of linear equations $$\begin{aligned}
P_u =&\, \rho Q_{\rho} + uQ_u,\\
P_{\rho} =&\, u Q_{\rho} + Q_u,\\
P_{x} =&\, -\rho P_m + u\rho Q_m - h'Q_u - Q_t.
\end{aligned}
\label{ODS}$$ The overdefined system of equations was investigated on compatibility.
According to the proposition 1, solutions of the system of equations can be compared to conservation laws of the equation . Below we provide functions $\widetilde{P}$ and $\widetilde{Q}$, which determine the basis of first-order conservation laws $\widetilde{D}_m(\widetilde{P}) + \widetilde{D}_t(\widetilde{Q})$ of the equation modulo additive trivial conservation laws for all possible bottom profiles $h(x)$.
**1. $h=h(x)$ is arbitrary function.** For any bottom profile $h(x)$, the equation has conservation laws with functions $$\begin{aligned}
&\widetilde{P}_1 = -\dfrac{x_t^2}{2} + \dfrac{1}{x_m} - h(x),\\
&\widetilde{P}_2 = x_t\Bigl(\dfrac{1}{x_m^2} - h^2(x)\Bigr),
\end{aligned}
\qquad
\begin{aligned}
&\widetilde{Q}_1 = x_tx_m,\\
&\widetilde{Q}_2 = x_t^2 + x_m\Bigl(\dfrac{1}{x_m} - h(x)\Bigr)^2.
\end{aligned}$$
**2. $h=a_1x+a_2$.** In this case additional conservation laws of the equation correspond to functions $$\begin{aligned}
&\widetilde{P}_3 = (a_1a_2t^2 - 2a_2x)r + ts^2 - a_2^2 t, \\
&\widetilde{Q}_3 = 2tr + \dfrac{2a_2x - a_1a_2t^2}{s} + a_1t^2 - 2x,\\
&\widetilde{P}_4 = 2mr^3 + 24tr^2s^2 - (18x - 9a_1t^2)rs^2 - 12mrs + 16ts^3,\\
&\widetilde{Q}_4 = 16tr(r^2 + 3s) + (9a_1t^2 - 18x)(r^2 + s) -
\dfrac{6mr^2}{s} - 12 m\ln s,\\
&\widetilde{P}_5 = 10trs^2 + 2mr^2 + (3a_1t^2 - 6x)s^2 - 4ms,\\
&\widetilde{Q}_5 = 10tr^2 + 10ts + (6a_1t^2 - 12x)r - \dfrac{4mr}{s},\\
&\widetilde{P}_{\infty} = p(r, s) - r q(r, s),\qquad
\widetilde{Q}_{\infty} = \dfrac{q(r,s)}{s},
\end{aligned}$$ where $r=x_t-a_1t$, $s=1/x_m$; functions $p(r, s)$, $q(r, s)$ are arbitrary solutions of the system of equations $$p_r = s q_{s} + rq_r\,, \qquad p_{s} = rq_{s} + q_r\,.$$
**3.1. $h=a_1x^2/2 + a_2x + a_3$, $a_1>0$.** In this case additional conservation laws of the equation correspond to functions $$\begin{aligned}
&\widetilde{P}_3 = e^{-t\sqrt{a_1}}\Bigl(x_thh' +
\dfrac{\sqrt{a_1}}{2}\Bigl(\dfrac{1}{x_m^2} - h^2\Bigr)\Bigr),\\
&\widetilde{Q}_3 = e^{-t\sqrt{a_1}}(\sqrt{a_1}x_t + (1 - x_mh)h'),\\
&\widetilde{P}_4 = e^{t\sqrt{a_1}}\Bigl(x_thh' -
\dfrac{\sqrt{a_1}}{2}\Bigl(\dfrac{1}{x_m^2} - h^2\Bigr)\Bigr),\\
&\widetilde{Q}_4 = e^{t\sqrt{a_1}}(-\sqrt{a_1}x_t + (1 - x_mh)h').
\end{aligned}$$
**3.2. $h=a_1x^2/2 + a_2x + a_3$, $a_1<0$.** In this case additional conservation laws of the equation correspond to functions $$\begin{aligned}
&\widetilde{P}_3 = \cos(t\sqrt{-a_1}\,)x_thh' +
\dfrac{\sqrt{-a_1}}{2}\sin(t\sqrt{-a_1}\,)\Bigl(\dfrac{1}{x_m^2} - h^2\Bigr),\\
&\widetilde{Q}_3 = \sin(t\sqrt{-a_1}\,)\sqrt{-a_1}x_t +
\cos(t\sqrt{-a_1}\,)(1 - x_mh)h',\\
&\widetilde{P}_4 = \sin(t\sqrt{-a_1}\,)x_thh' -
\dfrac{\sqrt{-a_1}}{2}\cos(t\sqrt{-a_1}\,)\Bigl(\dfrac{1}{x_m^2} - h^2\Bigr),\\
&\widetilde{Q}_4 = -\cos(t\sqrt{-a_1}\,)\sqrt{-a_1}x_t +
\sin(t\sqrt{-a_1}\,)(1 - x_mh)h'.
\end{aligned}$$
**4. $h=a_1(x + a_2)^{-4/3} + a_3$, $a_1\neq
0$, $x>-a_2$.** In this case additional conservation laws of the equation correspond to functions $$\begin{aligned}
&\widetilde{P}_3 = -\dfrac{5t x_t}{x_m^2} - mx_t^2 +
\dfrac{3(x + a_2)}{x_m^2} + 2m\Bigl(\dfrac{1}{x_m} - h\Bigr),\\
&\widetilde{Q}_3 = -5t\Bigl(x_t^2 + \dfrac{1}{x_m}\Bigr) +
6(x + a_2) x_t + (10 h - 8a_3)t + 2 mx_tx_m\,.
\end{aligned}$$
The conservation laws for cases 2–4 are additional conservation laws to the conservation laws of the general case 1.
Comparison of conservation laws in Euler’s and Lagrangian’s variables
=====================================================================
Conservation laws of the equation that do not correspond to the conservation laws of the system of equations are obtained from the conservation laws of the system of equations , which are depend on the Lagrangian variable $m$. The results of the section \[ConsL\] show that such a conservation laws are exist only in two cases. In the case of $h=a_1x+a_2$ conservation laws, which are defined by the functions $(\widetilde{P}_4, \widetilde{Q}_4)$ and $(\widetilde{P}_5,
\widetilde{Q}_5)$, are not correspond to the conservation laws of the system of equations ; in the case $h=a_1(x + a_2)^{-4/3} +
a_3$ ($a_1\neq 0$, $x>-a_2$) conservation laws, which are defined by the functions $(\widetilde{P}_3, \widetilde{Q}_3)$, also do not comply with the conservation laws of the system of equations . The system of shallow water equations in Eulerian variables has no additional conservation laws in this case [@AksDr]. All other first-order conservation laws of the equation are correspond to the conservation laws of the system of equations .
Acknowledgments
===============
The authors wish to express gratitude to V.A. Dorodnitsyn for constructive discussions and helpful remarks.
The work was supported by the Russian Science Foundation under grant 18-11-00238.
[99]{}
E. Noether, Invariante Variationsprobleme, Nachr. D. König. Gesellsch. D. Wissen. Zu Göttingen, Math.-Phys. Klasse, 235–257 (1918) (English translation: Transport Theory and Stat. Phys., **1** (3), 186–207 (1971)).
P.J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed., Springer (1993).
Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Editors: A.M. Vinogradov and I.S. Krasil’schik, American Mathematical Society, **182** (1999).
G.W. Bluman, A.F.Cheviakov, S.C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Springer (2010).
N.H. Ibragimov, A new conservation theorem,Journal of Mathematical Analysis and Applications, **333** (1), 311-328 (2007).
G.W. Bluman, A.F. Cheviakov, St.C. Anco, Construction of Conservation Laws: How the Direct Method Generalizes Noether’s Theorem, Proceedings of 4th Workshop ”Group Analysis of Differential Equations & Integrability”, 1–23 (2009).
Yu.D. Shmyglevski, Analytical Study of Gas Dynamics and Fluid, Editorial URSS, Moscow (1999) (in Russian).
P. Siriwat, Ch. Kaewmanee, S.V. Meleshko, Symmetries of the hyperbolics shallow water equations and the Green–Naghdi model in Lagrangian coordinates, International Journal of Non-Linear Mechanics, **86**, 185–195 (2016).
P. Siriwat and S.V. Meleshko, Group properties of the extended Green-Naghdi equations, Applied Mathematics Letters, **81**, 1–6 (2018).
A.D. Polyanin, V.F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, 2nd ed., CRC Press, Boca Raton (2012).
A.D. Polyanin, V.F. Zaitsev, A.I. Zhurov, Methods for solving nonlinear equations of mathematical physics and mechanics, Fizmatlit, Moscow (2005) (in Russian).
A.V. Aksenov and K.P. Druzhkov, Conservation laws and symmetries of the shallow water system above rough bottom, Journal of Physics: Conference Series, **722**, 1–7 (2016).
J.J. Stoker, The formation of breakers and bores. The Theory of Nonlinear Wave Propagation in Shallow Water and Open Channels, Comm. Pure Appl. Math., **1** (1), 1–87 (1948).
R. Courant and K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publ. (1948).
G.G. Chernyi, Gas dynamics, Nauka, Moscow (1988) (in Russian).
|
---
abstract: 'The recent BICEP2 B-mode polarization determination of an inflationary tensor-scalar ratio $r=0.2^{+0.07}_{-0.05}$ is in tension with simple scale-free models of inflation due to a lack of a corresponding low multipole excess in the temperature power spectrum which places a limit of $r_{0.002}<0.11$ (95% CL) on such models. Single-field inflationary models that reconcile these two observations, even those where the tilt runs substantially, introduce a scale into the scalar power spectrum. To cancel the tensor excess, and simultaneously explain the excess already present in $\Lambda$CDM, ideally the model should introduce this scale as a relatively sharp transition in the tensor-scalar ratio around the horizon at recombination. We consider models which generate such a step in this quantity and find that they can improve the joint fit to the temperature and polarization data by up to $2\Delta \ln{\cal L} \approx -14$ without changing cosmological parameters. Precision E-mode polarization measurements should be able to test this explanation.'
author:
- Vinícius Miranda
- Wayne Hu
- Peter Adshead
bibliography:
- 'bistep.bib'
title: Steps to Reconcile Inflationary Tensor and Scalar Spectra
---
Introduction
============
The recent BICEP2 measurement of a tensor-scalar ratio $r=0.2^{+0.07}_{-0.05}$ from degree scale B-mode polarization of the cosmic-microwave background (CMB) [@Ade:2014xna] is in “moderately-strong" tension with slow-roll inflation models that predict scale-free, albeit slightly tilted ($1-n_s \ll 1$) power-law power spectra. This tension is due to the implied excess in the temperature spectrum at low multipoles which is not observed and restricts $r_{0.002}< 0.11$ (95% CL) in this context [@Ade:2013uln].
These findings can be reconciled in the single-field inflationary paradigm by introducing a scale into the scalar power spectra to suppress power on these large-angular scales. For example a large running of tilt, $dn_s/d\ln k \sim -0.02$, is possible as a compromise [@Ade:2014xna]. Here the scale introduced is associated with the scalar spectrum transiently passing through a scale-invariant slope near observed scales. However, such a large running is uncomfortable in the simplest models of inflation which typically produce running of order $\mathcal{O}[\left(1-n_s\right)^2]$. Moreover, a large running also requires further additional parameters in order that inflation does not end too quickly after the observed scales leave the horizon [@Easther:2006tv].
The temperature anisotropy excess implied by tensors is also not a smooth function of scale, but rather cut off at the horizon at recombination. To counter this excess, a transition in the scalar power spectrum that occurs more sharply, though coincidentally near these scales, would be preferred. Such a transition can occur without affecting the tensor spectrum if there is a slow-roll violating step in the tensor-scalar ratio while the Hubble rate is left nearly fixed. In this work we consider the effects of placing such a feature near scales associated with the horizon at recombination, thereby suppressing the scalar spectrum on large scales.
This slow-roll violating behavior also produces oscillations in the power spectrum [@Adams:2001vc; @Peiris:2003ff; @Park:2012rh; @Miranda:2012rm] and generates enhanced non-Gaussianity [@Chen:2006xjb; @Chen:2008wn] if this transition occurs in much less than an efold. For transitions that alleviate the tensor-scalar tension, these oscillations would violate tight constraints on the acoustic peaks and hence only transitions that occur over at least an efold are allowed. The resulting non-Gaussianity is then undetectable [@Adshead:2011bw; @Adshead:2012xz]. Throughout, we work in natural units where the reduced Planck mass $M_{\rm Pl} = (8\pi G_N)^{-1/2} = 1$ as well as $c = \hbar = 1$.
Step Solutions {#sec:step}
==============
In slow roll inflation, the tensor power spectrum in each gravitational wave polarization state is directly related to the Hubble scale during inflation $$\Delta_{+,\times}^2 = \frac{H^2}{2\pi^2},$$ whereas the scalar or curvature power spectrum is given by $$\Delta_{{\cal R}}^2 = \frac{H^2 }{8\pi^2 \epsilon_H c_s},$$ where $\epsilon_H = -d\ln H/d\ln a$ and $c_s$ is the sound speed, yielding a tensor-scalar ratio $r= 4 \Delta_{+,\times}^2 / \Delta_{{\cal R}}^2 = 16\epsilon_H c_s$. The addition of a nearly scale invariant tensor spectrum to the CMB temperature anisotropy produces excess power below $\ell\approx 100$ which at $r=0.2$ is difficult to accommodate in slow roll inflation where the scalar spectrum is, to a good approximation, a scale-free power law (see Fig. \[plot:cl\]).
The scalar power spectrum can be changed largely without affecting the tensors if the quantity $\epsilon_H c_s$ changes while $\epsilon_H$ remains small. As shown in Fig. \[plot:cl\], the excess power resembles a step in this quantity on scales near the horizon at recombination. Hence to alleviate the tension between the tensor inference from the BICEP2 experiment, $r=0.2^{+0.07}_{-0.05}$, and the upper limits from the combined CMB temperature power spectrum $r_{0.002}< 0.11$ (95% CL), we examine models where there is a step in this quantity. In this paper we quote $r$ at the scalar pivot of $k=0.05$ Mpc$^{-1}$ where it is unaffected by changes to the scalar power spectrum that we introduce whereas the upper limit is quoted at $k=0.002$ Mpc$^{-1}$.
As an example, we consider a step in the warp $$\begin{aligned}
\label{eqn:T(phi)}
T(\phi) =& \frac{\phi^4}{\lambda_B} \left\{1 + b_T \left[ \tanh\Big(\frac{\phi-\phi_s}{d}\Big)-1 \right] \right\}\end{aligned}$$ of Dirac-Born-Infeld (DBI) inflation[^1] [@Silverstein:2003hf; @Alishahiha:2004eh] with the Lagrangian $${\cal L}= \left[
1-\sqrt{1 - 2 X/T(\phi)} \right] T(\phi)- V(\phi),
\label{eqn:DBI}$$ where the kinetic term $2X = - \nabla^{\mu} \phi \nabla_{\mu} \phi$, the sound speed $$c_s(\phi,X) =\sqrt{ 1 - 2 X/T(\phi)}.$$ Here $\{ b_T,\phi_s,d \}$ parameterize the height, field position and field width of the step while the underlying parameters $\lambda_B$ and the inflaton potential $V(\phi)$ are set to to fix $n_s$ and $A_s$ [@Adshead:2013zfa]. In Ref. [@Miranda:2012rm], we showed that such a model produces a step in the quantity $\epsilon_H c_s$ that controls the tensor-scalar ratio. To keep this discussion model independent, we follow Ref. [@Miranda:2013wxa] and quantify the amplitude of the step by the change in this quantity $$C_1 = -\ln \frac{ \epsilon_{Hb} c_{sb}}{ \epsilon_{Ha} c_{sa}},$$ where “$b$" and “$a$" denote the quantities before and after the step on the slow roll attractor. For definiteness, we take $c_{sb}\approx 1$. In place of $\phi_s$ we quote the sound horizon $$s = \int d{N} \frac{c_s }{a H}$$ at the step $s_s=s(\phi_s)$ and in place of the width in field space $d$, we take the inverse of the number of efolds $N$ the inflaton takes in traversing the step $$x_d = \frac{1}{\pi d} \frac{ d\phi}{d\ln s}.$$ See Ref. [@Adshead:2011jq; @Miranda:2013wxa] for details of this description. We utilize the generalized slow roll technique [@Stewart:2001cd; @Dvorkin:2009ne; @Hu:2011vr] to calculate the power spectra of these models since at the step the slow roll approximation is transiently violated.
$r$ $C_1$ $s_s$(Mpc) $x_d$ $A_s \times 10^9$ $n_s$ $-2\ln{\cal L}_P$ $-2\ln {\cal L}_B $ $-2\ln \Delta{\cal L}_{\rm tot}$
----- ------- ------------ ------- ------------------- ------- ------------------- --------------------- ----------------------------------
0 0 - - 2.1972 0.961 9802.7 89.1 40.1
0 -0.15 337.1 1.58 2.2003 0.957 9798.6 89.2 36.1
0.1 0 - - 2.1961 0.962 9806.5 47.9 2.7
0.1 -0.22 339.2 1.60 2.2000 0.958 9797.8 48.2 -5.7
0.2 0 - - 2.1939 0.963 9812.3 39.4 0
0.2 -0.31 351.8 1.47 2.2002 0.959 9798.1 39.9 -13.7
Joint Fit
=========
We jointly fit the Planck CMB temperature results, WMAP9 polarization results, and BICEP2 to models with and without steps in the tensor-scalar ratio parameter $\epsilon_H c_s$. We use the MIGRAD variable metric algorithm from the CERN Minuit2 code [@James:1975dr] and a modified version of CAMB [@Lewis:1999bs; @Howlett:2012mh] for model comparisons. The Planck likelihood includes the Planck low-$\ell$ spectrum (Commander, $\ell < 50$) and the high-$\ell$ spectrum (CAMspec, $50 <\ell < 2500$), whereas the BICEP2 likelihood[^2] includes both its $E$ and $B$ contributions.
We begin with the baseline best fit 6 parameter slow-roll flat $\Lambda$CDM model with $r=0$. This model sets the non-inflationary cosmological parameters to $\Omega_c h^2= 0.1200$, $\Omega_b h^2=0.02204$, $h=0.672$, $\tau=0.0895$ and the inflationary scalar amplitude at $k=0.05$Mpc$^{-1}$, $A_s=2.1972 \times 10^{-9}$, and spectral tilt, $n_s=0.961$. When considering alternate models we fix the non-inflationary parameters to these values while allowing the inflationary parameters, including $A_s$ and $n_s$ to vary.
As shown in Tab. \[tab:fits\], this $r=0$ model is strongly penalized by the BICEP2 data. Moving to the $r=0.2$ model with the same parameters removes this penalty at the expense of making the Planck likelihood worse by $2\Delta\ln {\cal L}=
9.6$ due to the excess in the $\ell \lesssim 100$ temperature power spectrum shown in Fig. \[plot:cl\].
Next we fit for a step with parameters $C_1$, $s_s$, $x_d$ controlling the amplitude, location and width of the step. The best fit model at $r=0.2$ more than removes the penalty from the temperature excess for Planck while fitting the BICEP2 $BB$ results equally well. The net result is a preference for a step feature at the level of $
2\Delta\ln {\cal L}_P=-14.2$ over no feature. The inclusion of BICEP2 results slightly degrades the fit to $
2\Delta\ln {\cal L}_{\rm tot}=-13.7$ due to changes in the $EE$ spectrum (see below). The $r=0.2$ model with a step is very close to the global maximum with further optimization in $r$ allowing only an improvement of $2\Delta\ln {\cal L}_{\rm tot}=-0.1$. With the addition of the step, there remains a small high-$\ell$ change in the vicinity of the first acoustic peak in Fig. \[plot:cl\] which is interestingly marginally favored by the data. Note that we have fixed the non-inflationary parameters to their values without the step, for example $\tau$. Thus the likelihood may in fact increase in a full fit (see Fig. \[plot:ee\]). Conversely, we do not consider any compromise solutions where non-inflationary cosmological parameters ameliorate the tension without a step. We leave these considerations to a future work.
The best fit step also predicts changes to the $EE$ polarization. Like the $TT$ spectrum, the excess power from the tensor contribution is partially compensated by the reduction in the scalar spectrum for $\ell \gtrsim 30$. This is a signature of the step model which requires only a moderate increase in data to test as witnessed by the change in the BICEP2 likelihood of $2\Delta \ln {\cal L}_B \sim 0.5$ it induces. Differences at $\ell \lesssim 30$, shown here at fixed $\tau$, are largely degenerate with changes in the ionization history [@Mortonson:2009qv]
Due to potential contributions from foregrounds in the BICEP2 data which may imply a shift to $r=0.16^{+0.06}_{-0.05}$ [@Ade:2014xna], we also test models at $r=0.1$ which would formally be in tension with the BICEP2 likelihood without foreground subtraction. Even in this case, the Planck portion of the likelihood improves with the inclusion of a step though the preference is weakened to $2\Delta\ln{\cal L}_P=-8.6$ versus no step. At $r=0$, the Planck data still prefers a step to remove power at a reduced improvement of $2\Delta\ln{\cal L}_P=-4.1$, a fact that was already evident in the Planck collaboration analysis of anticorrelated isocurvature perturbations [@Ade:2013rta]. Such an explanation should also help resolve the tensor-scalar tension albeit outside of the context of single-field inflation. Interestingly, the addition of tensors at both $r=0.1$ and $0.2$ in fact further helps step models fit the Planck data due to the changes shown in Fig. \[plot:cl\] independent of the BICEP2 result.
Discussion
==========
A transient violation of slow-roll which generates a step in the scalar power spectrum at scales near to the horizon size at recombination can alleviate problems of predicted excess power in the temperature spectrum, present already in the best fit $\Lambda$CDM spectrum, and greatly exacerbated by tensor contributions implied by the BICEP2 measurement. Such a step may be generated by a sharp change in the speed of the rolling of the inflaton $\epsilon_H$ or by a sharp change in the speed of sound $c_s$ over a period of around an efolding which combine to form the tensor-scalar ratio. Preference for a step from the temperature power spectrum is at a level of $2\Delta\ln {\cal L}_P = -14.2$ if $r=0.2$ and is still $-8.6$ at $r=0.1$, the lowest plausible value that would fit the BICEP2 data.
Such an explanation makes several concrete predictions. Since slow-roll is transiently violated in this scenario, there will be an enhancement in the associated three-point correlation function. However, we do not expect this signal to be observable as it impacts only a small number of modes [@Adshead:2011bw; @Adshead:2012xz]. $E$-mode fluctuations on similar scales would be predicted to have a smaller enhancement then with tensors alone. This prediction should soon be testable; in the BICEP2 data it brings down the total likelihood improvement to $2\Delta\ln {\cal L}_{\rm tot}=-13.7$ with a step at $r=0.2$.
While we have used a DBI type Lagrangian to illustrate the impact of a change in the tensor-scalar ratio parameter $\epsilon_H c_s$ due to a step in the sound speed, we do not expect that our results require this form, though precise details of the fit may change. Transient shifts in the speed of sound have been found to occur in inflationary models where additional heavy degrees of freedom have been integrated out [@Achucarro:2012yr]. We leave investigation of specific constructions to future work.
While this work was in preparation, the work [@Contaldi:2014zua] appeared which has some overlap with the work presented here. We thank Maurício Calvão, Cora Dvorkin, Dan Grin, Chris Sheehy and Ioav Waga for useful conversations. This work was supported in part by the Kavli Institute for Cosmological Physics at the University of Chicago through grants NSF PHY-1125897 and an endowment from the Kavli Foundation and its founder Fred Kavli. WH was additionally supported by U.S. Dept. of Energy contract DE-FG02-13ER41958 and VM by the Brazilian Research Agency CAPES Foundation and by U.S. Fulbright Organization.
[^1]: Of course, we are well outside the region of validity of UV complete versions of DBI inflation. However, this is merely a phenomenological proof of principle rather than a working construction.
[^2]: <http://bicepkeck.org/>
|
---
author:
- |
Daniel R. Gulotta\
Department of Physics, Princeton University\
Princeton, NJ 08544, USA\
bibliography:
- 'algorithm.bib'
title: 'Properly ordered dimers, $R$-charges, and an efficient inverse algorithm'
---
Introduction
============
The AdS-CFT correspondence [@Maldacena:1997re; @Gubser:1998bc; @Witten:1998qj] tells us that Type IIB string theory on $AdS_5 \times X_5$, where $X$ is a five-dimensional Sasaki-Einstein manifold, is dual to a four-dimensional $\mathcal{N}=1$ superconformal gauge theory. We can study the gauge theory by placing D3-branes at a singularity of $Y_6$, the cone over $X_5$, which is a Calabi-Yau threefold.
In the case where $Y_6$ is toric, dimer models [@Kasteleyn-1967; @Hanany:2005ve; @Franco:2005rj; @Franco:2005sm; @Hanany:2005ss; @Feng:2005gw; @stienstra-2007] are a convenient way of encoding the field content and superpotential of the CFT. One can try to compute the geometry from the dimer or vice versa. There are algorithms for solving the former problem by taking the determinant of the Kasteleyn matrix [@Kasteleyn-1967; @Hanany:2005ve; @Franco:2005rj; @Franco:2005sm; @Hanany:2005ss; @Feng:2005gw; @stienstra-2007] and by counting the windings of zigzag paths [@Hanany:2005ss; @Feng:2005gw; @stienstra-2007]. The latter problem can be solved by the “Fast Inverse Algorithm” [@Hanany:2005ss; @Feng:2005gw; @stienstra-2007], although the algorithm is computationally infeasible for all but very simple toric varieties due to the large amount of trial and error required. We resolve this problem by eliminating the need for trial and error. Our algorithm uses some ideas from the Fast Inverse Algorithm and the method of partial resolution of the toric singularity [@Douglas:1997de; @Morrison:1998cs; @Feng:2000mi; @GarciaEtxebarria:2006aq].
One difficulty in constructing dimers is that not every dimer describes a consistent field theory. One way of determining that a field theory is not consistent is by counting its faces. Each face represents a gauge group, and a consistent theory should have as many gauge groups as there are cycles for Type IIB D-branes to wrap in the AdS theory. Previously there was not a simple, easy to check criterion for determining that a dimer is consistent. We propose that any dimer that has the correct number of faces and that has no nodes of valence one is consistent. We will present several pieces of evidence to support our proposal.
If the dimer is consistent, then the cubic anomalies of the CFT should be equal to the Chern-Simons coefficients of the AdS dual [@Witten:1998qj; @Benvenuti:2006xg]. We show that equality holds in dimers that meet our two criteria.
In a four-dimensional SCFT the unitarity bound says that each gauge invariant scalar operator should have dimension at least one [@Mack-1977], and the $R$-charge of a chiral primary operator is two-thirds of its dimension [@Dobrev-1985]. However, when we try to compute the $R$-charge of a gauge invariant chiral primary operator in an inconsistent dimer theory, the answer is sometimes less than two-thirds. We will show that in dimers that meet our two criteria, the $R$-charges of chiral primary operators are always at least two-thirds if the number of colors is sufficiently large.
We also show that dimers that meet our two criteria have the properties that corner perfect matchings are unique, and that the zigzag path windings agree with the $(p,q)$-legs of the toric diagram.
While studying $R$-charges, we prove that $\frac{27N^2 K}{8 \pi^2} < a \le \frac{N^2 K}{2}$ for toric theories, where $a$ is the cubic ’t Hooft anomaly $\frac{3}{32}(3 \operatorname{Tr}R^3 - \operatorname{Tr}R)$, $N$ is the number of colors of each gauge group, and $K$ is the area of the toric diagram (which is half the number of gauge groups).
\[sec:defs\]Definitions
=======================
A *dimer model* [@Kasteleyn-1967; @Hanany:2005ve; @Franco:2005rj; @Franco:2005sm; @Hanany:2005ss; @Feng:2005gw; @stienstra-2007] consists of a graph whose vertices are colored black or white, and every edge connects a white vertex to a black vertex, i. e. the graph is bipartite. We will use dimer models embedded on the torus $T^2$ to describe toric quiver gauge theories.
A *perfect matching* of the dimer is a set of edges of the dimer such that each vertex is an endpoint of exactly one of the edges. The difference of two perfect matchings is the set of edges that belong to exactly one of the matchings.
The *Kasteleyn matrix* is a weighted adjacency matrix of the dimer. There is one row for each white vertex and one column for each black vertex. Let $\gamma_w$ and $\gamma_z$ be a pair of curves whose winding numbers generate the homology group $H^1(T^2)$. The weight of an edge is $c w^a z^b$ where $c$ is an arbitrary nonzero complex number[^1], $w$ and $z$ are variables, $a$ is the number of times $\gamma_w$ crosses the edge with the white edge endpoint on its left minus the number of times $\gamma_w$ crosses the edge with the white endpoint on its right and $b$ is defined similarly with $\gamma_w$ replaced by $\gamma_z$. The determinant of this matrix tells us the geometry of the field configuration.
The *Newton polygon* of a multivariate polynomial is the convex hull of the set of exponents of monomials appearing in the polynomial. The Newton polygon of the determinant is known as the *toric diagram*. If we choose a different basis for computing the Kasteleyn matrix, then the toric diagram changes by an affine transformation.
A *$(p,q)$-leg* of a toric diagram is a line segment drawn perpendicular to and proportional in length to a segment joining consecutive boundary lattice points of the diagram.
A *zigzag path* is a path of the dimer on which edges alternate between being clockwise adjacent around a vertex and being counterclockwise adjacent around a vertex. A zigzag path is uniquely determined by a choice of an edge and whether to turn clockwise or counterclockwise to find the next edge. Therefore each edge belongs to two zigzag paths. (These paths could turn out to be the same, although it will turn out that we want to work with models in which they are always different.)
In [@Hanany:2005ss] it is conjectured that in a consistent field theory, the toric diagram can also be computed by looking at the windings of the zigzag paths: they are in one-to-one correspondence with the $(p,q)$-legs. The conjecture was proved using mirror symmetry in [@Feng:2005gw].
The *unsigned crossing number* of a pair of closed paths on the torus is the number of times they intersect. The *signed crossing number* of a pair of oriented closed paths on the torus is the number of times they intersect with a positive orientation (the tangent vector to the second path is counterclockwise from the tangent to the first at the point of intersection) minus the number of times they intersect with a negative orientation. It is a basic fact from homology theory that the signed crossing number of a path with winding $(a,b)$ and a path with winding $(c,d)$ is $
(a,b)\wedge(c,d)=ad-bc$.
We will work with the zigzag path diagrams of [@Hanany:2005ss] (referred to there as rhombus loop diagrams). We obtain a zigzag path diagram from a dimer as follows. For each edge of the dimer we draw a vertex of the zigzag path diagram at a point on that edge. To avoid confusion between the vertices of this diagram and the vertices of the dimer we will call the latter nodes. We connect two vertices of the zigzag path diagram if the dimer edges they represent are consecutive along a zigzag path. (This is equivalent to them being consecutive around a node and also to them being consecutive around a face.) We orient the edges of the zigzag path diagram as follows. If the endpoints lie on dimer edges that meet at a white (resp. black) node, then the edge should go counterclockwise (resp. clockwise) as seen from that node. With this definition, each node of the dimer becomes a face of the zigzag path diagram, with all edges oriented counterclockwise for a white node, or clockwise for a black node. The other faces of the zigzag path diagram correspond to faces of the dimer, and the orientations of their edges alternate. Figure \[fig:exampledimer\] shows an example of a dimer and its corresponding zigzag path diagram.
Conversely, we can obtain a dimer from a zigzag path diagram provided that the orientations of the intersections alternate along each path. Around each vertex of such a zigzag path diagram, there is one face with all counterclockwise oriented edges, one face with all clockwise oriented edges, and two faces whose edge orientations alternate. Draw a white node at each counterclockwise oriented face and a black node at each clockwise oriented face, and connect nodes whose faces share a corner.
\[sec:proper\]Consistency of dimer field theories
=================================================
Criteria for consistency and inconsistency
------------------------------------------
One difficulty in dealing with dimer models is that not all of them produce valid field theories. While there are a number of ways of determining that a dimer produces an invalid field theory there has not yet been a simple criterion for showing that a dimer theory is valid.
One way of proving that a dimer produces an invalid field theory is by counting the number of faces of the dimer, i. e. the number of gauge groups. If the dimer theory is consistent, then the number of gauge groups should equal the number of 0, 2, and 4-cycles in the Calabi-Yau around which D3, D5, and D7-branes, respectively, can wrap [@Benvenuti:2004dy]. The Euler characteristic of the Calabi-Yau is the number of even dimensional cycles minus the number of odd dimensional cycles. There are no odd dimensional cycles, so the number of gauge groups should be equal to the Euler characteristic. The Euler characteristic of a toric variety equals twice the area of the toric diagram [@Fulton-1993].
We propose that a dimer will produce a valid field theory if the dimer has no nodes of valence one and it has a number of faces equal to twice the area of the lattice polygon whose $(p,q)$-legs are the winding numbers of the zigzag paths. (Recall that this polygon is the same as the Newton polygon of the determinant of the Kasteleyn matrix for consistent theories.) In this section, we will show that dimers satisfying our two criteria also have the properties that their cubic anomalies agree with the Chern-Simons coefficients of the AdS dual, the $R$-charges of gauge invariant chiral primary operators are greater than or equal to two-thirds, the windings of the zigzag paths are in one-to-one correspondence with the $(p,q)$-legs of the toric diagram, and the corner perfect matchings are unique.
It will be convenient to introduce a property that we call “proper ordering”, which will turn out to be equivalent to the property of having the correct number of faces and no valence one nodes. We call a node of the dimer *properly ordered* if the order of the zigzag paths around that node is the same as the circular order of the directions of their windings. (We do not allow two zigzag paths with the same winding to intersect, nor do we allow zigzag paths of winding zero, since these scenarios make the ordering ambiguous.) We call a dimer properly ordered if each of its nodes is properly ordered.
A connected dimer is properly ordered iff it has no valence one nodes and it has a number of faces equal to twice the area of the convex polygon whose $(p,q)$-legs are the winding numbers of the zigzag paths of the dimer.
A properly ordered dimer cannot have a valence one node, since such a node would be the endpoint of an edge that is an intersection of a zigzag path with itself. Therefore it suffices to prove that a dimer with no valence one nodes is properly ordered iff it has a number of faces equal to twice the area of the convex polygon whose $(p,q)$-legs are the winding numbers of the zigzag paths of the dimer.
Define the “winding excess” of a node $v$ of the dimer as follows. Let $\mathbf{w}_0, \mathbf{w}_1, ..., \mathbf{w}_{n-1}$ be the winding numbers of the zigzag paths passing through $v$ (in the order that the paths appear around $v$). Start at $\mathbf{w}_0$ and turn counterclockwise to $\mathbf{w}_1$, then counterclockwise to $\mathbf{w}_2$, etc., and finally counterclockwise back to $\mathbf{w}_0$. Then the winding excess is defined as the number of revolutions that we have made minus one. (In the special case where $\mathbf{w}_i$ and $\mathbf{w}_{i+1}$ are equal or one of them is zero, we count one-half of a revolution.) A node is properly ordered iff it has winding excess zero and none of the $\mathbf{w}_i$ are zero and no two consecutive windings are equal. A node with a $\mathbf{w}_i=0$ or $\mathbf{w}_i = \mathbf{w}_{i+1}$ can have winding excess zero only if it has exactly two edges (and hence two zigzag paths passing through it). There must be some other node that is an endpoint of one of the edges where the two zigzag paths intersect, and that has more than two edges (since the graph is connected). This node cannot have winding excess zero. So all nodes are properly ordered iff all nodes have winding excess zero. A node has negative winding excess iff it has just one edge, and we have assumed that the dimer has no such nodes. Therefore the dimer is properly ordered iff the sum of all of the winding excesses is zero.
If we choose a node and draw all of the wedges between the consecutive winding numbers, then the winding excess is the number of wedges containing any given ray minus one. (We can think of the special case of consecutive winding numbers being the same as the average of a full wedge and an empty wedge, and the case of a zero winding number as the average of wedges of all angles.) Now consider the sum of the winding excess over all vertices. A pair of oppositely oriented intersections between two zigzag paths forms two full wedges and therefore contributes two to the sum. The sum of the contributions from unpaired intersections can be computed as follows. Label the winding numbers $\mathbf{w}_0, \mathbf{w}_1, ..., \mathbf{w}_{n-1}$, ordered by counterclockwise angle from some ray $\mathcal{R}$. (A zigzag path with zero winding number has no unpaired intersections, so it is not included.) Then for $i<j$ the unpaired wedges formed by $\mathbf{w}_i$ and $\mathbf{w}_j$ will contain $\mathcal{R}$ iff $\mathbf{w}_i \wedge \mathbf{w}_j < 0$. There are $2 |\mathbf{w}_i \wedge \mathbf{w}_j|$ unpaired wedges ($|\mathbf{w}_i \wedge \mathbf{w}_j|$ unpaired crossings of the zigzag paths, and each appears in two vertices). So the number of unpaired wedges formed by $\mathbf{w}_i$ and $\mathbf{w}_j$ containing $\mathcal{R}$ equals $\max(- 2 \mathbf{w}_i \wedge \mathbf{w}_j, 0)=|\mathbf{w}_i \wedge \mathbf{w}_j|-\mathbf{w}_i \wedge \mathbf{w}_j$. Since $\sum_{i<j} |\mathbf{w}_i \wedge \mathbf{w}_j|$ is the number of unpaired edges, it follows that the number of wedges containing $\mathcal{R}$ is the number of paired edges plus the number of unpaired edges minus $\sum_{i<j} \mathbf{w}_i \wedge \mathbf{w}_j$, or $E-\sum_{i<j} \mathbf{w}_i \wedge \mathbf{w}_j$, where $E$ is the total number of edges of the dimer. The sum of the winding excesses is $E-V-\sum_{i<j} \mathbf{w}_i \wedge \mathbf{w}_j=
F-\sum_{i<j} \mathbf{w}_i \wedge \mathbf{w}_j$, where $V$ and $F$ are the number of nodes and faces of the dimer, respectively. We have $\sum_{i<j} \mathbf{w}_i \wedge \mathbf{w}_j = \sum_{i} \mathbf{w}_i \wedge
\sum_{j>i} \mathbf{w}_j$. If we lay the winding vectors tip-to-tail, then $\mathbf{w}_i \wedge \sum_{j>i} \mathbf{w}_j$ is twice the area of the triangle formed by the tail of $\mathbf{w}_0$ and the tip and tail of $\mathbf{w}_i$. Hence $\sum_{i} \mathbf{w}_i \wedge \sum_{j>i} \mathbf{w}_j$ is twice the area of the convex polygon formed by all the winding vectors. If we rotate the polygon 90 degrees then we get a polygon whose $(p,q)$-legs are the winding numbers. So the sum of the winding deficiencies of the nodes is zero iff $F$ equals twice the area of the lattice polygon whose $(p,q)$-legs are the zigzag path winding numbers.
Some perfect matchings of properly ordered dimers
-------------------------------------------------
\[fig:cornermatching\]
\[fig:rays\]
We will construct some perfect matchings that will turn out to correspond to the corners of the toric diagram. Our construction of the perfect matchings is similar to Theorem 7.2 of [@stienstra-2007]. Let $\mathcal{R}$ be any ray whose direction does not coincide with that of the winding number of any zigzag path. For any node $v$, consider the zigzag paths passing through $v$ whose winding numbers make the smallest clockwise and smallest counterclockwise angles with $\mathcal{R}$. (These paths are unique because the proper ordering condition requires that all paths through $v$ have different winding numbers.) By proper ordering, these two zigzag paths must be consecutive around $v$. Therefore they share an edge that has $v$ as an endpoint. Call this edge $e(v)$. Let $v'$ be the other endpoint of $e(v)$. The same two zigzag paths must be consecutive about $v'$ since they form the edge $e$. Since $v'$ is properly ordered it must then be the case that those two paths make the smallest clockwise and counterclockwise angles with $\mathcal{R}$ among all paths passing through $v'$. Hence $e(v)=e(v')$. So the pairing of $v$ with $v'$ is a perfect matching. We will call this matching $M(\mathcal{R})$. Figure \[fig:rays\] depicts the relationship between rays and perfect matchings.
The following characterization of the boundary perfect matchings containing a given edge follows immediately from our definition and will be useful later.
\[lem:edgeinpm\] For any edge $e$ of the dimer, let $Z_r$ and $Z_s$ be the zigzag paths such that $e$ is a positively oriented intersection of $Z_r$ with $Z_s$. (Equivalently, $e$ is a negatively oriented intersection of $Z_s$ with $Z_r$.) Let $\mathbf{w}_r$ and $\mathbf{w}_s$ be the windings of $Z_r$ and $Z_s$, respectively. Let $\mathcal{R}$ be a ray. Then $e$ is in $M(\mathcal{R})$ iff $\mathcal{R}$ is in the wedge that goes counterclockwise from $\mathbf{w}_r$ to $\mathbf{w}_s$.
In particular each edge is in at least one corner perfect matching.
\[sec:zzp\]Zigzag paths and $(p,q)$-legs
----------------------------------------
As we mentioned in Section \[sec:defs\], it is known [@Hanany:2005ss; @Feng:2005gw] that dimers that produce a consistent field theory have the property that the $(p,q)$-legs of the toric diagram are in one-to-one correspondence with the winding numbers of the zigzag paths.
\[thm:zzp\] In a dimer with properly ordered nodes, the zigzag paths are in one-to-one correspondence with the $(p,q)$-legs of the toric diagram.
Our proof of Theorem \[thm:zzp\] resembles that of Theorem 9.3 of [@stienstra-2007].
\[fig:gammaz\]
\[lem:coordfunc\] For any zigzag path $Z$ in any dimer, the number of intersections of a perfect matching with $Z$ is a degree one polynomial function of its coordinates.
In computing the Kasteleyn matrix we can choose the path $\gamma_z$ to follow $Z$, so that the number of times $\gamma_z$ intersects a perfect matching $M$ is just the number of edges that $M$ and $Z$ have in common. (See figure \[fig:gammaz\].) For this choice of $\gamma_z$, the point corresponding to $M$ has $y$-coordinate equal to $|M \cap Z|$. For a different choice of $\gamma_z$, the coordinates differ by an affine transformation.
\[lem:bdyiszover2\] Let $Z$ be a zigzag path of a properly ordered dimer, and let $\mathcal{R}_1$ and $\mathcal{R}_2$ be rays such that the winding direction of $Z$ lies between them and all of the other winding directions do not. Then there exists a boundary line of the toric diagram passing through $M(\mathcal{R}_1)$ and $M(\mathcal{R}_2)$ such that all perfect matchings on this line intersect $Z$ exactly $\frac{|Z|}{2}$ times, and all perfect matchings not on the line intersect $Z$ fewer than $\frac{|Z|}{2}$ times.
Since the winding number of $Z$ is adjacent to $\mathcal{R}_1$, $M(\mathcal{R}_1)$ must choose one of the two $Z$-edges of each node that has them. Hence $|M(\mathcal{R}_1) \cap Z|=\frac{|Z|}{2}$ and similarly $|M(\mathcal{R}_2) \cap Z|=\frac{|Z|}{2}$. No perfect matching can contain more than half of the edges of the path. Therefore the toric diagram lies in the half plane that, in the coordinate system of Lemma \[lem:coordfunc\], is given by the equation $y \le \frac{|Z|}{2}$. $M(\mathcal{R}_1)$ and $M(\mathcal{R}_2)$ are both on the boundary.
\[prop:corner\] The matchings $M(\mathcal{R})$ lie on the corners of the toric diagram. The order of the corners around the boundary is the same as the order of the ray directions.
The intersection of all half planes described in the proof of Lemma \[lem:bdyiszover2\] is the convex hull of all of the $M(\mathcal{R})$’s. Conversely, each $M(\mathcal{R})$ is in the toric diagram. So the toric diagram must be the convex hull of the $M(\mathcal{R})$’s.
Each $M(\mathcal{R})$ must be at a corner of the toric diagram since it is contained in two different boundary lines (one for the first counterclockwise zigzag path direction from $\mathcal{R}$ and another for the first clockwise zigzag path direction). Furthermore, if $\mathcal{R}_1$ and $\mathcal{R}_2$ have only one winding direction between them, then they share a boundary line and hence $M(\mathcal{R}_1)$ and $M(\mathcal{R}_2)$ lie on consecutive corners.
Let $\mathbf{w}$ be the winding number of a zigzag path, and let $n$ be the number of zigzag paths with that winding. Let $\mathcal{R}_1$ and $\mathcal{R}_2$ be rays such that $\mathbf{w}$ lies between them and all other winding directions do not. By Proposition \[prop:corner\], $M(\mathcal{R}_1)$ and $M(\mathcal{R}_2)$ lie on consecutive corners of the toric diagram. An edge belonging to one of the zigzag paths of winding $\mathbf{w}$ will be in either $M(\mathcal{R}_1)$ or $M(\mathcal{R}_2)$ but not both, while all other edges are in neither or both perfect matchings. Therefore the difference of the two perfect matchings is just the union of the zigzag paths with winding $\mathbf{w}$. Therefore the toric diagram points corresponding to $M(\mathcal{R}_1)$ and $M(\mathcal{R}_2)$ are separated by $-n \mathbf{w}^{\perp}$, where $-\mathbf{w}^{\perp}$ is the 90 degree clockwise rotation of $\mathbf{w}$. This proves the theorem.
\[sec:unique\]Unique corner perfect matchings
---------------------------------------------
It is generally believed that dimers that have more than one perfect matching at a corner of the toric diagram are inconsistent [@Hanany:2005ve; @Butti:2005vn; @Hanany:2005ss]. We show that properly ordered dimers have unique corner perfect matchings.
If a dimer is properly ordered, then each corner of the toric diagram has just one perfect matching.
Suppose there exists a perfect matching $M'$ that shares a toric diagram point with $M(\mathcal{R})$ but is not equal to $M(\mathcal{R})$. Consider the set of zigzag paths that contain an edge that is in $M(\mathcal{R})$ or $M'$ but not both. Let $Z$ be one with minimal counterclockwise angle from $\mathcal{R}$. Let $v$ be a node of the dimer through which $Z$ passes. If $v$ includes a zigzag path with winding between $\mathcal{R}$ and that of $Z$, then $M(\mathcal{R})$ and $M'$ are the same at that vertex. If not, then $M(\mathcal{R})$ chooses one of the edges of $Z$ at $v$. Recall that Lemma \[lem:coordfunc\] says that the number of intersections with $Z$ depends only on the toric diagram point. Therefore $M'$ has the same number of edges in $Z$ as $M(\mathcal{R})$. Since $M(\mathcal{R})$ chooses an edge of $Z$ at every node where $M(\mathcal{R})$ and $M'$ differ, equality can hold only if $M'$ chooses the other edge of $Z$ at every such node. If we start at an edge of $Z$ that is in $M(\mathcal{R})$ but not $M'$ and alternately follow edges of the $M(\mathcal{R})$ and $M'$, then we will traverse a cycle that lies entirely in $Z$. Since zigzag paths in properly ordered dimers do not intersect themselves, the cycle must be $Z$. Then both $M(\mathcal{R})$ and $M'$ contain half the edges of $Z$. So the winding number of $Z$ is either the closest or farthest from $\mathcal{R}$ in the counterclockwise direction. If $Z$ were the farthest, then $M(\mathcal{R})$ and $M'$ would have to be the same because every edge of the dimer would be in at least one zigzag path whose winding is closer to $\mathcal{R}$ in the counterclockwise direction than $Z$’s. So $Z$ must be the closest in the counterclockwise direction.
Now let $Z'$ be a zigzag path with minimal clockwise angle from $\mathcal{R}$ on which $M(\mathcal{R})$ and $M'$ differ. By the same reasoning as above, we find the winding direction of $Z'$ is the closest to $\mathcal{R}$ in the clockwise direction and that $M(\mathcal{R})$ and $M'$ have no edges of $Z'$ in common. Since $Z$ and $Z'$ represent consecutive sides of the toric diagram, the crossing number of $Z$ and $Z'$ must be nonzero. A node can have two edges belonging to both $Z$ and $Z'$ only if they have opposite orientations, i. e. they contribute zero to the signed crossing number. Therefore there must be a node with only one $Z$-$Z'$ intersection. $M'$ must include this edge because it includes an edge of $Z$ and $Z'$ at every node that has one, but it cannot include this edge because it does not share any edges of $Z$ with $M(\mathcal{R})$. Therefore our assumption that there existed a matching $M'$ differing from $M(\mathcal{R})$ but sharing the same toric diagram point must be false.
Once we know that the corner matchings are unique, we can also classify all of the boundary perfect matchings.
\[cor:bdymatchings\] Consider a point $A$ on the boundary of the toric diagram such that the nearest corner $B$ in the counterclockwise direction is $p$ segments away and the nearest corner $C$ in the clockwise direction is $q$ segments away. Then each perfect matching at $A$ may be obtained from the perfect matching associated to $B$ by flipping $p$ zigzag paths and from the perfect matching associated to $C$ by flipping $q$ zigzag paths. The number of perfect matchings at $A$ is $\binom{p+q}{q}$.
For any boundary perfect matching $M$ there exists a winding $\mathbf{w}$ such that $M$ contains half the edges of each zigzag path of winding $\mathbf{w}$. For any zigzag path $Z$ of winding $\mathbf{w}$, we can delete the half of the $Z$-edges that are in $M$ and add the other half. This operation moves the perfect matching one segment along the boundary of the toric diagram. There can be at most $p$ zigzag paths for which the operation moves the toric diagram point counterclockwise and at most $q$ zigzag paths for which the operation moves the point clockwise. But there are a total of $p+q$ zigzag paths of winding $\mathbf{w}$, so there must be exactly $p$ of the former and $q$ of the latter. Consequently we see that $M$ can be obtained from a corner perfect matching by flipping $p$ zigzag paths (or from a different corner perfect matching by flipping $q$ zigzag paths). The number of ways of choosing the paths to flip is $\binom{p+q}{p}$.
\[sec:rcharge\]$R$-charges and cubic anomalies
----------------------------------------------
The $R$-charges of the fields may be determined by *$a$-maximization* [@Intriligator:2003jj]. First, we impose the constraint that the $R$-charge of each superpotential term should be two. We also impose the constraint that the beta function of each gauge group should be zero. These conditions can be expressed as $$\begin{aligned}
\label{eq:rvert} \sum_{e \in v} R(e) & = & 2 \\
\label{eq:rface} \sum_{e \in f} \left[1-R(e)\right] & = & 2.\end{aligned}$$
Among all $U(1)$ symmetries satisfying these constraints, the $R$-symmetry is the one that locally maximizes the cubic ’t Hooft anomaly $$\label{eq:thooft}
a = \frac{9N^2}{32} \left[F+\sum_e (R(e)-1)^3 \right].$$
Butti and Zaffaroni [@Butti:2005vn] have proposed some techniques for simplifying the computation of the $R$-charge. For any perfect matching $M$ we can define a function $\delta_M$ that takes the value $2$ on all edges in the perfect matching and zero on all other edges. Any such $\delta_M$ automatically satisfies (\[eq:rvert\]). Butti and Zaffaroni noted that in some cases the perfect matchings on the boundary of the toric diagram yield functions that also satisfy (\[eq:rface\]), and these functions span the set of solutions to (\[eq:rvert\]) and (\[eq:rface\]). We will show that their observation is true for properly ordered dimers.
\[thm:bdysymmetry\] In a dimer with properly oriented nodes, the solutions to (\[eq:rvert\]) and (\[eq:rface\]) are precisely the linear combinations of $\delta_M$, for boundary perfect matchings $M$.
We first determine the dimension of the solution space of (\[eq:rvert\]) and (\[eq:rface\]), so that we will be able to show that there are not any more solutions beyond the boundary $\delta_M$.
\[lem:pathsminus1\] For any dimer in which the zigzag paths have winding numbers that are prime (i. e. their $x$ and $y$ components are relatively prime, or equivalently, they can each be sent to $(1,0)$ by an $SL_2(\mathbb{Z})$ transformation) and not all parallel and in which no zigzag path intersects itself, the set of solutions to (\[eq:rvert\]) and (\[eq:rface\]) has dimension equal to the number of zigzag paths minus one.
First we will show that the number of solutions depends only on the winding numbers of the zigzag paths. We will work with the zigzag path diagram. In this diagram, $R$ is a function on vertices. We can unify (\[eq:rvert\]) and (\[eq:rface\]) into a single equation as follows. We first define the function $\sigma_{v,f}(x)$, where $v$ is a vertex of the zigzag path diagram and $f$ is a face of the zigzag path diagram having $v$ as a corner. If the two zigzag paths containing $v$ are similarly oriented around $f$, then $\sigma_{v,f}(x)=x$; if they are oppositely oriented around $f$ then $\sigma_{v,f}(x)=1-x$. Then (\[eq:rvert\]) and (\[eq:rface\]) can be expressed as $$\label{eq:rcomb}
\sum_{v \in f} \sigma_{v,f}(R(v))=2.$$ (See Figure \[fig:rcontribution\]).
We can deform any zigzag path diagram with non-self-intersecting zigzag paths to any other zigzag path diagram with non-self-intersecting zigzag paths with the same winding numbers. As the diagram is deformed, it can change combinatorially in several ways: a pair of intersections between a pair of zigzag paths can be added or removed, or a zigzag path can be moved past the crossing of two other zigzag paths. Figure \[fig:rearrange\] illustrates these possibilities. Note that at intermediate steps, the zigzag path diagram may not correspond to a dimer, but we can still consider the set of solutions to (\[eq:rcomb\]).
First consider the case where a pair of intersections between a pair of zigzag paths is added or removed. If $C_1$ is not the same face as $C_2$, then the values of the two new crossings are constrained by the equations for $C_1$ and $C_2$ and the dimension of the set of solutions to (\[eq:rvert\]) and (\[eq:rface\]) remains unchanged. If the two zigzag paths have winding numbers that are not parallel, then they must intersect somewhere else, which implies $C_1 \ne C_2$. If the winding numbers are parallel, then there must be some other zigzag path whose winding number is not parallel to either and hence must intersect both. Again $C_1 \ne C_2$.
Now consider the case where a zigzag path is moved past the crossing of two other zigzag paths. We can check that any solution to (\[eq:rcomb\]) in the first diagram is also a solution to (\[eq:rcomb\]) in the second diagram, and vice versa. So performing the move depicted in the second diagram does not change the solution set. So we have shown that the dimension of the solution space to (\[eq:rcomb\]) depends only on the winding numbers of the zigzag paths.
In Lemma \[lem:pm1construct\] we will exhibit for any set of winding numbers a dimer for which the number of independent solutions to (\[eq:rvert\]) and (\[eq:rface\]) is the number of zigzag paths minus one.
Lemma \[lem:pathsminus1\] tells us how to solve (\[eq:rvert\]) and (\[eq:rface\]) for a large class of dimers, many of which are not properly ordered. It is interesting to note that the second move shown in Figure \[fig:rearrange\] does not change either $a$ or $\sum_e (1-R(e))$. The first move also leaves $a$ and $\sum_e (1-R(e))$ invariant in the case where the two zigzag paths are oppositely oriented (the charges of the introduced vertices sum to two and $(R_1-1)^3+(R_2-1)^3=(R_1+R_2-2)\left[ (R_1-1)^2 - (R_1-1)(R_2-1) + (R_2-1)^2 \right]$). When the zigzag path diagram corresponds to a dimer, $\sum_e (1-R(e))$ is the number of faces in the dimer.
First we will show that the $\delta_M$ are solutions to (\[eq:rface\]). Suppose a face $f$ with $2n$ sides had $n$ of those sides in a boundary perfect matching $M$. (A side of a face is an edge of the face along with a normal pointing into the face. If a face borders itself then the bordering edge is part of two different sides of the face. If a self-border edge is in a perfect matching, then we count two sides of $f$ in that perfect matching.) From Corollary \[cor:bdymatchings\], we know that we can get from $M$ to any other boundary perfect matching by flipping zigzag paths. Note that this operation leaves invariant the number of sides of each face in the perfect matching. Therefore every boundary perfect matching has $n$ sides of $f$. So every node of $f$ selects one of the two adjacent sides of $f$ for all boundary perfect matchings. By Lemma \[lem:edgeinpm\] we know that every edge is in some corner perfect matching. So the only edges belonging to any node of $f$ are the adjacent sides of $f$. Therefore, as we move along the boundary of the face we are following a zigzag path. But then we have a zigzag path with zero winding, which violates proper ordering. So the assumption that a face with $2n$ sides can have $n$ sides in a boundary perfect matching must be false. Therefore a face with $2n$ sides can have at most $n-1$ sides in a boundary perfect matching. Sum this inequality over all faces: $$\sum_f \sum_{s \in f \cap M} 1 \le \sum_{f} \left [\left( \sum_{s \in f} \frac{1}{2} \right) - 1 \right]$$ where $f$ runs over faces and $s$ runs over sidess. Now reverse the order of the sums: $$\begin{aligned}
\sum_{s \in M} \sum_{f \owns s} 1 & \le & \sum_s \sum_{f \owns s} \frac{1}{2} - F \\
V & \le & (2E)\left(\frac{1}{2}\right) - F \\
V & \le & E-F.\end{aligned}$$ Since we know $V=E-F$, equality must have held in each case. So (\[eq:rface\]) is satisfied by boundary perfect matchings.
The difference between any two boundary perfect matchings is a sum of functions $\delta_Z$, where $Z$ is a zigzag path and the value $\delta_Z$ alternates between $2$ and $-2$ on $Z$ and is zero outside of $Z$. The only relation obeyed by the $\delta_Z$ is that they sum to zero. So the dimension of the space of solutions to (\[eq:rvert\]) and (\[eq:rface\]) that we have found equals the number of zigzag paths minus one. By Lemma \[lem:pathsminus1\], there can be no more solutions.
When some of the boundary points of the toric diagram are not corners, there are many sets of perfect matchings that form a basis for the solutions to $(\ref{eq:rvert})$ and $(\ref{eq:rface})$. We will construct a basis by associating each segment of the boundary of the toric diagram with a zigzag path, and choosing one perfect matching at each boundary point so that the difference between two consecutive perfect matchings is the zigzag path corresponding to the segment between them. Write $R = \sum_i \lambda_i \delta_{M_i}$, where $M_i$ are the perfect matchings in the basis and the $\lambda_i$ are real numbers.
Butti and Zaffaroni [@Butti:2005vn] also noted that in many cases each edge that is a positively oriented intersection of a zigzag path $Z_r$ with another zigzag path $Z_s$ occurs in the perfect matchings in $cc(r,s)$, the counterclockwise segment from $r$ to $s$, while a negatively oriented intersection of $Z_r$ with $Z_s$ occurs in the perfect matchings not in $cc(r,s)$. In this case, the value of $R-1$ for a positively oriented intersection of $Z_r$ with $Z_s$ is $2\left(\sum_{i \in cc(r,s)} \lambda_i \right) -1$. For a negatively oriented intersection the value of $R-1$ is $2\left(\sum_{i \notin cc(r,s)} \lambda_i \right) -1$, which equals $-\left[2\left(\sum_{i \in cc(r,s)} \lambda_i \right) -1\right]$ since $\sum_i \lambda_i=1$. So then the total contribution to $\sum_e (R-1)^3$ from the intersections of $Z_r$ with $Z_s$ is $\left(\mathbf{w}_r \wedge \mathbf{w}_s \right)
\left[2\left(\sum_{i \in cc(r,s)} \lambda_i \right) -1 \right]^3$. Hence (\[eq:thooft\]) can be rewritten as $$\label{eq:buttia}
a = \frac{9N^2}{32} \left[ F + \sum_{r<s} \left(\mathbf{w}_r \wedge \mathbf{w}_s
\right)
\left( 2\left(\sum_{i \in cc(r,s)} \lambda_i \right) -1 \right)^3 \right].$$
\[prop:buttia\] If a dimer has properly oriented nodes, then it is the case that all positively (resp. negatively) oriented intersections of $Z_r$ with $Z_s$ are in precisely the perfect matchings that are in $cc(r,s)$ (resp. $cc(s,r)$). Hence \[eq:buttia\] holds for properly ordered dimers.
Assume that the dimer has properly ordered nodes. As we go around the toric diagram, the perfect matching switches from containing an edge $e$ to not containing it only if we changed the perfect matching by a zigzag path containing $e$. So each intersection of $Z_r$ with $Z_s$ occurs in either the perfect matchings in $cc(r,s)$ or the perfect matchings in its complement. From Lemma \[lem:edgeinpm\] we know that the positively oriented intersections are in the corners of $cc(r,s)$ and the negatively oriented intersections are not.
A particularly nice rearrangement of (\[eq:buttia\]) that we will find useful is [@Benvenuti:2006xg; @Lee:2006ru] $$\label{eq:area}
a = \frac{9N^2}{4} \sum_{ijk} \operatorname{area}(P_i P_j P_k) \lambda_i \lambda_j
\lambda_k.$$ where $P_i$ is the point on the toric diagram corresponding to the $i$th perfect matching. This formula tells us that the triangle anomaly of the three symmetries with respective charges $\delta_{M_i}$, $\delta_{M_j}$, and $\delta_{M_k}$ is $\frac{N^2}{2} \operatorname{area}(P_i P_j P_k)$. AdS-CFT predicts that the $U(1)$ symmetries of the CFT correspond to gauge symmetries in the AdS theory, and that the triangle anomalies of the CFT should equal the corresponding Chern-Simons coefficients in the AdS theory [@Witten:1998qj]. The Chern-Simons coefficients are indeed found to be $\frac{N^2}{2} \operatorname{area}(P_i P_j P_k)$ [@Benvenuti:2006xg]. So the field theory produced by a properly ordered dimer will have precisely the cubic anomalies predicted by the AdS theory. This is strong evidence that properly ordered dimers are consistent.
Unitarity bound
---------------
Gauge invariant scalar operators in a four-dimensional CFT must have dimension at least one [@Mack-1977]. We also have the BPS bound $\Delta \geq \frac{3}{2}|R|$, where $\Delta$ is the dimension of an operator and $R$ is its $R$-charge. Equality is achieved in the case of chiral primary operators [@Dobrev-1985]. So in order for the theory to be physically valid it is necessary that the gauge invariant chiral primary operators have $R$-charge at least $\frac{2}{3}$.
\[thm:r23\] If $a$ can be expressed in the form (\[eq:area\]), then there exists an $N$ such that in the dimer theory with $N$ colors, each gauge invariant chiral primary operator has $R$-charge at least $\frac{2}{3}$. In particular properly ordered dimers have this property.
\[lem:nonnegative\] At the point where $a$ is locally maximized, the weight of each corner perfect matching is positive, and the weight of the other boundary perfect matchings is zero.
This follows immediately from equation (4.2) of [@Butti:2005vn].
\[lem:muis3a\] If $a$ is given by (\[eq:area\]), then at the point where $a$ is locally maximized, $$\frac{\partial a}{\partial \lambda_i}=3a.$$
We can use Lagrange multipliers to find the local maximum of $a$. $$\frac{\partial a}{\partial \lambda_i}=\mu \frac{\partial}{\partial \lambda_i}
\sum_j \lambda_j = \mu$$ for some constant $\mu$. Since $a$ is homogeneous of degree three, $$\begin{aligned}
3 a & = & \sum_i \lambda_i \frac{\partial a}{\partial \lambda_i} \\
& = & \sum_i \lambda_i \mu \\ \label{eq:muis3a}
& = & \mu.\end{aligned}$$
\[lem:onethird\] At the point where $a$ is locally maximized, each $\lambda_i$ is at most $\frac{1}{3}$.
By Lemma \[lem:muis3a\], $3\lambda_i a=\lambda_i \frac{\partial a}{\partial \lambda_i}$. Since every term of $a$ is degree zero or one in $\lambda_i$, the right-hand side is simply the terms of $a$ containing $\lambda_i$. We can see from (\[eq:area\]) that the coefficient of each term of $a$ is nonnegative and from Lemma \[lem:nonnegative\] that each $\lambda_i$ is nonnegative when $a$ is maximized. Hence the sum of the terms of $a$ containing $\lambda_i$ is at most $a$. Therefore $3 \lambda_i a \le a$, so $\lambda_i \le \frac{1}{3}$.
First consider the mesonic operators, which arise as the trace of a product of of operators corresponding to the edges around a loop of the quiver. The number of signed crossings between a loop and a perfect matching of the dimer is an affine function of the perfect matching’s position in the toric diagram. If the loop has nonzero winding, then the function is not constant, and its zero locus is a line. This line can intersect the corners of the toric diagram at most twice. Therefore each loop intersects all but at most two of the corner perfect matchings. The sums of the weights of those two perfect matchings is at most $\frac{2}{3}$, and from Lemma \[lem:nonnegative\] we know that the non-corner matchings have weight zero. The sum of the weights of the perfect matchings that do intersect the loop is then at least $\frac{1}{3}$. So the loop has $R$-charge at least $\frac{2}{3}$. The $R$-charge of a loop with zero winding is twice the number of intersections it has with any perfect matching. Every edge is in at least one perfect matching so this number must be positive. So a loop with zero winding has $R$-charge at least $2$.
The theory also has baryonic operators. If the gauge groups are $SU(N)$ then these operators are the $N$th exterior powers the bifundamental fields. Each edge of the dimer is contained in at least one corner perfect matching by Lemma \[lem:edgeinpm\], and we know from Lemma \[lem:nonnegative\] that each corner of the toric diagram has a positive contribution to the $R$-charge. So each dimer edge has positive $R$-charge. For sufficiently large $N$, the corresponding baryonic operator will have $R$-charge at least $\frac{2}{3}$.
Bounds on $a$
=============
Bounds on $a$ for toric theories
--------------------------------
We can use (\[eq:area\]) to establish bounds for $a$. In this section we let the indices $ijk$ of the perfect matchings run over the corner perfect matchings only, since we know from Lemma \[lem:nonnegative\] that the non-corner perfect matchings have weight zero.
\[thm:bound\] Let $N$ be the number of colors of each gauge group, and let $K$ be the area of the toric diagram (which is half the number of gauge groups). Then $$\label{eq:bounds}
\frac{27N^2K}{8 \pi^2} < a \le \frac{N^2K}{2}.$$ The upper bound is achieved iff the toric diagram is a triangle, and the lower bound is approached as the toric diagram approaches an ellipse.
The polar body $X^*_R$ of a convex polygon $X$ with respect to the point $R$ is defined as the set of points $Q$ satisfying $\overrightarrow{RQ} \cdot \overrightarrow{RP} \le 1$ for all $P \in X$. Recall that maximizing $a$ is equivalent to minimizing the volume of a slice of the dual toric cone [@Martelli:2005tp; @Butti:2005vn]. More specifically, if $\vec{r}$ is the three-dimensional Reeb vector, then $\frac{9N^2}{8a}$ is the volume of the set of points $\vec{x}$ in the dual cone satisfying $\vec{r} \cdot \vec{x} \le 3$. The cross section of the dual cone in the plane $\vec{r} \cdot \vec{x} = 3$ is the polar body of the toric diagram with respect to the Reeb vector (considered as a point in the plane of the toric diagram). If we call the toric diagram $X$, then $\frac{27N^2}{8a} = \inf_{R \in X} \operatorname{area}(X^*_R)$. Then the statement of the lower bound is equivalent to $\operatorname{area}(X) \inf_{R \in X} \operatorname{area}(X^*_R) < \pi^2$. The result $\operatorname{area}(X) \inf_{R \in X} \operatorname{area}(X^*_R) \le \pi^2$ was proved by Blaschke [@Blaschke-1917; @Petty-1985]; equality occurs in the case of an ellipse. Since the toric diagram is a polygon, it cannot be perfectly elliptical and hence equality does not hold.
We will need to use the following results for the proof of the upper bound.
\[prop:localismax\] The local maximum of $a$ is the overall maximum of $a$ in the region $\lambda_i \ge 0, \sum_i \lambda_i = 1$.
\[prop:maxpoint\] Let $R$ be a point in the interior of the toric diagram. Define $$\begin{aligned}
\label{eq:reebf}
f_i & = & \frac{\operatorname{area}(P_{i-1} P_i P_{i+1})}{\operatorname{area}(P_{i-1} P_i R) \operatorname{area}(P_i P_{i+1} R)} \\ \label{eq:reebs}
S & = & \sum_i f_i \\ \label{eq:reeblambda}
\lambda_i & = & f_i/S.\end{aligned}$$ Then the following results hold: $$\begin{aligned}
R & = & \sum_i \lambda_i P_i \label{eq:reebcm} \\
a & = & \frac{27N^2}{2S} \label{eq:reebrecip}.\end{aligned}$$ Furthermore, when $R$ is the Reeb vector and the $\lambda_i$ are given by (\[eq:reeblambda\]), $a$ is locally maximized (over all choices of $\lambda_i$, not just those of the form (\[eq:reeblambda\])).
We use induction on the number of corners of the toric diagram. If the toric diagram is a triangle, then $a$ is maximized when each $\lambda_i$ is $\frac{1}{3}$. So $a=\frac{9N^2}{4}K(3!)\left(\frac{1}{3}\right)^3=\frac{N^2 K}{2}$.
Assume the toric diagram has more than three corners. Let $\lambda_i^M$ be the values of $\lambda_i$ for which $a$ is locally maximized. Choose a particular $i$ and let $\lambda_i^D=0$, $\lambda_{i+1}^D=\lambda_i^M+\lambda_{i+1}^M$, and $\lambda_j^D=\lambda_j^M$ for all other $j$. We will define $a^M=a|_{\lambda^M}$, $a^D=a|_{\lambda^D}$, and $\Delta a = a^D - a^M$. Since $a$ has degree one in each individual $\lambda_j$, $$\label{eq:dalinear}
\Delta a =
\left.\frac{\partial a}{\partial \lambda_i}\right|_{\lambda^M} (-\lambda_i^M) +
\left.\frac{\partial a}{\partial \lambda_{i+1}}\right|_{\lambda^M} \lambda_i^M +
\left.\frac{\partial^2 a}{\partial \lambda_i \partial \lambda_{i+1}}\right|_{\lambda^M} (-\lambda_i^M)
(\lambda_i^M)$$ Recall that since $a$ is initially maximized, $\frac{\partial a}{\partial \lambda_i}|_{\lambda^M}=
\frac{\partial a}{\partial \lambda_{i+1}}|_{\lambda^M}$ and hence the first two terms of (\[eq:dalinear\]) cancel. Now use (\[eq:area\]) to expand the last term: $$\Delta a = -\frac{27N^2}{2} (\lambda_i^M)^2 \sum_j \lambda_j^M \operatorname{area}(P_i P_{i+1} P_j).$$ Since all of the $P_j$ are on the same side of the line $P_i P_{i+1}$, $$\Delta a = -\frac{27N^2}{2}(\lambda_i^M)^2 \operatorname{area}(P_i P_{i+1} R)$$ where $R$ is the weighted center of mass of the $P_j$ with weights $\lambda_j^M$. Now apply Proposition \[prop:maxpoint\]. We can write $$\begin{aligned}
\Delta a & = & -\lambda_i^M \frac{27N^2\operatorname{area}(P_{i-1} P_i P_{i+1})}{2S \operatorname{area}(P_{i-1} P_i R)} \\
& = & -\lambda_i^M a^M \frac{\operatorname{area}(P_{i-1} P_i P_{i+1})}{\operatorname{area}(P_{i-1} P_i R)}.\end{aligned}$$ Since $\sum_i \lambda_i^M = 1$ and $\sum_i \operatorname{area}(P_{i-1} P_i R)=K$, there must be some $i$ for which $\frac{\lambda_i^M}{\operatorname{area}(P_{i-1} P_i R)} \le \frac{1}{K}$. For such an $i$, $$-\frac{\Delta a}{a^M} \le \frac{\operatorname{area}(P_{i-1} P_i P_{i+1})}{K}$$ Note that $\operatorname{area}(P_{i-1} P_i P_{i+1})$ is the amount by which $K$ would decrease if we removed $P_i$ from the toric diagram. Since $\lambda_i^D=0$, the $\lambda_j^D$ are a valid choice of weights for the toric diagram with $P_i$ removed. Then $$-\frac{\Delta a}{a^M} \le -\frac{\Delta K}{K}.$$ Therefore $\frac{a^D}{K+\Delta K} \ge \frac{a^M}{K}$. By Proposition \[prop:localismax\] the local maximum value of $a$ for the new toric diagram is at least as large as $a^D$. We want to show that it is strictly larger, or equivalently, that $\lambda^D_j$ do not locally maximize $a$ for the new toric diagram. Recall from Lemma \[lem:muis3a\] that $a$ is locally maximized when $\frac{\partial a}{\partial \lambda_{i+1}}=3a$. Hence $a$ will continue to be maximized only if $\Delta \frac{\partial a}{\partial \lambda_{i+1}}=3 \Delta a$. Once again we use the fact that $a$ is degree one in each individual $\lambda_j$: $$\begin{aligned}
\Delta \frac{\partial a}{\partial \lambda_{i+1}} & = &
\left.\frac{\partial^2 a}{\partial \lambda_i \partial \lambda_{i+1}}\right|_{\lambda^M} (-\lambda_i^M) \\
& = & \frac{\Delta a}{\lambda_i^M}.\end{aligned}$$ Hence $a$ can continue to be maximized only if $\lambda_i^M=\frac{1}{3}$. But $\lambda_{i+1}^M$ is positive (since we chose to let our indices enumerate corner perfect matchings only), so $\lambda_{i+1}^D=\lambda_i^M + \lambda_{i+1}^M>\frac{1}{3}$. By Lemma \[lem:onethird\], $\lambda_{j}^D$ cannot be the local maximum point. By the induction hypothesis, the new $\frac{a}{K}$ is at most $\frac{1}{2}$, so the old $\frac{a}{K}$ must be smaller than $\frac{1}{2}$.
Comparison to non-toric field theories
--------------------------------------
Let us consider how we might formulate a similar bound for non-toric CFTs. We need to decide how to interpret $K$ in the non-toric case. If seems natural to replace $2N^2 K$ with the sum of the squares of the numbers of colors of each gauge group.
\[tab:karpov\]
Equation in [@Karpov-1997] (x,y,z) $(9-n)(\alpha x^2 + \beta y^2 + \gamma z^2)$
---------------------------- ----------- ---------------------------------------------- -- --
(1) $(1,1,1)$ 27
(1) $(1,1,2)$ 54
(1) $(1,2,5)$ 270
(2) $(1,1,1)$ 32
(3) $(1,1,1)$ 36
(4) $(1,2,1)$ 50
(5) $(1,1,1)$ 32
(6.1) $(1,1,1)$ 27
(6.2) $(2,1,1)$ 36
(7.1) $(2,2,1)$ 32
(7.2) $(2,1,1)$ 32
(7.3) $(3,1,1)$ 36
(8.1) $(3,3,1)$ 27
(8.2) $(4,2,1)$ 32
(8.3) $(3,2,1)$ 36
(8.4) $(5,2,1)$ 50
Let’s look at the values of $\frac{a}{N^2K}$ for a cone over a del Pezzo surface. Reference [@Karpov-1997] lists some quiver gauge theories that are dual to these Calabi-Yaus. In their notation, the sum of the squares of the number of colors is $\alpha x^2 + \beta y^2 + \gamma z^2$. We can compute $a$ by looking at the $AdS$ dual theory. References [@Gubser:1998vd; @Henningson:1998gx] tell us that $\frac{\pi^3}{4a}$ is the volume of the horizon, and [@Bergman:2001qi] tells us that the volume of the real cone over $dP_n$ is $\frac{\pi^3(9-n)}{27}$. So $a=\frac{27}{4(9-n)}$. So then $\frac{54N^2K}{a}=(9-n)(\alpha x^2 + \beta y^2 + \gamma z^2)$, and the bound ($\ref{eq:bounds}$) for toric theories is then equivalent to $27 \le (9-n)(\alpha x^2 + \beta y^2 + \gamma z^2) < 4 \pi^2$. From Table \[tab:karpov\] we see that the toric upper bound on $\frac{54 N^2 K}{a}$ is not true for all quiver gauge theories. In fact, $\frac{N^2 K}{a}$ can be arbitrarily large. Equation (1) of [@Karpov-1997] is $x^2+y^2+z^2=3xyz$. If we set $z=1$ then we have a Pell’s equation in $x$ and $y$ and there are infinitely many solutions. On the other hand, $27 \le (9-n)(\alpha x^2 + \beta y^2 + \gamma z^2)$ still holds for all of the theories considered in [@Karpov-1997]. It would be interesting to know if the inequality holds more generally.
\[sec:deleting\]Merging zigzag paths
====================================
Deleting an edge of the dimer
-----------------------------
Theorem \[thm:zzp\] says that, if a dimer is properly ordered, then we can determine its toric diagram from the windings of its zigzag paths. As we mentioned in section \[sec:zzp\], Hanany and Vegh [@Hanany:2005ss] and Stienstra [@stienstra-2007] have previously made proposals for drawing a dimer with given zigzag winding numbers, but their procedures are impractical for large dimers because of the large amount of trial and error required.
Partial resolution [@Douglas:1997de; @Morrison:1998cs; @Feng:2000mi; @GarciaEtxebarria:2006aq] has previously been suggested as a method of determining the dimer from the quiver [@Feng:2000mi; @Feng:2002zw; @Hanany:2005ve]. It involves starting with a toric diagram whose dimer model is known and introducing Fayet-Iliopoulos terms that Higgs some of the fields and remove part of the toric diagram to create a new diagram. However, as is the case with the Fast Inverse Algorithm, the previous proposals involving partial resolution suffered from being computationally infeasible.
In this section, we will explore how certain operations on the dimer affect its zigzag paths. These operations can be interpreted as partial resolutions. We will later use these operations to construct an algorithm for drawing a properly ordered dimer with given winding numbers that requires no trial and error.
\[fig:merging\]
One operation that we can perform is to remove an intersection of two zigzag paths (or equivalently, delete an edge of the dimer). The operation has the effect of merging the two paths into a single path. An example is shown in figure \[fig:merging\]. In physical terms, we are Higgsing away the edge by turning on Fayet-Iliopoulos parameters for the adjacent faces. This is an example of partial resolution of the toric singularity [@Douglas:1997de; @Morrison:1998cs; @Feng:2000mi; @GarciaEtxebarria:2006aq]. We will always merge paths that intersect just once. In the following we will sometimes assume that the windings of the paths are $(1,0)$ and $(0,1)$; any other case is $SL_2(\mathbb{Z})$ equivalent to this one.
Making multiple deletions
-------------------------
\[fig:maketwo\]
Suppose we want to make $n>1$ $(1,1)$ edges from $(1,0)$ and $(0,1)$ edges. If we make them one at a time, then we would violate the proper ordering of nodes because we would have $(1,1)$ paths intersecting each other. We should instead delete all $n^2$ edges between the $n$ $(1,0)$ edges and the $n$ $(0,1)$ edges. We will refer to this procedure as Operation I.
\[sec:extra\]Extra crossings
----------------------------
\[fig:extracrossing\]
We mentioned in section \[sec:defs\] that the number of oriented crossings between a pair of paths is a function only on their windings. The number of unoriented crossings is greater than or equal to the absolute value of the number of oriented crossings. If equality does not hold then we say that the pair of paths has “extra crossings” . We say that a diagram has extra crossings if any pair of its paths does. There is nothing inherently wrong with extra crossings, but we may find it desirable to produce diagrams without them.
The edge deletion procedure mentioned in the previous section sometimes introduces extra crossings. An example of this is shown in Figure \[fig:extracrossing\]. We combine a $(1,0)$ zigzag path and a $(0,1)$ to make a $(1,1)$ zigzag path, and we also combine $(-1,0)$ and $(0,-1)$ paths to make a $(-1,-1)$ path. The $(1,1)$ path and $(-1,-1)$ path have a positively oriented intersection coming from the $(0,1)-(-1,0)$ intersection and a negatively oriented intersection coming from the $(1,0)-(0,-1)$ intersection. Note that we can get rid of these crossings by moving the two paths past each other. In terms of the dimer, moving the paths past each other merges the two vertices adjacent to a valence two node. Physically, we are integrating out a mass term.
\[fig:orientations\]
\[fig:movepast\]
{height="3.125in"}
We define a pair of zigzag paths to be an “opposite pair” if they have opposite winding numbers, they do not intersect, and they bound a region containing no crossings. Also, we define the orientation of an opposite pair to be positive if the area containing no crossings is to the left of an observer traveling along one of the paths, and negative if the area is on the right. (See figure \[fig:orientations\].) We have just seen how to take a pair negatively oriented horizontal paths and a pair of negatively oriented vertical paths and turn them into a pair of negatively oriented diagonal paths. Similarly we can turn a pair of positively oriented horizontal paths and a pair of positively oriented vertical paths into a pair of positively oriented diagonal paths. In terms of dimers, this operation takes a node of valence four, deletes two opposite edges, and merges the other endpoints of the two remaining edges. Figure \[fig:movepast\] shows the operation in terms of both zigzag paths and dimers.
\[fig:multipledeletions\]
{width="4.5in"}
More generally, we can make $n$ $(1,1)$ paths and $n$ $(-1,-1)$ paths and get rid of their crossings. An example is given in figure \[fig:multipledeletions\]. We have to untangle each $(1,1)$ path from each $(-1,-1)$ path. Note that all $2n$ paths must have the same orientation. We will call this procedure Operation II.
\[fig:uneven\]
{width="4.5in"}
\[fig:unevendimer\]
{width="3.5in"}
If we want to create differing numbers of $(1,1)$ and $(-1,-1)$ paths, then we run into the problem that we cannot pair them all. We will need to do something more complicated. Let $m$ be the number of $(1,1)$ paths we want to make, and let $n$ be the number of $(-1,-1)$ paths we want to make. Assume $m>n$. We first make $m-n$ $(1,1)$ paths. Now we completely remove $n$ pairs of adjacent $(1,0)$ and $(-1,0)$ paths and $n$ pairs of adjacent $(0,1)$ and $(0,-1)$ paths. Because the pairs are adjacent, the condition that intersection orientations alternate along a path is preserved. Now we want to insert $n$ pairs of adjacent $(1,1)$ and $(-1,1)$ paths, and we want to make sure that there are no extra crossings. This can be accomplished by making them follow one of the $m-n$ already existing $(1,1)$ paths. An example is given in figure \[fig:uneven\]. Figure \[fig:unevendimer\] shows what removing or adding a pair of zigzag paths does to the dimer. This procedure will be called Operation III.
\[sec:algorithm\]An efficient inverse algorithm
===============================================
Description of the algorithm
----------------------------
In describing the algorithm we find it useful to draw toric diagrams rotated 90 degrees counterclockwise from their usual presentation. Our convention will make the algorithm easier to visualize, because it makes the windings of the zigzag paths equal to, rather than perpendicular to, the vectors of the toric diagram edges.
Let $X$ be a toric diagram for which we would like to construct a dimer. Let $Y$ be the smallest rectangle with horizontal and vertical sides that contains $X$. Since $Y$ represents an orbifold of the conifold, we know a dimer for $Y$. We will modify this dimer until we get a dimer for $X$.
\[fig:tangents\]
{height="1.25in"}
Before we begin, we need to make the following definition. A tangent line to a convex polygon $P$ is a line $\ell$ such that $\ell \cap P \subseteq \partial P$ and $\ell \cap P \ne \emptyset$. Note that a convex polygon has exactly two tangent lines with a given slope.
\[fig:farey\]
{width="4.5in"}
We begin by finding the slope one tangent lines to $X$ and cutting $Y$ along these lines to produce some $(1,1)$ and $(-1,-1)$ paths. We use Operation I if the number of $(1,1)$ or $(-1,-1)$ paths desired is zero, Operation II if the numbers are equal, and Operation III if the numbers are both nonzero and unequal. Next we want to cut along the slope $1/2$ tangent lines to $X$ to produce $(2,1)$ and $(-2,-1)$ paths. In fact we already know how to do this, because $SL_2(\mathbb{Z})$ equivalence reduces the problem of making $(2,1)$ and $(-2,-1)$ paths from $(1,0)$, $(-1,0)$, $(1,1)$, and $(-1,-1)$ paths to the problem of making $(1,1)$ and $(-1,1)$ paths from $(1,0)$, $(-1,0)$, $(0,1)$ and $(0,-1)$ paths. Hence we can now cut $Y$ along the slope $1/2$ tangent lines to $X$. Similarly, we can cut $Y$ along the slope $2$ tangent lines to $X$. After this, we can make $(3,1)$ paths by combining $(1,0)$ and $(2,1)$ paths, $(3,2)$ paths by combining $(1,1)$ and $(2,1)$ paths, etc. We can eventually make paths of all slopes, with the order in which we make the paths determined by the Farey tree. (See figure \[fig:farey\].) We can then enumerate over all negative slopes, starting with $-1$. When we are finished, we will have a dimer for $X$.
{height="1.25in"}
$
\left(
\begin{array}{cccccccccccc}
0 & 0 & z & 0 & 0 & 0 & 0 & 0 & -w & 0 & 0 & wz \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & w & -w & 0 & 0 \\
1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & w & -w & 0 \\
0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & w & -w \\
-1 & 0 & z & 0 & 0 & 0 & z & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & -1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & -1 & 0 & 0 & z & z & 0 & 0 & 0 & 0 \\
0 & -1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & z & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 1 & 1 \\
\end{array}
\right)
$
$
\det = (w^2 - w) z^4 + (-w^4 - 37w^3 - 137w^2 - 35w - 1)z^3 +
(3w^4 - 175w^3 + 146w^2 - 2w)z^2$
$+ (-3w^4-40w^3-w^2)z+w^4$
Figure \[fig:example\] shows an example case of the algorithm.
Proof of the algorithm
----------------------
We need to prove that we have the paths necessary to perform each step, and that the finished dimer has properly ordered nodes and has no extra crossings.
\[thm:algorithm\] At each step of the algorithm, the following are true:
1. If there are $m$ zigzag paths with winding $(a,b)$ and $n$ zigzag paths with winding $(-a,-b)$, then there are $\min(m,n)$ negatively oriented pairs of $(a,b)$ and $(-a,-b)$ paths. (This condition ensures that we can always perform the next step of the algorithm.)
2. There are no extra crossings.
3. All nodes are properly ordered.
\[fig:pairtopair\]
{height="1.75in"}
It is clear that all of these conditions hold for the initial dimer. Now let’s look at whether the first condition will be preserved. Operation I will preserve the condition for the winding of the paths being merged provided that we merge unpaired paths when possible. It will also satisfy the condition for the windings of the newly created paths since there are no $(-a,-b)$ paths. Operation II will preserve condition 1 for the windings of the paths being merged since it only deletes negatively oriented pairs. Figure \[fig:pairtopair\] illustrates why Operation II creates negatively oriented pairs of opposite paths. For Operation III we should again merge unpaired paths when possible. It is clear that the reinserted paths form pairs, and we can make these pairs negatively oriented if we desire.
Now consider whether extra crossings are introduced. Let the windings of the paths being merged be $(a,b)$, $(-a,-b)$, $(c,d)$, and $(-c,-d)$, where $ad-bc=1$. A path of winding $(e,f)$ will have extra crossings with the new $(a+c,b+d)$ paths if $af-be$ and $cf-de$ have opposite signs. Equivalently, there will be extra crossings if $f/e$ is between $b/a$ and $d/c$. But because of the Farey fraction ordering, there are no windings $(e,f)$ with this property. So extra crossings are not introduced.
Finally consider whether proper ordering is preserved. Again let the windings of the paths being merged be $(a,b)$, $(-a,-b)$, $(c,d)$, and $(-c,-d)$, $ad-bc=1$. In Operation I, some nodes will see an $(a,b)$ path or a $(c,d)$ path become an $(a+c,b+d)$ path. Therefore proper ordering is preserved provided there are no windings between $(a,b)$ and $(c,d)$. This is always the case because of the Farey fraction ordering. In Operation II, in addition to deletion we also need to move paths past each other. Some nodes are deleted and the others remain unchanged, so proper ordering is preserved. In Operation III, the process of making the lone paths is the same as Operation I, so it preserves proper ordering. Removing pairs also preserves proper ordering. Inserting pairs of paths preserves proper ordering if each intersection between a path in the pair and another path has the same sign as their crossing number, i. e. the paths in the pair do not have extra crossings. Since we are inserting them along an existing path, they will not have extra crossings if the existing path does not have any. We have already showed that we never introduce extra crossings.
\[sec:algextra\]Allowing extra crossings
----------------------------------------
If we want to produce diagrams with extra crossings, we can always just skip the steps for removing the extra crossings. When we want to create $(a,b)$ and $(-a,-b)$ paths, we just perform Operation I twice. There is one potential issue in that we have always assumed that the zigzag paths that we join have just one crossing. We always join paths with oriented crossing number $\pm 1$, but now the unoriented crossing number can be larger than the absolute value of the oriented number. But we recall that the only extra crossings we create are between paths with windings of the form $(a,b)$ and $(-a,-b)$. We may later merge these paths with some other paths, but the extra crossings will always be between paths with oppositely signed $x$-coordinates and oppositely signed $y$-coordinates. We never merge such pairs of paths.
\[sec:algunitarity\]The number of independent solutions to the $R$-charge equations
-----------------------------------------------------------------------------------
We now exhibit the dimers required by Lemma \[lem:pathsminus1\].
\[lem:pm1construct\] The algorithm described in section \[sec:algextra\] produces dimers for which the set of all solutions to equations (\[eq:rvert\]) and (\[eq:rface\]) has dimension equal to the number of zigzag paths minus one.
Our proof is by induction. Our algorithm starts with a dimer that is a diamond-shaped grid. We denote the position of an edge in the grid by $(i,j)$. We can see (e. g. by Fourier analysis) that the general solution to (\[eq:rvert\]) and (\[eq:rface\]) is $\frac{1}{2}+(-1)^i f(j) +(-1)^j g(i)$ for arbitrary functions $f,g$. The number of independent solutions is the number of rows plus the number of columns minus one (the minus one come from the fact that $f(j)=(-1)^j, g(j)=-(-1)^i$ produces the same solution as $f(j)=0, g(j)=0$), which is the number of zigzag paths minus one.
Now consider what happens when our algorithm deletes an edge of the toric diagram. If we have a solution to the equations (\[eq:rvert\]) and (\[eq:rface\]) in the new dimer, we can construct a solution to the equations in the old dimer by assigning a value of zero to the deleted edge. Conversely, if we have a solution in the old dimer in which the deleted edge has value zero, then we have solution in the new dimer as well. We know that there exists a solution in the old dimer where the deleted edge is nonzero, since the deleted edge is contained in some boundary perfect matching. So deleting the edge reduces the dimension of the solution space of (\[eq:rvert\]) and (\[eq:rface\]) by one, and also reduces the number of zigzag paths by one.
Conclusions
===========
We showed that dimers that have the number of faces predicted by the AdS dual theory and that have valence one nodes will have many nice properties: they are “properly ordered”, their cubic anomalies are in agreement with the Chern-Simons coefficients of the AdS dual, gauge-invariant chiral primary operators satisfy the unitarity bound, corner perfect matchings are unique, and zigzag path windings are in one-to-one correspondence with the $(p,q)$-legs of the toric diagram.
We derived some simple bounds for the cubic anomaly $a$ in terms of the area of the toric diagram (and hence in terms of the number of gauge groups).
We provided a precise, computationally feasible algorithm for producing a dimer model for a given toric diagram based on previous partial resolution techniques and the Fast Inverse Algorithm.
It would be interesting to see if our results could apply to the three-dimensional dimers discussed in [@Lee:2006hw] and the orientifold dimers discussed in [@Franco:2007ii].
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank Christopher Herzog for suggesting this problem to me and for providing many helpful discussions. I would like to thank Daniel Kane for helpful discussions regarding Lemmas \[lem:pathsminus1\] and \[lem:pm1construct\] and Alberto Zaffaroni for discussion. I would also like to thank Amihay Hanany for introducing me to dimer models. This work was supported in part by the NSF Graduate Fellowship Program and NSF Grant PHY-075696.
[^1]: The original definition of the Kasteleyn matrix imposes constraints on $c$ for the purpose of counting perfect matchings [@Hanany:2005ve; @Franco:2005rj; @Franco:2005sm; @Hanany:2005ss]. However, these constraints are not necessary for determining the Newton polygon. We follow the convention of [@Feng:2005gw], which points out that it is useful for the purposes of mirror symmetry to allow arbitrary nonzero coefficients.
|
---
abstract: 'For a Brylinski-Deligne covering group of a general linear group, we calculate some values of unramified Whittaker functions for certain representations that are analogous to the theta representations.'
address: 'Department of Mathematics, Weizmann Institute of Science, Rehovot, 7610001, Israel'
author:
- Yuanqing Cai
title: 'Unramified Whittaker functions for certain Brylinski-Deligne covering groups'
---
Introduction
============
The unramified Whittaker functions and their analogues play an important role in modern number theory, arising naturally as terms in the Fourier coefficients of automorphic forms. It is generally desirable to calculate explicit values for these functions, as the information proves useful in many aspects of study related to the automorphic form (for example, in the construction of associated $L$-functions). When an automorphic representation possesses a Whittaker model or another suitable unique model, the method described in [@CS80] may be used to compute an explicit formula (the Casselman-Shalika formula) for the values of the unramified Whittaker function (or the analogous function). In this paper, we consider representations of Brylinski-Deligne covering groups. For these groups, the uniqueness of Whittaker models fails in general. This causes obstructions to some advancement of the theory. Nevertheless, in the past decades, it is discovered that Fourier coefficients of Eisenstein series on covering groups are closely tied to the Weyl group multiple Dirichlet series. This leads to several generalizations of the Casselman-Shalika formula to the covering group setup. One is to interpret the value of an unramified Whittaker function as a weighted sum over a crystal graph. In this vein, this beautiful idea is realized in [@BBF11; @McNamara11; @FZ15] for root systems of type $A$ and $C$. The other description is to express the value as the average of a Weyl group action. This approach is closer to the one of Casselman-Shalika and is successful for all types of root systems (see [@CO13; @McNamara16; @CG10]). In the linear case, the equivalence of these two descriptions is a formula of Tokuyama.
However, the formulas mentioned above are not explicit to work with. To seek applications towards the theory of automorphic forms on covering groups, we would like to have a formula analogous to the original Casselman-Shalika formula. At the moment, we believe that this is impossible in general. Thus, in this paper, we would like to consider the following weaker question:
- For representations on covering groups with additional features (for example, theta representations), is it possible to give a simple formula for some values of the unramified Whittaker functions?
In this paper, we address this question for Brylinski-Deligne covering groups of general linear groups. We give an answer to this question for a family of representations, that can be viewed as analogues of the theta representations. Such representations were also studied in [@Suzuki97; @Suzuki98], and a formula was successfully obtained in some cases. Our results generalize part of Suzuki’s results.
Let ${\mathbb{G}}={\mathbb{GL}}_r$ over a local non-Archimedean field $F$ and ${\overline{G}}$ be the degree $n$ Brylinski-Deligne covering group arising from a ${\mathbb K}_2$-extension ${\overline{{\mathbb{G}}}}$ of ${\mathbb{G}}$. Let ${\mathbb M}$ be a Levi subgroup of ${\mathbb{G}}$. Let $I({\overline{\chi}})$ be an unramified principle series representation of ${\overline{G}}$. Suppose that ${\overline{\chi}}$ is an “anti-exceptional character in ${\mathbb M}$” (Definition \[def:exceptional\]). Let ${\mathbf{w}}_M$ be the long element in the Weyl group $W({\mathbb M})$. Define $\Theta({\overline{G}}/{\overline{M}},{\overline{\chi}})$ as the image of the intertwining operator $I({\overline{\chi}})\to I({}^{{\mathbf{w}}_M}{\overline{\chi}})$ (Sect. \[sec:relative theta\]). Let ${\mathcal{W}}_0({\overline{g}},{\overline{\chi}})$ be an unramified Whittaker function in a certain Whittaker model of $\Theta({\overline{G}}/{\overline{M}},{\overline{\chi}})$. Let $e$ be the identity element in ${\overline{G}}$.
With the above notations and certain assumptions on the rank of ${\mathbb M}$ and the degree of ${\overline{G}}$, ${\mathcal{W}}_0(e,{\overline{\chi}})$ is a product of a certain Gauss sum and a polynomial in terms of ‘Satake parameters’ of $I({\overline{\chi}})$.
When ${\mathbb M}={\mathbb{G}}$, then $\Theta({\overline{G}}/{\overline{M}},{\overline{\chi}})$ is the theta representation studied in [@KP84; @Gao17]. When ${\mathbb M}$ has up to two factors, such results are obtained in [@Suzuki97; @Suzuki98]. Our proof uses ideas in these two papers.
To generalize the results in Suzuki’s papers to our setup, another idea is required. That is to utilize the crystal graph description as a key input. This idea was already used in [@Kaplan] Theorem 43. Here we extend it to a slightly more general setup.
For small rank symplectic groups, similar formulas were obtained in [@Gao]. It will be interesting to see whether the method in this paper can be extended to other groups.
We now give an outline of this paper. Sect. \[sec:preliminaries\] gives preliminary results on the Brylinski-Deligne covering groups. We introduce the unramified principal series representations and the Casselman-Shalika formula in Sect. \[sec:unramified principal\]. We then prove an inductive formula for unramified Whittaker functions in Sect. \[sec:inductive\]. Such results were obtained by Suzuki in type $A$ and here we extend it to all types. We then introduce the representation $\Theta({\overline{G}}/{\overline{M}},{\overline{\chi}})$, which we call the relative theta representation (Sect. \[sec:relative theta\]). In Sect. \[sec:general linear\], we specialize our results to the case of general linear groups. We calculate a crucial local matrix coefficient in Sect. \[sec:calculation of local matrix\]. This is where the ideas of Suzuki are used. In Sect. \[sec:main result\], we state our main results and give a proof. We also add simple examples to help the reader understand the ideas. As the area of covering groups is of deep nature, we either give reliable references or reproduce the necessary proofs here. We also try to fill gaps in past literatures as much as possible.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The author would like to thank Solomon Friedberg and Eyal Kaplan for explaining to him that his original approach did not work. The author would also like to thank the Institute for Mathematical Sciences at the National University of Singapore, where part of this work was done during a visit from December 2018 to January 2019.
Preliminaries {#sec:preliminaries}
=============
We first recall some structural facts on the Brylinski-Deligne covering groups [@BD01; @GG18]. In this paper, we concentrate exclusively on unramified Brylinski-Deligne covering groups. We use [@Gao17] as our main reference.
${\mathbb K}_2$-extensions
--------------------------
Let $F$ be a non-Archimedean field of characteristic $0$, with residual characteristic $p$. Let $O_F$ be the ring of integers. Fix a uniformizer $\varpi$ of $F$. Let ${\mathbb{G}}$ be a split connected linear algebraic group over $F$ with maximal split torus ${\mathbb T}$. Let $$\{
X, \ \Phi, \ \Delta; \ Y, \ \Phi^\vee, \ \Delta^\vee
\}$$ be the based root datum of ${\mathbb{G}}$. Here $X$ (resp. $Y$) is the character lattice (resp. cocharacter lattice) for $({\mathbb{G}},{\mathbb T})$. Choose a set $\Delta\subset \Phi$ of simple roots from the set of roots $\Phi$, and $\Delta^\vee$ the corresponding simple coroots from $\Phi^\vee$. Write $Y^{sc}\subset Y$ for the sublattice generated by $\Phi^\vee$. Let ${\mathbb B}={\mathbb T}{\mathbb U}$ be the Borel subgroup associated with $\Delta$. Denote by ${\mathbb U}^-\subset {\mathbb{G}}$ the unipotent subgroup opposite to ${\mathbb U}$.
Fix a Chevalley system of pinnings for $({\mathbb{G}},{\mathbb T})$, that is, we fix a set of compatible isomorphisms $$\{e_{\alpha}:{\mathbb{G}}_a\to {\mathbb U}_{\alpha}\}_{{\alpha}\in\Phi},$$ where ${\mathbb U}_{\alpha}\subset {\mathbb{G}}$ is the root subgroup associated with ${\alpha}$. In particular, for each ${\alpha}\in\Phi$, there is a unique morphism $\varphi_{\alpha}:{\mathbb{SL}}_2\to{\mathbb{G}}$ which restricts to $e_{\pm {\alpha}}$ on the upper and lower triangular subgroup of unipotent matrixes of ${\mathbb{SL}}_2$.
Denote by $W=W({\mathbb{G}})$ the Weyl group of $({\mathbb{G}},{\mathbb T})$, which we identify with the Weyl group of the coroot system. In particular, $W$ is generated by simple reflections $\{{\boldsymbol{\sigma}}_{\alpha}:{\alpha}^\vee\in\Delta^\vee\}$ for $Y\otimes {\mathbb Q}$. Let $\ell:W\to {\mathbb N}$ be the length function. Let ${\mathbf{w}}_G$ be the longest element in $W$.
Consider the algebro-geometric ${\mathbb K}_2$-extension ${\overline{{\mathbb{G}}}}$ of ${\mathbb{G}}$, which is categorically equivalent to the pairs $\{(D,\eta)\}$ (see [@GG18] Section 2.6). Here $\eta:Y^{sc}\to F^\times$ is a homomorphism. On the other hand, $$D:Y\times Y\to {\mathbb Z}$$ is a bisector associated to a Weyl-invariant quadratic form $Q:Y\to {\mathbb Z}$. That is, let $B_Q$ be the Weyl-invariant bilinear form associated to $Q$ such that $$B_Q(y_1,y_2)=Q(y_1+y_2)-Q(y_1)-Q(y_2),$$ then $D$ is a bilinear form on $Y$ satisfying $$D(y_1,y_2)+D(y_2,y_1)=B_Q(y_1,y_2).$$ The bisection $D$ is not necessarily symmetric. Any ${\overline{{\mathbb{G}}}}$ is, up to isomorphism, incarnated by (i.e. categorically associated to) a pair $(D,\eta)$ for a bisector $D$ and $\eta$.
Topological covering
--------------------
Let $n\geq 1$ be a natural number. Assume that $F^\times$ contains the full group $\mu_{2n}$ of $2n$-th roots of unity and $p\nmid n$. With this assumption, $(\varpi,\varpi)_n=1$ for the Hilbert symbol $(\cdot,\cdot)_n$. This fact is crucial for several results later.
Let ${\overline{{\mathbb{G}}}}$ be incarnated by $(D,\eta)$. One naturally obtains degree $n$ topological covering groups ${\overline{G}},{\overline{T}},{\overline{B}}$ of rational points $G:={\mathbb{G}}(F), T:={\mathbb T}(F), B:={\mathbb B}(F)$, such as $$\mu_n\hookrightarrow {\overline{G}}\twoheadrightarrow G.$$ We may write ${\overline{G}}^{(n)}$ for ${\overline{G}}$ to emphasize the degree of covering. For any subset $H\subset G$, we write ${\overline{H}}\subset {\overline{G}}$ for the preimage of $H$ with respect to the quotient map ${\overline{G}}\to G$. The Bruhat-Tits theory gives a maximal compact subgroup $K\subset G$, which depends on the fixed pinnings. We assume that ${\overline{G}}$ splits over $K$ and fixes such a splitting; the group ${\overline{G}}$ is called an unramified Brylinski-Deligne covering group in this case. We remark that if the derived group of ${\mathbb{G}}$ is simply connected, then ${\overline{G}}$ splits over $K$ (see [@GG18] Theorem 4.2). On the other hand, there is a certain double cover of $\mathrm{PGL}_2$ where the splitting does not exist (see [@GG18], Sect. 4.6).
The data $(D,\eta)$ play the following role for the structural fact on ${\overline{G}}$:
- The group ${\overline{G}}$ splits canonically over any unipotent element of $G$. In particular, we write $\overline{e}_{\alpha}(u)\in{\overline{G}}, \ {\alpha}\in \Phi, \ u\in F$ for the canonical lifting of $e_{\alpha}(u)\in G$. For any ${\alpha}\in\Phi$, there is a natural representative $\sigma_{\alpha}:=e_{\alpha}(1)e_{-{\alpha}}(-1)e_{\alpha}(1)\in K$ (and therefore $\overline{\sigma}_{\alpha}\in{\overline{G}}$ by the splitting of $K$) of the Weyl element ${\boldsymbol{\sigma}}_{\alpha}\in W$. For a general Weyl group element ${\mathbf{w}}$, one can find a lift $w\in{\overline{G}}$ based on a reduced decomposition of ${\mathbf{w}}$. This lift does not depend on the choice of reduced decomposition. We refer to [@Gao18] Sect. 6.1 for a detailed discussion on this matter. Moreover, for $h_{\alpha}(a):={\alpha}^\vee(a)\in G, {\alpha}\in\Phi, a\in F^\times$, there is a natural lifting $\overline{h}_{\alpha}(a)\in{\overline{G}}$ of $h_{\alpha}(a)$, which depends only on the pinning and the canonical unipotent splitting ([@GG18] Sect. 4.6).
- There is a section ${\mathbf{s}}$ of ${\overline{T}}$ over $T$ such that the group law on ${\overline{T}}$ is given by $${\mathbf{s}}(y_1(a))\cdot {\mathbf{s}}(y_2(b))=(a,b)_n^{D(y_1,y_2)}\cdot {\mathbf{s}}(y_1(a)\cdot y_2(b)).$$ Moreover, for the natural lifting $\overline{h}_{\alpha}(a)$, one has $$\overline{h}_{\alpha}(a)=(\eta({\alpha}^\vee),a)_n \cdot {\mathbf{s}}(h_{\alpha}(a))\in{\overline{T}}.$$
- Let ${\boldsymbol{\sigma}}_{\alpha}\in G$ be the natural representative of ${\boldsymbol{\sigma}}_{\alpha}\in W$. For any $\overline{y(a)}\in{\overline{T}}$, $${\boldsymbol{\sigma}}_{\alpha}\cdot \overline{y(a)} \cdot {\boldsymbol{\sigma}}_{\alpha}^{-1}=\overline{y(a)}\cdot \overline{h}_{\alpha}(a^{-{\langle}y,{\alpha}{\rangle}}),$$ where ${\langle}\ , \ {\rangle}$ is the pairing between $Y$ and $X$.
We recall the following lemma.
For all $y\in Y$, $$w\cdot {\mathbf{s}}_y \cdot w^{-1}={\mathbf{s}}_{{\mathbf{w}}(y)}.$$
Define the sublattice $$Y_{Q,n}:=
\{
y\in Y: B_Q(y,y')\in n{\mathbb Z}\}$$ of $Y$. For every ${\alpha}^\vee\in\Phi^\vee$, define $$n_{\alpha}:=n/\gcd(n,Q({\alpha}^\vee)).$$ Write ${\alpha}_{Q,n}^\vee:=n_{\alpha}{\alpha}^\vee, {\alpha}_{Q,n}:=n_{\alpha}^{-1}{\alpha}$. Let $Y_{Q,n}^{sc}\subset Y$ be the sublattice generated by $\{{\alpha}_{Q,n}^\vee\}_{{\alpha}\in\Phi}$. The complex dual group ${\overline{G}}^\vee$ for ${\overline{G}}$ as given in [@FL10; @McNamara12; @Reich12] has root data $$(Y_{Q,n}, \ \{ {\alpha}_{Q,n}^\vee \}, \ \operatorname{Hom}(Y_{Q,n},{\mathbb Z}), \ \{{\alpha}_{Q,n}\}).$$ In particular, $Y_{Q,n}^{sc}$ is the root lattice for ${\overline{G}}^\vee$.
Gauss sum
---------
Consider the Haar measure $\mu$ of $F$ such that $\mu(O_F)=1$. Thus, $$\mu(O_F^\times)=1-1/q.$$ The Gauss sum is given by $$G_\psi(a,b)=\int_{O_F^\times} (u,\varpi)_n^a \cdot \psi(\varpi^b u) \ \mu(u), \qquad a,b\in{\mathbb Z}.$$ It is known that $$G_\psi(a,b)=
\begin{cases}
0, & \mbox{if } b<-1 \\
1-1/q, & \mbox{if } n\mid a, \ b\geq 0 \\
0, & \mbox{if } n\nmid a, \ b\geq 0 \\
-1/q, & \mbox{if } n\mid a, \ b=-1 \\
G_\psi(a,-1) \text{ with } |G_\psi(a,-1)|=q^{-1/2}, & \mbox{if }n\nmid a, \ b=-1.
\end{cases}$$ Let $\varepsilon:=(-1,\varpi)_n\in{\mathbb C}^\times$. One has $\overline{G_\psi(a,b)}=\varepsilon^a\cdot G_\psi(-a,b)$. For any $k\in {\mathbb Z}$, we write $${\mathbf g}_\psi(k):=G_\psi(k,-1).$$
Actions
-------
Let $\rho=\frac{1}{2}\sum_{{\alpha}\in\Phi^+}{\alpha}^\vee$. We define an action of $W$ on $Y\otimes {\mathbb Q}$, which we denote by ${\mathbf{w}}[y]$ by $${\mathbf{w}}[y]:={\mathbf{w}}(y-\rho)+\rho.$$ If we write $y_\rho:=y-\rho$ for any $y\in Y$, then ${\mathbf{w}}[y]-y={\mathbf{w}}(y_\rho)-y_\rho$. From now on, by Weyl orbits in $Y$ or $Y\otimes {\mathbb Q}$ we always refer to the ones with respect to the action ${\mathbf{w}}[y]$. Note that here $0\in Y$ is a vector. The size of this vector is always clear in the context, and we hope that this does not arise any confusion.
We now list some other notations which appear frequently in the text:
- $\psi$: a fixed additive character of $F\to {\mathbb C}^\times$ with conductor $O_F$. For any $a\in F^\times$, the twisted character $\psi_a$ is given by $$\psi_a:x\mapsto \psi(ax).$$
- ${\mathbf{s}}_y:$ for any $y\in Y$, we write ${\mathbf{s}}_y:={\mathbf{s}}(\varpi^y)\in{\overline{T}}$.
- $\lceil x \rceil$: the minimum integer such that $\lceil x \rceil \geq x$ for a real number $x$.
- $\lfloor x \rfloor$: the maxmial integer such that $\lfloor x \rfloor \leq x$ for a real number $x$.
- ${\overline{\chi}}_{\alpha}$: for an unramified character ${\overline{\chi}}$, we sometimes write ${\overline{\chi}}_{\alpha}={\overline{\chi}}({\overline{h}}_{\alpha}(\varpi^{n_{\alpha}}))$.
- $y\sim y'$: if $y,y'\in Y$, we write $y\sim y'$ if there exists ${\mathbf{w}}\in W$ such that $y'={\mathbf{w}}[y]$.
- ${\mathrm{id}}\in W$: the identity element in $W$.
- ${\overline{\chi}}\sim ({\overline{\chi}}_1,\cdots,{\overline{\chi}}_k)$: see Sect. \[sec:local mat coeff for Levi\].
Unramified principal series representations {#sec:unramified principal}
===========================================
Fix an embedding $\iota:\mu_n\hookrightarrow {\mathbb C}^\times$. A representation of ${\overline{G}}$ is called $\iota$-genuine if $\mu_n$ acts via $\iota$. We consider throughout the paper $\iota$-genuine (or simply genuine) representations of ${\overline{G}}$.
Let $U$ be the unipotent subgroup of $B=TU$. As $U$ splits canonically in ${\overline{G}}$, we have ${\overline{B}}={\overline{T}}U$. The covering torus ${\overline{T}}$ is a Heisenberg group with center $Z({\overline{T}})$. The image of $Z({\overline{T}})$ in $T$ is equal to the image of the isogeny $Y_{Q,n}\otimes F^\times \to T$ induced from $Y_{Q,n}\to Y$.
Let ${\overline{\chi}}\in \operatorname{Hom}_\iota(Z({\overline{T}}),{\mathbb C}^\times)$ be a genuine character of $Z({\overline{T}})$. Write $i({\overline{\chi}}):=\operatorname{Ind}_{{\overline{A}}}^{{\overline{T}}}{\overline{\chi}}'$ for the induced representation on ${\overline{T}}$, where ${\overline{A}}$ is any maximal abelian subgroup of ${\overline{T}}$, and ${\overline{\chi}}'$ is any extension of ${\overline{\chi}}$. By the Stone-von Neumann theorem (see [@Weissman09] Theorem 3.1, [@McNamara12] Theorem 3), the construction ${\overline{\chi}}\mapsto i({\overline{\chi}})$ gives a bijection between isomorphism classes of genuine representations of $Z({\overline{T}})$ and ${\overline{T}}$. Since we consider an unramified covering group ${\overline{G}}$ in this paper, we take ${\overline{A}}$ to be $Z({\overline{T}})\cdot (K\cap T)$ from now on.
The choice of this maximal abelian group here is crucial for our calculation in Sect. \[sec:main result\].
Definition
----------
View $i({\overline{\chi}})$ as a genuine representation of ${\overline{B}}$ by inflation from the quotient map ${\overline{B}}\to {\overline{T}}$. We now define the unramified principal series representation $I(\chi):=\operatorname{Ind}_{{\overline{B}}}^{{\overline{G}}}i(\chi)$. The induction is normalized. One knows that $I({\overline{\chi}})$ is unramified (i.e. $I({\overline{\chi}})^K\neq 0$) if and only if ${\overline{\chi}}$ is unramified (i.e. ${\overline{\chi}}$ is trivial on $Z({\overline{T}})\cap K$). We only consider unramified genuine representations in this paper. One has the natually arising abelian extension $$\mu_n\hookrightarrow \overline{Y}_{Q,n}\twoheadrightarrow Y_{Q,n}$$ such that unramified genuine characters ${\overline{\chi}}$ of $Z({\overline{T}})$ correspond to genuine characters of $\overline{Y}_{Q,n}$. Here $\overline{Y}_{Q,n}=Z({\overline{T}})/Z({\overline{T}})\cap K$. Since ${\overline{A}}/(T\cap K)\simeq \overline{Y}_{Q,n}$ as well, there is a canonical extension (also denoted by ${\overline{\chi}}$) of an unramified character ${\overline{\chi}}$ of $Z({\overline{T}})$ to ${\overline{A}}$, by composing ${\overline{\chi}}$ with ${\overline{A}}\twoheadrightarrow \overline{Y}_{Q,n}$. Therefore, we will identity $i({\overline{\chi}})$ as $\operatorname{Ind}_{{\overline{A}}}^{{\overline{T}}}{\overline{\chi}}$ with this ${\overline{\chi}}$.
The following result appears in the proof of [@McNamara12] Lemma 2.
\[lem:supp of spherical\] An unramified principal series representation $I(\chi)$ has a one-dimensional space of $K$-fixed vectors. There is an isomorphism $$i(\chi)^{{\overline{T}}\cap K}\simeq I({\overline{\chi}})^{K}.$$ Given $f\in i(\chi)^{{\overline{T}}\cap K}$, the support of $f$ is in ${\overline{A}}$.
For any ${\mathbf{w}}\in W$, the intertwining operator $T_{{\mathbf{w}},{\overline{\chi}}}:I({\overline{\chi}})\to I({}^{{\mathbf{w}}}{\overline{\chi}})$ is defined by $$(T_{{\mathbf{w}},{\overline{\chi}}}f)({\overline{g}})=\int_{U_w} f(w^{-1} u {\overline{g}}) \ du$$ when it is absolutely convergent. Here, $U_w=U\cap wU^-w^{-1}$. Moreover, it can be meromorphically continued for all ${\overline{\chi}}$ ([@McNamara12] Sect. 7). For $I({\overline{\chi}})$ unramified and ${\mathbf{w}}={\boldsymbol{\sigma}}_{\alpha}$ with ${\alpha}\in\Delta$, $T_{{\boldsymbol{\sigma}}_{\alpha},{\overline{\chi}}}$ is determined by $$T_{{\boldsymbol{\sigma}}_{\alpha},{\overline{\chi}}}(\phi_K)=c({\boldsymbol{\sigma}}_{\alpha},{\overline{\chi}})\cdot \phi_K^{{\boldsymbol{\sigma}}_{\alpha}}$$ where $$c({\boldsymbol{\sigma}}_{\alpha},{\overline{\chi}})=\dfrac{1-q^{-1}{\overline{\chi}}({\overline{h}}_{\alpha}(\varpi^{n_{\alpha}}))}{1-{\overline{\chi}}({\overline{h}}_{\alpha}(\varpi^{n_{\alpha}}))}.$$ Here $\phi_K\in I({\overline{\chi}})$ and $\phi_K^{{\boldsymbol{\sigma}}_{\alpha}}\in I({}^{{\boldsymbol{\sigma}}}{\overline{\chi}})$ are the normalized unramified vectors ([@McNamara12; @Gao18]).
For a general ${\mathbf{w}}\in W$, denote $$\Phi({\mathbf{w}}):=\{{\alpha}\in \Phi:{\alpha}>0 \text{ and }{\mathbf{w}}({\alpha})<0\}.$$ Then the Gindikin-Karpelevich coefficient $c({\mathbf{w}},\chi)$ associated with $T_{{\mathbf{w}},\chi}$ is $$c({\mathbf{w}},\chi)=\prod_{{\alpha}\in\Phi({\mathbf{w}})} c({\boldsymbol{\sigma}}_{\alpha},\chi)$$ such that $T_{{\mathbf{w}},\chi}(\phi_K)=c({\mathbf{w}},{\overline{\chi}})\phi'_K$.
Whittaker functional
--------------------
Let ${\mathrm{Ftn}}(i({\overline{\chi}}))$ be the vector space of functions ${\mathbf{c}}$ on ${\overline{T}}$ satisfying $${\mathbf{c}}({\overline{t}}\cdot {\overline{z}})={\mathbf{c}}({\overline{t}})\cdot {\overline{\chi}}({\overline{z}}), \qquad {\overline{t}}\in{\overline{T}}\text{ and } {\overline{z}}\in{\overline{A}}.$$ The support of any ${\mathbf{c}}\in{\mathrm{Ftn}}(i({\overline{\chi}}))$ is a disjoint union of cosets in ${\overline{T}}/{\overline{A}}$. Moreover $\dim({\mathrm{Ftn}}(i({\overline{\chi}})))=|Y/Y_{Q,n}|$ since ${\overline{T}}/{\overline{A}}$ has the same size as $Y/Y_{Q,n}$.
There is a natural isomorphism of vector spaces ${\mathrm{Ftn}}(i({\overline{\chi}}))\simeq i({\overline{\chi}})^\vee$, where $i({\overline{\chi}})^\vee$ is the complex dual space of functionals of $i({\overline{\chi}})$. Explicitly, let $\{\gamma_i\}\subset {\overline{T}}$ be a set of representatives of ${\overline{T}}/{\overline{A}}$. Consider ${\mathbf{c}}_{\gamma_i}\in{\mathrm{Ftn}}(i({\overline{\chi}}))$ which has support $\gamma_i\cdot{\overline{A}}$ and ${\mathbf{c}}_{\gamma_i}(\gamma_i)=1$. It gives rise to a linear functional ${\lambda}_{\gamma_i}^{{\overline{\chi}}}\in i({\overline{\chi}})^\vee$ such that $${\lambda}_{\gamma_i}^{{\overline{\chi}}}(f_{\gamma_j})=\delta_{ij},$$ where $f_{\gamma_j}\in i({\overline{\chi}})$ is the unique element such that $\mathrm{supp}(f_{\gamma_j})={\overline{A}}\cdot \gamma_j^{-1}$ and $f_{\gamma_j}(\gamma_j^{-1})=1$. That is, $$f_{\gamma_j}=i({\overline{\chi}})(\gamma_j)\phi_K.$$ The isomorphism ${\mathrm{Ftn}}(i({\overline{\chi}}))\simeq i({\overline{\chi}})^\vee$ is given explicitly by $${\mathbf{c}}\mapsto {\lambda}_{{\mathbf{c}}}^{{\overline{\chi}}}:=\sum_{\gamma_i\in{\overline{T}}{\backslash}{\overline{A}}} {\mathbf{c}}(\gamma_i){\lambda}_{\gamma_i}^{{\overline{\chi}}}.$$
Consider the principal series $I({\overline{\chi}}):=I(i({\overline{\chi}}))$ for an unramified character ${\overline{\chi}}\in \operatorname{Hom}(Z({\overline{T}}),{\mathbb C}^\times)$. We define a space of Whittaker functionals on $I({\overline{\chi}})$.
Let $\psi_U:U\to{\mathbb C}^\times$ be the character on $U$ such that its restriction to every $U_{\alpha},{\alpha}\in\Delta$ is given by $\psi\circ e_{\alpha}^{-1}$. We may write $\psi$ for $\psi_U$ if no confusion arises.
For any genuine representation $(\overline{\sigma}, V_{\overline{\sigma}})$ of ${\overline{G}}$, a linear functional ${\lambda}:V_{\overline{\sigma}}\to {\mathbb C}$ is called a $\psi$-Whittaker functional if ${\lambda}(\overline{\sigma}(u)v)=\psi(u)\cdot v$ for all $u\in U$ and $v\in V_{\overline{\sigma}}$. Write $\mathrm{Wh}_\psi(\overline{\sigma})$ for the space of $\psi$-Whittaker functionals for $\overline{\sigma}$.
Consider the following integral $$\int_{U} f(w_G ug)\overline{\psi(u)} \ du$$ for $f\in I({\overline{\chi}})$. This is a $i({\overline{\chi}})$-valued functional. To obtain a Whittaker functional, we need to apply an element in $i({\overline{\chi}})^\vee$. By [@McNamara16] Sect. 6, there is an isomorphism between $i({\overline{\chi}})^\vee$ and the space ${\mathrm{Wh}}_\psi(I({\overline{\chi}}))$ of $\psi$-Whittaker functionals on $I({\overline{\chi}})$, given by ${\lambda}\mapsto W_{\lambda}$ with $$\label{eq:whittaker functional}
W_{\lambda}:I({\overline{\chi}})\to {\mathbb C}, \qquad f\mapsto {\lambda}\left( \int_U f(w_G u)\overline{\psi(u)} \ du \right),$$ where $f\in I({\overline{\chi}})$ is an $i({\overline{\chi}})$-valued function on ${\overline{G}}$; $w_G\in K$ is a representative of ${\mathbf{w}}_G$.
For ${\mathbf{c}}\in {\mathrm{Ftn}}(i({\overline{\chi}}))$, by abuse of notation, we will write ${\lambda}_{{\mathbf{c}}}^{{\overline{\chi}}}\in {\mathrm{Wh}}_\psi (I({\overline{\chi}}))$ for the resulting $\psi$-Whittaker functional of $I({\overline{\chi}})$ from the isomorphism ${\mathrm{Ftn}}(i({\overline{\chi}}))\simeq i({\overline{\chi}})^\vee \simeq {\mathrm{Wh}}_{\psi}(I({\overline{\chi}}))$. As a consequence, $\dim {\mathrm{Wh}}_\psi (I({\overline{\chi}})) \simeq |Y/Y_{Q,n}|$.
Local coefficient matrix
------------------------
Let $J({\mathbf{w}},{\overline{\chi}})$ be the image of $T_{{\mathbf{w}},{\overline{\chi}}}$. The operator $T_{{\mathbf{w}},{\overline{\chi}}}$ induces a homomorphism $T_{{\mathbf{w}},{\overline{\chi}}}^{\ast}$ of vector spaces with image ${\mathrm{Wh}}_{\psi}(J({\mathbf{w}},\chi))$: $$T_{{\mathbf{w}},{\overline{\chi}}}^{\ast}:{\mathrm{Wh}}_{\psi}(I({}^{{\mathbf{w}}}{\overline{\chi}}))\to {\mathrm{Wh}}_{\psi}(I({\overline{\chi}}))$$ which is given by $${\langle}\lambda_{{\mathbf{c}}}^{{}^{{\mathbf{w}}}{\overline{\chi}}}, - {\rangle}\mapsto {\langle}\lambda_{{\mathbf{c}}}^{{}^{{\mathbf{w}}}{\overline{\chi}}}, T_{{\mathbf{w}},{\overline{\chi}}}(-) {\rangle}$$ for any ${\mathbf{c}}\in{\mathrm{Ftn}}(i({}^{{\mathbf{w}}}{\overline{\chi}}))$. Let $\{\lambda_{\gamma}^{{}^{{\mathbf{w}}}{\overline{\chi}}}\}_{\gamma\in {\overline{T}}/{\overline{A}}}$ be a basis for ${\mathrm{Wh}}_{\psi}(I({}^{{\mathbf{w}}}{\overline{\chi}}))$, and ${\lambda}_{\gamma'}^{{\overline{\chi}}}$ a basis for ${\mathrm{Wh}}_{\psi}(I({\overline{\chi}}))$. The map $T_{{\mathbf{w}},{\overline{\chi}}}^{\ast}$ is then determined by the square matrix $[\tau({\overline{\chi}},{\boldsymbol{\sigma}},\gamma,\gamma')]_{\gamma,\gamma'\in{\overline{T}}/{\overline{A}}}$ of size $|Y/Y_{Q,n}|$ such that $$T_{{\mathbf{w}},{\overline{\chi}}}^{\ast}({\lambda}_{\gamma}^{{}^{{\mathbf{w}}}{\overline{\chi}}}) =\sum_{\gamma'\in{\overline{T}}/{\overline{A}}}\tau({\mathbf{w}},{\overline{\chi}},\gamma,\gamma') {\lambda}_{\gamma'}^{{\overline{\chi}}}.$$
The local coefficient matrix satisfies the following properties.
\[lem：local coefficient matrix cocycle\] For ${\mathbf{w}}\in W$ and $\bar{z},\bar{z}'\in \bar{A}$, the following identity holds: $$\tau({\mathbf{w}},{\overline{\chi}},\gamma\cdot {\overline{z}},\gamma'\cdot {\overline{z}}')
=({}^{{\mathbf{w}}}{\overline{\chi}}^{-1}(\bar{z}))\cdot \tau( {\mathbf{w}},{\overline{\chi}}, \gamma,\gamma')\cdot {\overline{\chi}}(\bar{z}').$$ Moreover, for ${\mathbf{w}}_1,{\mathbf{w}}_2\in W$ such that $\ell({\mathbf{w}}_1{\mathbf{w}}_2)=\ell({\mathbf{w}}_1)+\ell({\mathbf{w}}_2)$, one has $$\tau( {\mathbf{w}}_1{\mathbf{w}}_2,{\overline{\chi}}, \gamma,\gamma')=\sum_{\gamma''\in{\overline{T}}/{\overline{A}}} \tau({\mathbf{w}}_1,{}^{{\mathbf{w}}_2}{\overline{\chi}}, \gamma,\gamma'')\cdot \tau({\mathbf{w}}_2,{\overline{\chi}}, \gamma'',\gamma'),$$ which is referred to as the cocycle relation.
This fact is standard. For example, it follows from [@Gao17] Lemma 3.2.
Thus the calculation of the local coefficient matrix $[\tau({\overline{\chi}},{\mathbf{w}},\gamma,\gamma')]_{\gamma,\gamma'}$ is reduced to the case when ${\mathbf{w}}$ is a simple reflection.
We now would like to compute the matrix $[\tau({\overline{\chi}},{\boldsymbol{\sigma}}_{\alpha},\gamma,\gamma')]_{\gamma,\gamma'}$ for any unramified character ${\overline{\chi}}$ and simple reflection ${\boldsymbol{\sigma}}_{\alpha}, {\alpha}\in\Delta$.
\[thm:local coefficient matrix\] Suppose that $\gamma={\mathbf{s}}_{y_1}$ and $\gamma'={\mathbf{s}}_y$ by $y$. Then we can write $$\tau({\boldsymbol{\sigma}}_{\alpha},{\overline{\chi}},\gamma,\gamma') =\tau^1({\boldsymbol{\sigma}}_{\alpha},{\overline{\chi}},\gamma,\gamma')+\tau^2({\boldsymbol{\sigma}}_{\alpha},{\overline{\chi}},\gamma,\gamma')$$ with the following properties:
- $\tau^i({\boldsymbol{\sigma}}_{\alpha},{\overline{\chi}},\gamma\cdot \bar{z},\gamma'\cdot\bar{z}') =({}^{{\boldsymbol{\sigma}}_{\alpha}}{\overline{\chi}})^{-1}(\bar{z})\cdot\tau^i({\boldsymbol{\sigma}}_{\alpha},{\overline{\chi}},\gamma,\gamma')\cdot {\overline{\chi}}(\bar{z}'),\qquad \bar{z},\bar{z}'\in\bar{A}$;
- $\tau^1({\boldsymbol{\sigma}}_{\alpha},{\overline{\chi}},\gamma,\gamma')=0$ unless $y_1\equiv y \mod Y_{Q,n}$;
- $\tau^2({\boldsymbol{\sigma}}_{\alpha},{\overline{\chi}},\gamma,\gamma')$ unless $y_1\equiv {\boldsymbol{\sigma}}_{\alpha}[y]\mod Y_{Q,n}$.
Moreover,
- If $y_1=y$, then $$\tau^1({\boldsymbol{\sigma}}_{\alpha},{\overline{\chi}},\gamma,\gamma')
=(1-q^{-1})\dfrac{{\overline{\chi}}(\bar{h}_{\alpha}(\varpi^{n_{\alpha}}))^{k_{y,{\alpha}}}}{1-{\overline{\chi}}({\overline{h}}_{\alpha}(\varpi^{n_{\alpha}}))}, \text{ where }k_{y,{\alpha}}=\left\lceil\dfrac{{\langle}y,{\alpha}{\rangle}}{n_{\alpha}}\right\rceil$$
- If $y_1={\boldsymbol{\sigma}}_{\alpha}[y]$, then $$\tau^2({\boldsymbol{\sigma}}_{\alpha},{\overline{\chi}},\gamma,\gamma')={\mathbf g}_{\psi^{-1}}({\langle}y_\rho,{\alpha}{\rangle}Q({\alpha}^\vee)).$$
Explicit calculation of the local coefficient matrix
----------------------------------------------------
Lemma \[lem：local coefficient matrix cocycle\] and Theorem \[thm:local coefficient matrix\] determine the local coefficient matrix completely. However, it is too complicated to obtain a general formula as one has to analyze the sum over ${\overline{T}}/{\overline{A}}$ inductively. In this section, we highlight some observations that will be useful for our calculation.
Notations: for $y,y'\in Y$, we write $$\tau({\mathbf{w}},{\overline{\chi}},y,y'):=\tau({\mathbf{w}},{\overline{\chi}},{\mathbf{s}}_{y},{\mathbf{s}}_{y'}).$$
Let ${\mathbf{w}}={\mathbf{w}}_1\cdots {\mathbf{w}}_k$ be a reduced decompositioin of ${\mathbf{w}}$ by simple reflections.
\[lem:calcualation of tau 1\] The coefficient $\tau({\mathbf{w}},{\overline{\chi}},y,y')=0$ unless $y'\equiv {\mathbf{w}}_k^{a_k}\cdots {\mathbf{w}}_1^{a_1}[y] \mod Y_{Q,n}$ for some $a_1,\cdots, a_k\in\{0,1\}$.
We can prove this by induction on $k$. When $k=1$, this follows from Theorem \[thm:local coefficient matrix\]. We now assume that the result is true for $k-1$. Then $$\tau({\mathbf{w}},{\overline{\chi}},y,y')=\sum_{y''\in Y/Y_{Q,n}} \tau({\mathbf{w}}_1\cdots {\mathbf{w}}_{k-1},{}^{{\mathbf{w}}_k}{\overline{\chi}}, y,y'')\cdot \tau({\mathbf{w}}_k,{\overline{\chi}}, y'',y').$$ If this is nonzero, then $\tau({\mathbf{w}}_1\cdots {\mathbf{w}}_{k-1},{}^{{\mathbf{w}}_k}{\overline{\chi}}, y,y'')\neq 0$ and $\tau({\mathbf{w}}_k,{\overline{\chi}}, y'',y')\neq 0$ for some $y''$. This implies that $$y''\equiv {\mathbf{w}}_{k-1}^{a_{k-1}}\cdots {\mathbf{w}}_1^{a_1}[y] \mod Y_{Q,n}, \text{ for some }a_1,\cdots, a_{k-1}\in \{0,1\},$$ and $y'\equiv {\mathbf{w}}_{k}^{a_k}[y''] \mod Y_{Q,n}$ for some $a_k\in\{0,1\}$. This proves the result.
We have an immediate corollary.
The coefficient $\tau({\mathbf{w}}_G,{\overline{\chi}},y,y')=0$ unless ${\mathbf{w}}[y]\equiv y' \mod Y_{Q,n}$ for some ${\mathbf{w}}\in W$.
The next result is very useful for calculation.
\[lem:tau unique decomposition\] Assume that ${\mathbf{w}}={\mathbf{w}}_1\cdots {\mathbf{w}}_k$ is a reduced decomposition of ${\mathbf{w}}$, and for any two subexpressions ${\mathbf{w}}_1^{a_1}\cdots {\mathbf{w}}_{k}^{a_k}={\mathbf{w}}_1^{a'_1}\cdots {\mathbf{w}}_{k}^{a'_k}$, $a_1,\cdots, a_k, a'_1,\cdots,a'_k\in \{0,1\}$, we have $a_i=a_i'$ for $i=1,\cdots, k$. If the orbit of $y$ is free, then $$\begin{aligned}
&\tau({\mathbf{w}},{\overline{\chi}},y,{\mathbf{w}}_k^{a_k}\cdots {\mathbf{w}}_{1}^{a_1}[y])\\
=&\tau({\mathbf{w}}_1,{}^{{\mathbf{w}}_2\cdots {\mathbf{w}}_k}{\overline{\chi}},y,{\mathbf{w}}_1^{a_1}[y]) \tau({\mathbf{w}}_2,{}^{{\mathbf{w}}_3\cdots {\mathbf{w}}_k}{\overline{\chi}},{\mathbf{w}}_1[y],{\mathbf{w}}_2^{a_2}{\mathbf{w}}_1^{a_1}[y])\\
&\cdots \tau({\mathbf{w}}_k,{\overline{\chi}},{\mathbf{w}}_{k-1}^{a_{k-1}}\cdots {\mathbf{w}}_{1}^{a_1}[y],{\mathbf{w}}_k^{a_k}\cdots {\mathbf{w}}_{1}^{a_1}[y]).
\end{aligned}$$ In other words, only one term in the summation is nonzero.
The assumption implies that ${\mathbf{w}}_k^{a_k}\cdots {\mathbf{w}}_{1}^{a_1}[y]$ are all distinct in $Y/Y_{Q,n}$ for $a_1,\cdots,a_k\in \{0,1\}$.
We prove it by induction on $k$. If $k=1$, there is nothing to prove. Assume the result is true for ${\mathbf{w}}_1\cdots {\mathbf{w}}_{k-1}$. Then $$\tau({\overline{\chi}},{\mathbf{w}},y,{\mathbf{w}}_k^{a_k}\cdots {\mathbf{w}}_{1}^{a_1}[y])=\sum_{y''\in Y/Y_{Q,n}} \tau({}^{{\mathbf{w}}_k}{\overline{\chi}}, {\mathbf{w}}_1\cdots {\mathbf{w}}_{k-1}, y,y'')\cdot \tau({\overline{\chi}}, {\mathbf{w}}_k, y'',{\mathbf{w}}_k^{a_k}\cdots {\mathbf{w}}_{1}^{a_1}[y]).$$ For a nonzero term in the summation, we have $$y''\equiv {\mathbf{w}}_{k-1}^{a'_{k-1}}\cdots {\mathbf{w}}_{1}^{a'_1}[y] \mod Y_{Q,n} \text{ for some }a'_1,\cdots,a'_{k-1}\in \{0,1\}$$ and ${\mathbf{w}}_k^{a_k}\cdots {\mathbf{w}}_{1}^{a_1}[y]\equiv {\mathbf{w}}_k^{a'_k}[y''] \mod Y_{Q,n}$ for some $a_k\in \{0,1\}$. As the orbit of $y$ is free, this implies that $${\mathbf{w}}_k^{a_k}\cdots {\mathbf{w}}_{1}^{a_1} ={\mathbf{w}}_k^{a'_k}\cdots {\mathbf{w}}_{1}^{a'_1}$$ and therefore $a_i=a'_i$ for $i=1,\cdots,k$. We now conclude that only the term $y''={\mathbf{w}}_k^{a_k}\cdots {\mathbf{w}}_{1}^{a_1}[y]$ has nonzero contribution in the summation and therefore $$\begin{aligned}
&\tau({\overline{\chi}},{\mathbf{w}},y,{\mathbf{w}}_k^{a_k}\cdots {\mathbf{w}}_{1}^{a_1}[y])\\
=&\tau({}^{{\mathbf{w}}_k}{\overline{\chi}}, {\mathbf{w}}_1\cdots {\mathbf{w}}_{k-1}, y,{\mathbf{w}}_{k-1}^{a_{k-1}}\cdots {\mathbf{w}}_{1}^{a_1}[y])\cdot \tau({\overline{\chi}}, {\mathbf{w}}_k, {\mathbf{w}}_{k-1}^{a_{k-1}}\cdots {\mathbf{w}}_{1}^{a_1}[y]),{\mathbf{w}}_k^{a_k}\cdots {\mathbf{w}}_{1}^{a_1}[y]).
\end{aligned}$$ By induction we obtain the desired formula.
The conditions in the lemma are satisfied in the following example: ${\mathbb{G}}={\mathbb{GL}}_r$ and ${\mathbf{w}}={\boldsymbol{\sigma}}_{{\alpha}_1}\cdots {\boldsymbol{\sigma}}_{{\alpha}_r}$. We will use it later.
Notice that $Y_{Q,n}$ is not well-behaved with respect to Levi subgroup so it is better to work with the lattice $Y_{Q,n}^{sc}$. Observe that $Y_{Q,n}\cap Y^{sc}=Y_{Q,n}^{sc}$.
\[lem:several lattices\] If $y\equiv {\mathbf{w}}[y]\mod Y_{Q,n}$ for some ${\mathbf{w}}\in W$, then $y\equiv {\mathbf{w}}[y]\mod Y_{Q,n}^{sc}$.
By [@BBF08] Lemma 2, $y-{\mathbf{w}}[y]\in Y^{sc}$. If $y-{\mathbf{w}}[y]\in Y_{Q,n}$, then it is in $Y_{Q,n}^{sc}$.
Let ${\mathbb T}_{Q,n}^{sc}$ be the split torus with cocharacter group $Y_{Q,n}^{sc}$, and $T_{Q,n}^{sc}:={\mathbb T}_{Q,n}^{sc}(F)$.
\[lem:dependence on chi\] The coefficient $\tau({\mathbf{w}},{\overline{\chi}},y,{\mathbf{w}}'[y])$ depends only on ${\overline{\chi}}|_{{\overline{T}}_{Q,n}^{sc}}$ for ${\mathbf{w}},{\mathbf{w}}'\in W$.
If ${\mathbf{w}}={\mathrm{id}}$, this result follows from Lemma \[lem:several lattices\] and Lemma \[lem：local coefficient matrix cocycle\].
We now consider a nontrivial Weyl group element with reduced decomposition ${\mathbf{w}}={\mathbf{w}}_1\cdots {\mathbf{w}}_k$. If $\tau({\mathbf{w}},{\overline{\chi}},y,{\mathbf{w}}'[y])\neq 0$, then by Lemma \[lem:calcualation of tau 1\], ${\mathbf{w}}'[y]\equiv {\mathbf{w}}_k^{a_k}\cdots {\mathbf{w}}_1^{a_1}[y] \mod Y_{Q,n}$ for some $a_1,\cdots, a_k\in\{0,1\}$. By Lemma \[lem:several lattices\], ${\mathbf{w}}'[y]\equiv {\mathbf{w}}_k^{a_k}\cdots {\mathbf{w}}_1^{a_1}[y] \mod Y_{Q,n}^{sc}$. So it suffices to prove the result for elements of the form ${\mathbf{w}}'={\mathbf{w}}_k^{a_k}\cdots {\mathbf{w}}_1^{a_1}$.
We now argue by induction on the length of ${\mathbf{w}}$. If ${\mathbf{w}}={\boldsymbol{\sigma}}_{\alpha}$, then the result is straightforward when $y\not\equiv {\boldsymbol{\sigma}}_{\alpha}[y]\mod Y_{Q,n}$. If $y\equiv {\boldsymbol{\sigma}}_{\alpha}[y]\mod Y_{Q,n}$, then $y\equiv {\boldsymbol{\sigma}}_{\alpha}[y]\mod Y_{Q,n}^{sc}$. The same argument above applies. The same argument again applies in the induction argument. This proves the result.
Unramified Whittaker functions
------------------------------
For an unramified principal series representation $I({\overline{\chi}})$, let ${\mathcal{W}}$ be the image of $\phi_K$ in the Whittaker model defined by . In other words, $${\mathcal{W}}_{\lambda}({\overline{t}},{\overline{\chi}})=\delta_B^{-1/2}({\overline{t}})W_{{\lambda}}({\overline{t}}\cdot \phi_K).$$ Note that our definition here is slightly different from [@Gao18b]. We divide by the modular quasi-character $\delta_B^{-1/2}$ to make our calculation slightly easier. If ${\lambda}$ is defined by $\gamma$, we write ${\mathcal{W}}_\gamma={\mathcal{W}}_{{\lambda}_\gamma}$. We also define ${\mathcal{W}}_{y}(y',{\overline{\chi}})={\mathcal{W}}_{{\mathbf{s}}_y}({\mathbf{s}}_{y'},{\overline{\chi}})$.
An element ${\overline{t}}\in{\overline{T}}$ is called dominant if ${\overline{t}}\cdot (U\cap K)\cdot {\overline{t}}^{-1}\subset K$.
\[thm:unramified whittaker function\] Let $I({\overline{\chi}})$ be an unramified principal series of ${\overline{G}}$ and $\gamma\in{\overline{T}}$. Let ${\mathcal{W}}_\gamma$ be the unramified Whittaker function associated to $\phi_K$. Then, ${\mathcal{W}}_\gamma({\overline{t}})=0$ unless ${\overline{t}}\in{\overline{T}}$ is dominant. Moreover, for dominant ${\overline{t}}$, one has $${\mathcal{W}}_\gamma({\overline{t}},{\overline{\chi}})=\sum_{{\mathbf{w}}\in W} c({\mathbf{w}}_G {\mathbf{w}},{\overline{\chi}}) \cdot \tau({\mathbf{w}}^{-1},{}^{{\mathbf{w}}}{\overline{\chi}},\gamma,w_G\cdot {\overline{t}}\cdot w_{G}^{-1}).$$
The proof in [@Gao18b] Proposition 3.3 works without essential change.
An inductive formula {#sec:inductive}
====================
As a consequence of Theorem \[thm:unramified whittaker function\], we now prove an inductive formula for unramified Whittaker function. The main result in this section is a generalization of the material presented in [@Suzuki98] Section 7.1.
For certain types of root systems, our formula might admit simplification – we discuss this in Sect. \[sec:general linear\]. See also [@Suzuki97] Lemma 4.1 and [@Suzuki98] Section 7.1. Note that there are some typos in the proofs of these two papers. We give full details here.
Basic setup
-----------
Let $\Delta'$ be a subset of $\Delta$. Let ${\mathbb P}={\mathbb M}{\mathbb N}$ be the parabolic subgroup of ${\mathbb{G}}$ associated with $\Delta'$. We write $$(X, \ \Phi_M, \ \Delta_M; \ Y, \ \Phi_M^\vee, \ \Delta_M^\vee)$$ for the root datum of $M$. Since ${\mathbb T}\subset {\mathbb M}$, the character and cocharacter lattices $X$ and $Y$ respectively are unchanged. However, we have $\Delta_M= \Delta'$ and $\Delta_M^\vee =\{\beta^\vee:\beta\in\Delta'\}$. Let ${\mathbb B}_M={\mathbb T}{\mathbb U}_M$ be the Borel subgroup of ${\mathbb M}$ corresponding to $\Delta_M$. Denote by $W({\mathbb M})\subset W({\mathbb{G}})$ the Weyl group of $({\mathbb M},{\mathbb T})$.
The functorial properties with respect to restriction is studied in [@GG18] Sect. 5.5. The cover $\overline{M}$ is associated to the pair $(D,\eta|_{Y_M^{\mathrm{sc}}})$, where the quadratic form $Q(x)=D(x,x)$ carries only the $W({\mathbb M})$-invariance by applying the “forgetful” functor from $W$-invariance.
Given a genuine character ${\overline{\chi}}:{\overline{T}}\to{\mathbb C}^\times$, one can define an unramified principal series representation $I_{{\overline{M}}}({\overline{\chi}})$ on ${\overline{M}}$. By induction in stages, $I({\overline{\chi}})=\operatorname{Ind}_{{\overline{P}}}^{{\overline{G}}}I_{{\overline{M}}}(\chi)$. Here $I_{{\overline{M}}}({\overline{\chi}})$ is inflated to a representation on ${\overline{P}}$ in the usual way. The study of Whittaker models and Whittaker functions applies to representations on ${\overline{M}}$. We add subscript ${\overline{M}}$ to indicate the ambient group.
We have the following observations:
- The section ${\mathbf{s}}_{{\overline{M}},y}={\mathbf{s}}_{{\overline{G}},y}$ for $y\in Y$. So the notation ${\mathbf{s}}_y$ does not arise any confusion.
- For ${\mathbf{w}}\in W({\mathbb M})$, one can calculate the local coefficient matrix $\tau_{{\overline{M}}}({\mathbf{w}},{\overline{\chi}},y,y')$. It is easy to check that $\tau_{{\overline{M}}}({\mathbf{w}},{\overline{\chi}},y,y')=\tau_{{\overline{G}}}({\mathbf{w}},{\overline{\chi}},y,y')$. Thus we can safely drop the subscript.
Let $W^M$ be the set of minimal representatives in $W({\mathbb M}){\backslash}W$. A element ${\mathbf{w}}\in W$ can be uniquely written as ${\mathbf{w}}={\mathbf{w}}_1{\mathbf{w}}_2$, where ${\mathbf{w}}_1\in W({\mathbb M})$ and ${\mathbf{w}}_2\in W^M$. The long element ${\mathbf{w}}_G$ is written as ${\mathbf{w}}_M{\mathbf{w}}^M$.
We have $$\Phi({\mathbf{w}}_G {\mathbf{w}})=\Phi({\mathbf{w}}^{M,-1}{\mathbf{w}}_2)\sqcup {\mathbf{w}}_2^{-1}(\Phi_M({\mathbf{w}}_M {\mathbf{w}}_1)).$$
Observe that $$\Phi({\mathbf{w}}_G{\mathbf{w}})=\{{\alpha}>0: {\mathbf{w}}({\alpha})>0\}.$$ and any element in this set satisfies ${\mathbf{w}}_2({\alpha})>0$. We have $$\{{\alpha}>0: {\mathbf{w}}({\alpha})>0\}=\{{\alpha}>0: {\mathbf{w}}_2({\alpha})\in \Phi^+-\Phi_M^+,{\mathbf{w}}({\alpha})>0\}\sqcup \{{\alpha}>0: {\mathbf{w}}_2({\alpha})\in\Phi_M^+,{\mathbf{w}}({\alpha})>0\}.$$ We now show that the first set is $\Phi({\mathbf{w}}^{M,-1}{\mathbf{w}}_2)$ and the second set is ${\mathbf{w}}_2^{-1}(\Phi_M({\mathbf{w}}_M {\mathbf{w}}_1))$.
Note that $\Phi({\mathbf{w}}^{M,-1})=\Phi^+-\Phi_M^+$. Thus $$\label{eq:the first set}
\Phi({\mathbf{w}}^{M,-1}{\mathbf{w}}_2)=\{{\alpha}>0:{\mathbf{w}}^{M,-1}{\mathbf{w}}_2({\alpha})<0\}=\{{\alpha}>0:{\mathbf{w}}_2({\alpha})\in \Phi^+ - \Phi_M^+\}.$$ Note that if ${\mathbf{w}}_2({\alpha}) \in \Phi^+ - \Phi_M^+$, then ${\mathbf{w}}({\alpha})>0$. Thus is the first set.
Let $\beta={\mathbf{w}}_2({\alpha})$. Then the second set is $$\{{\alpha}>0: {\mathbf{w}}_2({\alpha})\in\Phi_M^+,{\mathbf{w}}({\alpha})>0\}=\{{\mathbf{w}}_2^{-1}(\beta)\in \Phi_M^+:{\mathbf{w}}_1(\beta)>0\}={\mathbf{w}}_2^{-1}(\Phi_M({\mathbf{w}}_M {\mathbf{w}}_1)).$$ Now the result follows.
The inductive formula
---------------------
We now give the inductive formula.
\[prop:inductive formula\] We have $$\begin{aligned}
{\mathcal{W}}_\gamma({\overline{t}},{\overline{\chi}})=&\sum_{{\mathbf{w}}_2\in W^M}\sum_{\gamma'\in {\overline{T}}/{\overline{A}}} \tau({\mathbf{w}}_2^{-1},{}^{{\mathbf{w}}_2}{\overline{\chi}},\gamma,\gamma') {\mathcal{W}}_{M,\gamma'}(w^{M}\cdot {\overline{t}}\cdot w^{M,-1},{}^{{\mathbf{w}}_2}{\overline{\chi}})\\
&\cdot \left(\prod_{{\alpha}>0:{\mathbf{w}}^{M,-1}{\mathbf{w}}_2({\alpha})<0} \dfrac{1-{\overline{\chi}}_{\alpha}q^{-1}}{1-{\overline{\chi}}_{\alpha}} \right).
\end{aligned}$$
Recall that $${\mathcal{W}}_\gamma({\overline{t}},{\overline{\chi}})=\sum_{{\mathbf{w}}\in W} c({\mathbf{w}}_G {\mathbf{w}},\chi) \cdot \tau({\mathbf{w}}^{-1},{}^{{\mathbf{w}}}\chi,\gamma,w_G\cdot {\overline{t}}\cdot w_{G}^{-1})$$ Given ${\mathbf{w}}\in W$, it can be uniquely written as ${\mathbf{w}}={\mathbf{w}}_1{\mathbf{w}}_2$ as above. By the cocycle relation in Lemma \[lem：local coefficient matrix cocycle\], we deduce that $$\tau({\mathbf{w}}^{-1},{}^{{\mathbf{w}}}{\overline{\chi}},\gamma,w_G\cdot \bar{t}\cdot w_{G}^{-1})
=
\sum_{\gamma'\in {\overline{T}}/{\overline{A}}} \tau({\mathbf{w}}_2^{-1},{}^{{\mathbf{w}}_2}{\overline{\chi}},\gamma,\gamma')
\tau({\mathbf{w}}_1^{-1},{}^{{\mathbf{w}}}\chi,\gamma',w_G\cdot \bar{t}\cdot w_{G}^{-1})$$ On the other hand, $$\begin{aligned}
c({\mathbf{w}}_G{\mathbf{w}},{\overline{\chi}})=&\prod_{{\alpha}>0, \ {\mathbf{w}}_G{\mathbf{w}}({\alpha})<0}\dfrac{1-{\overline{\chi}}_{\alpha}q^{-1}}{1-{\overline{\chi}}_{\alpha}}\\
=&\left(\prod_{{\alpha}>0:{\mathbf{w}}^{M,-1}{\mathbf{w}}_2({\alpha})<0} \dfrac{1-{\overline{\chi}}_{\alpha}q^{-1}}{1-{\overline{\chi}}_{\alpha}} \right)
\left(\prod_{{\mathbf{w}}_2^{-1} \{ {\alpha}>0:{\mathbf{w}}_M{\mathbf{w}}_1({\alpha})<0 \}}\dfrac{1-{\overline{\chi}}_{{\mathbf{w}}_2^{-1}({\alpha})} q^{-1}}{1-{\overline{\chi}}_{{\mathbf{w}}_2^{-1}({\alpha})}}\right)\\
=& \left(\prod_{{\alpha}>0:{\mathbf{w}}^{M,-1}{\mathbf{w}}_2({\alpha})<0} \dfrac{1-{\overline{\chi}}_{\alpha}q^{-1}}{1-{\overline{\chi}}_{\alpha}} \right)
\left(\prod_{{\alpha}>0:{\mathbf{w}}_M{\mathbf{w}}_1({\alpha})<0 }\dfrac{1-({}^{{\mathbf{w}}_2}{\overline{\chi}})_{\alpha}q^{-1}}{1-({}^{{\mathbf{w}}_2}{\overline{\chi}})_{\alpha}}\right).
\end{aligned}$$ Here, we use the following fact: ${\overline{\chi}}_{{\mathbf{w}}^{-1}({\alpha})}=({}^{{\mathbf{w}}}{\overline{\chi}})_{\alpha}$. This can be seen from the following identity: ${\langle}x,{\mathbf{w}}^{-1}({\alpha}^\vee){\rangle}={\langle}{\mathbf{w}}(x),{\alpha}^\vee{\rangle}$ for any $x\in X$.
From this we deduce that $$\begin{aligned}
&\sum_{{\mathbf{w}}\in W} c({\mathbf{w}}_G {\mathbf{w}},\chi) \cdot \tau({\mathbf{w}}^{-1},{}^{{\mathbf{w}}}{\overline{\chi}},\gamma,w_G\cdot {\overline{t}}\cdot w_{G}^{-1})\\
=&\sum_{{\mathbf{w}}_2\in W^M}\sum_{{\mathbf{w}}_1\in W({\mathbb M})}\sum_{\gamma'\in {\overline{T}}/{\overline{A}}} \tau({\mathbf{w}}_2^{-1},{}^{{\mathbf{w}}_2}{\overline{\chi}},\gamma,\gamma')
\tau({\mathbf{w}}_1^{-1},{}^{{\mathbf{w}}}\chi,\gamma',w_G\cdot \bar{t}\cdot w_{G}^{-1})\\
&\cdot \left(\prod_{{\alpha}>0:{\mathbf{w}}^{M,-1}{\mathbf{w}}_2({\alpha})<0} \dfrac{1-{\overline{\chi}}_{\alpha}q^{-1}}{1-{\overline{\chi}}_{\alpha}} \right)
\left(\prod_{{\alpha}>0:{\mathbf{w}}_M{\mathbf{w}}_1({\alpha})<0 }\dfrac{1-({}^{{\mathbf{w}}_2}{\overline{\chi}})_{\alpha}q^{-1}}{1-({}^{{\mathbf{w}}_2}{\overline{\chi}})_{\alpha}}\right).\\
\end{aligned}$$ Note that $$\sum_{{\mathbf{w}}_1\in W({\mathbb M})} \tau({\mathbf{w}}_1^{-1},{}^{{\mathbf{w}}}\chi,\gamma',w_G\cdot \bar{t}\cdot w_{G}^{-1})
\left(\prod_{{\alpha}>0:{\mathbf{w}}_M{\mathbf{w}}_1({\alpha})<0 }\dfrac{1-({}^{{\mathbf{w}}_2}{\overline{\chi}})_{\alpha}q^{-1}}{1-({}^{{\mathbf{w}}_2}{\overline{\chi}})_{\alpha}}\right)
={\mathcal{W}}_{M,\gamma'}(w^{M}\cdot {\overline{t}}\cdot w^{M,-1},{}^{{\mathbf{w}}_2}{\overline{\chi}}).$$ Thus we deduce that ${\mathcal{W}}_\gamma({\overline{t}},{\overline{\chi}})$ equals $$\sum_{{\mathbf{w}}_2\in W^M}\sum_{\gamma'\in {\overline{T}}/{\overline{A}}} \tau({\mathbf{w}}_2^{-1},{}^{{\mathbf{w}}_2}{\overline{\chi}},\gamma,\gamma') {\mathcal{W}}_{M,\gamma'}(w^{M}\cdot {\overline{t}}\cdot w^{M,-1},{}^{{\mathbf{w}}_2}{\overline{\chi}})
\left(\prod_{{\alpha}>0:{\mathbf{w}}^{M,-1}{\mathbf{w}}_2({\alpha})<0} \dfrac{1-{\overline{\chi}}_{\alpha}q^{-1}}{1-{\overline{\chi}}_{\alpha}} \right).$$
Local coefficient matrix {#sec:local mat coeff for Levi}
------------------------
We end this section with a useful result on the local coefficient matrix. We now write ${\mathbb M}={\mathbb M}_1\times \cdots \times {\mathbb M}_k$. Let ${\mathbb T}_i={\mathbb T}\cap {\mathbb M}_i$. Let $Y_i$ be the cocharacter lattice of ${\mathbb T}_i$. Let $W({\mathbb M}_i)$ be the Weyl group of $({\mathbb M}_i,{\mathbb T}_i)$.
Let ${\mathbf{w}}=({\mathbf{w}}_1,\cdots,{\mathbf{w}}_k)\in W({\mathbb M})$ with ${\mathbf{w}}_i\in W({\mathbb M}_i)$. Let ${\mathbf{w}}'=({\mathbf{w}}'_1,\cdots,{\mathbf{w}}'_k)\in W({\mathbb M})$ with ${\mathbf{w}}'_i\in W({\mathbb M}_i)$. Let $y=(y_1,\cdots,y_k)$ where $y_i\in Y_i$. Let $y'_i={\mathbf{w}}_i'[y_i]$.
We now consider ${\overline{\chi}}|_{{\overline{T}}_{Q,n}^{sc}}$. Let ${\overline{\chi}}_i$ be a character of $Z({\overline{T}}_i)$ so that its restriction to ${\overline{T}}_{i,Q,n}^{sc}$ agrees with ${\overline{\chi}}_i|_{{\overline{T}}_{i,Q,n}^{sc}}$. In such situations, we write ${\overline{\chi}}\sim ({\overline{\chi}}_1,\cdots,{\overline{\chi}}_k)$. Recall from Lemma \[lem:dependence on chi\] that $\tau({\overline{\chi}}_i,{\mathbf{w}}_i,y,{\mathbf{w}}_i'[y'])$ only depends on the choice of ${\overline{\chi}}_i|_{{\overline{T}}_{i,Q,n}^{sc}}$ but not on the choice of ${\overline{\chi}}_i$.
\[lem:10\] With notations as above, $$\tau({\mathbf{w}},{\overline{\chi}},y,y')=\prod_{i=1}^k \tau({\mathbf{w}}_i,{\overline{\chi}}_i,y_i,y'_i).$$
By induction, it suffices to prove the case $k=2$. So we assume $k=2$ from now on.
For the case $k=2$, we prove it by induction on the length of ${\mathbf{w}}$. If ${\mathbf{w}}={\mathrm{id}}$, the result is trivial.
We now assume the result is true for ${\mathbf{w}}$ and prove it for ${\boldsymbol{\sigma}}_{\alpha}{\mathbf{w}}$ where $\ell({\boldsymbol{\sigma}}_{\alpha}{\mathbf{w}})=\ell({\mathbf{w}})+1$ and ${\boldsymbol{\sigma}}_{\alpha}$ is in either $W({\mathbb M}_1)$ or $W({\mathbb M}_2)$. We assume that ${\boldsymbol{\sigma}}_{\alpha}\in W({\mathbb M}_1)$ without loss of generality. We have $$\tau({\overline{\chi}},{\boldsymbol{\sigma}}_{\alpha}{\mathbf{w}},y,{\mathbf{w}}'[y])
=\sum_{y''\in Y/Y_{Q,n}} \tau({}^{{\mathbf{w}}}{\overline{\chi}},{\boldsymbol{\sigma}}_{\alpha},y,y'')\tau({\overline{\chi}},{\mathbf{w}},y'',{\mathbf{w}}'[y]).$$ The first term is nonzero only when $y''=y$ or $\sigma_{\alpha}[y]$. We write $y''=(y_1'',y_2'')$. By induction, we have $$\tau({\overline{\chi}},{\mathbf{w}},y'',{\mathbf{w}}'[y])=\tau({\overline{\chi}}_1,{\mathbf{w}}_1,y''_1,{\mathbf{w}}'_1[y_1]) \tau({\overline{\chi}}_2,{\mathbf{w}}_2,y''_2,{\mathbf{w}}'_2[y_2]).$$
Note that ${\boldsymbol{\sigma}}_{\alpha}[y]=({\boldsymbol{\sigma}}_{\alpha}[y_1],y_2)$ and $\tau({}^{{\boldsymbol{\sigma}}_{\alpha}}{\overline{\chi}},{\boldsymbol{\sigma}}_{\alpha},y,{\boldsymbol{\sigma}}_{\alpha}[y])=\tau({}^{{\boldsymbol{\sigma}}_{\alpha}}{\overline{\chi}}_1,{\boldsymbol{\sigma}}_{\alpha},y_1,{\boldsymbol{\sigma}}_{\alpha}[y_1])$. By Lemma \[lem:several lattices\], it is easy to verify that $y\equiv{\boldsymbol{\sigma}}_{\alpha}[y]\mod Y_{Q,n}$ if and only if $y_1\equiv {\boldsymbol{\sigma}}_{\alpha}[y_1] \mod Y_{1,Q,n}$. If $y\not\equiv{\boldsymbol{\sigma}}_{\alpha}[y] \mod Y_{Q,n}$, then $$\begin{aligned}
&\tau({\overline{\chi}},{\boldsymbol{\sigma}}_{\alpha}{\mathbf{w}},y,y')\\
=&\tau({\boldsymbol{\sigma}}_{\alpha},{}^{{\mathbf{w}}}{\overline{\chi}},y,y)\tau({\overline{\chi}}_1,{\mathbf{w}}_1,y_1,y'_1)\tau({\overline{\chi}}_2,{\mathbf{w}}_2,y_2,y'_2)
+\tau({\boldsymbol{\sigma}}_{\alpha},{}^{{\mathbf{w}}}{\overline{\chi}},y,{\boldsymbol{\sigma}}_{\alpha}[y])\tau({\overline{\chi}}_1,{\mathbf{w}}_1,{\boldsymbol{\sigma}}_{\alpha}[y_1],y'_1)\tau({\overline{\chi}}_2,{\mathbf{w}}_2,y_2,y'_2)\\
=&(\tau({\boldsymbol{\sigma}}_{\alpha},{}^{{\mathbf{w}}}{\overline{\chi}},y,y)\tau({\overline{\chi}}_1,{\mathbf{w}}_1,y_1,y'_1)
+\tau({\boldsymbol{\sigma}}_{\alpha},{}^{{\mathbf{w}}}{\overline{\chi}},y,{\boldsymbol{\sigma}}_{\alpha}[y])\tau({\overline{\chi}}_1,{\mathbf{w}}_1,{\boldsymbol{\sigma}}_{\alpha}[y_1],y'_1))\tau({\overline{\chi}}_2,{\mathbf{w}}_2,y_2,y'_2)\\
=&\tau({\overline{\chi}}_1,{\boldsymbol{\sigma}}_{\alpha}{\mathbf{w}}_1,y_1,y_1')\tau({\overline{\chi}}_2,{\mathbf{w}}_2,y_2,y'_2)\\
\end{aligned}$$ If $y\equiv {\boldsymbol{\sigma}}_{\alpha}[y] \mod Y_{Q,n}$, $$\begin{aligned}
&\tau({\overline{\chi}},{\boldsymbol{\sigma}}_{\alpha}{\mathbf{w}},y,y')\\
=&\tau({\boldsymbol{\sigma}}_{\alpha},{}^{{\mathbf{w}}}{\overline{\chi}},y,y)\tau({\overline{\chi}}_1,{\mathbf{w}}_1,y_1,y'_1)\tau({\overline{\chi}}_2,{\mathbf{w}}_2,y_2,y'_2)\\
=&\tau({\overline{\chi}}_1,{\boldsymbol{\sigma}}_{\alpha}{\mathbf{w}}_1,y_1,y_1')\tau({\overline{\chi}}_2,{\mathbf{w}}_2,y_2,y'_2).\\
\end{aligned}$$
Relative theta representations {#sec:relative theta}
==============================
We first recall the definition of theta representations and discuss its generalization given in [@Suzuki98] and [@Gao].
Definition
----------
We start with the following definition.
An unramified genuine character ${\overline{\chi}}$ of $Z({\overline{T}})$ is called *exceptional* if $${\overline{\chi}}({\overline{h}}_{\alpha}(\varpi^{n_{\alpha}}))=q^{-1} \text{ for all }{\alpha}\in\Delta.$$ The theta representation $\Theta({\overline{G}},{\overline{\chi}})$ associated to an exceptional character ${\overline{\chi}}$ is the unique Langlands quotient (see [@BJ13]) of $I({\overline{\chi}})$, which is also equal to the image of the intertwining operator $T_{{\mathbf{w}}_G,{\overline{\chi}}}:I({\overline{\chi}})\to I({}^{{\mathbf{w}}_G}{\overline{\chi}})$.
To make our discussion more flexible, we introduce the following definition. It can be viewed as a generalization of [@Suzuki98] and [@Gao].
\[def:exceptional\] For any subset $\Delta'\subset \Delta$, a genuine character $\chi$ is called $\Delta'$-exceptional (resp. $\Delta'$-anti-exceptional) if ${\overline{\chi}}({\overline{h}}_{\alpha}(\varpi^{n_{\alpha}}))=q^{-1}$ (resp. ${\overline{\chi}}({\overline{h}}_{\alpha}(\varpi^{n_{\alpha}}))=q$) for every ${\alpha}\in\Delta'$. In the case $\Delta'=\Delta$, it is simply called exceptional or anti-exceptional, respectively.
Let ${\mathbb M}$ be the Levi subgroup corresponding to $\Delta'$. Then a $\Delta'$-exceptional character can be viewed as an exceptional character for ${\overline{M}}$. In other words, we obtain a representation $\Theta({\overline{M}},{\overline{\chi}})$ of ${\overline{M}}$ as the image of the intertwining operator $$T_{{\mathbf{w}}_{M},{\overline{\chi}}}:I_{{\overline{M}}}({\overline{\chi}})\to I_{{\overline{M}}}({}^{{\mathbf{w}}_M}{\overline{\chi}}).$$ Here ${\mathbf{w}}_M$ is the longest element in the Weyl group of $M$. We also add subscript ‘${\overline{M}}$’ to indicate the ambient group. We will do so in the rest of this section.
We can now define a representation on ${\overline{G}}$ by normalized induction: $$\Theta({\overline{G}}/{\overline{M}},{\overline{\chi}}):=\operatorname{Ind}_{{\overline{P}}}^{{\overline{G}}}\Theta({\overline{M}},{\overline{\chi}}).$$ We call it a *relative Theta representation*. The representation $\Theta({\overline{G}}/{\overline{M}},{\overline{\chi}})$ can also be defined as the image of the intertwining operator $$T_{{\mathbf{w}}_M,{\overline{\chi}}}:I({\overline{\chi}})\to I({}^{{\mathbf{w}}_{M}}{\overline{\chi}}).$$ Note that $\Theta({\overline{G}}/{\overline{M}},{\overline{\chi}})$ might be reducible.
Some properties
---------------
We discuss some properties of $\Theta({\overline{G}}/{\overline{M}},{\overline{\chi}})$. The intertwining operator $T_{{\mathbf{w}}_M,{\overline{\chi}}}:I({\overline{\chi}})\to I({}^{{\mathbf{w}}_{M}}{\overline{\chi}})$ induces a map on the space of Whittaker functional $$T_{{\mathbf{w}}_M,{\overline{\chi}}}^{\ast}: {\mathrm{Wh}}_{\psi}(I({}^{{\mathbf{w}}_{M}}{\overline{\chi}}))\to {\mathrm{Wh}}_{\psi}(I({\overline{\chi}})).$$ The matrix is defined by $$T_{{\mathbf{w}}_M,{\overline{\chi}}}^{\ast}(\lambda_{\gamma}^{{}^{{\mathbf{w}}_M}{\overline{\chi}}})=\sum_{\gamma'\in {\overline{T}}/{\overline{A}}} \tau({\mathbf{w}}_M,{\overline{\chi}},\gamma,\gamma')\cdot \lambda_{\gamma'}^{{\overline{\chi}}}.$$
A function ${\mathbf{c}}\in {\mathrm{Ftn}}(i({\overline{\chi}}))$ gives rise to a functional in ${\mathrm{Wh}}_{\psi}(\Theta({\overline{G}}/{\overline{M}},{\overline{\chi}}))$ if and only if for all ${\alpha}\in \Delta'$, $$\sum_{\gamma\in{\overline{T}}/{\overline{A}}} {\mathbf{c}}(\gamma) \tau({\boldsymbol{\sigma}}_{\alpha},{}^{{\boldsymbol{\sigma}}_{\alpha}}{\overline{\chi}},\gamma,\gamma')=0 \text{ for all }\gamma'.$$ The left-hand side is independent of the choice of representatives for ${\overline{T}}/{\overline{A}}$.
The same proof in [@KP84] page 76 works here as well.
Let ${\overline{\chi}}$ be an unramified $\Delta'$-exceptional character. Let $\lambda_{{\mathbf{c}}}^{{\overline{\chi}}}\in {\mathrm{Wh}}_\psi(I({\overline{\chi}}))$ be the $\psi$-Whittaker functional of $I({\overline{\chi}})$ associated to some ${\mathbf{c}}\in {\mathrm{Ftn}}(i({\overline{\chi}}))$. Then, $\lambda_{{\mathbf{c}}}^{{\overline{\chi}}}$ lies in ${\mathrm{Wh}}_{\psi}(\Theta({\overline{G}}/{\overline{M}},{\overline{\chi}}))$ if and only for any simple root ${\alpha}\in \Delta'$ one has $${\mathbf{c}}({\mathbf{s}}_{{\boldsymbol{\sigma}}_{\alpha}[y]})=q^{k_{y,{\alpha}}-1}{\mathbf g}_{\psi^{-1}}({\langle}y,{\alpha}^\vee{\rangle}Q({\alpha}^\vee))^{-1}\cdot {\mathbf{c}}({\mathbf{s}}_y) \text{ for all }y.$$
The proof in [@Gao17] Corollary 3.7 works the same here.
We now state some basic properties of these coefficients. See also [@Suzuki97] Sect. 3.6.
\[prop:tau on weyl orbit\] Let ${\overline{\chi}}$ be an unramified $\Delta'$-exceptional character.
1. $\tau({\mathbf{w}}_M,{\overline{\chi}},y,y')=0$ unless $y'={\mathbf{w}}[y]$ for some ${\mathbf{w}}\in W({\mathbb M})$.
2. $\tau({\mathbf{w}}_M,{\overline{\chi}},y,{\mathbf{w}}[y'])=R({\mathbf{w}},y)\tau({\mathbf{w}}_M,{\overline{\chi}},y,y')$ for ${\mathbf{w}}\in W({\mathbb M})$, where $R({\mathbf{w}},y)$ is some function of $W$ and $Y$.
The first one is obvious. The second one follows from the above lemma. In fact, $\tau({\overline{\chi}},{\mathbf{w}}_M,\gamma,-)\in {\mathrm{Ftn}}(i({\overline{\chi}}))$ gives rise to a functional in ${\mathrm{Wh}}_\psi(\Theta({\overline{G}}/{\overline{M}},{\overline{\chi}}))$. When ${\mathbf{w}}={\boldsymbol{\sigma}}_{\alpha}$ is a simple reflection, then $$\tau({\mathbf{w}}_M,{\overline{\chi}},y,{\boldsymbol{\sigma}}_{\alpha}[y'])=q^{k_{y,{\alpha}}-1}{\mathbf g}_{\psi^{-1}}({\langle}y,{\alpha}{\rangle}Q({\alpha}^\vee))^{-1}\tau({\mathbf{w}}_M,{\overline{\chi}},y,y').$$ The rest follows by induction.
\[cor:proportional\] Let ${\mathbf{w}}\in W({\mathbb M})$. Then $T_{{\mathbf{w}}_M,{\overline{\chi}}}^{\ast}(\lambda_{y}^{{}^{{\mathbf{w}}_M}{\overline{\chi}}})$ and $T_{{\mathbf{w}}_M,{\overline{\chi}}}^{\ast}(\lambda_{{\mathbf{w}}[y]}^{{}^{{\mathbf{w}}_M}{\overline{\chi}}})$ are proportional on $I({\overline{\chi}})$, and $$T_{{\mathbf{w}}_M,{\overline{\chi}}}^{\ast}(\lambda_{{\mathbf{w}}[y]}^{{}^{{\mathbf{w}}_M}{\overline{\chi}}}) =\dfrac{\tau({\mathbf{w}}_M,{\overline{\chi}},{\mathbf{w}}[y],y)}{\tau({\mathbf{w}}_M,{\overline{\chi}},y,y)} T_{{\mathbf{w}}_M,{\overline{\chi}}}^{\ast}(\lambda_{y}^{{}^{{\mathbf{w}}_M}{\overline{\chi}}})$$
This is an immediate consequence of Proposition \[prop:tau on weyl orbit\]. In fact, $$T_{{\mathbf{w}}_M,{\overline{\chi}}}^{\ast}(\lambda_{y}^{{}^{{\mathbf{w}}_M}{\overline{\chi}}}) =\sum_{y'\in Y/Y_{Q,n}}\tau({\mathbf{w}}_M,{\overline{\chi}},y,y') {\lambda}_{y'}^{{\overline{\chi}}} =\sum_{\tilde {\mathbf{w}}\in W({\mathbb M}) }R(\tilde{\mathbf{w}},y)\cdot \tau({\mathbf{w}}_M,{\overline{\chi}},y,y) {\lambda}_{\tilde{\mathbf{w}}[y']}^{{\overline{\chi}}}$$ and similarly $$T_{{\mathbf{w}}_M,{\overline{\chi}}}^{\ast}(\lambda_{{\mathbf{w}}[y]}^{{}^{{\mathbf{w}}_M}{\overline{\chi}}}) =\sum_{y'\in Y/Y_{Q,n}}\tau({\mathbf{w}}_M,{\overline{\chi}},{\mathbf{w}}[y],y') {\lambda}_{y'}^{{\overline{\chi}}}
=\sum_{\tilde {\mathbf{w}}\in W({\mathbb M}) }R(\tilde{\mathbf{w}},y)\cdot \tau({\mathbf{w}}_M,{\overline{\chi}},{\mathbf{w}}[y],y) {\lambda}_{\tilde{\mathbf{w}}[y']}^{{\overline{\chi}}}.$$ This gives the desired result.
Rodier’s lemma {#sec:rodier}
--------------
We end this section with a generalization of a lemma of Rodier. This will be useful later. Recall that Rodier’s result says that when the inducing data is generic, then so is the induced representation.
\[prop:rodier\] The representation $\operatorname{Ind}_{{\overline{P}}}^{{\overline{G}}} (\pi)$ is generic if and only if $\pi$ is generic. Moreover, $$\dim \operatorname{Hom}_{U}(\operatorname{Ind}_{{\overline{P}}}^{{\overline{G}}} (\pi),\psi) =\dim \operatorname{Hom}_{U_M}(\pi,\psi|_{U_M}).$$
This follows from [@BZ77] Theorem 5.2 and [@CS80] Lemma 1.5.
The case of general linear groups {#sec:general linear}
=================================
From now on, we focus on the case of ${\mathbb{G}}={\mathbb{GL}}_{r+1}$. We now introduce some notations in this setup. Write $\Delta=\{{\alpha}_1,\cdots,{\alpha}_r\}$ with the standard enumeration and the Weyl group is generated by $\{{\boldsymbol{\sigma}}_1,\cdots,{\boldsymbol{\sigma}}_r\}$. The root system is simply-laced, and we write $n_Q=n_{\alpha}$ for any ${\alpha}\in \Delta$. For ${\alpha}={\alpha}_i+\cdots+{\alpha}_{j-1}$, write ${\overline{\chi}}_{ij}={\overline{\chi}}_{\alpha}$.
Inductive formula
-----------------
The inductive formula in Proposition \[prop:inductive formula\] admits a refinement in the case of ${\mathbb M}={\mathbb{GL}}_{r}\times {\mathbb{GL}}_1$ in ${\mathbb{GL}}_{r+1}$.. This is similar to [@Suzuki97] Lemma 4.1. In this case, the $W^M$ is $\{ {\boldsymbol{\sigma}}_r,\ {\boldsymbol{\sigma}}_{r}{\boldsymbol{\sigma}}_{r-1},\ \cdots,\ {\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_1 \}$.
Recall that ${\mathcal{W}}_y(y',{\overline{\chi}})\neq 0$ if and only if $y$ and ${\mathbf{w}}_G(y')$ lies in the same orbit under the Weyl group action. (Note that this is not ${\mathbf{w}}_G[y']$.) We now assume that ${\mathbf{w}}_G(y')={\mathbf{w}}'[y]$ for some ${\mathbf{w}}'\in W$. Note that this is true identity instead of $\mod Y_{Q,n}$. Any ${\mathbf{w}}'\in W$ can be uniquely written as ${\mathbf{w}}'={\mathbf{w}}{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_{r_0}$ for an integer $r_0$ and ${\mathbf{w}}\in W({\mathbb M})$. We have arrived at $${\mathbf{w}}_G(y') = {\mathbf{w}}{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_{r_0} [y]$$ with ${\mathbf{w}}\in W({\mathbb M})$.
\[prop:inductive 1 in general linear\] Assume that the orbit of $y$ under $W$ is free. We have $${\mathcal{W}}_{y}(y',{\overline{\chi}})=
\sum_{i=1}^{r_0}\sum_{y''} \tau({\boldsymbol{\sigma}}_i\cdots{\boldsymbol{\sigma}}_r,{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_i}{\overline{\chi}},y,y'') {\mathcal{W}}_{M,y''}({\mathbf{w}}^{M}(y'),{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_i}{\overline{\chi}}) \left(\prod_{j=1}^{i}\dfrac{1-{\overline{\chi}}_{ji}q^{-1}}{1-{\overline{\chi}}_{ji}} \right),$$ where the second sum is over the set $$\begin{aligned}
&\left\{
{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_{r_0} {\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_i^{a_i}[y]: a_i,\cdots,a_{r_0-2}\in \{0,1\}
\right\}\\
=&\left\{
{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_i^{a_i}{\mathbf{w}}^{-1}[{\mathbf{w}}_G(y')]: a_i,\cdots,a_{r_0-2}\in \{0,1\}
\right\}.\\
\end{aligned}$$
This result is probably true in general. But we only prove what we need here.
Note ${\mathbf{w}}^M={\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_1$. If ${\mathbf{w}}_2={\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_i$, then ${\mathbf{w}}^{M,-1}{\mathbf{w}}_2={\boldsymbol{\sigma}}_1\cdots {\boldsymbol{\sigma}}_{i-1}$. Thus, $$\{{\alpha}>0:{\mathbf{w}}^{M,-1}{\mathbf{w}}_2({\alpha})<0\}=
\{
{\alpha}_{i-1}, \ {\alpha}_{i-2}+{\alpha}_{i-1},\cdots, {\alpha}_1+\cdots + {\alpha}_{i-1}
\}.$$ In this case, $$\prod_{{\alpha}>0:{\mathbf{w}}^{M,-1}{\mathbf{w}}_2({\alpha})<0} \dfrac{1-{\overline{\chi}}_{\alpha}q^{-1}}{1-{\overline{\chi}}_{\alpha}} =\prod_{j=1}^{i}\dfrac{1-{\overline{\chi}}_{ji}q^{-1}}{1-{\overline{\chi}}_{ji}}.$$ We may rewrite the formula as $$\begin{aligned}
&{\mathcal{W}}_y(y',{\overline{\chi}})\\
=&\sum_{i=1}^r\sum_{y''\in Y/Y_{Q,n}} \tau({\boldsymbol{\sigma}}_i\cdots{\boldsymbol{\sigma}}_r,{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_i}{\overline{\chi}},y,y'') {\mathcal{W}}_{M,y''}({\mathbf{w}}^{M}(y'),{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_i}{\overline{\chi}}) \left(\prod_{j=1}^{i}\dfrac{1-{\overline{\chi}}_{ji}q^{-1}}{1-{\overline{\chi}}_{ji}} \right).
\end{aligned}$$ We now analyze when both $\tau({\boldsymbol{\sigma}}_i\cdots{\boldsymbol{\sigma}}_r,{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_i}{\overline{\chi}},y,y'')$ and ${\mathcal{W}}_{M,y''}({\mathbf{w}}^{M}(y'),{}^{{\mathbf{w}}_2}{\overline{\chi}})$ are nonzero.
We know that ${\mathbf{w}}_G(y')={\mathbf{w}}{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_{r_0} [y]$ with ${\mathbf{w}}\in W({\mathbb M})$. If ${\mathcal{W}}_{M,y''}({\mathbf{w}}^{M}(y'),{}^{{\mathbf{w}}_2}{\overline{\chi}}) \neq 0$, then for some $y''\in Y$ and ${\mathbf{w}}' \in W$, $$y''\equiv {\mathbf{w}}'[{\mathbf{w}}_G(y')] \mod Y_{Q,n}.$$ This condition implies that $$y''\equiv {\mathbf{w}}''{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_{r_0} [y] \mod Y_{Q,n}$$ for some ${\mathbf{w}}''\in W({\mathbb M})$. If $\tau({\boldsymbol{\sigma}}_i\cdots{\boldsymbol{\sigma}}_r,{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_i}{\overline{\chi}},y,y'')\neq 0$, then $${\boldsymbol{\sigma}}_r^{a_r}\cdots {\boldsymbol{\sigma}}_i^{a_i}[y]\equiv y'' \mod Y_{Q,n}$$ for some $a_i,\cdots,a_r\in \{0,1\}$.
We now use the assumption that the orbit of $y$ is free. This implies that $$\label{eq:equality of Weyl}
{\boldsymbol{\sigma}}_r^{a_r}\cdots {\boldsymbol{\sigma}}_i^{a_i}={\mathbf{w}}''{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_{r_0}$$ for some $a_i,\cdots,a_r\in \{0,1\}$ and ${\mathbf{w}}''\in W({\mathbb M})$. By considering the images of both sides in $W({\mathbb M}){\backslash}W$, we know that this is possible only when $1\leq i\leq r_0$. The same argument shows that we must have $a_{r_0-1}=0$. Thus we conclude that elements in is of the form $${\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_{r_0} {\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_i^{a_i}.$$ So finally, we have arrived at $${\mathcal{W}}_y(y',{\overline{\chi}})=\sum_{i=1}^{r_0}\sum_{y''} \tau({\boldsymbol{\sigma}}_i\cdots{\boldsymbol{\sigma}}_r,{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_i}{\overline{\chi}},y,y'') {\mathcal{W}}_{M,y''}({\mathbf{w}}^M(y'),{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_i}{\overline{\chi}}) \left(\prod_{j=1}^{i}\dfrac{1-{\overline{\chi}}_{ji}q^{-1}}{1-{\overline{\chi}}_{ji}} \right)$$ where the second sum is over the set $$\begin{aligned}
&\left\{
{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_{r_0} {\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_i^{a_i}[y]\mid a_i,\cdots,a_{r_0-2}\in \{0,1\}
\right\}\\
=&\left\{
{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_i^{a_i}{\mathbf{w}}^{-1}[{\mathbf{w}}_G(y')]\mid a_i,\cdots,a_{r_0-2}\in \{0,1\}
\right\}.\\
\end{aligned}$$
We now discuss the covering group obtained by $${\mathbb{GL}}_r\hookrightarrow {\mathbb{GL}}_{r+1},\qquad g\mapsto \mathrm{diag}(g,1).$$ Write $Y=Y_1\oplus Y_2$ where $Y_1$ (resp. $Y_2$) be the cocharacter lattice of ${\mathbb{GL}}_r$ (resp. ${\mathbb{GL}}_1$). Then we have embeddings $\iota:Y_1\hookrightarrow Y$ and $\iota:Y_1^{sc}\hookrightarrow Y^{sc}$. The cover of $\overline{\operatorname{GL}}_r$ is associated with $(D|_{Y'},\eta|_{Y_1^{sc}})$. We also observe that for $y\in Y_1$, ${\mathbf{s}}_{{\mathbb{GL}}_r,\iota(y)}={\mathbf{s}}_{{\mathbb{GL}}_{r-1},y}$. Thus, the notation ${\mathbf{s}}_y$ as this does not arise any confusion.
\[lem:m to gl on w\] Let
- $y=(y_1,y_2), y'=(y'_1,y'_2)$ with $y_1,y_1'\in Y_1$ and $y_2,y_2'\in Y_2$,
- ${\overline{\chi}}_1$ be an unramified character for $\overline{\operatorname{GL}_r}$ such that ${\overline{\chi}}_{1}|_{Y_{1,Q,n}^{sc}}={\overline{\chi}}|_{Y_{Q,n}^{sc}}$.
If $y_2=y'_2$, then ${\mathcal{W}}_{M,y}(y',{\overline{\chi}})={\mathcal{W}}_{\operatorname{GL}_r,y_1}(y'_1,{\overline{\chi}}_1)$.
Recall that $${\mathcal{W}}_{M,y}(y',{\overline{\chi}})=\sum_{{\mathbf{w}}\in W(M)} c_M({\mathbf{w}}_M {\mathbf{w}},{\overline{\chi}}) \cdot \tau_M({\mathbf{w}}^{-1},{}^{{\mathbf{w}}}{\overline{\chi}},y,{\mathbf{w}}_M(y')),$$ It is straightforward to see that $c_M({\mathbf{w}}_M {\mathbf{w}},{\overline{\chi}})=c_{\operatorname{GL}_r}({\mathbf{w}}_M {\mathbf{w}},{\overline{\chi}}_1)$. By Lemma \[lem:10\], $$\tau_M({\mathbf{w}}^{-1},{}^{{\mathbf{w}}}{\overline{\chi}},y,{\mathbf{w}}_M(y')) =\tau_{\operatorname{GL}_r}({\mathbf{w}}^{-1},{}^{{\mathbf{w}}}{\overline{\chi}}_1,y_1,{\mathbf{w}}_{\operatorname{GL}_r}(y_1')).$$ (Note that the condition $y_2=y_2'$ does not appear explicitly in the proof but must be satisfied.) Therefore, ${\mathcal{W}}_{M,y}(y',{\overline{\chi}})={\mathcal{W}}_{\operatorname{GL}_r,y_1}(y'_1,{\overline{\chi}}_1)$.
Relative theta representations {#relative-theta-representations}
------------------------------
We now discuss the Whittaker models for the relative theta representations. In particular, we determine when these representations are non-generic and possess a unique Whittaker model. The main ingredient here is [@Gao17] Theorem 1.1, which is a generalization of [@KP84] Theorem I.3.5.
The root system spanned by $\Delta'$ is of type $A_{r_1-1}\times \cdots \times A_{r_k-1}$, where $r_1+\cdots+r_k=r$. In this way, we obtain a bijection between subsets of $\Delta$ and ordered partitions $(r_1\cdots r_k)$ of $r$. The following result can be proved along the same line as in [@Gao17].
1. If $r_i>n_Q$ for some $i$, then the representation $\Theta({\overline{M}},{\overline{\chi}})$ is non-generic.
2. If $r_i\leq n_Q$ for all $i$, then the representation $\Theta({\overline{M}},{\overline{\chi}})$ is generic.
3. If $r_i=n_Q$ for all $i$, then the representation $\Theta({\overline{M}},{\overline{\chi}})$ has a unique Whittaker model.
The proof in [@Gao17] Example 3.16 and [@KP84] Corollary I.3.6 applies without essential change.
By combining this result with Proposition \[prop:rodier\], we deduce the following result, in analogy with [@Suzuki98] Corollary 3.3.
\[thm:whittaker for relative theta\]
1. If $r_i>n_Q$ for some $i$, then the representation $\Theta({\overline{G}}/{\overline{M}},{\overline{\chi}})$ is non-generic.
2. If $r_i\leq n_Q$ for all $i$, then the representation $\Theta({\overline{G}}/{\overline{M}},{\overline{\chi}})$ is generic.
3. If $r_i=n_Q$ for all $i$, then the representation $\Theta({\overline{G}}/{\overline{M}},{\overline{\chi}})$ has a unique Whittaker model.
In the rest of this paper, we would like find a formula for some values of the unramified Whittaker functions in some special instances.
Some calculation of $c({\mathbf{w}},{\overline{\chi}})$
-------------------------------------------------------
In this section, we carry out some calculation of $c({\mathbf{w}},{\overline{\chi}})$ for exceptional and anti-exceptional characters. In particular, we show that $\Theta({\overline{G}}/{\overline{M}},{\overline{\chi}})$ contains a spherical vector.
\[lem:gk for exceptiona\] Let ${\overline{\chi}}$ be an exceptional character. Then for $1\leq i\leq r-1$ $$c({\boldsymbol{\sigma}}_i\cdots {\boldsymbol{\sigma}}_1,{\overline{\chi}})\neq 0.$$
By direct calculation, $$c({\boldsymbol{\sigma}}_i\cdots {\boldsymbol{\sigma}}_1,{\overline{\chi}})=\dfrac{1-q^{-2}}{1-q^{-1}}\cdots \dfrac{1-q^{-(i+1)}}{1-q^{-i}}\neq 0.$$
Let ${\overline{\chi}}$ be an exceptional character. Then $$c({\mathbf{w}}_G,{\overline{\chi}})\neq 0.$$
This follows the above lemma and induction.
The representation $\Theta({\overline{G}}/{\overline{M}},{\overline{\chi}})$ contains a spherical vector.
The representation $\Theta({\overline{G}}/{\overline{M}},{\overline{\chi}})$ is defined as the image of $T_{{\mathbf{w}}_M,{\overline{\chi}}}:I({\overline{\chi}})\to I({}^{{\mathbf{w}}_M}{\overline{\chi}})$. The image contains the vector $c({\mathbf{w}}_G,{\overline{\chi}})\phi_K^{{}^{{\mathbf{w}}}{\overline{\chi}}}$, which is nonzero.
\[cor:vanishing of unramified whittaker\] If $r_i> n_Q$ for some $i$, then $${\mathcal{W}}_{\gamma}(\gamma',{}^{{\mathbf{w}}_M}{\overline{\chi}})=0$$ for any $\gamma,\gamma'$.
If ${\mathcal{W}}_{\gamma}(\gamma',{}^{{\mathbf{w}}_M}{\overline{\chi}})\neq 0$, then the Whittaker functional is nonzero on a spherical vector $\phi_K$.
We already know that $\phi_K\in\Theta({\overline{G}}/{\overline{M}},{\overline{\chi}})$. Thus this implies that $\Theta({\overline{G}}/{\overline{M}},{\overline{\chi}})$ has a nonzero Whittaker functional. This contradicts with our assumption and Theorem \[thm:whittaker for relative theta\].
\[lem:gk for anti 1\] Let ${\overline{\chi}}$ be an anti-exceptional character. Then $$c({\boldsymbol{\sigma}}_i\cdots {\boldsymbol{\sigma}}_1,{}^{{\boldsymbol{\sigma}}_1\cdots {\boldsymbol{\sigma}}_i}{\overline{\chi}})\neq 0.$$
The proof is the same as Lemma \[lem:gk for exceptiona\]. We have $$c({\boldsymbol{\sigma}}_i\cdots {\boldsymbol{\sigma}}_1,{}^{{\boldsymbol{\sigma}}_1\cdots {\boldsymbol{\sigma}}_i}{\overline{\chi}})=\dfrac{1-q^{-(i+1)}}{1-q^{-i}}\cdots \dfrac{1-q^{-2}}{1-q^{-1}}\neq 0.$$
\[lem:gk for anti 2\] Let ${\overline{\chi}}$ be an anti-exceptional character. Let $${\mathbf{w}}=({\boldsymbol{\sigma}}_{r-i+1}\cdots {\boldsymbol{\sigma}}_r)\cdots ({\boldsymbol{\sigma}}_2\cdots {\boldsymbol{\sigma}}_{i+1})({\boldsymbol{\sigma}}_1\cdots {\boldsymbol{\sigma}}_i)$$ Then $c({\mathbf{w}}^{-1},{}^{{\mathbf{w}}}{\overline{\chi}})\neq 0$.
This follows from induction and Lemma \[lem:gk for anti 2\].
Calculation of certain local matrix coefficients {#sec:calculation of local matrix}
================================================
Assume that $n_Q \geq r$. The goal in this section is to calculation $\tau({\mathbf{w}}_G,{}^{{\mathbf{w}}_G}{\overline{\chi}},y,0)$ where ${\overline{\chi}}$ is an anti-exceptional character for $\overline{\operatorname{GL}}_{r}$. Recall that $\tau({\mathbf{w}}_G,{}^{{\mathbf{w}}_G}{\overline{\chi}},y,0)\neq 0$ unless $y={\mathbf{w}}[0]$ for some ${\mathbf{w}}\in W$. As $n_Q\geq r$, the orbit of $0$ under the action of $W$ is free. The theta representation $\Theta({\overline{G}},{}^{{\mathbf{w}}_G}{\overline{\chi}})$ is realized as a subrepresentation of $I({\overline{\chi}})$. Recall that $\phi_K\in \Theta({\overline{G}},{}^{{\mathbf{w}}_G}{\overline{\chi}})$.
\[lem:3\] For ${\mathbf{w}}\in W$, $${\mathcal{W}}_{{\mathbf{w}}[0]}(0,{\overline{\chi}})= \tau({\mathbf{w}}_G,{}^{{\mathbf{w}}_G}{\overline{\chi}},{\mathbf{w}}[0],0).$$
This follows from Theorem \[thm:unramified whittaker function\]. Note that $c({\mathbf{w}}_G{\mathbf{w}},{\overline{\chi}})=0$ unless ${\mathbf{w}}={\mathbf{w}}_G$.
Let $y\in Y$. We define the Gauss sum $g({\mathbf{w}},y)$ for ${\mathbf{w}}\in W$ as follows:
1. $g({\mathrm{id}},y)=1$;
2. For a simple reflection ${\boldsymbol{\sigma}}_{\alpha}$, $$g({\boldsymbol{\sigma}}_{\alpha},y)={\mathbf g}_{\psi^{-1}}({\langle}y_\rho,{\alpha}{\rangle}Q({\alpha}^\vee)).$$
3. If ${\mathbf{w}}_1,{\mathbf{w}}_2\in W$ such that $\ell({\mathbf{w}}_1{\mathbf{w}}_2)=\ell({\mathbf{w}}_1)+\ell({\mathbf{w}}_2)$, then $$g({\mathbf{w}}_1{\mathbf{w}}_2,y)=g({\mathbf{w}}_1,{\mathbf{w}}_2[y])g({\mathbf{w}}_2,y).$$
We have to verify that this is well-defined.
We have $$g({\mathbf{w}},y)=\prod_{{\alpha}\in \Phi({\mathbf{w}})} {\mathbf g}_{\psi^{-1}}({\langle}y_\rho,{\alpha}{\rangle}Q({\alpha}^\vee)).$$ Therefore, $g({\mathbf{w}},y)$ is well-defined.
Recall that ${\mathbf{w}}[y]_\rho={\mathbf{w}}(y-\rho)$ for any ${\mathbf{w}}\in W$. Fix a reduced decomposition ${\mathbf{w}}={\boldsymbol{\sigma}}_{i_1}\cdots{\boldsymbol{\sigma}}_{i_k}$. Then $$\begin{aligned}
g({\mathbf{w}},y)=&g({\boldsymbol{\sigma}}_{i_1},{\boldsymbol{\sigma}}_{i_2}\cdots {\boldsymbol{\sigma}}_{i_k}[y])\cdots g({\boldsymbol{\sigma}}_{i_{k-1}},{\boldsymbol{\sigma}}_{i_k}[y])g({\boldsymbol{\sigma}}_{i_k},y)\\
=&\prod_{j=1}^k {\mathbf g}_{\psi^{-1}}({\langle}{\boldsymbol{\sigma}}_{i_{j+1}}\cdots{\boldsymbol{\sigma}}_{i_k}(y-\rho),{\alpha}_{i_j}{\rangle}Q({\alpha}^\vee))\\
=&\prod_{j=1}^k {\mathbf g}_{\psi^{-1}}({\langle}y-\rho,{\boldsymbol{\sigma}}_{i_{k}}\cdots{\boldsymbol{\sigma}}_{i_{j+1}}({\alpha}_{i_j}){\rangle}Q({\alpha}^\vee))\\
=&\prod_{{\alpha}\in \Phi({\mathbf{w}})} {\mathbf g}_{\psi^{-1}}({\langle}y-\rho,{\alpha}{\rangle}Q({\alpha}^\vee)).
\end{aligned}$$ The last equality follows from [@Bump13] Proposition 20.10.
We can now state the main result of this section.
\[prop:tau in the stable range\] For ${\mathbf{w}}\in W$, $$\tau({\mathbf{w}}_G,{}^{{\mathbf{w}}_G}{\overline{\chi}},{\mathbf{w}}[0],0)=g({\mathbf{w}},0).$$
The rest of this section is devoted to proving this result. Before the proof, we need some preparation.
Two lemmas
----------
\[lem:1\] We have $\tau({\boldsymbol{\sigma}}_{\alpha},{\overline{\chi}},y,{\boldsymbol{\sigma}}_{\alpha}[y])={\mathbf g}_{\psi^{-1}}(-{\langle}y_\rho,{\alpha}{\rangle}Q({\alpha}^\vee))$.
This is done by direct calculation. Recall ${\mathbf{w}}[y]={\mathbf{w}}(y-\rho)+\rho$. The left-hand side is $$\begin{aligned}
&\tau({\boldsymbol{\sigma}}_{\alpha},{\overline{\chi}},{\boldsymbol{\sigma}}_{\alpha}[{\boldsymbol{\sigma}}_{\alpha}[y]],{\boldsymbol{\sigma}}_{\alpha}[y])\\
=&{\mathbf g}_{\psi^{-1}}({\langle}{\boldsymbol{\sigma}}_{\alpha}[y]-\rho,{\alpha}{\rangle}Q({\alpha}^\vee))
={\mathbf g}_{\psi^{-1}}({\langle}y-\rho,{\boldsymbol{\sigma}}_{\alpha}({\alpha}){\rangle}Q({\alpha}^\vee))\\
=&{\mathbf g}_{\psi^{-1}}({\langle}y-\rho,-{\alpha}{\rangle}Q({\alpha}^\vee))
={\mathbf g}_{\psi^{-1}}(-{\langle}y_\rho,{\alpha}{\rangle}Q({\alpha}^\vee)).
\end{aligned}$$
\[lem:5\] If $\ell({\boldsymbol{\sigma}}_{\alpha}{\mathbf{w}})=\ell({\mathbf{w}})+1$, then ${\langle}{\mathbf{w}}[0],{\alpha}{\rangle}\leq 0$; if $\ell({\boldsymbol{\sigma}}_{\alpha}{\mathbf{w}})=\ell({\mathbf{w}})-1$, then ${\langle}{\mathbf{w}}[0],{\alpha}{\rangle}>0$.
If $\ell({\boldsymbol{\sigma}}_{\alpha}{\mathbf{w}})=\ell({\mathbf{w}})+1$, then ${\mathbf{w}}^{-1}({\alpha})$ is a positive root ([@Bump13] Proposition 20.2). This implies that ${\langle}{\mathbf{w}}(\rho),{\alpha}{\rangle}={\langle}\rho,{\mathbf{w}}^{-1}({\alpha}){\rangle}\geq 1$. Note that ${\langle}\rho,{\alpha}^\vee{\rangle}=1$. Thus $${\langle}\rho-{\mathbf{w}}(\rho),{\alpha}{\rangle}={\langle}\rho,{\alpha}{\rangle}-{\langle}{\mathbf{w}}(\rho),{\alpha}{\rangle}\leq 0.$$
We now consider the other case. Note $$({\boldsymbol{\sigma}}_{\alpha}{\mathbf{w}})[0]={\boldsymbol{\sigma}}_{\alpha}{\mathbf{w}}(-\rho)+\rho={\boldsymbol{\sigma}}_{\alpha}({\mathbf{w}}(-\rho)+\rho)-{\boldsymbol{\sigma}}_{\alpha}(\rho)+\rho ={\boldsymbol{\sigma}}_{\alpha}({\mathbf{w}}[0])-{\boldsymbol{\sigma}}_{\alpha}[0].$$ If $\ell({\boldsymbol{\sigma}}_{\alpha}{\mathbf{w}})=\ell({\mathbf{w}})-1$ or $\ell({\boldsymbol{\sigma}}_{\alpha}{\mathbf{w}})+1=\ell({\mathbf{w}})$, then ${\langle}({\boldsymbol{\sigma}}_{\alpha}{\mathbf{w}})[0],{\alpha}{\rangle}\leq 0$. But $${\langle}({\boldsymbol{\sigma}}_{\alpha}{\mathbf{w}})[0],{\alpha}{\rangle}={\langle}{\boldsymbol{\sigma}}_{\alpha}({\mathbf{w}}[0])-{\boldsymbol{\sigma}}_{\alpha}[0],{\alpha}{\rangle}=-{\langle}{\mathbf{w}}[0],{\alpha}{\rangle}+2.$$ Here we use ${\langle}{\boldsymbol{\sigma}}_{\alpha}[0],{\alpha}{\rangle}=-2$. This gives the desired result.
Proof of Proposition \[prop:tau in the stable range\]
-----------------------------------------------------
We first check some small rank cases. If $r=1$, then both sides are $1$. If $r=2$, we only have two Weyl group elements two consider. If ${\mathbf{w}}={\mathrm{id}}$, then $$\tau({\mathbf{w}}_G,{}^{{\mathbf{w}}_G}{\overline{\chi}},{\mathbf{w}}[0],0)=\dfrac{1-q^{-1}}{1-q^{-1}}=1;$$ if ${\mathbf{w}}={\mathbf{w}}_G$, then clearly $$\tau({\mathbf{w}}_G,{}^{{\mathbf{w}}_G}{\overline{\chi}},{\mathbf{w}}_G[0],0)=g({\mathbf{w}}_G,0).$$
Our proof is a simplified version of the proof of [@Suzuki97] Lemma 4.2. We now assume that the result is true for $r$ and prove it for $r+1$. We first apply the inductive formula in Proposition \[prop:inductive 1 in general linear\]. Observe the following:
- Recall that ${\mathbf{w}}^{-1}[{\mathbf{w}}[0]]=0$ and we write ${\mathbf{w}}^{-1}={\mathbf{w}}^{'-1}{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_{r_0}$ for a unique integer $r_0$ and ${\mathbf{w}}'\in W({\mathbb M})$.
- We are working with the exceptional representation. If $i\neq 1$, then $\prod_{j=1}^{i-1}\frac{1-q^{-1}{\overline{\chi}}_{ji}}{1-{\overline{\chi}}_{ji}}=0$ since $1-q^{-1}{\overline{\chi}}_{i-1,i}=0$. Thus only one term $(i=1)$ in the outer summation is nonzero.
We obtain $${\mathcal{W}}_{y}(0,{\overline{\chi}})=\sum_{y'}
\tau({\boldsymbol{\sigma}}_1\cdots {\boldsymbol{\sigma}}_r,{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}},y,y')
{\mathcal{W}}_{M,y'}(0,{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}}),$$ where the sum is over the set $$\begin{aligned}
&\left\{
{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_{r_0} {\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_i^{a_1}[y]: a_1,\cdots,a_{r_0-2}\in \{0,1\}
\right\}\\
=&\left\{
{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_i^{a_1}{\mathbf{w}}'[0]: a_1,\cdots,a_{r_0-2}\in \{0,1\}
\right\}.\\
\end{aligned}$$ Since the orbit of $0$ (hence $y$) is free, this set has no repetition for different $(a_1,\cdots,a_{r_0-2})$. Thus we now obtain $${\mathcal{W}}_{y}(0,{\overline{\chi}})=\sum_{a_1\in \{0,1\}}\cdots \sum_{a_{r_0-2}\in \{0,1\}} \tau({\boldsymbol{\sigma}}_1\cdots {\boldsymbol{\sigma}}_r,{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}}, y,y')
{\mathcal{W}}_{M,y'}(0,{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}})$$ where $$y'={\boldsymbol{\sigma}}_{r}\cdots {\boldsymbol{\sigma}}_{r_0}{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0]={\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}'[0].$$
For $y'={\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}'[0]$, using Lemma \[lem:tau unique decomposition\], we see that $\tau({\boldsymbol{\sigma}}_1\cdots {\boldsymbol{\sigma}}_r,{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}}, y,y')$ is the product of the following three terms:
- $\tau({\boldsymbol{\sigma}}_1\cdots {\boldsymbol{\sigma}}_{r_0-2},~ {}^{{\boldsymbol{\sigma}}_{r_0-2}\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}},~ y,~{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0])$,
- $\tau({\boldsymbol{\sigma}}_{r_0-1},~ {}^{{\boldsymbol{\sigma}}_{r_0-1}\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}}, ~{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0], ~{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0])$,
- $\tau({\boldsymbol{\sigma}}_{r_0}\cdots {\boldsymbol{\sigma}}_{r}, ~ {}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}},~ {\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0], ~ y')$.
We now analyze each term. We start with ${\mathcal{W}}_{M,y'}(0,{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}})$.
We have $$\begin{aligned}
&{\mathcal{W}}_{M,y'}(0,{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}})\\
=&g({\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}',0).\\
\end{aligned}$$
Here we apply Lemma \[lem:m to gl on w\]. We observe that the character ${}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}}$ restricted to ${\overline{T}}_{\operatorname{GL}_r,Q,n}^{sc}$ is again an anti-exceptional character. So we can apply induction to calculate the value. It is $g({\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}',0)$.
We have $$\tau({\boldsymbol{\sigma}}_{r_0-1},{}^{{\boldsymbol{\sigma}}_{r_0-1}\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}},{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0],{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0])
=\dfrac{1-q^{-1}}{1-q^{-(r_0-1)}}.$$
Note ${\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}$ is of the form ${\boldsymbol{\sigma}}_{r_0}\cdots {\boldsymbol{\sigma}}_{r}{\mathbf{w}}''$ with ${\mathbf{w}}''\in W({\mathbb{GL}}_r)$. Lemma \[lem:5\] shows that ${\langle}{\boldsymbol{\sigma}}_{r_0}\cdots {\boldsymbol{\sigma}}_{r-1}{\mathbf{w}}''[0],{\alpha}_{r_0-1}{\rangle}<0$. Therefore $k_{{\boldsymbol{\sigma}}_{r_0}\cdots {\boldsymbol{\sigma}}_{r-1}{\mathbf{w}}''[0],{\alpha}_{r_0-1}}=0$ and $$\tau({\boldsymbol{\sigma}}_{r_0-1},{}^{{\boldsymbol{\sigma}}_{r_0-1}\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}},{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0],{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0])
=\dfrac{1-q^{-1}}{1-q^{-(r_0-1)}}.$$
We have $$\tau({\boldsymbol{\sigma}}_{r_0}\cdots {\boldsymbol{\sigma}}_{r},{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}},{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0],y')=g({\boldsymbol{\sigma}}_{r_0}\cdots {\boldsymbol{\sigma}}_{r},{\mathbf{w}}'[0]).$$
Since the action of ${\boldsymbol{\sigma}}_{r_0}\cdots {\boldsymbol{\sigma}}_{r}$ and ${\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1}$ are disjoint, we have $$\begin{aligned}
&\tau({\boldsymbol{\sigma}}_{r_0}\cdots {\boldsymbol{\sigma}}_{r},{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}},{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0],y')\\
=&\tau({\boldsymbol{\sigma}}_{r_0}\cdots {\boldsymbol{\sigma}}_{r},{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}},{\boldsymbol{\sigma}}_{r_0}\cdots {\boldsymbol{\sigma}}_{r}{\mathbf{w}}'[0],{\mathbf{w}}'[0]).
\end{aligned}$$ We can now use Lemma \[lem:tau unique decomposition\] to calculate $$\begin{aligned}
&\tau({\boldsymbol{\sigma}}_{r_0}\cdots {\boldsymbol{\sigma}}_{r},{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}},{\boldsymbol{\sigma}}_{r_0}\cdots {\boldsymbol{\sigma}}_{r}{\mathbf{w}}'[0],{\mathbf{w}}'[0])\\
=&\tau({\boldsymbol{\sigma}}_{r_0},{}^{{\boldsymbol{\sigma}}_{r_0}\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}},{\boldsymbol{\sigma}}_{r_0}\cdots {\boldsymbol{\sigma}}_{r}{\mathbf{w}}'[0],{\boldsymbol{\sigma}}_{r_0-1}\cdots {\boldsymbol{\sigma}}_{r}{\mathbf{w}}'[0])
\cdots \tau({\boldsymbol{\sigma}}_{r},{}^{{\boldsymbol{\sigma}}_r\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}},{\boldsymbol{\sigma}}_{r}{\mathbf{w}}'[0],{\mathbf{w}}'[0])\\
=&g({\boldsymbol{\sigma}}_{r_0},{\boldsymbol{\sigma}}_{r_0-1}\cdots {\boldsymbol{\sigma}}_{r}{\mathbf{w}}'[0])\cdots g({\boldsymbol{\sigma}}_{r},{\mathbf{w}}'[0])\\
=&g({\boldsymbol{\sigma}}_{r_0}\cdots {\boldsymbol{\sigma}}_{r},{\mathbf{w}}'[0]).
\end{aligned}$$
Let us summarize what we have done so far. Let us rewrite $$g({\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}',0)
= \dfrac{g({\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}',0)}{g({\mathbf{w}}',0)}g({\mathbf{w}}',0).$$ and use $$g({\mathbf{w}},0)=g({\boldsymbol{\sigma}}_{r_0}\cdots {\boldsymbol{\sigma}}_{r},{\mathbf{w}}'[0])g({\mathbf{w}}',0).$$ By the above results, we deduce that $$\label{eq:rewrite as a sum}
\begin{aligned}
&{\mathcal{W}}_{y}(0,{\overline{\chi}})=g({\mathbf{w}},0)\dfrac{1-q^{-1}}{1-q^{-(r_0-1)}}\\
\cdot&\sum_{a_1\in \{0,1\}}\cdots \sum_{a_{r_0-2}\in \{0,1\}} \tau ({\boldsymbol{\sigma}}_1\cdots {\boldsymbol{\sigma}}_{r_0-2},{}^{{\boldsymbol{\sigma}}_{r_0-2}\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}},y,{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0])
\dfrac{g({\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}',0)}{g({\mathbf{w}}',0)}.
\end{aligned}$$
Thus, it remains to the summation in the second line. Notice that $$\dfrac{g({\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}',0)}{g({\mathbf{w}}',0)}=\dfrac{g({\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}},0)}{g({\mathbf{w}},0)}.$$ We now rewrite the summation as $$\label{eq:break the sum}
\begin{aligned}
&\sum_{a_1\in \{0,1\}}\cdots \sum_{a_{r_0-3}\in \{0,1\}}
\tau({\boldsymbol{\sigma}}_1\cdots {\boldsymbol{\sigma}}_{r_0-3},{}^{{\boldsymbol{\sigma}}_{r_0-3}\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}},y,{\boldsymbol{\sigma}}_{r_0-3}^{i_{r_0-3}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0]) \dfrac{g({\boldsymbol{\sigma}}_{r_0-3}^{a_{r_0-3}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}},0)}{g({\mathbf{w}},0)} \\
\cdot &\sum_{a_{r_0-2}\in \{0,1\}}
\tau({\boldsymbol{\sigma}}_{r_0-2},{}^{{\boldsymbol{\sigma}}_{r_0-2}\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}}, {\boldsymbol{\sigma}}_{r_0-3}^{i_{r_0-3}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0],{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0])
\dfrac{g({\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}},0)}{g({\boldsymbol{\sigma}}_{r_0-3}^{a_{r_0-3}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}},0)}.
\end{aligned}$$ We first calculate the inner sum.
The inner sum in is equal to $\dfrac{1-q^{-(r_0-1)}}{1-q^{-(r_0-2)}}$.
To calculate the inner sum, there are two cases to consider. (Note that this discussion does not appear in [@Suzuki97] Sect. 4.2.)
For ease of notation, we write $\tilde y={\boldsymbol{\sigma}}_{r_0-3}^{i_{r_0-3}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0]$. Clearly, $k_{\tilde y,{\alpha}_{r_0-2}}$ is either $0$ or $1$. We have two cases to consider.
*Case 1*: $\ell({\boldsymbol{\sigma}}_{r_0-2}{\boldsymbol{\sigma}}_{r_0-3}^{i_{r_0-3}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}})=\ell({\boldsymbol{\sigma}}_{r_0-3}^{i_{r_0-3}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}})+1$.
When $a_{r_0-2}=0$, $$\dfrac{g({\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}},0)}{g({\boldsymbol{\sigma}}_{r_0-3}^{a_{r_0-3}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}},0)}=1.$$ By Lemma \[lem:5\], ${\langle}\tilde y,{\alpha}_{r_0-2}{\rangle}\leq 0$ and $k_{\tilde y,{\alpha}_{r_0-2}}=0$. Thus, $$\tau({\boldsymbol{\sigma}}_{r_0-2},{}^{{\boldsymbol{\sigma}}_{r_0-2}\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}},{\boldsymbol{\sigma}}_{r_0-3}^{a_{r_0-3}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0],{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0])
=\dfrac{1-q^{-1}}{1-q^{-(r_0-2)}}.$$ When $a_{r_0-2}=1$, $$\dfrac{g({\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}},0)}{g({\boldsymbol{\sigma}}_{r_0-3}^{a_{r_0-3}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}},0)}=g({\boldsymbol{\sigma}}_{r_0-2},{\boldsymbol{\sigma}}_{r_0-3}^{i_{r_0-3}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0])={\mathbf g}_{\psi^{-1}}({\langle}\tilde y_\rho,{\alpha}_{r_0-2}{\rangle}Q({\alpha}^\vee)),$$ and $$\tau({\boldsymbol{\sigma}}_{r_0-2},{}^{{\boldsymbol{\sigma}}_{r_0-2}\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}},{\boldsymbol{\sigma}}_{r_0-3}^{i_{r_0-3}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0],{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0]
)={\mathbf g}_{\psi^{-1}}(-{\langle}\tilde y_\rho,{\alpha}_{r_0-2}{\rangle}Q({\alpha}^\vee)).$$ Thus the inner sum in is $$\begin{aligned}
&\dfrac{1-q^{-1}}{1-q^{-(r_0-2)}}\cdot 1 +{\mathbf g}_{\psi^{-1}}({\langle}\tilde y_\rho,{\alpha}_{r_0-2}{\rangle}Q({\alpha}^\vee))\cdot {\mathbf g}_{\psi^{-1}}(-{\langle}\tilde y_\rho,{\alpha}_{r_0-2}{\rangle}Q({\alpha}^\vee))\\
=&\dfrac{1-q^{-1}}{1-q^{-(r_0-2)}}+q^{-1}=\dfrac{1-q^{-(r_0-1)}}{1-q^{-(r_0-2)}}.
\end{aligned}$$
*Case 2*: $\ell({\boldsymbol{\sigma}}_{r_0-2}{\boldsymbol{\sigma}}_{r_0-3}^{i_{r_0-3}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}})=\ell({\boldsymbol{\sigma}}_{r_0-3}^{i_{r_0-3}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}})-1$. When $a_{r_0-2}=0$, $$\dfrac{g({\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}},0)}{g({\boldsymbol{\sigma}}_{r_0-3}^{a_{r_0-3}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}},0)}=1.$$ By Lemma \[lem:5\], ${\langle}\tilde y,{\alpha}_{r_0-2}{\rangle}>0$ and $k_{\tilde y,{\alpha}_{r_0-2}}=1$. Thus, $$\tau({\boldsymbol{\sigma}}_{r_0-2},{}^{{\boldsymbol{\sigma}}_{r_0-2}\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}},{\boldsymbol{\sigma}}_{r_0-3}^{a_{r_0-3}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0],{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0])
=\dfrac{1-q^{-1}}{1-q^{-(r_0-2)}}q^{-(r_0-2)}.$$ When $a_{r_0-2}=1$, $$\tau({\boldsymbol{\sigma}}_{r_0-2},{}^{{\boldsymbol{\sigma}}_{r_0-2}\cdots {\boldsymbol{\sigma}}_1}{\overline{\chi}},{\boldsymbol{\sigma}}_{r_0-3}^{i_{r_0-3}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0],{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0])={\mathbf g}_{\psi^{-1}}({\langle}\tilde y_\rho,{\alpha}_{r_0-2}{\rangle}Q({\alpha}^\vee)),$$ and $$\dfrac{g({\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}},0)}{g({\boldsymbol{\sigma}}_{r_0-3}^{a_{r_0-3}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}},0)}=\dfrac{1}{g({\boldsymbol{\sigma}}_{r_0-2},{\boldsymbol{\sigma}}_{r_0-2}^{a_{r_0-2}}\cdots {\boldsymbol{\sigma}}_1^{a_1} {\mathbf{w}}[0])}=\dfrac{1}{{\mathbf g}_{\psi^{-1}}({\langle}\tilde y_\rho,{\alpha}_{r_0-2}{\rangle}Q({\alpha}^\vee))}.$$ Then the inner sum in is $$\begin{aligned}
&\dfrac{1-q^{-1}}{1-q^{-(r_0-2)}}q^{-(r_0-2)}+{\mathbf g}_{\psi^{-1}}({\langle}\tilde y_\rho,{\alpha}_{r_0-2}{\rangle}Q({\alpha}^\vee))\cdot \dfrac{1}{{\mathbf g}_{\psi^{-1}}({\langle}\tilde y_\rho,{\alpha}_{r_0-2}{\rangle}Q({\alpha}^\vee))}\\
=&\dfrac{1-q^{-1}}{1-q^{-(r_0-2)}}q^{-(r_0-2)}+1=\dfrac{1-q^{-(r_0-1)}}{1-q^{-(r_0-2)}}.
\end{aligned}$$
This finishes the calculation of the inner sum in . We can now proceed for the other summations and deduce that is $$\dfrac{1-q^{-2}}{1-q^{-1}}\cdots \dfrac{1-q^{-(r_0-2)}}{1-q^{-(r_0-3)}}\dfrac{1-q^{-(r_0-1)}}{1-q^{-(r_0-2)}} =\dfrac{1-q^{-(r_0-1)}}{1-q^{-1}}.$$ By , this implies that ${\mathcal{W}}_{y}(0,{\overline{\chi}})=g({\mathbf{w}},0)$. The proof of Proposition \[prop:tau in the stable range\] is complete.
Main result {#sec:main result}
===========
Statement
---------
We now state our main result. We work with the group $\overline{\operatorname{GL}}_{r}$. Let $\Delta'\subset \Delta$ so that the corresponding Levi subgroup is ${\mathbb M}={\mathbb{GL}}_{r_1}\times \cdots \times {\mathbb{GL}}_{r_k}$. Let ${\overline{\chi}}$ be an $\Delta'$-anti-exceptional character for $\overline{\operatorname{GL}}_r$. Define $e_i=r_1+\cdots+r_{i-1}$ and $x_{ij}={\overline{\chi}}_{e_i+1,e_j+1}$.
\[thm:final formula\] Assume that
- for all $1\leq i\leq k$, $r_i\leq n_Q$,
- for all $1\leq i,j\leq k$, $r_i+r_j>n_Q$
- for $i=1,\cdots,k-1$, $$\label{eq:strange condition}
\left\lfloor \dfrac{(\sum_{j=1}^i r_j)(\sum_{j=i+1}^k r_j)}{n_Q}\right\rfloor=
(k-i)\left(\sum_{j=1}^i r_j\right)+
i\left(\sum_{j=i+1}^k r_j\right)
-i(k-i)n_Q.$$
Then $${\mathcal{W}}_{{\mathbf{w}}[0]}(0,{\overline{\chi}})=g({\mathbf{w}},0)\prod_{1\leq i<j\leq k}\ \prod_{l=n_Q-r_j+1}^{r_i}(1-x_{ij}q^{-l})$$ for ${\mathbf{w}}\in W({\mathbb M})$.
We will first prove the result for ${\mathbf{w}}={\mathrm{id}}$ and then for the general case. We begin with some remarks in the case of ${\mathbf{w}}={\mathrm{id}}$.
1. A result of [@McNamara11] says that ${\mathcal{W}}_0(0,{\overline{\chi}})$ is a weighted sum over a finite crystal graph and is therefore a polynomial in $x_{12},\cdots,x_{k-1,k}$. Note that everything stated here is done in $\overline{\operatorname{SL}}_r$ so McNamara’s result does apply.
2. When ${\mathbf{w}}={\mathrm{id}}$, we can rewrite the right-hand side as a polynomial in $x_{12},\cdots, x_{k-1,k}$. Let $f(x_{12},\cdots,x_{k-1,k})$ be this polynomial. The monomial with highest total degree is $$\prod_{1\leq i<j\leq k}x_{ij}^{r_i+r_j-n}=x_{12}^{b_1}\cdots x_{k-1,k}^{b_{k-1}}$$ where $$b_i=(k-i)\left(\sum_{j=1}^i r_j\right)+i\left(\sum_{j=i+1}^k r_j\right)-i(k-i)n_Q$$ is the right-hand side of .
3. The condition in does seem strange and this is not satisfied for all tuples. However, it is easy to check that holds when $(r_1,\cdots,r_k)=(n_Q,\cdots,n_Q,n')$ where $1\leq n'\leq n_Q$.
4. We expect the result to be true without the condition in . But we do not know how to extend it at the moment.
Proof of Theorem \[thm:final formula\]: the base case
-----------------------------------------------------
For the base case, the proof presented here is adapted from [@Kaplan] Theorem 43. We will give also examples to explain some ideas and give the reader some flavor of the proof.
We now give an outline of the proof. We first observe that, by the results in [@McNamara11], ${\mathcal{W}}_0(0,{\overline{\chi}})$ is weight sum over certain Gelfand-Tsetlin patterns and is therefore a polynomial in $x_{12},\cdots,x_{k-1,k}$. It is sufficient to prove the following three things:
1. Every factor of $f(x_{12},\cdots,x_{k-1,k})$ divides ${\mathcal{W}}_0(0,{\overline{\chi}})$.
2. The monomial of the highest total degree of ${\mathcal{W}}_{0}(0,{\overline{\chi}})$ is the same as $f(x_{12},\cdots,x_{k-1,k})$, up to a scalar.
3. The constant coefficient of $f(x_{12},\cdots,x_{k-1,k})$ is $1$. So it is enough to prove that the constant coefficient of ${\mathcal{W}}_{0}(0,{\overline{\chi}})$ is $1$.
The first one is proved by a representation-theoretic argument. The last two are proved using the formula of [@McNamara11] Sect. 8, which is based on the Gelfand-Tsetlin description of [@BBF11] Sect. 8. Note that the proof in [@Kaplan] Theorem 43 does not use uniqueness of Whittaker models.
We start with the representation-theoretic argument.
\[ex:main section\] We assume that $n=3$, $r=8$, $(r_1,r_2,r_3)=(3,3,2)$, $Q({\alpha}^\vee)=1$. Let $\Delta'=\{{\alpha}_1,{\alpha}_2,{\alpha}_4,{\alpha}_5,{\alpha}_7\}$ and ${\overline{\chi}}$ be a $\Delta'$-anti-exceptional character. Therefore, ${\overline{\chi}}_{{\alpha}}=q$ for ${\alpha}\in \Delta'$. Let $x_1={\overline{\chi}}_{{\alpha}_1+{\alpha}_2+{\alpha}_3}$ and $x_2={\overline{\chi}}_{{\alpha}_4+{\alpha}_5+{\alpha}_6}$. It is easy to check, for instance, ${\overline{\chi}}_{{\alpha}_3}=q^{-2}x_1$.
Clearly if $x_1=q^3$, then ${\overline{\chi}}$ is a $\Delta'\cup \{{\alpha}_3\}$-anti-exceptional character and ${\mathcal{W}}_0(0,{\overline{\chi}})=0$ by Corollary \[cor:vanishing of unramified whittaker\]. In other words, as a function of $x_1,x_2$, ${\mathcal{W}}_0(0,{\overline{\chi}})$ is zero along the hyperplane $1-q^{-3}x_1=0$.
Now let us consider the following question: is ${\mathcal{W}}_0(0,{\overline{\chi}})$ along other hyperplanes? A quick examination shows that $1-q^{-2}x_1=0$ does the job. In fact, under this assumption, ${\overline{\chi}}_{{\alpha}_3+{\alpha}_4}=q$. Thus ${}^{{\boldsymbol{\sigma}}_4}{\overline{\chi}}$ is an $\{{\alpha}_1,{\alpha}_2,{\alpha}_3\}$-anti-exceptional character. We consider the following intertwining operator $T_{{\boldsymbol{\sigma}}_4,{}^{{\boldsymbol{\sigma}}_4}{\overline{\chi}}}:I({}^{{\boldsymbol{\sigma}}_4}{\overline{\chi}})\to I({\overline{\chi}})$. Using Lemma \[lem:gk for anti 2\], it is easy to check $$T_{{\boldsymbol{\sigma}}_4,{}^{{\boldsymbol{\sigma}}_4}{\overline{\chi}}}(\phi_K)=\dfrac{1-q^{-1}}{1-q^{-2}}\phi'_K.$$ If ${\mathcal{W}}_0(0,{\overline{\chi}})\neq 0$, then by composing this Whittaker functional with $T_{{\boldsymbol{\sigma}}_4,{}^{{\boldsymbol{\sigma}}_4}{\overline{\chi}}}$, we obtain a nonzero Whittaker functional on $I({}^{{\boldsymbol{\sigma}}_4}{\overline{\chi}})$. However, this contradicts with Corollary \[cor:vanishing of unramified whittaker\] as ${}^{{\boldsymbol{\sigma}}_4}{\overline{\chi}}$ is an $\{{\alpha}_1,{\alpha}_2,{\alpha}_3\}$-anti-exceptional character.
The same argument shows that ${\mathcal{W}}_0(0,{\overline{\chi}})=0$ if $1-q^{-1}x_1=0$. The same argument can be applied for $x_2$.
We now consider the hyperplane $1-{\overline{\chi}}_{{\alpha}_1+\cdots+{\alpha}_6}q^{-2}=1-x_1x_2q^{-2}=0$. With this assumption, ${\overline{\chi}}_{{\alpha}_3+\cdots+{\alpha}_7}=q$. Therefore, ${}^{{\boldsymbol{\sigma}}_4\cdots{\boldsymbol{\sigma}}_7}{\overline{\chi}}$ is $\{{\alpha}_1,{\alpha}_2,{\alpha}_3\}$-anti-exceptional. The intertwining operator $$\label{eq:intertwining in examples}
T_{{\boldsymbol{\sigma}}_7\cdots{\boldsymbol{\sigma}}_4,{}^{{\boldsymbol{\sigma}}_4\cdots{\boldsymbol{\sigma}}_7}{\overline{\chi}}}:I({}^{{\boldsymbol{\sigma}}_4\cdots{\boldsymbol{\sigma}}_7}{\overline{\chi}})\to I({\overline{\chi}})$$ could have zeros. But $c({\boldsymbol{\sigma}}_7\cdots{\boldsymbol{\sigma}}_4,{}^{{\boldsymbol{\sigma}}_4\cdots{\boldsymbol{\sigma}}_7}{\overline{\chi}})$ has two types of factors: the first of the form $1-x_2q^{-l}$ for some integer $l$ and the factor as in the statement of Lemma \[lem:gk for anti 2\]. In any case, is nonzero on spherical vectors along $1-x_1x_2q^{-2}=0$. Now the same argument as above shows that ${\mathcal{W}}_0(0,{\overline{\chi}})=0$.
By repeating this argument, one can find $3+2+2$ factors of ${\mathcal{W}}_0(0,{\overline{\chi}})$. They are exactly the factors appearing in the statement of Theorem \[thm:final formula\].
If $1-x_{ij}q^{-l}=0$ for $i< j$ and $n_Q-r_j+1\leq l\leq r_i$, then ${\mathcal{W}}_{0}(0,{\overline{\chi}})=0$.
We write ${\overline{\chi}}\sim({\overline{\chi}}_1,\cdots, {\overline{\chi}}_k)$ where ${\overline{\chi}}_m$ is anti-exceptional for the group $\overline{\operatorname{GL}}_{r_m}$. We further write ${\overline{\chi}}_j\sim ({\overline{\chi}}_j^\dagger, {\overline{\chi}}_j^\ddagger)$ where the size of ${\overline{\chi}}_j^\dagger$ is $r_i-l$ (which could be $0$). Let ${\mathbf{w}}$ be the Weyl group element so that $${}^{{\mathbf{w}}}{\overline{\chi}}\sim ({\overline{\chi}}_1, \cdots, {\overline{\chi}}_i, {\overline{\chi}}_j^\ddagger,\cdots, {\overline{\chi}}_{j-1}, {\overline{\chi}}_j^\dagger, {\overline{\chi}}_{j+1}, \cdots, {\overline{\chi}}_k).$$ Observe that since $1-x_{ij}q^{-1}=0$, $({\overline{\chi}}_i,{\overline{\chi}}_j^\ddagger)$ is an anti-exceptional character of size $r_i+r_j-r_i+l=r_j+l\geq n_Q+1$. Thus ${}^{{\mathbf{w}}}{\overline{\chi}}$ is an anti-exceptional character that satisfies the condition in Corollary \[cor:vanishing of unramified whittaker\].
We now check that the intertwining operator $$T_{{\mathbf{w}}^{-1},{}^{{\mathbf{w}}}{\overline{\chi}}}:I({}^{{\mathbf{w}}}{\overline{\chi}})\to I({\overline{\chi}})$$ is nonzero on spherical vectors along $1-x_{ij}q^{-l}=0$. Indeed, it is enough check that $c({\mathbf{w}}^{-1},{}^{{\mathbf{w}}}{\overline{\chi}})\neq 0$ along $1-x_{ij}q^{-l}=0$. A quick calculation shows that the denominator of $c({\mathbf{w}}^{-1},{}^{{\mathbf{w}}}{\overline{\chi}})$ is either of the form $1-x_{jj'}q^{-l'}$ for $j'\neq i,j$, or a factor of the form as in Lemma \[lem:gk for anti 2\]. In either case, this is nonzero when $1-x_{ij}q^{-l}=0$.
Suppose now that ${\mathcal{W}}_0(0,{\overline{\chi}})\neq 0$ along $1-x_{ij}q^{-l}=0$. We then have a Whittaker functional on $I({}^{{\mathbf{w}}}{\overline{\chi}})$ via $$I({}^{{\mathbf{w}}}{\overline{\chi}})\to I({\overline{\chi}})\to {\mathbb C}.$$ This is nonzero since it is nonzero on the spherical vector $\phi_K$. However, by our discussion above, this contradicts Corollary \[cor:vanishing of unramified whittaker\].
Thus we know that $(1-\chi_{ij}q^{-l})\mid {\mathcal{W}}_0(0,{\overline{\chi}})$. As the factors of $f(x_{12},\cdots,x_{k-1,k})$ are distinct and ${\mathbb C}[x_{12},\cdots,x_{k-1,k}]$ is a unique factorization domain, $f(x_{12},\cdots,x_{k-1,k})$ divides ${\mathcal{W}}_{0}(0,{\overline{\chi}})$.
We now use the formula in [@McNamara11] to estimate the degree of ${\mathcal{W}}_0(0,{\overline{\chi}})$. We briefly recall how this is derived. In [@McNamara11], with a choice of a reduced decomposition of the longest element in the Weyl group, McNamara introduces an algorithm, called *explicit Iwasawa decomposition*. This is to write an element $u\in U^-$ as $u=tnk$ where $t\in T$, $n\in U$ and $k\in K$. Equivalently, this is to write $U^-$ as a cell decomposition $U^{-}=\bigsqcup_{\mathbf{m}}C_{\mathbf{m}}$, where $\mathbf{m}$ is a tuple of integers. Thus one can write $$\int_{U^-}=\sum_{\mathbf{m}}\int_{C_{\mathbf{m}}}$$ and this yields a combinatorial sum of these integrals. The main result in [@McNamara11] says that unramified Whittaker functions can be calculated in this way, and the tuples $\mathbf{m}$ with nonzero contributions are in bijection with a set of Gelfand-Tsetlin patterns. The contribution can be calculated in terms of Gauss sums.
Recall the a strict Gelfand-Tsetlin pattern is a triangular array $\{a_{i,j}\}$ of non-negative integers, such that each row is strictly decreasing and $a_{i,j} \geq a_{i+1,j}\geq a_{i_{j+1}}$ for all $i,j$ such that all entries exist. For each $i$, define $$d_i:=\sum_{j=1}^{r-i+1}a_{i,j}-a_{1,i+j-1}=\sum_{j=1}^{r-i+1}a_{i,j}-(r-i-j+1).$$
Here ${\mathcal{W}}_{0}(0,{\overline{\chi}})$ is expressed as a sum over the set of Gelfand-Tsetlin patterns with the first row $a_{1,j}=r-j, 1\leq j\leq r$. The resulting monomial for such a pattern is of the form $$C\prod_{l=1}^{k-1}x_{l,l+1}^{d_{e_{l+1}+1}/n_Q},$$ where $C$ is a certain product of powers of $q$ and Gauss sums.
In this paper, we choose a particular maximal abelian subgroup ${\overline{A}}$. This imposes another condition on the patterns we need to consider. With this choice of torus, by Lemma \[lem:supp of spherical\], the torus elements that lies in the support of $\phi_K$ are in ${\overline{A}}$. Also the torus elements appearing in the calculation are in $\overline{\operatorname{SL}}_r$. Recall that $Y_{Q,n}\cap Y^{sc}=Y_{Q,n}^{sc}=n_QY^{sc}$. As a consequence, the only patterns to consider are those where $d_i\equiv 0 \mod n_Q$. (See also [@Kaplan] Theorem 43.)
\[lem:bounding the degree\] The monomial of the highest total degree in ${\mathcal{W}}_0(0,{\overline{\chi}})$ is at most the same as in $f(x_{12},\cdots,x_{k-1,k})$, up to a scalar.
\[ex:main section 2\] We continue with the set up in Example \[ex:main section\]. A quick calculation shows that the monomial of the highest total degree should be $x_1^5x_2^4$.
Now let us check it from the Gelfand-Tsetlin description. We require that the $4$th and $7$th row to be as large as possible. So the maximal possible $4$th row is $(7 ~ 6 ~ 5 ~ 4 ~ 3 )$ and the maximal possible $7$th row is $(7~6)$. This gives $d_4\leq 3\times 5=15$ and $d_7\leq 6\times 2=12$. Therefore, the monomial with the highest total degree is again $x_1^5x_2^4$. $$\begin{pmatrix}
7 & & 6 & & 5 & & 4 & & 3 & & 2 & & 1 & & 0 \\
& \ast & & \ast & & \ast & & \ast & & \ast & & \ast & & \ast & \\
& & \ast & & \ast & & \ast & & \ast & & \ast & & \ast & & \\
& & & 7 & & 6 & & 5 & & 4 & & 3 & & & \\
& & & & \ast & & \ast & & \ast & & \ast & & & & \\
& & & & & \ast & & \ast & & \ast & & & & & \\
& & & & & & 7 & & 6 & & & & & & \\
& & & & & & & \ast & & & & & & & \\
\end{pmatrix}.$$
We now seek the monomial of highest total degree. We consider patterns with the maximal entries $a_{e_{l+1}+1,j}$ for $1\leq l\leq k-1$ and $1\leq j\leq r-(e_{l+1}+1)$ possible. We now fix $l$. Note that $a_{e_{l+1}+1,1}\leq r-1$ as $a_{1,1}=r-1$. The maximal possible choice of row $e_{l+1}+1$ is $$(r-1,r-2,\cdots)$$ Therefore, $d_{e_{l+1}+1}\leq e_{l+1}(\sum_{j=l+1}^{k}r_j)= (\sum_{j=1}^lr_j)(\sum_{j=l+1}^{k}r_j)$. As $d_{e_{l+1}+1}/n_Q$ must be an integer, this implies that $$d_{e_{l+1}+1}/n_Q\leq\left\lfloor\dfrac{(\sum_{j=1}^l r_j)(\sum_{j=l+1}^k r_j)}{n}\right\rfloor.$$ The result now follows from .
By the above two results, we know that ${\mathcal{W}}_0(0,{\overline{\chi}})=cf(x_{12},\cdots,x_{k-1,k})$ for some constant $c$. It remains to compute a single coefficient of $f$. In [@Kaplan], the highest monomial is used for this purpose. Here, we calculate the constant coefficient. We claim that only the lowest pattern contributes to the constant coefficient and the contribution is therefore $1$.
We are again in the setup of Example \[ex:main section\]. We would like to show that only the lowest pattern contributes to the constant coefficient. First of all, we have $d_4=d_7=0$. This determines the $4$th and $7$th rows. These entries are as small as possible. Thus some entries in the other rows are determined. So far we have: $$\begin{pmatrix}
7 & & 6 & & 5 & & 4 & & 3 & & 2 & & 1 & & 0 \\
& a_{21} & & a_{22} & & 4 & & 3 & & 2 & & 1 & & 0 & \\
& & a_{31} & & 4 & & 3 & & 2 & & 1 & & 0 & & \\
& & & 4 & & 3 & & 2 & & 1 & & 0 & & & \\
& & & & a_{51} & & a_{52} & & 1 & & 0 & & & & \\
& & & & & a_{61} & & 1 & & 0 & & & & & \\
& & & & & & 1 & & 0 & & & & & & \\
& & & & & & & a_{81} & & & & & & & \\
\end{pmatrix}.$$ We next show that $d_i=0$ for all $i$. This determines the pattern completely. For instance, $$d_2=(a_{21}-6)+(a_{22}-5)\leq (a_{11}-6)+(a_{12}-5)=2.$$ As $d_2\equiv 0\mod 3$, we must have $d_2=0$. The other cases can be proved similarly.
Only the lowest pattern contributes to the constant coefficient of ${\mathcal{W}}_0(0,{\overline{\chi}})$.
To find the term contributing to the constant coefficient, we must have $$d_{e_{l+1}+1}=0,\qquad l=1,\cdots, d-1.$$ Given a fixed $l$, this determines row $e_{l+1}+1$, which is $$\label{eq:entries in certain row}
(r-e_{l+1}-1,\cdots,1,0).$$ The last $r-e_{l+1}$ entries from row $e_{l}+2$ to row $e_{l+1}$ are also determined. They are as well. We now determine the remaining coefficients. We now fix $1\leq l\leq k$. We argue by induction to show $d_{e_l+i}=0$ for $i=1,\cdots, r_l$. The case $i=1$ follows from our discussion above. We now assume that $d_{e_l+i}=0$. Then row $e_l+i$ is $$(r-e_l-i,\cdots, 1,0).$$ In other words, $a_{e_l+i,j}=r-e_l-i-j+1$. Thus, $$\begin{aligned}
d_{e_l+i+1}=&\sum_{j=1}^{r_l-i} a_{e_l+i+1,j}-(r-e_l-i-j)\\
\leq& \sum_{j=1}^{r_l-i} a_{e_l+i,j}-(r-e_l-i-j)\\
=& \sum_{j=1}^{r_l-i} (r-e_l-i-j+1)-(r-e_l-i-j)=r_l-i<n_Q.\\
\end{aligned}$$ As $d_{e_l+i+1}\equiv 0 \mod n_Q$, we deduce that $d_{e_l+i+1}=0$.
We now conclude that $d_i=0$ for $i$ and the pattern must be the lowest pattern. This completes the proof.
It is straightforward to see that the contribution of the lowest pattern is $1$. The proof of the base case is now complete.
Proof of Theorem \[thm:final formula\]: the general case
--------------------------------------------------------
We now prove the general case of Theorem \[thm:final formula\]. It remains to show that for ${\mathbf{w}}\in W({\mathbb M})$, $${\mathcal{W}}_{{\mathbf{w}}[0]}(0,{\overline{\chi}})=g({\mathbf{w}},0){\mathcal{W}}_{0}(0,{\overline{\chi}}).$$ By Corollary \[cor:proportional\], it suffices to show that $$\dfrac{\tau({\mathbf{w}}_M,{}^{{\mathbf{w}}_M^{-1}}{\overline{\chi}},{\mathbf{w}}[0],0)}{\tau({\mathbf{w}}_M,{}^{{\mathbf{w}}_M^{-1}}{\overline{\chi}},0,0)} =g({\mathbf{w}},0).$$ We now use Lemma \[lem:10\] to calculate the left-hand side. Suppose ${\overline{\chi}}\sim ({\overline{\chi}}_1,\cdots, {\overline{\chi}}_k)$ for some anti-exceptional characters ${\overline{\chi}}_1,\cdots,{\overline{\chi}}_k$. We have $$\begin{aligned}
&\tau({\mathbf{w}}_M,{}^{{\mathbf{w}}_M^{-1}}{\overline{\chi}},{\mathbf{w}}[0],0)\\
=&\prod_{i=1}^k \tau\left({\mathbf{w}}_{\operatorname{GL}_{r_i}},{}^{{\mathbf{w}}_{\operatorname{GL}_{r_i}}^{-1}}{\overline{\chi}},{\mathbf{w}}_i[0],0\right)\\
=&\prod_{i=1}^k g({\mathbf{w}}_i,0)=g({\mathbf{w}},0).
\end{aligned}$$ This finishes the proof.
[Gao18b]{}
Dubravka Ban and Chris Jantzen. The [L]{}anglands quotient theorem for finite central extensions of [$p$]{}-adic groups. , 48(68)(2):313–334, 2013.
I. N. Bernstein and A. V. Zelevinsky. Induced representations of reductive [${\mathfrak p}$]{}-adic groups. [I]{}. , 10(4):441–472, 1977.
Ben Brubaker, Daniel Bump, and Solomon Friedberg. Twisted [W]{}eyl group multiple [D]{}irichlet series: the stable case. In [*Eisenstein series and applications*]{}, volume 258 of [ *Progr. Math.*]{}, pages 1–26. Birkhäuser Boston, Boston, MA, 2008.
Ben Brubaker, Daniel Bump, and Solomon Friedberg. Weyl group multiple [D]{}irichlet series, [E]{}isenstein series and crystal bases. , 173(2):1081–1120, 2011.
Jean-Luc Brylinski and Pierre Deligne. Central extensions of reductive groups by [$\bold K_2$]{}. , (94):5–85, 2001.
Daniel Bump. , volume 225 of [*Graduate Texts in Mathematics*]{}. Springer, New York, second edition, 2013.
W. Casselman and J. Shalika. The unramified principal series of [$p$]{}-adic groups. [II]{}. [T]{}he [W]{}hittaker function. , 41(2):207–231, 1980.
Gautam Chinta and Paul E. Gunnells. Constructing [W]{}eyl group multiple [D]{}irichlet series. , 23(1):189–215, 2010.
Gautam Chinta and Omer Offen. A metaplectic [C]{}asselman-[S]{}halika formula for [${\rm GL}_r$]{}. , 135(2):403–441, 2013.
Michael Finkelberg and Sergey Lysenko. Twisted geometric [S]{}atake equivalence. , 9(4):719–739, 2010.
Solomon Friedberg and Lei Zhang. Eisenstein series on covers of odd orthogonal groups. , 137(4):953–1011, 2015.
Wee Teck Gan and Fan Gao. The [L]{}anglands-[W]{}eissman program for [B]{}rylinski-[D]{}eligne extensions. , (398):187–275, 2018. L-groups and the Langlands program for covering groups.
Fan Gao. Distinguished theta representations for certain covering groups. , 290(2):333–379, 2017.
Fan Gao. Generalized [B]{}ump-[H]{}offstein conjecture for coverings of the general linear groups. , 499:183–228, 2018.
Fan Gao. Hecke [L]{}-functions and [F]{}ourier coefficients of covering [E]{}isenstein series. , 2018.
Fan Gao. The [L]{}anglands-[S]{}hahidi [$L$]{}-functions for [B]{}rylinski-[D]{}eligne extensions. , 140(1):83–137, 2018.
Eyal Kaplan. Doubling constructions and tensor product [$L$]{}-functions: coverings of the symplectic group. , 2019.
D. A. Kazhdan and S. J. Patterson. Metaplectic forms. , (59):35–142, 1984.
Peter J. McNamara. Metaplectic [W]{}hittaker functions and crystal bases. , 156(1):1–31, 2011.
Peter J. McNamara. Principal series representations of metaplectic groups over local fields. In [*Multiple [D]{}irichlet series, [L]{}-functions and automorphic forms*]{}, volume 300 of [*Progr. Math.*]{}, pages 299–327. Birkhäuser/Springer, New York, 2012.
Peter J. McNamara. The metaplectic [C]{}asselman-[S]{}halika formula. , 368(4):2913–2937, 2016.
Ryan Cohen Reich. Twisted geometric [S]{}atake equivalence via gerbes on the factorizable [G]{}rassmannian. , 16:345–449, 2012.
Toshiaki Suzuki. Metaplectic [E]{}isenstein series and the [B]{}ump-[H]{}offstein conjecture. , 90(3):577–630, 1997.
Toshiaki Suzuki. Distinguished representations of metaplectic groups. , 120(4):723–755, 1998.
Martin H. Weissman. Metaplectic tori over local fields. , 241(1):169–200, 2009.
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abstract: 'Three different cases of magnetic field reentrant ground states in heavy fermion systems, URhGe, UGe$_2$ and CeRhIn$_5$ recently studied in Grenoble, are discussed. URhGe is a ferromagnetic superconductor with reentrace of superconductivity under magnetic field ($H$) which is associated to the spin reorientation field. UGe$_2$ is a ferromagnetic superconductor with an enhancement of the upper critical field $H_{c2}$ of superconductivity at its metamagnetic transition between 2 ferromagnetic phases. CeRhIn$_5$ is a superconductor with $H$ reentrant antiferromagnetism. We analyze the links between the $H$ enhancement of the different contributions to the effective mass and the field reentrant phase.'
address: |
$^1$ Commissariat à l’ Énergie Atomique, INAC, SPSMS, 17 rue des Martyrs, 38054 Grenoble, France\
$^2$ KYOKUGEN, Osaka University, Osaka, Japan
author:
- 'A. Miyake$^{1, 2}$, D. Aoki$^1$, G. Knebel$^1$, V. Taufour$^1$, J. Flouquet$^1$'
title: Mass Enhancement and Reentrant Ground State in Magnetic Field
---
The unconventional superconductivity (SC) of heavy fermion compounds often occurs close to quantum singularities driven by magnetic or valence instabilities. Here we will present three cases of field reentrant phenomena linked to the interplay between field changes of their magnetic and superconducting properties.
URhGe is a very nice example of ferromagnetic (FM) superconductor; i.e. i) its critical superconducting transition temperature $T_{\rm sc} \sim$0.27 K is far below its Curie temperature $T_{\rm Curie}\sim$ 9.5 K [@Aoki2001], ii) applying a pressure drives the system deeper in the FM domain as shown in Fig. \[URhGe\](a) [@Hardy2005; @miyake2009]; thus, one escapes from a FM quantum singularity where $T_{\rm sc}\sim T_{\rm Curie}$; iii) at $T_{\rm Curie}$, the Fermi surface is well formed and thus the Fermi liquid regime as $AT^2$ resistivity law is also well established; iv) furthermore, the rather weak magnetic anisotropy leads to the remarked phenomena that the sublattice magnetization $M_0$ changes its alignment from the initial $c$-axis to the $b$-axis at $H_{\rm R} \sim$12 T along one of the hard axis (the $b$-axis) without any drastic change of $M_0\sim 0.4~\mu_{\rm B}$ [@Levy2005]. The surprising result is that when $H$ increases along the $b$-axis, the reentrant superconductivity (RSC) is observed in a rather large $H$ window sticked to $H_{\rm R}$ [@Levy2005]. By a careful study of the resistivity under pressure and magnetic field, we have recently demonstrated that the SC properties can be well understood via a McMillan type formula, $T_{\rm sc} = T_0\exp(-m^{\ast}/m^{\ast\ast})$, where the effective mass $m^{\ast}$ is the sum of a renormalized band mass $m_{\rm B}$ and of a correlated mass $m^{\ast\ast}$ source of the SC pairing [@miyake2009]. Field enhancement of $m^{\ast\ast}(H)$ will occur in the vicinity of $H_{\rm R}$, while $m_{\rm B}$ appears as field and pressure invariant. Under pressure, $m^{\ast\ast}(0)$ decreases thus $T_{\rm sc}(m^{\ast\ast}(0))$ and the upper critical field ($H_{c2}(0, m^{\ast\ast}(0))$ decrease slowly. The RSC is strongly affected by the concomitant decrease of $T_{\rm sc}(m^{\ast\ast}(H))$ and of $H_{\rm c2}\sim (m^{\ast}_H)^2T_{\rm sc}^2(m^{\ast}_H)$. As predicted by our simple model, the RSC collapses at a pressure $P_{\rm RSC}\sim$ 1.5 GPa far lower than the pressure $P_{\rm sc}$ where the low field SC will disappear, i.e. $m^{\ast\ast}(0)$ will collapse. The low field SC seems to disappear only near 4 GPa (Fig. \[URhGe\](b)) [@miyake2009].
 ($P, T$) phase diagram of URhGe at $H = 0$ T. (b) Pressure dependence of $T_{\rm sc}$ at $H = 0$ T (circle) and $H_{\rm R}$ (square).](PTPD_URhGe.eps){width="40pc"}
In UGe$_2$, the SC dome is not linked to the proximity of the FM quantum singularity where FM will disappear but to the first order singularity at $P_x\sim$ 1.3 GPa where the system switches from a low pressure high sublattice magnetization FM2 phase ($M_0 = 1.5~\mu_{\rm B}$) to a high pressure low sublattice magnetization FM1 phase ($M_0 = 1~\mu_{\rm B}$). As shown in Fig.\[UGe2\](a), the new features are revealed that i) the first order line ($T_x, P_x$) between FM1 and FM2 terminates at a critical end point ($T_{\rm cr}$ =15 K, $P_{\rm cr}$=1 GPa) and ii) SC appears just above $P_{\rm cr}$. The crossover from FM1 to FM2 due to a continuous mixing of both phase below $P_{\rm cr}$ was drawn from the broadening of the thermal expansion and the resistivity. At ambient pressure, this definition corresponds to the upper limit of 10% of emerging FM2 content in FM1 and the lower limit to 10% remaining FM1 phase in FM2 as determined by combined specific heat and thermal expansion measurements [@Hardy]. At a pressure $P_x + \epsilon\sim$1.35 GPa applying a magnetic field along the easy axis leads to switch from the initial FM1 to the FM2 state via a metamagnetic transition; the consequence for SC shown in Fig.\[UGe2\](b) is that the system reenters in the SC+FM2 phase, which has quite different parameters from SC+FM1; i.e. $m_{\rm B}$ and $m^{\ast\ast}$ (and thus $T_{\rm sc}$ and $H_{c2}$) [@sheikin2001rcs]. In contrast to URhGe, as $M_0$ is not preserved a drastic change of the Fermi surface occurs [@terashima2001eqp]. The concomitant variation of $m_{\rm B}$ and of $m^{\ast\ast}$ are marked by the fact above $P_x$ that its Sommerfeld coefficient is almost pressure invariant while $T_{\rm sc}(P)$ in the FM1 phase changes drastically [@Tateiwa2004]. This simultaneous ($P, H$) variation of $m_{\rm B}$ and $m^{\ast\ast}$ was predicted in the theoretical model of Ref. [@Sandeman2003] with a double peaked structure in the density of states.
These two examples of FM-SC corresponds to the case where $T_{\rm sc} \ll T_{\rm Curie}$. With the recent case of UCoGe having $T_{\rm Cuire}\sim$ 3 K and $T_{\rm sc}\sim$ 0.6 K [@Huy2007], the hope is to get new coexisting states when $T_{\rm Curie}$ and $T_{\rm sc}$ will cross under pressure as the possibility of a cascade from paramagnetic, superconducting and mixed SC+FM states [@mineev].
![\[UGe2\] (a) ($P, T$) phase diagram of UGe$_{2}$ at $H = 0$ T. The full diamond represents the $T_x (P)$ line, the star indicates a critical end point at $P_{\rm cr}$ = 1 GPa and $T_{\rm cr}$ = 15 K. The crossover regime below $P_x$ is delimited by the width of the resistivity and thermal expansion bumps created by the proximity of the critical end point. (b) Superconducting phase diagrams of URhGe at ambient pressure (open circle) and UGe$_2$ at 1.35 GPa (open square) after [@sheikin2001rcs]. ](UGe2.eps){width="40pc"}
 ($P, T$) phase diagram of CeRhIn$_{5}$ at $H = 0$ T from ac calorimetry (circles) and from recent NQR experiments (square) [@Knebel2006; @Yashima2007; @Yashima2009]. (b) ($T, H$) phase diagram at a pressure close to $P_c$ of $P =$ 2.4 GPa.](CeRhIn5.eps){width="22pc"}
As a guide line, it is interesting to look to careful studies realized under $(P, H)$ on CeRhIn$_5$ [@Knebel2004; @Park2006; @Knebel2008]. As shown in Fig.\[CeRhIn5\](a), antiferromagnetism (AF) and SC are robust at low and high pressure, respectively. A coexisting SC+AF phase at zero field occurs only between $P\sim$ 1.7 GPa and $P_c^{\ast}\sim$ 2 GPa, where $T_{\rm N}$ and $T_{\rm sc}$ merge into one point [@Knebel2008]. The simple idea is a competition or interplay between an AF pseudogap $\Delta_{\rm AF}$ and the SC gap $\Delta_{\rm sc}$ assuming that both the AF and the SC phase will have the same (1/2, 1/2, 1/2) hot spot in the coexisting domain which seems to be shown by recent NQR results.[@Yashima2007; @Yashima2009]. Between $P^{\ast}_{c}<P<P_{c}\sim$ 2.5 GPa, the new event is the field reentrance of AF under magnetic field and the associated creation of vortices [@Park2006; @Knebel2008], $T_{\rm N}(H)$ can exceed $T_{\rm sc}(H)$ as the critical magnetic field $H_{M}\sim$ 40 T at $T\to$ 0 K is much larger than the value of the upper critical field $H_{c2}(0)\sim 10$ T, so far $T_{\rm N}$ does not collapse, i.e. $P$ is lower than $P_c$. As indicated by the arrows in Fig.\[CeRhIn5\](b), the reentrant AF sticked to $H_{c2}(0)$ will collapse at $P_c$. Up to now, attempts to fit the upper critical field curves $H_{c2}(T)$ via the orbital and Pauli competition plus additional possibility of strong coupling constant ($\lambda = m^{\ast}/m_{\rm B}-1$) have failed [@Knebel2008]; as discussed previously on FM-SC, for SC of CeRhIn$_5$ there is clearly the necessity to incorporate $H$ dependence of $m^{\ast}$ which is not only pressure dependent.
It is worthwhile to mention that CeCoIn$_5$ at $P$ = 0 may be just above $P^{\ast}_c$, and $H_{\rm M}$ must be lower than $H_{c2}(0)$; then the SC gap will allow to stick AF to $H_{c2}(0)$, so far $\Delta_{\rm sc}(H)$ is below a critical value $\Delta_c$ to allow the establishment of itinerant AF (see [@Knebel2008]). Taking into account the strong coupling correlation, with this picture we arrive to the conclusion that increasing the pressure and thus decreasing the strength of $\lambda$, the AF phase will expand in a larger $H(P)/H_{c2}(P)$ domain than in zero pressure in good agreement with the experiment [@Knebel2008]. Furthermore, when the SC gap at zero field will go down to $\Delta_c$, reentrant magnetism must disappear.
The comparison of the three cases where reentrant phenomena exists show clearly the importance to take into account the $P$ and also $H$ dependence of the effective mass. For FM-SC of URhGe the simplicity is that only the field dependence of the correlated mass $m^{\ast\ast}$ must be considered; in UGe$_2$ the $P$ and $H$ variation affect both contributions of the effective mass. With the case of CeRhIn$_5$ and CeCoIn$_5$, the interplay between an AF pseudogap and the SC gap appears the appealing key parameter. Next interesting example to reveal the relation between magnetism and SC is a careful study of the FM superconductor UCoGe in the critical regime near 1 GPa where $T_{\rm sc} = T_{\rm Curie}$ [@Hassinger2008].
References {#references .unnumbered}
==========
[9]{}
Aoki D, Huxley A, Ressouche E, Braithwaite D, Flouquet J, Brison J P, Lhotel E and Paulsen C 2001 [*Nature*]{} [**413**]{} 613
Hardy F, Huxley A, Flouquet J, Salce B, Knebel G, Braithwaite D, Aoki D, Uhlarz M and Pfleiderer C 2005 [*Physica B*]{} [**359-361**]{} 1111
Miyake A, Aoki D and Flouquet J 2009 [*J. of Phys. Soc. of Jpn*]{} [**78**]{} 063703
Lévy F, Sheikin I, Grenier B and Huxley A D 2005 [*Science*]{} [**309**]{} 1343
Hardy F, Meingast C, Taufour V, Flouquet J, v Löhneysen H, Fisher R A, Phillips N E, Huxley A and Lashley J C [*to be published*]{}
Sheikin I, Huxley A, Braithwaite D, Brison J, Watanabe S, Miyake K and Flouquet J 2001 [*Phys. Rev. B*]{} [**64**]{} 220503
Terashima T, Matsumoto T, Terakura C, Uji S, Kimura N, Endo M, Komatsubara T and Aoki H 2001 [*Phys. Rev. Lett.*]{} [**87**]{} 166401
Tateiwa N, Kobayashi T C, Amaya K, Haga Y, Settai R and Ōnuki Y 2004 [*Phys. Rev. B*]{} [**69**]{} 180513
Sandeman K G, Lonzarich G G and Schofield A J 2003 [*Phys. Rev. Lett.*]{} [**90**]{} 167005
Huy N T, Gasparini A, de Nijs D E, Huang Y, Klaasse J C P, Gortenmulder T, de Visser A, Hamann A, Görlach T and v Löhneysen H 2007 [*Phys. Rev. Lett.*]{} [**99**]{} 067006
Mineev V P 2006 [*C. R. Physique*]{} [**7**]{} 35
Knebel G, Méasson M A, Salce B, Aoki D, Braithwaite D, Brison J and Flouquet J 2004 [*Journal of Physics: Condensed Matter*]{} [**16**]{} 8905
Knebel G, Aoki D, Braithwaite D, Salce B, and Flouquet J 2006 [*Phys. Rev. B*]{} [**74**]{} 020501(R)
Park T, Ronning F, Yuan H Q, Salamon M B, Movshovich R, Sarrao J L and Thompson J D 2006 [*Nature*]{} [**440**]{} 65
Knebel G, Aoki D, Brison J P and Flouquet J 2008 [*J. of Phys. Soc. of Jpn.*]{} [**77**]{} 114704
Yashima M, Kawasaki S, Mukuda H, Kitaoka Y, Shishido H, Settai R and Ōnuki Y 2007 [*Phys. Rev. B*]{} [ **76**]{} 020509 Yashima M, Mukuda H, Kitaoka Y, Shishido H, Settai R and Ōnuki Y [*arXiv:0906.0078v1*]{} Hassinger E, Aoki D, Knebel G and Flouquet J 2008 [*J. of Phys. Soc. of Jpn.*]{} [**77**]{} 073703
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abstract: 'Soit $F_0$ un corps local non archimédien de caractéristique nulle et de caractéristique résiduelle impaire. On décrit explicitement les changements de base des représentations supercuspidales de $U(1,1)(F_{0})$. C’est une étape vers la description du changement de base des paquets endoscopiques supercuspidaux de $U(2,1)(F_{0})$.'
address: 'Département de Mathématiques et U.M.R. 8628 du C.N.R.S., Université Paris-Sud, Bâtiment 425, 91405 Orsay cedex, France'
author:
- Laure Blasco
date: 7 septembre 2009
title: 'Changements de base explicites des représentations supercuspidales de $U(1,1)(F_0)$'
---
Bien que le titre n’en dise rien, cet article fait suite à [@Bl2] dans lequel nous décrivons explicitement le changement de base stable de certaines représentations supercuspidales du groupe unitaire $U(2,1)(F_{0})$ relativement à $F$ au groupe linéaire $GL(3,F)$ où $F_{0}$ est un corps $p$-adique de caractéristique résiduelle impaire et $F$ une extension quadratique de $F_{0}$. Cette description repose sur la classification des représentations supercuspidales par la théorie des types de C. Bushnell et Ph. Kutzko et sur les travaux de J. Rogawski [@Ro].
Plus précisément, dans [@Bl2], n’est explicité que le changement de base stable de paquets cuspidaux de cardinal 1 (et on pense qu’ils y sont tous). Il reste donc à faire l’analogue pour les paquets cuspidaux endoscopiques, ce qui se réalise en deux temps [@Ro Ch.4, §2] :
1. déterminer les images par l’application de transfert des représentations de carré intégrable du groupe $U(1,1)(F_{0})\x U(1)(F_{0})$ au groupe $U(2,1)(F_{0})$ ;
2. décrire le changement de base “labile” (ou “instable”) des représentations supercuspidales de $U(1,1)(F_{0})$ au groupe $GL(2,F)$.
On conclut alors grâce aux résultats de J. Rogawski [@Ro prop. 13.2.2(c)]. Notons que pour la représentation de Steinberg de $U(1,1)(F_{0})$ et ses tordues par un caractère, la deuxième étape est faite dans [@Ro] (prop. 11.4.1 et démonstration de la prop. 12.4.1).
Ce texte présente la réalisation du point (2). Elle repose sur la liste des types de $U(1,1)(F_{0})$ obtenue dans l’annexe de [@Bl1], sur les résultats concernant $GL(2,F)$ exposés dans les chapitres 5 et 8 de [@BH3] et sur ceux du chapitre 11 de [@Ro]. Comme dans le cas de $U(2,1)(F_{0})$, on détermine le changement de base stable dont l’identité de caractères ne dépend que des classes de conjugaison et conjugaison tordue stables, le changement de base labile s’en déduisant aisément puisqu’il est explicitement relié au précédent ([@Ro §11.4] ou voir §\[notations\]). La compatibilité du changement de base à la torsion par un caractère permet de se restreindre aux représentations supercuspidales de niveau minimal parmi leurs tordues par un caractère.
Dans le cas de $U(1,1)(F_{0})$, les paquets supercuspidaux sont bien connus : on en donne une description “typique” dans le paragraphe \[paquets\]. Les paquets endoscopiques sont précisément les paquets de cardinal 2 [@Ro prop. 11.1.1(a)] et sont formés, en niveau strictement positif, de représentations [*cuspidales scindées*]{} (c’est-à-dire celles dont le type provient d’une strate gauche fondamentale scindée suivant un sous-groupe de Levi non rationnel sur $F_{0}$). Les représentations supercuspidales de niveau 0 sont toutes dans des paquets de cardinal 2 si $F$ est non ramifiée sur $F_{0}$ et toutes sauf deux dans des paquets singletons si $F$ est ramifiée sur $F_{0}$.
Par contre, il est plus difficile de distinguer les images des deux changements de base, toutes deux formées de représentations admissibles, invariantes sous l’action du groupe de Galois de $F/F_{0}$ et de caractère central trivial sur $F_{0}^\x$. Pour un paquet endoscopique, emprunter la “voie” par l’application transfert de $U(1)(F_{0})\x U(1)(F_{0})$ à $U(1,1)(F_{0})$ et utiliser le changement de base de $U(1)(F_{0})$ à $F^\x$ permet d’ignorer cette question. Mais pour un paquet singleton où l’on établit le changement de base en vérifiant une identité de caractères entre représentations de $U(1,1)(F_{0})$ et de $GL(2,F)$ (voir (\[id\])), repérer les représentations supercuspidales de $GL(2,F)$ qui appartiennent à l’image du changement de base stable est un point crucial.
Dans le paragraphe \[paquetsH\], on détermine le changement de base stable des paquets endoscopiques en empruntant la méthode de Y. Flicker [@Fl] puis on conclut grâce à [@Ro prop. 11.4.1(a)]. Dans le paragraphe \[paquetssing\], on décrit les changements de base des paquets non endoscopiques en suivant la même démarche que celle exposée dans [@Bl2] dont on ne reprend pas tous les détails. L’ensemble des résultats est présenté aux corollaire \[BC-endoscopiques\] et théorème \[BCdim2\].
Le dernier paragraphe expose un calcul technique permettant de comparer les caractères de représentations d’un groupe et de l’un de ses sous-groupes. Il est utilisé à plusieurs reprises dans les paragraphes précédents.
Les méthodes utilisées n’ont pas d’originalité mais fournissent les résultats désirés, non écrits jusque-là, sous une forme que nous pourrons exploiter ultérieurement et sous un minimum d’hypothèses ($F_{0}$ de caractéristique nulle et de caractéristique résiduelle impaire). Pour ces raisons, on inclut les représentations cuspidales de niveau 0 déjà étudiées par J. Adler et J. Lansky ([@AL] et [@AL2]).
Je remercie Corinne Blondel pour ses encouragements patients et sa relecture critique.
Notations
=========
Soient $F_0$ un corps local non archimédien, de caractéristique nulle et de caractéristique résiduelle différente de deux et $F$ une extension quadratique (séparable) de $F_0$ d’indice de ramification $e_0$ et dont le groupe de Galois est noté $\Gamma$ : $\Gamma=\{1, {}^- \}$.
On désigne par ${\goth o}_0$ (resp. ${\goth o}$) l’anneau des entiers de $F_0$ (resp. $F$), ${\goth p}_0$ (resp. ${\goth p}$) l’idéal maximal de ${\goth o}_0$ (resp. ${\goth o}$) et $\varpi _0$ (resp. $\varpi$) une uniformisante de ${\goth p}_0$ (resp. ${\goth p}$). On choisit les uniformisantes $\varpi$ et $\varpi _0$ telles que : $\varpi =\varpi _0$ si $e_0=1$, $\varpi $ est de trace nulle et de norme $\varpi _0$ si $e_0=2$. On note $k_0$ et $k$ les corps résiduels de $F_0$ et $F$ respectivement et $q$ le cardinal de $k_0$.
Pour une extension $E$ de $F_0$, on conserve les mêmes notations que pour $F$, cette fois indexées par $E$. Si $L$ est une sous-extension de $E$, on désigne par $E_{\vert L}^1$ le groupe des éléments de $E$ dont la norme dans $L$ est 1.
On fixe un caractère additif $\psi_0$ de $F_0$, de conducteur ${\goth
p}_{0}$. Sa composée avec la trace $\tr _{F/F_0}$ est un caractère $\psi$ de $F$ de conducteur ${\goth p}$. On fixe également un prolongement à $F^\times$ du caractère $\omega_{F/F_0}$ de $F_0^\times$ associé à $F/ F_0$ par la théorie du corps de classes. On choisit ce prolongement, noté $\mu$, égal à $x\ffl (-1)^{\val_F(x)}$ si $F$ n’est pas ramifiée sur $F_0$ ; trivial sur $1+{\goth p}$ et égal à la constante de Langlands $\lambda_{F_{\vert F_{0}}}(\psi_{0})$ en $\varpi$ si $F$ est ramifiée sur $F_0$ (voir par exemple [@BH3 §34.3].
Si $\chi$ est un caractère de $F_{\vert F_{0}}^1$, on désigne par $\tl \chi$ le caractère de $F^\x$ défini par : $\tl \chi (x)=\chi(\frac{x}{\ol x}), x\in F^\x$. L’application $\chi \mapsto \tl \chi$ n’est autre que le changement de base stable de $U(1)(F_{0})$ à $GL(1,F)$.
Soient $V$ un $F$-espace vectoriel de dimension 2 muni d’une forme hermitienne “isotrope” non dégénérée $< , >$, $G=U(1,1)(F_0)$ son groupe d’isométries et $\widetilde G=GL (2,F)$ son groupe d’automorphismes. On note $Z$ le centre de $U(1,1)$ : $Z(F_{0})$ s’identifie à $F_{\vert F_{0}}^1$ et $Z(F)$ à $F^\x$.\
Le groupe $\Gamma$ agit sur $\widetilde G$ par : l’élément non trivial de $\Gamma$ transforme un élément $g$ de $\widetilde G$ en $\tau
(g):= \sigma(g)^{-1}$ où $\sigma$ désigne l’involution définie sur $\End
_FV$ et associée à $<,>$. Alors, $G$ n’est autre que le groupe des points de $\widetilde G$ fixes sous $\Gamma$ : $G=\widetilde G^\tau$.
On fixe une base hyperbolique ${\scr B}=(e_{-1}, e_{1})$ de $V$ et on note $g_{0}$ la similitude de matrice $\left(\begin{array}{cc} 1&0\\0&\alpha_{0}\end{array}\right)$ où $\alpha_{0}=\varpi_{0}$ si $F$ n’est pas ramifiée, $\alpha_{0}\in {\goth o}_{0}^\x$ n’est pas une norme si $F$ est ramifiée. Alors le groupe $GU(1,1)(F_{0})$ des similitudes unitaires de $V$ est la réunion de $Z(F) G$ et $g_{0}Z(F) G$.
On fixe également un élément $\varepsilon$ de $F$ : $\varepsilon$ est une unité de trace nulle si $F$ n’est pas ramifiée sur $F_{0}$, $\varpi$ sinon. On note également $\varepsilon$ l’élément de $\tl G$ dont la matrice dans ${\scr B}$ est $\left( \begin{array}{cc} 1&0\\0&\varepsilon \end{array}\right)$. Alors l’application $\Phi$ définie par : $$g \mapsto {}^\varepsilon g:=\left( \begin{array}{cc} 1&0\\0&\varepsilon \end{array}\right) g \left( \begin{array}{cc} 1&0\\0&\varepsilon \end{array}\right)^{-1}$$ est un isomorphisme entre $GZ(F)$ et $GL(2,F_{0})^+Z(F)$ qui identifie $SU(1,1)(F_{0})$ et $SL_{2}(F_{0})$. Rappelons que : $GL(2,F_{0})^+=\{ g\in GL(2,F_{0}) \vert \Det g\in N_{F\vert F_{0}}(F^\x)\}$.
Le groupe endoscopique elliptique $H=U(1)(F_{0})\x U(1)(F_{0})$ de $G$ s’identifie au sous-groupe des éléments de $G$ dont la matrice dans la base orthogonale ${\scr B}^o=(e_{-1}+\frac{1}{2}e_{1},e_{-1}-\frac{1}{2}e_{1})$ est diagonale. A conjugaison près dans $G$, il existe au plus deux plongements de $H$. Ils sont alors conjugués par $g_{0}$.
Les définitions de l’application de transfert de $H$ à $G$ (cf. prop. \[donnees\]) et du changement de base labile dépendent du choix d’un caractère de $F^\x$ prolongeant $\omega_{F/F_{0}}$. On choisit le caractère $\mu$ défini précédemment. On a alors [@Ro §11.4] : [*si $\pi$ est une représentation admissible de $G$, son image par le changement de base stable est $\tl \pi$ si et seulement si son image par le changement de base labile est $\tl \pi \cdot \mu\circ \Det$.*]{}
Pour établir le changement de base stable des paquets supercuspidaux singletons de $G$, on vérifie l’identité de caractères qui le caractérise et qui s’exprime grâce à la [*norme cyclique*]{}, notée $\Nc $ (\[id\]). Il s’agit d’une bijection de l’ensemble des classes de $\tau$-conjugaison stable de $\tl G$ dans l’ensemble des classes de conjugaison stable de $G$, qui associe à la classe de $\tau$-conjugaison stable de $g$, l’intersection de la classe de $\tl G$-conjugaison de $\N (g)=g\tau(g)$ avec $ G$ [@Ko].\
Rappelons que la classe de $\tau$-conjugaison d’un élément $g$ de $\tl G$, ${\scr Cl}_\tau (g)$, est l’ensemble des éléments de $\tl G$ de la forme $h^{-1}g\tau(h)$ avec $ h\in {\tl G}$ et que sa classe de $\tau$-conjugaison stable $\Cl_{\tau}^{st} (g)$ est l’ensemble des éléments $g'$ de $\tl G$ tel que $\N (g')$ soit $\tl G$-conjugué à $\N (g)$.\
De façon analogue, pour un élément $x$ de $G$, on désigne par $\Cl (x)$ sa classe de conjugaison et par $\Cl^{st}(x)$ sa classe de conjugaison stable (c’est-à-dire de conjugaison sous $\tl G$).
Les paquets supercuspidaux de $U(1,1)(F_{0})$. {#paquets}
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Le groupe $GU(1,1)(F_{0})$ agit par conjugaison sur l’ensemble des représentations lisses irréductibles de $U(1,1)(F_{0})$. Les orbites sont exactement les paquets de $U(1,1)(F_{0})$ [@Ro §11.1]. On en déduit :
\(i) Les représentations irréductibles très cuspidales de $U(1,1)(F_{0})$ de niveau strictement positif sont seules dans leur paquet. Les autres représentations cuspidales de niveau strictement positif appartiennent à des paquets de cardinal $2$.\
(ii) Si $F$ est non ramifiée sur $F_{0}$, tous les paquets formés par des représentations irréductibles cuspidales de niveau $0$ sont de cardinal 2.\
(iii) Si $F$ est ramifiée sur $F_{0}$, tous les paquets formés par des représentations irréductibles cuspidales de niveau $0$ sont de cardinal $1$ sauf un. Ce dernier est $\{ \ind_{K}^G\sigma^+,\ind_{K}^G\sigma^-\}$ où $K$ est le sous-groupe parahorique maximal et $\sigma^\pm$ les deux représentations de $SL_{2}(k_{0})$ de dimension $\frac{q-1}{2}$ [@Sp].
La suite du paragraphe justifie ces assertions.
Soient $\pi$ une représentation irréductible cuspidale de $G$ et $(J,\lambda)$ un type pour cette représentation : $\pi=\ind_{J}^G\lambda$. Il suffit d’étudier si $\pi$ et $^{g_{0}}\pi$, sa conjuguée par $g_{0}$, sont isomorphes. On distingue deux cas selon le niveau.
$\pi$ est de niveau strictement positif.
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Alors $(J,\lambda)$ provient d’une strate gauche semi-simple ${\goth s}=(\scr L, n,\left[\frac{n}{2}\right],b)$ (au sens de [@St déf. 4.9]). On note $Z_{\tl G}(b)$ le centralisateur de $b$ dans $\tl G$ et $Z_{G}(b)$ celui de $b$ dans $G$ : $Z_{G}(b)=Z_{\tl G}(b)\cap G$.
Si $\pi$ est très cuspidale [@Bl1 §A.5], $Z_{\tl G} (b)\cup \{0\}$ est l’extension quadratique $F[b]$ de $F$. L’élément $c=b-\frac{1}{2}\tr_{F[b]_{\vert F}} b$ engendre aussi $F[b]$ sur $F$, appartient à l’algèbre de Lie de $G$ et engendre une extension quadratique $F_{0}[c]$ sur $F_{0}$, autre que $F$. On vérifie que les éléments de $F_{0}[c]^\x$ sont des éléments de $GU(1,1)(F_{0})$ dont le rapport est égal à leur norme dans $F_{0}$. Il en existe donc un, $c_{0}$, qui appartient à $g_{0}F_{0}^\x U(1,1)(F_{0})$. Alors : $^{g_{0}}\pi\simeq {}^{c_{0}}\pi\simeq \pi$.
Si $\pi$ est cuspidale scindée [@Bl1 §A.6], $Z_{\tl G}(b)$ est isomorphe à $F^\x \x F^\x$. On montre par l’absurde que l’entrelacement ${\scr E}({\goth s},{}^{g_{0}}{\goth s})$ des strates semi-simples ${\goth s}$ et ${}^{g_{0}}{\goth s}=(g_{0 }\scr L, n,\left[\frac{n}{2}\right],g_{0}^{-1}bg_{0})$ contenues dans $\lambda$ et ${}^{g_{0}}\lambda$ respectivement, est vide (d’où l’on déduit que $\lambda$ et ${}^{g_{0}}\lambda$ ne sont pas entrelacées dans $G$).\
L’entrelacement $\tl {\scr E}({\goth s},{}^{g_{0}}{\goth s})$ de ${\goth s}$ et ${}^{g_{0}}{\goth s}$ dans $\tl G$ est égal à $\tl {\scr E}({\goth s})g_{0}^{-1}$ où $\tl {\scr E}({\goth s})$ est l’entrelacement de ${\goth s}$ dans $\tl G$. De plus, par le théorème 4.10 précisé par (4.15) de [@St], $\tl {\scr E}({\goth s})$ est égal à $\tl U_{m}(\scr L)Z_{\tl G}(b) \tl U_{m}(\scr L)$ avec $m=\left[\frac{n+1}{2}\right]$. On déduit que : $$\begin{split}
{\scr E}({\goth s},{}^{g_{0}}{\goth s})&=\tl {\scr E}({\goth s},{}^{g_{0}}{\goth s})^\tau=\left(\tl U_{m}(\scr L) Z_{\tl G}(b)g_{0}^{-1}\tl U_{m}(g_{0} {\scr L})\right)^\tau \\
&=\cup_{z\in Z_{\tl G}(b)}\left(\tl U_{m}(\scr L) zg_{0}^{-1}\tl U_{m}(g_{0} {\scr L})\right)^\tau.
\end{split}$$ Si ${\scr E}({\goth s},{}^{g_{0}}{\goth s}) \neq \emptyset$, il existe $z\in Z_{\tl G}(b)$ tel que : $\left(\tl U_{m}(\scr L) zg_{0}^{-1}\tl U_{m}(g_{0} {\scr L})\right)^\tau\neq \emptyset$. Fixons un tel $z$. Comme $\scr L$ et $g_{0}\scr L$ sont des chaînes autoduales ($g_{0}$ appartient à $GU(1,1)(F_{0})$), les deux pro-$p$-sous-groupes $\tl U_{m}(\scr L)$ et $\tl U_{m}(g_{0}\scr L)$ sont invariants par $\tau$. Par le lemme 2.2 de [@St], généralisé sans encombre au cas de deux pro-$p$-groupes, $\tl U_{m}(\scr L) zg_{0}^{-1}\tl U_{m}(g_{0} {\scr L})$ est invariant par $\tau$. En particulier $\tau(zg_{0}^{-1})$, qui est égal à $\alpha_{0}\tau(z)g_{0}^{-1}$, appartient à $\tl U_{m}(\scr L) zg_{0}^{-1}\tl U_{m}(g_{0} {\scr L})=\tl U_{m}(\scr L) z\tl U_{m}({\scr L})g_{0}^{-1}$ d’où : $\alpha_{0}z^{-1}\tau(z)= z^{-1}uz\cdot u'$ avec $u, u'\in \tl U_{m}(\scr L)$.\
Exprimés dans une base orthogonale formée de vecteurs propres de $b$ et adaptée à $\scr L$, le terme de gauche est une matrice diagonale dont les termes diagonaux appartiennent à $\alpha_{0}N_{F/F_{0}}(F^\x)$ tandis que le terme de droite a au moins un de ses coefficients diagonaux dans $1+{\goth p}^m$. L’égalité entraîne donc que $\alpha_{0}$ doit être une norme de $F^\x$ dans $F_{0}$ ce qui est absurde.
$\pi$ est de niveau $0$.
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Il existe un sous-groupe parahorique $K$ maximal (au sens de Bruhat-Tits) et une représentation irréductible $\sigma$ de $K$, triviale sur le radical pro-unipotent $K_{1}$ et définissant une représentation cuspidale du quotient réductif (connexe) $K/K_{1}$, tels que $\pi$ soit isomorphe à $\ind_{K}^G\sigma$ (dans ce cas, $K$ est son propre normalisateur) [@Mo §§2 et 3].
Quand $F$ est non ramifiée sur $F_{0}$, il existe, à conjugaison près, deux sous-groupes parahoriques maximaux : $K=U_{0}(\scr L)$ où $\scr L$ est une chaîne de réseaux autoduale de période 1, d’invariant pair ou impair, et $K'$ son conjugué par $g_{0}$. Dans ce cas, $\pi=\ind_{K}^G\sigma$ et ${}^{g_{0}}\pi=\ind_{K'}^G{}^{g_{0}}\sigma$ ne sont pas isomorphes.\
En effet, puisque $U(1,1)(F_{0})=K SU(1,1)(F_{0})$, les restrictions à $SU(1,1)(F_{0})$ de $\pi$ et ${}^{g_{0}}\pi$ sont deux représentations irréductibles de $SU(1,1)(F_{0})$ dont les conjuguées par $\varepsilon$, notées ${}^\varepsilon \pi$ et ${}^\varepsilon ({}^{g_{0}}\pi)$ respectivement, sont deux représentations de $SL_{2}( F_{0})$, cuspidales, irréductibles et de niveau 0 : ${}^\varepsilon \pi=\ind_{{}^\varepsilon K}^{SL_{2}(F_{0})} {}^\varepsilon \sigma$ et ${}^\varepsilon ({}^{g_{0}}\pi)={}^{g_{0}}({}^\varepsilon \pi)$ (puisque $\varepsilon $ et $g_{0}$ commutent). Elles sont donc toutes deux sous-représentations de la restriction à $SL_{2}(F_{0})$ d’une représentation irréductible cuspidale de niveau 0 de $GL_{2}(F_{0})$ et ne sont donc pas équivalentes [@KS th. 4.4].
Lorsque $F$ est ramifiée, il existe une unique classe de conjugaison de sous-groupes parahoriques maximaux représentée par $U_{0}(\scr L)$ où $\scr L$ est la chaîne de réseaux autoduale de période 1 et d’invariant impair. Le quotient $U_{0}(\scr L)/U_{1}(\scr L)$ est isomorphe à $SL_{2}(k_{0})$.\
Ainsi, $\sigma$ définit une représentation cuspidale irréductible de $SL_{2}(k_{0})$. D’après T. A. Springer [@Sp II §3], $\sigma$ est de dimension $q-1$ ou $\frac{q-1}{2}$ et est décrite par son caractère. Il est alors aisé de montrer que si $\sigma$ est de dimension $q-1$, elle est isomorphe à sa conjuguée par $g_{0}$ tandis que si sa dimension est $\frac{q-1}{2}$, sa conjuguée est l’autre représentation cuspidale de même dimension, notée $\sigma'$. Dans le premier cas, $\pi$ est isomorphe à sa conjuguée par $g_{0}$. Dans le deuxième cas, montrons que $\sigma$ et $\sigma'$ ne sont pas entrelacées.\
Grâce à la décomposition de Cartan, $K$ étant un bon compact dans ce cas, il suffit de montrer que $g=\left(\begin{array}{cc}\varpi&0\\ 0&{\ol \varpi} ^{-1}\end{array}\right)^m$, $m\in {\Bbb N}$, n’entrelace pas $\sigma$ et $\sigma'$. Si $m=0$, c’est clair car $\sigma$ et $\sigma'$ ne sont pas isomorphes. Si $m>0$, $K\cap {}^gK$ contient le sous-groupe $\left(\begin{array}{cc}1&0\\ \varpi{\goth o}_{0}&1\end{array}\right)$. Or la restriction de $\sigma'$ à ce sous-groupe ne contient pas le caractère trivial tandis que celle de ${}^g\sigma$ est triviale.
Les paquets endoscopiques cuspidaux et les caractères de $H$. {#paquetsH}
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Il s’agit d’identifier le caractère $\theta$ de $H$ associé à un paquet endoscopique cuspidal de $G$ par l’application de transfert.
Remarquons que lorsque $F$ est ramifiée sur $F_{0}$, il existe un unique paquet endoscopique de niveau 0 et deux caractères réguliers de $H$ de niveau 0 : $\theta= 1\otimes \chi$ et $\chi\otimes 1={}^\tau \theta$ où $\chi$ est le caractère de $F_{\vert F_{0}}^1$ d’ordre 2 trivial sur $(1+{\goth p})_{\vert F_{0}}^1$. Nécessairement, le paquet endoscopique est l’image de $\theta$ (et ${}^\tau \theta$) par l’application de transfert. Par la suite, on suppose donc : $$n>0 \quad \text{ ou } \quad F \text{ n'est pas ramifi\'ee sur } F_{0}.$$
{#donnees}
Soit $\Pi$ un paquet endoscopique cuspidal de $G$. Il est entièrement déterminé par la donnée d’un type simple $(J,\lambda)$ :
1. $J= HU_{m}(\scr L)$ où $\scr L$ est la chaîne de ${\goth o}$-réseaux autoduale stable par $H$, c’est-à-dire celle de période 1, d’invariant pair, et $m=[\frac{n+1}{2}]$, $n\in \Bbb N$ ;
2. [*Cas de niveau 0, $F$ non ramifiée*]{} : $\lambda$ est le relèvement à $U_{0}(\scr L)$ d’une représentation irréductible cuspidale $\sigma$ de $U(1,1)(k_{0})$ vérifiant :
3. $\quad \quad \forall x\in H(k_{0}), \tr \sigma(x)=\left\{ \begin{array}{ll} (q-1) \cdot \chi(x) & \text{ si } x\in Z(k_{0})\\ -(\chi (x)+\chi(wxw^{-1}))& \text{ si } x\not \in Z(k_{0})\\ \end{array}\right.$
4. pour un caractère régulier $\chi$ de $H(k_{0})$ [@En §6]. L’élément $w$ appartient au normalisateur de $H(k_{0})$ mais non à $H(k_{0})$. Notons $\theta_{1}\otimes \theta_{2}$ le caractère de $H$ relevant $\chi$.
5. [*Cas de niveau strictement positif*]{} : $\lambda$ est une représentation irréductible de $J$ dont la restriction à $U_{m}(\scr L)$ est multiple d’un caractère $\psi_{b}$ avec $b\in {\goth a}_{-n}(\scr L)^-\setminus {\goth a}_{-n+1}(\scr L)^-$ et dont la restriction à $H$ vérifie :
6. $\quad \quad \forall x\in H, \tr \lambda(x)=\left\{ \begin{array}{ll} \dim \lambda \cdot \theta_{1}\otimes \theta_{2}(x) & \text{ si } x\in Z(F_{0})\\ -\theta_{1}\otimes \theta_{2}(x) & \text{ si } x\not \in Z(F_{0})\\ \end{array}\right.$
7. pour un caractère régulier $\theta_{1}\otimes \theta_{2}$ de $H$ prolongeant ${\psi_{b}}_{\vert H\cap U_{m}(\scr L)}$ (voir §\[Traces\] et en particulier §\[precisions\], (1)). Précisons que $b$ est de la forme $b_{1}\oplus b_{2}$ dans la décomposition de $F^2$ définie par ${\scr B}^o$ avec $b_{1}-b_{2}\not \in {\goth p}^{-n+1}$ (quitte à tordre $\Pi$ par un caractère de $G$).
On a alors : $\Pi=\{ \pi, {}^{g_{0}}\pi\}$ où $\pi=\ind_{J}^G \lambda$.
Soit $\theta$ le caractère de $H$ correspondant à $\Pi$ par l’application de transfert. Il existe un unique $\epsilon_{\theta}\in \{\pm 1\}$ tel que pour tout $h\in H_{G-reg}$, $h=(h_{1}, h_{2})$ (c’est-à-dire $h$ de matrice $\diag(h_{1}, h_{2})$ dans ${\scr B}^o$), $$\label{IC-tr}
\tr{\pi}(h)-\tr{{}^{g_{0}}\pi}(h)=\epsilon_{\theta}\frac{\iota(h)}{D_{G}(h)}(\theta(h)+\theta(whw^{-1}))$$ où $\iota(h)=\mu(h_{1}-h_{2})$, $D_{G}(h)=\left\vert \frac{(h_{1}-h_{2})^2}{h_{1}h_{2}}\right\vert_{F_{0}}^{\frac{1}{2}}$ et $w=\left(\begin{array}{cc} 0&1\\1&0Ê\end{array}\right)$ dans ${\scr B}^o$.
{#section}
Evaluons le membre de gauche de (\[IC-tr\]) en suivant la méthode de Y. Flicker [@Fl], c’est-à-dire en se ramenant aux résultats de [@LL §2], grâce à l’isomorphisme $\Phi$ entre $GZ(F)$ et $GL(2,F_{0})^+Z(F)$ (cf. §\[notations\]). Précisons que dorénavant, toutes les matrices sont exprimées relativement à la base $\scr B$.\
On note $i$ le plongement de $F^\times$ dans $GL(2,F_{0})^+$ qui à $x=\alpha+\varepsilon \beta\in F^\x$ ($\alpha,\beta\in F_{0}$) associe $i(x)=\left( \begin{array}{cc} \alpha&2\beta\\ \frac{\varepsilon^2}{2}\beta&\alpha\end{array}\right)$.
\[Phi\]\
(i) Soient $h=(h_{1}, h_{2})\in H$, $x\in F^\x$ et $z\in Z(F)$. Alors : $$\begin{aligned}
&\Phi (hz)=i(\ol a h_{1})\cdot \ol a^{-1}z \text{ o\`u } a\in F^\x \text{ tel que } a\ol a^{-1}=\Det h=h_{1}h_{2}\\\text{ et }\quad &\Phi^{-1} (xz)=h_{x}\cdot \ol xz \text{ o\`u } h_{x}=(x\ol x^{-1},1)\in H.\end{aligned}$$ En particulier, $\Phi$ identifie $HZ(F)$ et $i(F^\x) Z(F)$.\
(ii) Soient $k\in {\Bbb N}$ et $\scr L^0$ la chaîne de ${\goth o}_{0}$-réseaux dans $F_{0}^2$ stable par $F^\x$. Lorsque $k=0$, on suppose en outre que $F$ n’est pas ramifiée sur $F_{0}$. Alors :\
$\Phi(U_{k}({\scr L})Z(F_{0}))= \{ uz, u\in U_{k}({\scr L}^0), z\in Z(F) \text{ tels que } \Det u\cdot N_{F_{\vert F_{0}}}(z)=1\}$.\
(iii) $\Phi( JZ(F))\cap GL(2,F_{0})^+=i(F^\x) U_{m}({\scr L}^0)$.
L’assertion [*(i)*]{} est obtenue par de simples calculs tandis que [*(iii)*]{} est une conséquence immédiate de [*(i)*]{} et [*(ii)*]{}.\
[*(ii)*]{} Supposons d’abord $k>0$. Soient $x\in U_{k}(\scr L)$ et $z\in Z(F_{0})$. Il existe $y\in 1+{\goth p}^k$ tel que : $y\ol y^{-1}=\Det x$, puis $x'\in U_{k}(\scr L)\cap SU(1,1)(F_{0})$ tel que : $x=\left( \begin{array}{cc} y&0\\ 0&\ol y^{-1}\end{array}\right)x'$. Alors : $\Phi(xz)=\left( \begin{array}{cc} y\ol y&0\\ 0&1\end{array}\right)\Phi(x')\cdot \ol yz$.\
Or $x'$ est de la forme $\left( \begin{array}{cc} a&\varepsilon b\\ \varepsilon^{-1}c& d\end{array}\right)$ avec $a,d\in (1+{\goth p}^k)\cap F_{0}=1+{\goth p}_{0}^{[\frac{k+1}{e_{0}}]}$ et $\varepsilon b, \varepsilon^{-1}c\in {\goth p}^k \cap F_{0}={\goth p}_{0}^{[\frac{k}{e_{0}}]}$ donc $\Phi(x')$ appartient à $$\left\{\begin{aligned}
\left( \begin{array}{cc} 1+{\goth p}_{0}^k&{\goth p}_{0}^k\\{\goth p}_{0}^k&1+{\goth p}_{0}^k \end{array}\right)\cap SL(2)(F_{0})\subset U_{k}({\scr L}^0) & \text{ si } e_{0}=1 \\
\left( \begin{array}{cc} 1+{\goth p}_{0}^{[\frac{k+1}{2}]}&{\goth p}_{0}^{[\frac{k}{2}]}\\{\goth p}_{0}^{[\frac{k}{2}]+1}&1+{\goth p}_{0}^{[\frac{k+1}{2}]} \end{array}\right)\cap SL(2)(F_{0})\subset U_{k}({\scr L}^0) & \text{ si } e_{0}=2.
\end{aligned}\right.$$ De plus, $y\ol y\in 1+\tr {\goth p}^k$ donc $\left( \begin{array}{cc} y\ol y&0\\ 0&1\end{array}\right)\in U_{k}({\scr L}^0)$. Ainsi, $\Phi(x)$ appartient à $U_{k}({\scr L}^0) Z(F)$ et $\Det\left(\left( \begin{array}{cc} y\ol y&0\\ 0&1\end{array}\right)\Phi(x')\right)\cdot N_{F_{\vert F_{0}}}(\ol y^{-1}z)=1$.\
L’inclusion inverse se montre de façon analogue.\
Lorsque $k$ est nul, $F$ est non ramifiée sur $F_{0}$ : les mêmes arguments fournissent les mêmes résultats.
{#section-1}
Pour poursuivre, on choisit un caractère $\Omega$ de $Z(F)$ prolongeant le caractère central $\omega_{\pi}$ de $\pi$ et tel que $$\label{Omega}
\left\{ \begin{aligned}
\forall u\in {\goth a}_{[\frac{n}{2}]+1}(\scr L)\cap Z(F), \quad \Omega(1+u)=\psi((b_{1}+b_{2})u) &\text{ si } n>0\\
\Omega=\Theta_{1}\Theta_{2}\quad \text{o\`u $\Theta_{i}$ est un prolongement de $\theta_{i}$ \`a $F^\x$} & \text{ si } n=0.\end{aligned}\right.$$ On prolonge alors $\pi$ et ${}^{g_{0}}\pi$ à $GZ(F)$ en faisant agir $Z(F)$ via $\Omega$. On note encore $\pi$ et ${}^{g_{0}}\pi$ ces prolongements puis : $$\pi_{0}^+=\pi\circ \Phi^{-1}_{\vert GL_{2}(F_{0})^+}, \quad \pi_{0}^-={}^{g_{0}}\pi\circ \Phi^{-1}_{\vert GL_{2}(F_{0})^+}={}^{g_{0}}\pi_{0}^+ , \quad \pi_{0}=\ind_{GL_{2}(F_{0})^+}^{GL_{2}(F_{0})} \pi_{0}^+.$$
Posons : $J_{0}=F^\x U_{m}({\scr L}^0)$, $J_{0,c}={\goth o}_{0}^\x U_{m}({\scr L}^0)$. Notons $\Lambda$ la représentation $(\lambda\Omega)\circ \Phi^{-1}_{\vert J_{0}}$ de $J_{0}$ et $\Lambda_{c}$ sa restriction à $J_{0,c}$.\
(i) La paire $(J_{0,c}, \Lambda_{c})$ est un type simple maximal de $\pi_{0}$ et $\pi_{0}=\ind_{J_{0}}^{GL_{2}(F_{0})} \Lambda$. La représentation $\pi_{0}$ est donc irréductible et cuspidale.\
(ii) Cas $n>0$. La restriction de $\Lambda$ à $U_{[\frac{n}{2}]+1}(\scr L ^0)$ est un multiple du caractère $\psi_{0, \alpha}$ où $\alpha=i(b_{1}-b_{2})
\in {\goth a}_{-n}({\scr L}^0)\setminus {\goth a}_{-n+1}({\scr L}^0)$.\
La restriction de $\Lambda$ à $F^\x$ vérifie pour tout $x\in F^\x$ :
- si $\dim \Lambda=1$, $\quad \Lambda (x)= \tl \theta_{1}(x)\Omega (\ol x)$ ;
- si $\dim \Lambda>1$, $\quad \tr \Lambda (x)=\left\{ \begin{array}{ll} \dim \Lambda \cdot \tl \theta_{1}(x)\Omega (\ol x)&\text{ si } x\in F_{0}^\x\\ - \tl \theta_{1}(x)\Omega (\ol x)&\text{ sinon.}\end{array}\right.$.
\(iii) Cas $n=0$ ($F$ n’est pas ramifiée). La restriction de $\Lambda$ à $U_{0}(\scr L ^0)$ est le relèvement d’une représentation $\sigma_{0}$ de $GL_{2}(k_{0})$ caractérisée par : $$\left\{
\begin{aligned}
\tr \sigma_{0}(z)=& (q-1)\Omega(z), \quad z\in k_{0}^\x\\
\tr \sigma_{0}(zn)=&-\Omega(z), \quad z\in k_{0}^\x, n\in N(k_{0})-\{ \id\} \\
\tr \sigma_{0}(x)=& -\Omega(\ol x)(\tl \theta_{1}(x)+\tl \theta_{2}(x)), \quad x\in k^\x-k_{0}^\x
\end{aligned}
\right.$$ et $\Lambda (\varpi_{0})=\Omega (\varpi_{0})$.
La première assertion est une conséquence des deux suivantes et ces dernières proviennent des propriétés de $\lambda$ et de $\Omega$ via $\Phi$.\
Plus précisément, pour établir [*(ii)*]{}, on étudie d’abord la restriction de $\Lambda$ à $U_{[\frac{n}{2}]+1}(\scr L ^0)\cap SU(1,1)(F_{0})$. Un calcul du même style que dans le lemme \[Phi\] montre que cette restriction est multiple de $\psi_{0,\alpha}$ où $\alpha=i(b_{1}-b_{2})+D$ avec $D$ une matrice diagonale. Ensuite, si $x=\diag(z,\ol z^{-1})\in U_{[\frac{n}{2}]+1}(\scr L ^0)$, $\Lambda (x)$ est multiple de $\psi_{{}^\varepsilon b}(x)\Omega(\ol z)$, qui vaut 1 par choix de $\Omega$ (\[Omega\]). On peut donc prendre $D=0$. L’élément $\alpha$ ainsi défini appartient à ${\goth a}_{-n}({\scr L}^0)\setminus {\goth a}_{-n+1}({\scr L}^0)$ par \[donnees\] (b’). La suite est immédiate, tout comme l’assertion [*(iii)*]{}.
Soit $\Delta_{\theta_{1}\otimes \theta_{2}}$ le caractère de $F^\x$ de niveau $0$ défini par :
- quand $F$ n’est pas ramifiée sur $F_{0}$, $\Delta_{\theta_{1}\otimes \theta_{2}}=\mu$ ;
- quand $F$ est ramifiée sur $F_{0}$,
${\Delta_{\theta_{1}\otimes \theta_{2}}}_{\vert F_{0}^\x}=\omega_{F/F_{0}}$ et $\Delta_{\theta_{1}\otimes \theta_{2}} (\varpi)=\omega_{F/F_{0}}(\varpi(b_{1}-b_{2}))\lambda_{F_{\vert F_{0}}}(\psi_{0})^{-1}$.
Posons : $\Theta=\Delta_{\theta_{1}\otimes \theta_{2}}^{-1}\tl\theta_{1}\ol \Omega $ si $n>0$ et $\Theta =\mu^{-1}\Theta_{1}\ol \Theta_{2}$ si $n=0$. Alors $\pi_{0}$ est la représentation de $GL_{2}(F_{0})$ associée au caractère $\Theta$ par [@JL].
C’est une application des résultats de [@BH3], §§19 et 34.
{#section-2}
D’après la théorie de Mackey, la restriction de $\pi_{0}$ à $GL_{2}(F_{0})^+$ est la somme des représentations $\pi_{0}^+$ et $\pi_{0}^-$. D’après [@LL p.738], pour tout $x\in F^\x$ régulier, $$\tr{\pi_{0}^+}(x)-\tr{\pi_{0}^-}(x)=\pm \lambda_{F_{\vert F_{0}}}(\psi_{0})\omega_{F/ F_{0}}\left(\frac{x-\ol x}{\varepsilon}\right) \frac{\Theta(x)+\Theta(\ol x)}{D(x)},$$ où $D(x)= \left\vert \frac{ (x-\ol x)^2}{x\ol x}\right\vert_{F_{0}}^{1/2}$. Mais, si $h=(h_{1}, h_{2})\in H$ est $G$-régulier et $a\in F^\x$ tel que $a\ol a=h_{1}h_{2}$, $\ol a h_{1}$ est un élément de $F^\x$ régulier dont le conjugué est $\ol a h_{2}$ et l’on a : $$\begin{aligned}
&\tr{\pi}(h)=\tr{\pi}(\Phi^{-1}(i(\ol a h_{1})\cdot \ol a^{-1}))= \tr{\pi_{0}^+}(i(\ol a h_{1}))\Omega(\ol a^{-1}),\\
\text{et \quad } &\tr{{}^{g_{0}}\pi}(h)= \tr{\pi_{0}^-}(i(\ol a h_{1}))\Omega(\ol a^{-1}).
\end{aligned}$$ Par suite (on note $\Delta=\Delta_{\theta_{1}\otimes \theta_{2}}$) : $$\begin{aligned}
\tr{\pi}(h)-\tr{{}^{g_{0}}\pi}(h)&= \pm \lambda_{F_{\vert F_{0}}}(\psi_{0})\omega_{F/ F_{0}}\left(\frac{\ol ah_{1}-\ol ah_{2}}{\varepsilon}\right) \Omega(\ol a ^{-1}) \frac{\Theta(\ol ah_{1})+\Theta(\ol ah_{2})}{\Delta(\ol ah_{1})}\\
&= \pm \lambda_{F_{\vert F_{0}}}(\psi_{0})\mu(h_{1}-h_{2})\mu\left(\frac{\ol a}{\varepsilon}\right)\Delta^{-1}(\ol a)\Omega(\ol a ^{-1}) \\
&\quad \quad \quad \quad \cdot
\frac{\Delta^{-1}(h_{1})\tl \theta_{1}(\ol ah_{1})\Omega(\ol ah_{2})+\Delta^{-1}(h_{2})\tl \theta_{1}(\ol ah_{2})\Omega(\ol ah_{1})}{D_{G}(h)}\\
&=\pm \lambda_{F_{\vert F_{0}}}(\psi_{0})\mu(\varepsilon)^{-1}\frac{\iota(h)}{D_{G}(h)}\mu(\ol a)\Delta^{-1} (\ol a)\\
&\quad \quad \quad \quad \quad \quad \quad \quad
\cdot \left(\mu^{-1}\theta_{1}\otimes \theta_{2}(h)+\mu^{-1}\theta_{1}\otimes \theta_{2}(whw^{-1})\right)\\
\end{aligned}$$ en remarquant que : $\Delta (x)=\mu(x)$ pour tout $x\in F_{\vert F_{0}}^1$ et $\tl \theta_{1}(\ol ah_{1})=\theta_{1}(h_{1}h_{2}^{-1})$ tandis que $\Omega(\ol a^{-1})\Omega (\ol ah_{2})=\Omega(h_{1}h_{2})\Omega(h_{1}^{-1})=\theta_{1}(h_{2})\theta_{2}(h_{2})$ car $h_{2}\in F_{\vert F_{0}}^1$. De plus, $\lambda_{F_{\vert F_{0}}}(\psi_{0})\mu(\varepsilon)^{-1}$ est égal à $-1$ si $F$ est non ramifiée et à 1 si $F$ est ramifiée (par choix de $\varepsilon$ et $\mu$) et le caractère de $F^\x$, $ a\mapsto \mu(\ol a)\Delta^{-1} (\ol a)$, est trivial si $F$ n’est pas ramifiée et égal à $(\omega_{F/F_{0}}(\varpi(b_{1}-b_{2}))^{\val(a)}$ si $F$ est ramifiée. Dans ce dernier cas, $a$ est de valuation paire si et seulement si $h_{1}h_{2}\in (1+{\goth p})_{F_{0}}^1$. En comparant à (\[IC-tr\]), on conclut :
Soit $\Pi=\{\pi, {}^{g_{0}}\pi\}$ un paquet endoscopique cuspidal de $G$ décrit en \[donnees\] (dont on reprend les notations). On définit un caractère $\delta _{\theta_{1}\otimes \theta_{2}}$ de $F_{\vert F_{0}}^1$ par : $\delta _{\theta_{1}\otimes \theta_{2}}$ est trivial si $F$ n’est pas ramifiée sur $F_{0}$ ; $\delta _{\theta_{1}\otimes \theta_{2}}$ est trivial sur $(1+{\goth p})^1_{\vert F_{0}}$ et $\delta _{\theta_{1}\otimes \theta_{2}}(-1)= \omega_{F/F_{0}}(\varpi (b_{1}-b_{2}))$ si $F$ est ramifiée.\
Alors le paquet $\Pi$ est l’image par l’application de transfert du caractère $\theta$ de $H$ défini par : $$\forall h\in H, \quad \theta (h)=
\delta _{ _{\theta_{1}\otimes \theta_{2}}}(\Det h)\cdot \mu^{-1}\theta_{1}\otimes\theta_{2}(h).$$
En application de la proposition 11.4.1(a) de [@Ro], on obtient :
\[BC-endoscopiques\] L’image par le changement de base stable du paquet $\Pi$ décrit en \[donnees\] (dont on reprend les notations) est $\tl \pi =\indu_{\tl P}^{GL_{2}(F)} \mu(\theta)$ où $\tl P$ est un sous-groupe parabolique de $GL_{2}(F)$ de facteur de Levi $H(F)$ et $\mu(\theta)$ le caractère de $H(F)$ défini par : $$\forall h\in H(F), \quad\mu(\theta) (h)=
\mu^{-1}\tl \delta _{ _{\theta_{1}\otimes \theta_{2}}}(\Det h)\cdot \tl \mu^{-1}\tl \theta_{1}\otimes\tl \theta_{2}(h).$$ Lorsque $F$ est ramifiée sur $F_{0}$ et que $\Pi$ est l’unique paquet endoscopique de niveau $0$, l’image de $\Pi$ est $\tl \pi =\indu_{\tl P}^{GL_{2}(F)} \mu^{-1}\otimes \tl \chi \mu^{-1}$.
Changement de base stable des paquets cuspidaux singletons. {#paquetssing}
===========================================================
On procède en trois étapes : la première est consacrée à la construction de représentations irréductibles cuspidales de $\tl G=GL_{2}(F)$, $\tau$-invariantes et de caractère central trivial sur $F_{0}^\x$ à partir de représentations très cuspidales de $G$. Par [@Ro prop. 11.4.1(c)], ces représentations appartiennent à l’image d’un des deux changements de base, le “stable” ou le “labile”. Dans la deuxième étape, on distingue parmi les représentations construites celles qui appartiennent à l’image du changement de base stable. La dernière étape décrit les changements de base stable et labile des paquets singletons de $G$.
{#donnees2}
Soit $(J,\lambda)$ un type simple maximal de $G$, c’est-à-dire :
1. [*Cas de niveau 0 ($F$ ramifiée sur $F_{0}$*]{}) : $J$ est le sous-groupe parahorique maximal de $G$, c’est-à-dire $J=U_{0}(\scr L)$ avec $\scr L$ la chaîne autoduale de $F^2$, de période 1 et d’invariant impair. La représentation $\lambda$ est une représentation irréductible de $J$, triviale sur le sous-groupe pro-unipotent $J_{1}$ de $J$ et dont la factorisation $\ol \lambda$ par $J/J_{1}\simeq SL_{2}(k_{0})$ est une représentation cuspidale de dimension $q-1$. Elle est associée à un caractère $\theta$ régulier d’ordre différent de 2 du groupe des éléments de norme 1 de l’extension quadratique $\ell$ de $k_{0}$ par :
2. $\quad \quad \left\{ \begin{array}{ll}
\tr \ol \lambda(x)= (q-1)\theta(x) & \text{ si } x\in \{\pm 1\}\\
\tr \ol \lambda(xn)=-\theta(x) & \text{ si } x\in \{\pm 1\}, n\in N(k_{0})-\{\id\}\\
\tr \ol \lambda(x)=-(\theta(x)+\theta(\gamma(x)))& \text{ si } x\in\ell_{\vert k_{0}}^1 - \{\pm 1\}
\end{array}\right.$
3. où $\gamma$ est l’élément non trivial de $\Gal (\ell/k_{0})$ [@Sp].
4. [*Cas de niveau strictement positif*]{} : $(J,\lambda)$ provient d’une strate gauche très cuspidale $(\scr L, n,n-1,b)$ avec $n>0$, c’est-à-dire, en notant $E=F[b]$ l’extension de $F$ engendrée par $b$ et $L=E^\sigma$ la sous-extension de $E$ formée des points fixes par $\sigma$, $$\begin{split}
J={\goth o}^1_{E|L}U_{[{n+1\over 2}]}({\scr L})\supset &
J_1=(1+{\goth p}_E)^1_{|L}U_{[{n+1\over 2}]}({\scr L})\\
&\quad \quad \quad \quad \quad
\supset H_1=(1+{\goth p}_E)^1_{|L}U_{[{n\over 2}]+1}({\scr L})
\end{split}$$ et $\lambda$ est obtenue à partir d’un caractère $\theta$ de $H_1$ prolongeant le caractère $\psi_{b}$ de $U_{[\frac{n}{2}]+1}(\scr L)$, comme un prolongement de l’unique représentation irréductible $\eta_{\theta}$ de $J_1$ contenant $\theta$ [@Bl1 annexe]. On note $\omega_{\lambda}$ le caractère central de $\lambda$.
On note $(\pi, \scr V)$ l’induite compacte de $J$ à $G$ de $\lambda$. C’est une représentation irréductible très cuspidale de $G$ (et toute représentation irréductible très cuspidale de $G$ s’obtient ainsi).
Construction dans le cas de niveau 0. {#construction0}
-------------------------------------
On considère le type simple $(J,\lambda)$ décrit en \[donnees2\] (a) (dont on reprend les notations) et on construit deux types simples maximaux $\tau$-invariants de $\tl G$, $(\tl J, \tl \Lambda)$ et $(\tl J, \tl \Lambda')$.
On choisit $\tl J= F^\x \tl U_{0}(\scr L)=\varpi^{\Bbb Z} \tl U_{0}(\scr L)$. La représentation $\tl \Lambda$ est un prolongement d’une représentation $\tau$-invariante $\tl \lambda$ de $\tl U_{0}(\scr L)$ triviale sur $\tl U_{1}(\scr L)$ qui se factorise en une représentation cuspidale $\ol {\tl \lambda}$ de $U_{0}(\scr L)/U_{1}(\scr L)\simeq GL_{2}(k_{0})$. On définit donc ${\tl \lambda}$ comme suit.\
On considère le caractère $\tl \theta$ de $\ell^\x$ défini par : $\tl \theta(x)=\theta(x\gamma(x)^{-1})$, $x\in \ell^\x$. Il est régulier donc associé à une représentation cuspidale $\ol {\tl \lambda}(\tl \theta)$ de $GL_{2}(k_{0})$, caractérisée par [@BH3 (6.4.1)] : $$\left\{ \begin{array}{ll}
\tr \ol {\tl \lambda}(\tl \theta)(x)= (q-1)\tl \theta(x) & \text{ si } x\in k_{0}^\x\\
\tr \ol {\tl \lambda}(\tl \theta)(xn)=-\tl \theta(x) & \text{ si } x\in k_{0}^\x, n\in N(k_{0})-\{\id\}\\
\tr \ol {\tl \lambda}(\tl \theta)(x)=-(\tl \theta(x)+\tl\theta(\gamma(x)))& \text{ si } x\in\ell^\x - k_{0}^\x.
\end{array}\right.$$ Alors $\tl \lambda$ est le relèvement de $\ol{\tl \lambda}(\tl \theta)$ à $\tl U_{0}(\scr L)$. Elle est de caractère central trivial sur ${\goth o}_{0}^\x$ et est $\tau$-invariante : $$\label{tau0}
\forall g\in \tl U_{0}(\scr L),\quad \tl \lambda (\tau(g))=\tl \lambda((\Det g)^{-1}g)=\tl \lambda(g)$$ Elle possède deux prolongements à $\tl J$ $\tau$-invariants, $\tl \Lambda$ et $\tl \Lambda'$, qui diffèrent par leur valeur en $\varpi$ : $\tl \Lambda (\varpi)= \omega_{\lambda}(-1)=-\tl \Lambda'(\varpi)$.
On note alors $$\label{construction-choix0}
\tl \pi =\ind_{\tl J}^{\tl G}\tl \Lambda \quad \text{ et } \quad \tl \pi' =\ind_{\tl J}^{\tl G}\tl \Lambda'$$ On prolonge $\tl \pi$ et $\tl \pi'$ à $\tl G\Gamma$ en imposant $\tl \Lambda(\tau)=\tl \Lambda'(\tau)=\tl \lambda(\tau)=1$ (\[tau0\]). En notant $E$ l’extension quadratique non ramifiée de $F$, dont le groupe multiplicatif $E^\x$ est plongé dans le normalisateur de ${\goth a}_{0}(\scr L)$, on a immédiatement : $$\label{tr-lambda-0}
\forall x\in E^\x, x\not \in F^\x, \quad \tr \tl \Lambda(x)=\tr \lambda(\N(x)).$$
Construction dans le cas de niveau strictement positif. {#construction}
-------------------------------------------------------
C’est l’analogue de [@Bl2] dans le cas de dimension deux.
On considère les sous-groupes ouverts compacts modulo le centre de $\widetilde G$, $\Gamma$-invariants : $$\aligned
&\widetilde H_1=(1+{\goth p}_E)\widetilde U_{[{n\over 2}]+1}({\scr L}),
\quad
\widetilde J_1=(1+{\goth p}_E)\widetilde U_{[{n+1\over 2}]}({\scr L}),
\cr
&\widetilde J_c={\goth o}_E^\times \widetilde U_{[{n+1\over 2}]}({\scr
L}), \quad
\widetilde J=E^\times \widetilde U_{[{n+1\over 2}]}({\scr L})=E^\times
\widetilde J_c.\endaligned$$ Sur $\widetilde H_1$, on considère le caractère $\tl \theta=\theta\circ \Nc$ [@Bl2 cor. 3.2]. Si $n$ est pair, il existe une unique représentation $\tl \eta$ de $\tl J_{1}$ contenant $\tl \theta$, nécessairement $\tau$-invariante. Si $n$ est impair, on pose $\tl \eta=\tl \theta$.\
Puisque la dimension de $\tl \eta$ et l’ordre de $\tl J_{c}/\tl J_{1}$ sont premiers entre eux, il existe des prolongements de $\tl \eta$, et en particulier des prolongements $\tau$-invariants de caractère central trivial sur ${\goth o}_{0}^\x$. Deux tels prolongements ont même restriction au sous-groupe $\tl J_{0}={\goth o}_{0}^\x N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J_{1}$. On distingue alors trois cas :
- [*$F$ n’est pas ramifiée sur $F_{0}$*]{} : l’extension $E$ est alors ramifiée et $\tl \eta$ est de dimension 1. De plus, $\mu$ est d’ordre 2 et les deux changements de base de $\pi$ ont même caractère central égal à $\tl{\omega_{\lambda}}$. On impose donc que le caractère central de $\tl \Lambda$ soit $\tl{\omega_{\lambda}}$. Il existe alors deux prolongements de $ \tl{\omega_{\lambda}}\tl \eta$ à $\tl J$ : ils ont même restriction $\tl \lambda$ à $\tl J_{c}$ et sont égaux à $\pm 1$ en $\varpi_{L}$.
- [*$F$ est ramifiée sur $F_{0}$ et $n$ est impair*]{} : l’extension $E$ n’est pas ramifiée sur $F$ et $\varpi^{-n}b$ est un élément de ${\goth o}_{E}^\x$ invariant par $\sigma$ : $L=F_{0}[\varpi^{-n}b]$ n’est pas ramifiée sur $F_{0}$ et $E$ est ramifiée sur $L$. Le groupe $\tl J_{0}$ est d’indice 2 dans $\tl J_{c}$ et $\tl J=F^\x\tl J_{c}$. Il existe quatre prolongements $\tau$-invariants de $\tl \eta=\tl \theta$ à $\tl J$ de caractère central trivial sur $F_{0}^\x$, deux de caractère central $\tl{\omega_{\lambda}}$ et deux de caractère central $(-1)^{\val}\tl{\omega_{\lambda}}$. Mais, si l’on considère l’autre prolongement de $\eta$ à $J$, on retrouve ces mêmes prolongements de $\tl \eta$.
- [*$F$ est ramifiée sur $F_{0}$ et $n$ est pair*]{} : l’extension $E$ n’est pas ramifiée sur $F$ et $L$ est engendrée sur $F_{0}$ par $\varpi^{-n+1}b$, une uniformisante de $E$. Donc $E$ n’est pas ramifiée sur $L$. Le groupe $\tl J_{0}$ est d’indice $q+1$ dans $\tl J_{c}$ et $\tl J=F^\x\tl J_{c}$. Il existe $2(q+1)$ prolongements $\tau$-invariants de $\tl \eta$ à $\tl J$ de caractère central trivial sur $F_{0}^\x$, une moitié de caractère central $\tl{\omega_{\lambda}}$ et l’autre de caractère central $(-1)^{\val}\tl{\omega_{\lambda}}$. Mais, si l’on considère les $(q+1)$ prolongements de $\eta$ à $J$ on retrouve les mêmes prolongements de $\tl \eta$.
Ces trois cas, auxquels on ne cesse de faire référence par la suite, se repèrent par la ramification de $F$ sur $F_{0}$ puis celle de $E$ sur $E^\sigma$. Les deux premiers cas correspondent à des paires $(E/F,\sigma)$ “paire” au sens de la définition 1.2 de [@HM] tandis que le dernier cas correspond à une paire $(E/F,\sigma)$ “impaire”, tout comme le cas de niveau 0 vu précédemment. Cette différence se reflète dans les énoncés des proposition \[stable\] et théorème \[BCdim2\].
\[construction-choix\] Il existe une unique représentation $\tl \Lambda$ de $\tl J\cdot \Gamma$ vérifiant :
- $\tl \Lambda_{\vert \tl J}$ est un prolongement $\tau$-invariant de $\tl \eta$, de caractère central $\tl{\omega_{\lambda}}$,
- pour tout $x\in E^\x$, $\tr \tl \Lambda (x\tau)=\tr \lambda(\Nc (x))$.
On note alors $$\label{construction-choix2}
\tl \pi =\ind_{\tl J}^{\tl G}\tl \Lambda_{\vert \tl J}$$ que l’on prolonge à $\tl G\Gamma$ par $\ind_{\tl J \Gamma}^{\tl G \Gamma}\tl \Lambda$.
On rappelle [@Bl2 lemme 3.3] qu’il existe un unique prolongement de $\tl \eta$ à $\tl J_{1}\Gamma$ tel que : $$\forall g\in \tl J_1, \tr \tl \eta (g\tau)= \tr \eta (\Nc (g)).$$ Soit $\tl \Lambda$ un prolongement de $\tl \eta$ de caractère central $\tl{\omega_{\lambda}}$. On le prolonge à $\tl J\cdot \Gamma$ en imposant : $\tl \Lambda(\tau)=\tl \eta (\tau)$. Dans les cas (nr-nr) et (r-r), $\tl \Lambda$ est de dimension 1 et la condition (ii) du lemme équivaut à : $\tl \Lambda(\varpi_{E})=1$ dans le cas (nr-nr) ; $ \tl \Lambda(\zeta)=1$ où $\zeta$ est une racine primitive $(q^2-1)$-ième de $1$ contenue dans $E$ dans le cas (r-r). Ceci définit un et un seul prolongement de la liste.\
Dans le cas (r-nr), seule la restriction $\tl \lambda$ de $\tl \Lambda$ à $\tl J_{c}$ importe. Le calcul de la trace de $\lambda$ sur les éléments de ${\goth o}_{E_{\vert L}}^1$ est effectué au paragraphe \[Traces\] (\[precisions\], (2)). Il existe donc un unique caractère $\xi$ de ${\goth o}_{E_{\vert L}}^1H_{1}$, prolongeant $\theta$ tel que : $$\label{tr-lambda}
\forall x\in {\goth o}_{E_{\vert L}}^1, x\not \in{\goth o}_{\vert F_{0}}^1, \tr \lambda(x)=\epsilon \xi(x),$$ où $\epsilon$ est égal à -1 si $\dim \eta =q$, 1 sinon.\
Pour le calcul de $\tr \tl \lambda$, on distingue deux cas suivant la dimension de $\tl \lambda$. Si $\tl \lambda$ est de dimension 1, $\lambda$ l’est également et le résultat est immédiat. Supposons donc $\tl \lambda$ de dimension $q$. On introduit alors le groupe ${\scr X}$ des caractères de $\tl J_{c}/{\goth o}^\x\tl J_{1}$ qui s’identifie au groupe des caractères de $k_{E}^\x$ triviaux sur $k^\x$, groupe cyclique d’ordre $q+1$ engendré par un caractère $\tau$-invariant $\tl \kappa$. Grâce aux calculs effectués en \[Tr32\] avec les notations précisées au (3) de \[precisions\], on est assuré de l’existence de $q+1$ entiers $m_{i}$ de somme $q$ tels que : $$\forall x\in {\goth o}_{E}^\x, \tr \tl \lambda(x)=\oplus_{i=0}^q m_{i}(\tl \kappa^i\tl \xi)(x).$$ Chaque composante isotypique étant $\tau$-invariante, il existe $q+1$ entiers $n_{i}$ tels que : $$\forall x\in {\goth o}_{E}^\x, \tr \tl \lambda(x\tau)=\oplus_{i=0}^q n_{i}(\tl \kappa^i\tl \xi)(x)\quad \text{ et } \quad \sum_{i=0}^q n_{i}=\tr \tl \eta (\tau)=\dim \eta.$$ On raisonne alors comme dans la démonstration de la proposition 3.5 de [@Bl2], et on obtient :
- si $\dim \eta =1$, tous les entiers $n_{i}$ sont nuls sauf un égal à $1$ ;
- si $\dim \eta =q$, tous les entiers $n_{i}$ sont égaux à $1$ sauf un qui est nul.
Il existe donc un et un seul prolongement $\tl \lambda$ de $\tl \eta$ à $\tl J_{c}$ tel que : $$\forall x\in {\goth o}_{E}^\x, x\not \in{\goth o}^\x, \tr \tl \lambda(x\tau)=\varepsilon \tl \xi(x)=\tr \lambda(\Nc (x)).$$ Ceci termine la démonstration du lemme.
Stabilité.
----------
\[stable\] Soit $\tl \pi=\ind_{\tl J}^{\tl G} \tl \Lambda$ la représentation de $GL_{2}(F)$ définie en $(\ref{construction-choix0})$ ou $( \ref{construction-choix2})$.\
(i) Elle appartient à l’image du changement de base stable si et seulement si
1. elle est de niveau $0$ ou
2. elle est de niveau strictement positif et l’extension $E$ associée est non ramifiée sur $F$ et sur $L:=E^\sigma$.
\(ii) Lorsque $\tl \pi$ appartient à l’image du changement de base labile, $\tl \pi \cdot \mu^{-1}\circ \Det$ appartient à l’image du changement de base stable.
L’assertion (ii) n’est qu’une redite de [@Ro §11.4]. Pour l’assertion (i), on distingue deux cas.\
On suppose d’abord que $\tl \Lambda$ est de dimension 1. Alors par [@BH3 §19] $\tl \pi$ est associée à la paire admissible $(E, \tl \Lambda_{\vert E^\x})$. Mais $E$ est soit d’indice de ramification 2 sur $F$ , soit non ramifiée sur $F$ mais ramifiée sur $L$. Autrement dit la paire $(E/F,\sigma)$ est paire au sens de [@HM déf. 1.2]. Comme $\tl \Lambda$ est triviale sur $L^\x$ (lemme \[construction-choix\] (ii)), $\tl \pi$ est $GL_{2}(F_{0})$-distinguée [@HM Théorème 1.1] donc $\tl \pi$ appartient au changement de base labile [@Fl2 Théorème 7].\
On suppose maintenant que $\tl \Lambda$ est de dimension au moins 2. Il n’y a plus de rapport simple entre la trace et la trace tordue de $\tl \Lambda$ ce qui rend l’emploi du critère précédent moins adapté que celui de la définition. Grâce à [@Ro], il n’y a que deux possibilités : soit le caractère tordu de $\tl \pi$, soit celui de $\tl \pi\cdot \mu\circ \Det^{-1}$ est constant sur les classes de $\tau$-conjugaison stable de $\tl G$. On peut donc déduire si $\tl \pi$ appartient à l’image du changement de base stable en évaluant son caractère tordu en deux éléments $g$ et $g'$ stablement $\tau$-conjugués et tels que $\tr \tl \pi(g\tau)\neq 0$ et $\mu\circ \Det (g^{-1}g')\neq 1$.\
On choisit $g=\zeta$ ($\zeta$ est une racine primitive $(q^2-1)$-ième de 1 contenue dans ${\goth o}_{E}^\x$) et $g'=\varpi_{L}g$ où $\varpi_{L}=\zeta^{\frac{q+1}{2}}\varpi$. Tous deux sont de norme cyclique $x=\zeta^{1-q}$ et $\mu\circ \Det (\varpi_{L})=\mu\circ N_{L_{\vert F_{0}}}(\varpi_{L})=-1$. Notons que $x\in \tl U_{0}(\scr L)$ est très régulier : $\forall h\in \tl G, \quad h^{-1}xh\in \tl U_{0}(\scr L) \impl h\in F^\x \tl U_{0}(\scr L)$.\
Par conséquent, si $\tl \pi$ est de niveau 0, la formule de Mackey donne : $$\begin{split}
&\tr \tl \pi(\zeta\tau)=\sum_{\begin{subarray}{c} h\in \tl G/\tl J\\ h^{-1}\zeta\tau(h)\in \tl J\end{subarray}} \tr \tl \Lambda (h^{-1}\zeta \tau(h)\tau)= \tr \tl \Lambda (\zeta\tau)=\tr \lambda(x) \\
\text{ et de m\^eme, } & \tr \tl \pi(\zeta\varpi_{L}\tau)=\omega_{\lambda}(-1)\tr \tl \pi(\zeta^\frac{q+3}{2}\tau)=\omega_{\lambda}(-1)\tr \lambda(-x)=\tr \tl \pi(\zeta\tau).
\end{split}$$ D’après la définition de $\lambda$ (cf. \[donnees2\] (a)), $\tr \lambda(x)$ n’est pas nul dès que le caractère $\theta$ associé n’est pas d’ordre 4. Si $\theta$ est d’ordre 4 ($q\equiv -1 \mod 4$), on refait le même raisonnement avec $g=\zeta^2$ sachant que $\tr \tl \pi(\zeta^2\tau)\neq0$.\
Quand $\tl \pi$ est de niveau 0, elle appartient à l’image du changement de base stable.
Supposons maintenant que $\tl \pi$ est de niveau strictement positif. Notons $\tl K(\scr L)$ le normalisateur dans $\tl G$ de l’ordre associé à $\scr L$, $\tl K(\scr L)=F^\x\tl U_{0}(\scr L)$, et $\tl \sigma=\ind_{\tl J\Gamma}^{\tl K(\scr L)\Gamma}\tl \Lambda$. Alors $\tl \pi$ est $\ind_{\tl K(\scr L)\Gamma}^{\tl G\Gamma}\tl \sigma$ et en appliquant deux fois la formule de Mackey, on a : $$\label{tr-tlpi(iii)}
\begin{split}
&\tr \tl \pi(\zeta\tau)=\tr \tl \sigma(\zeta\tau)=\sum_{\begin{subarray}{c} h\in \tl U_{0}(\scr L)/\tl J_{c}\\ h^{-1}\zeta\tau(h)\in \tl J_{c}\end{subarray}} \tr \tl \Lambda (h^{-1}\zeta \tau(h)\tau)\\ \text{ et } \quad
&\tr \tl \pi(\zeta\varpi_{L}\tau)=\omega_{\lambda}(-1)\sum_{\begin{subarray}{c} h\in \tl U_{0}(\scr L)/\tl J_{c}\\ h^{-1}\zeta^\frac{q+3}{2}\tau(h)\in \tl J_{c}\end{subarray}} \tr \tl \Lambda (h^{-1}\zeta^\frac{q+3}{2} \tau(h)\tau)
\end{split}$$ On reprend alors le raisonnement de [@Bl2 §4.7] en remarquant que $\Gal (E/F_{0})$ est abélien (car d’ordre 4). On obtient : $$\begin{split}
\tr \tl \pi(\zeta\tau)&=\sum_{\gamma\in \Gal (E/F)}\tr\tl \lambda(\gamma(\zeta)\tau) =\tr \lambda(x)+\tr \lambda(x^{-1})\\&=\omega_{\lambda}(-1)(\tr \lambda(-x)+\tr\lambda(-x^{-1}))=\tr \tl \pi(\zeta\varpi_{L}\tau)
\end{split}$$ Or $\lambda$ est associé à un caractère $\xi$ par (\[tr-lambda\]) : ou bien $\xi(x)+\xi(x^{-1})\neq 0$, ou bien $\xi(x^2)+\xi(x^{-2})\neq 0$. On procède comme dans le cas de niveau 0.
Description des changements de base.
------------------------------------
\[BCdim2\] Soit $(\pi, {\scr V})$ une représentation irréductible très cuspidale de $G$ et $(J,\lambda)$ un type simple maximal définissant $\pi$ comme au paragraphe $\ref{donnees2}$, (a) ou (b). On note $\tl \pi=\ind_{\tl J}^{\tl G}\tl \Lambda$ la représentation irréductible cuspidale de $\tl G$ définie en $(\ref{construction-choix0})$ ou $(\ref{construction-choix2})$ selon que $\pi$ est de niveau $0$ ou non.\
(i) [$F$ n’est pas ramifiée sur $F_{0}$ :]{} l’image de $\pi$ par le changement de base stable est $\tl \pi\cdot \mu^{-1}\!\circ\! \Det $ tandis que celle par le changement de base labile est $\tl \pi$.\
(ii) [$F$ est ramifiée sur $F_{0}$ et $E$ est ramifiée sur $E^\sigma$ :]{} on note $\chi$ le caractère d’ordre $2$ de $J$ trivial sur $J_{1}$ et on pose : $\pi'= \ind_{J}^G \lambda\cdot \chi\;$ et $\; \tl \pi'=\ind_{\tl J}^{\tl G}\tl \Lambda\cdot \tl \chi$.\
L’image de $\pi$ par le changement de base stable est $\tl \pi\cdot \mu^{-1}\circ \Det $ si $q\equiv 1 \mod 4$, $\tl \pi'\cdot \mu^{-1}\circ \Det $ sinon. L’image de $\tl \pi$ par le changement de base labile est alors $\tl \pi$ si $q\equiv 1 \mod 4$ et $\tl \pi'$ sinon.\
(iii) [$F$ est ramifiée sur $F_{0}$ et $E$ n’est pas ramifiée sur $E^\sigma$ (y compris cas de niveau $0$) :]{} l’image de $\pi$ par le changement de base stable est $\tl \pi$, celle par le changement de base labile $\tl \pi\cdot \mu\circ \Det$.
Suit la démonstration de ce théorème.
{#section-3}
Grâce à [@Ro §11.4] il suffit d’établir le résultat pour le changement de base stable. Notons $\tl \pi_{st}$ la représentation de $\tl G$ préssentie être l’image de $\pi$ par le changement de base stable dans le théorème et montrons qu’elle vérifie bien l’identité de caractères décrivant ce dernier [@Ro §§ 4.11, 12.5 et 11.4], c’est-à-dire : pour tout $g\in \tl G$ dont la norme cyclique $\Nc (g)$ est régulière elliptique et tout $x\in \Nc (g)$, $$\label{id}
\tr \tl \pi_{st} (g\tau)=c_\tau(\tl \pi_{st} ) \tr
\pi (x),$$ où $\tl \pi_{st} (\tau)$ est un opérateur d’entrelacement d’ordre 2 entre $\tl \pi_{st} $ et $\tl \pi_{st} ^\tau$ et $c_\tau(\tl \pi_{st} )$ un signe ne dépendant que du choix de $\tl \pi_{st} (\tau)$. Dans la construction précédente, on a choisi $\tl \pi_{st}(\tau)$ pour que $c_\tau(\tl \pi_{st} )$ soit égal à 1.
Soit $g\in \tl G$. On suppose que $x=\N(g)$ est un élément de $G$, régulier et elliptique. On note $T(F_{0})$ le centralisateur de $x$ dans $G$. C’est un tore compact de $G$ et $T(F)$ est $\tau$-invariant et isomorphe soit au groupe multiplicatif d’une extension quadratique $E_{x}$ de $F$, soit à $F^\x \x F^\x$.\
La formule de Mackey fournit une expression des traces de $\pi (x)$ et $\tl \pi (g\tau)$ : $$\label{tr1}
\tr \pi (x)= \sum_{\smallmatrix y\in G/J \cr y^{-1}xy\in
J\cr\endmatrix}
\tr \lambda(y^{-1}xy) \quad \text{ et } \quad
\tr \tl \pi_{st} (g\tau)=\sum_{\smallmatrix h\in \tl G/\tl J\cr
h^{-1}g\tau(h)\in \tl J\cr\endmatrix} \tr \tl \Lambda_{st} (h^{-1}g\tau(h)\tau).$$
{#section-4}
On suppose d’abord que $\pi$ est de niveau strictement positif. La démonstration est semblable à celle du théorème 3.7 de [@Bl2].\
On introduit le sous-groupe distingué de $\tl G$, noté $\tl G^+$, défini comme le noyau du caractère $\mu\circ \Det$. Pour tout sous-groupe $H$ de $\tl G$, on abrège $H\cap \tl G^+$ en $H^+$.
Soient $g\in \tl G$ et $x\in G$ comme ci-dessus. On note ${\scr Cl}_{\tau}^{st}(g)$ la classe de $\tau$-conjugaison stable de $g$ et ${\scr Cl}^{st}(x)$ la classe de conjugaison stable de $x$.\
(i) Le groupe $\tl G^+$ est réunion de classes de $\tau$-conjugaison.\
(ii) Si $\mu\circ \Det$ est d’ordre $2$, $\tl G^+$ rencontre la classe de $\tau$-conjugaison stable de $g$ et ${\scr Cl}_{\tau}^{st}(g)^+$ contient la moitié des classes de $\tau$-conjugaison contenues dans $ {\scr Cl}_{\tau}^{st}(g)$.\
(iii) Si $\mu\circ \Det$ est d’ordre $4$, ou bien ${\scr Cl}_{\tau}^{st}(g)^+$ n’est pas vide et alors ${\scr Cl}_{\tau}^{st}(g)^+$ contient la moitié des classes de $\tau$-conjugaison contenues dans $ {\scr Cl}_{\tau}^{st}(g)$ ; ou bien ${\scr Cl}_{\tau}^{st}(g)$ ne rencontre pas $\tl G^+$, et alors :
$ {\scr Cl}_{\tau}^{st}(g)\cap \tl J=\emptyset\quad $ et $\quad {\scr Cl}^{st}(x)\cap J=\emptyset$.\
(iv) Chaque classe de $\tau$-conjugaison dans ${\scr Cl}_{\tau}^{st}(g)^+$ contient deux classes de $\tau$-$\tl G^+$-conjugaison.
Pour la première assertion, il suffit de remarquer que $\mu\circ \Det$ est constant sur les classes de $\tau$-conjugaison.\
Pour la suite, on sait que les classes de $\tau$-conjugaison de $g$ dans $ {\scr Cl}_{\tau}^{st}(g)$ sont paramétrées par $H^1(\Gamma, T(F))$ : à un cocycle $c$, on associe la classe de $\tau$-conjugaison de $c(\tau)g$.\
Si $T(F)\simeq E_{x}^\x$, il y a deux classes de $\tau$-conjugaison, celle de $g$ et celle de $ag$ où $a\in L_{x}^\x, a\not \in N_{{E_{x}}_{\vert L_{x}}}(E_{x}^\x)$ et $L_{x}={E_{x}^\sigma}$. Notons que $L_{x}$ est une extension quadratique de $F_{0}$, nécessairement distincte de $F$. Mais alors, $\Det a$ appartient à $N_{{L_{x}}_{\vert F_{0}}}(L_{x}^\x)$ sans être un carré de $F_{0}$ donc $\mu\circ \Det (a)$ est égal à $-1$. Ainsi, $\mu\circ \Det$ prend deux valeurs opposées sur les deux classes de $\tau$-conjugaison.\
Si $T(F)\simeq F^\x \x F^\x$, il y a quatre classes de $\tau$-conjugaison, celles de $g$, $(1, \alpha_{0})g$, $(\alpha_{0}, 1)g$ et $(\alpha_{0}, \alpha_{0})g$. Encore une fois $\mu\circ \Det$ prend deux valeurs opposées.\
Par conséquent, si $\mu \circ \Det$ est d’ordre 2, la moitié des classes de $\tau$-conjugaison dans $ {\scr Cl}_{\tau}^{st}(g)$ sont contenues dans $\tl G^+$.\
Supposons que $\mu\circ \Det$ est d’ordre 4. Ou bien $\mu\circ\Det (g)\in \{ \pm 1\}$ et la moitié des classes de $\tau$-conjugaison dans $ {\scr Cl}_{\tau}^{st}(g)$ sont contenues dans $\tl G^+$ ; ou bien $\mu\circ \Det (g)\in \{ \pm i\}$ et ${\scr Cl}_{\tau}^{st}(g)^+$ est vide. On termine la démonstration en remarquant que : $\mu\circ \Det (\tl J)=\{ \pm 1\}$ et $\Det (J)\subset 1+{\goth p}$. En effet, l’hypothèse sur $\Det g$ implique que celui-ci est de valuation impaire et par suite, que $\Det x$ appartient à $-1+{\goth p}$.\
(iv) Si $g_{1}$ et $g_{2}$ sont deux éléments de $\tl G^+$ $\tau$-$\tl G$-conjugués par deux éléments $h_{1}$ et $h_{2}$ alors $h_{1}h_{2}^{-1}$ appartient au $\tau$-centralisateur de $g_{1}$, en particulier la norme de son déterminant est 1. Par conséquent, si $\mu$ est d’ordre 2, $h_{1}$ et $h_{2}$ diffèrent d’un élément de $\tl G^+$. Si $\mu$ est d’ordre 4, le $\tau$-centralisateur de $g_{1}$ contient un élément $a$ tel que $\mu\circ \Det (a)=-1$ (cf. (ii) et (iii)) et $h_{1}$ et $h_{2}$ diffèrent d’un élément de $\tl G^+$ ou de $\tl G^+a$. Dans les deux cas, il y a deux classes de $\tau$-$\tl G^+$-conjugaison dans une classe de $\tau$-conjugaison.
Dans le cas où ${\scr Cl}_{\tau}^{st}(g)^+$ est vide, l’égalité (\[id\]) est satisfaite puisque les deux sommes de (\[tr1\]) sont nulles. On peut donc supposer que ${\scr Cl}_{\tau}^{st}(g)^+$ n’est pas vide et, quitte à changer $g$ dans sa classe de $\tau$-conjugaison stable sans changer de $x$, que $g\in \tl G^+$.
Notons ${\scr C}_{J}^{st}(x)$ l’ensemble des classes de $J$-conjugaison contenues dans ${\scr Cl}^{st}(x)\cap J$ et ${\scr C}_{\tau,\tl J^+}^{st}(g)$ celui des classes de $\tau$-$\tl J^+$-conjugaison contenues dans ${\scr Cl}_{\tau}^{st}(g)\cap \tl J^+$. Notons également $n(x)$ (resp. $\tl n(g)$) le nombre de classes de conjugaison (resp. $\tau$-conjugaison) contenues dans ${\scr Cl}^{st}(x)$ (resp. ${\scr Cl}_{\tau}^{st}(g)$). En suivant le raisonnement de [@Bl2 §4.2], on a : $$\label{tr2}
\begin{aligned}
&\tr \pi (x)=\frac{1}{n(x)}\sum_{x'\in
{\scr C}_{J}^{st}(x)} c(x')\tr \lambda(x') \\
Ê\text{ et } \quad&
\tr \tl \pi_{st} (g\tau)=\frac{1}{\tl n(g)} \sum_{g'\in {\scr C}_{\tau,\tl J^+}^{st}(g)} \tl
c(g')\tr \tl \Lambda_{st}(g'\tau)
\end{aligned}$$ où $\quad c(x')=[T(F_0) : T(F_0)\cap yJy^{-1}] $ si $x'=y^{-1}xy,\,\, y\in G$\
et $\quad\tl c(g')=[T(F_0) : T(F_0)\cap h\tl J{h}^{-1}] $ si $g'=h^{-1}g\tau (h),\,\, h\in \tl G$.
{#section-5}
On introduit les sous-groupes $J'$ et $\tl J'$ de $J$ et $\tl J^+$ respectivement, en distinguant deux cas :
- [*$F/F_{0}$ est non ramifiée ou $q\equiv 1\mod 4$*]{}. Dans ce cas, $J'=F_{\vert F_{0}}^1J_{1}$ et $\tl J'=F^\x\tl J_{1}$ ;
- [*$F/F_{0}$ est ramifiée et $q\equiv -1 \mod 4$*]{}. Alors $J'=J_{1}$ et $\tl J'=F_{0}^\x\tl J_{1}$.
Dans les cas où $J$ est plus grand que $J'$ ou $\tl J^+$ plus grand que $N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'$, c’est-à-dire lorsqu’on se trouve dans le cas (r-nr) ou le cas (r-r) et $q\equiv -1 \mod 4$, on considére les représentations par “paquets” fabriqués de la façon suivante.
Du côté de $G$, on considère tous les prolongements à $J$ de la restriction à $J'$ de $\lambda$. Ils sont au nombre de $d=[J:J']$ et de la forme $\lambda \otimes \kappa^r$, $0\leq r\leq d-1$, où $\kappa$ est un caractère de $J$, trivial sur $J'$ et d’ordre $d$ (qui s’identifie à un caractère du groupe engendré par $\zeta^{q-1}$ dans le cas (r-nr), -1 dans le cas (r-r)).\
Notons $\tl \kappa$ le caractère de $\tl J$ que l’on obtient sur $\tl J$ en relevant le caractère $\kappa\circ \N$ de $\tl J/N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'$. On remplace $\pi$ et $\tl \pi_{st}$ par : $$\begin{aligned}
\pi=\oplus_{r=0}^{d-1}\pi_{r} \quad& \text{ o\`u } \quad \pi_{r}=\ind_{J}^G \lambda\otimes \kappa^r \\
\tl \pi_{st}=\oplus_{r=0}^{d-1}\tl \pi_{r} \quad& \text{ o\`u } \quad \tl \pi_{r}=\ind_{\tl J}^{\tl G}\tl \Lambda_{st}\otimes\tl \kappa^r
\end{aligned}$$ Alors, en sommant les expressions (\[tr2\]) des caractères des $\pi_{r}$ d’une part, et des $\tl \pi_{r}$ d’autre part, on obtient : $$\begin{aligned}
&\tr \pi (x)=\frac{1}{n(x)}\sum_{x'\in
{\scr C}_{J}^{st}(x)\cap J'}d c(x')\tr \lambda(x')=\frac{1}{n(x)}\sum_{x'\in
{\scr C}_{J'}^{st}(x)}\frac{d}{d(x')} c(x')\tr \lambda(x')Ê\\
&
\begin{aligned}\text{ et} \quad\tr \tl \pi_{st} (g\tau)=
&\frac{1}{\tl n(g)} \sum_{g'\in {\scr C}_{\tau,\tl J^+}^{st}(g)\cap N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'} d\tl c(g')\tr \tl \Lambda_{st}(g'\tau)\\
=&\frac{1}{\tl n(g)} \sum_{g'\in {\scr C}_{\tau,N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'}^{st}(g)}\frac{d}{\tl d(g')}\tl c(g')\tr \tl \Lambda_{st}(g'\tau)\end{aligned}\end{aligned}$$ où $d(x')$ est le nombre de classes de $J'$-conjugaison dans l’intersection de la classe de $J$-conjugaison de $x'$ avec $J'$ et $\tl d(g')$ celui des classes de $\tau$-$N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'$-conjugaison dans l’intersection de la classe de $\tau$-$\tl J^+$-conjugaison de $g'$ avec $N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'$.\
Remarquons que les nombres $d(x')$ et $\tl d(g')$ sont invariants quand on conjugue $x'$ par un élément de $J$, respectivement $\tau$-conjugue $g'$ par un élément de $N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'$.
On uniformise les notations en posant $\tl d(g')=d(x')=1$ dans les cas non mentionnés dans ce paragraphe.
{#section-6}
L’étape suivante consiste à remplacer $N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'$ par $\tl J'$. Elle ne concerne que le cas (r-r).
Dans le cas (r-r), $$\label{sansnom}
\sum_{g'\in {\scr C}_{\tau,N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'}^{st}(g)}\frac{d}{\tl d(g')} \tl c(g')\tr \tl \Lambda_{st}(g'\tau)=
\sum_{g'\in {\scr C}_{\tau,\tl J'}^{st}(g)}\frac{d}{\tl d(g')} \tl c(g')\tr \tl \Lambda_{st}(g'\tau).$$
Considérons l’application $\Psi$ de ${\scr C}_{\tau,\tl J'}^{st}(g)$ dans ${\scr C}_{\tau,N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'}^{st}(g)$ qui à la classe de $\tau$-$\tl J'$-conjugaison de $g'\in {\scr Cl}_{\tau}^{st}(g)\cap \tl J'$ associe la classe de $\tau$-$N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'$-conjugaison de $g'$. Etudions d’abord son image.\
Soit $g'\in N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'$. L’ensemble des classes à droite modulo $\tl J'$ dans $N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'$ est représenté par $\{ \zeta^{2r}, r\in \[ 0,\frac{q-1}{2}\]\}$. Il existe donc $r\in \[ 0,\frac{q-1}{2}\]$ et $h\in \tl J'$ tels que : $g'=\zeta^{2r} h=\zeta^r(\zeta^r h\zeta^{-r})\tau(\zeta^{-r})$. L’élément $\zeta^r h\zeta^{-r}$ appartient à $\tl J'$ tandis que $\zeta^r$ appartient à $N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'$ si et seulement si $r$ est pair. Donc si $r$ est pair, $g'$ appartient à $\im \Psi$. Réciproquement, les classes de $\tau$-$\tl J'$-conjugaison dans la classe de $\tau$-$N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'$-conjugaison de $g'$ ont un représentant de la forme $\zeta^{-2s}g'\tau(\zeta^{2s})=\zeta^{2r-4s}(\zeta^{2s}h\zeta^{-2s})$ pour un $s\in \[ 0,\frac{q-1}{2}\]$. L’une d’entre elles est contenue dans $\tl J'$ si et seulement il existe $s\in \[ 0,\frac{q-1}{2}\]$ tel que $\zeta^{2r-4s}\in F_{0}^\x$, c’est-à-dire $r-2s\equiv 0 \mod (\frac{q+1}{2})$.\
Si $q\equiv 1 \mod 4$, $g'$ est donc $\tau$-$N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'$-conjugué à un élément de $\tl J'$ : $\Psi$ est surjective.\
Si $q\equiv -1 \mod 4$, $g'$ est $\tau$-$N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'$-conjugué à un élément de $\tl J'$ si et seulement si $r$ est pair.\
Etudions les fibres de $\Psi$. Soient $g_{1}, g_{2}$ deux éléments de $\tl J'$ qui sont $\tau$-$N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'$-conjugués. Quitte à $\tau$-$\tl J'$-conjugué $g_{1}$, on peut supposer qu’il existe $r\in\[ 0,\frac{q-1}{2}\]$ tel que $\zeta^{2r}$ $\tau$-conjugue $g_{2}$ en $g_{1}$. Mais alors $\zeta^{-4r}$ appartient à $\tl J'$, c’est-à-dire $\frac{q+1}{2}$ divise $2r$.\
Si $q\equiv 1 \mod 4$, $\frac{q+1}{2}$ divise $r$ et $g_{1}=g_{2}$ : $\Psi$ est injective. En remarquant que $\Psi$ conserve la valeur de $\frac{d}{\tl d(g')}\tl c(g')\tr \tl \Lambda_{st}(g'\tau)$, on établit l’égalité (\[sansnom\]).\
Si $q\equiv -1 \mod 4$, $\frac{q+1}{4}$ divise $r$ et $g_{2}$ est égal à $g_{1}$ ou $\zeta^{-\frac{q+1}{2}}g_{1}\tau(\zeta^{\frac{q+1}{2}})$ : les fibres de $\Psi$ sont de cardinal 2.\
Dans ce cas, on remarque que l’application $g' \mapsto \zeta^{-1}g'\tau(\zeta)$ définit une bijection de $\im \Psi$ sur son complémentaire dans ${\scr C}_{\tau,N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'}^{st}(g)$ qui préserve la valeur de $\frac{d}{\tl d(g')}\tl c(g')\tr \tl \Lambda_{st}(g'\tau)$. On retrouve donc l’égalité (\[sansnom\]).
{#section-7}
Il reste à étudier la restriction de l’application $\N$ à ${\scr C}_{\tau,\tl J'}^{st}(g)$.
\[cle\] Soient $g$ et $x$ comme précédemment.\
(i) Soit $g'\in \tl J'$. Il existe $h\in \tl J'$ tel que $\N(h^{-1}g'\tau(h))\in J'$.\
(ii) Soient $x_{1}, x_{2}$ deux éléments de $J'$ conjugués sous $\tl J'$. Alors ils sont conjugués sous $J'$.\
(iii) Soit $x'\in {\scr Cl}^{st}(x)\cap J'$. Il existe $g'\in {\scr Cl}_{\tau}^{st}(g)\cap \tl J'$ tel que : $\N (g')=x'$. Il en existe deux à $\tau$-$\tl J'$-conjugaison près dans le cas 1 et $4$ dans le cas 2.
On se place dans le cas 1.\
(i) On a $g'=zj$ où $z\in F^\x$, $j\in \tl J_{1}$. Par le lemme 3.1 de [@Bl2], il existe $h\in \tl J_{1}$ tel que $\N(h^{-1}j\tau(h))\in J_{1}$. Alors $\N(h^{-1}g'\tau(h))\in J'$.\
(ii) Notons pour $i=1$ ou 2, $x_{i}=u_{i}j_{i}$ où $u_{i}\in F_{\vert F_{0}}^1$, $j_{i}\in J_{1}$. Soit $h\in \tl J'$ tel que $x_{2}=h^{-1}x_{1}h$. On peut prendre $h\in \tl J_{1}$. Alors : $$x_{2}=h^{-1}x_{1}h \ssi u_{1}^{-1}u_{2}=h^{-1}j_{1}hj_{2}^{-1} \in F_{\vert F_{0}}^1\cap \tl J_{1}\ssi u_{1}=u_{2} \text{ et } j_{2}=h^{-1}j_{1}h.$$ Par le lemme 3.1(ii) déjà cité, on peut choisir $h\in J_{1}$.\
(iii) Notons $x'=uj$ avec $u\in F_{\vert F_{0}}^1$ et $j\in J_{1}$. Par le théorème de Hilbert 90 et encore le lemme 3.1(ii), il existe $g'=z\tl j$, $z\in F^\x$ et $\tl j\in \tl J_{1}$ tel que : $\N(g')=x'$. De plus, $g'\in {\scr Cl}_{\tau}^{st}(g)$.\
Soient maintenant $g'_{1}, g'_{2}\in {\scr Cl}_{\tau}^{st}(g)\cap \tl J'$ tels que : $\N(g_{1})=\N(g_{2})=x'$. Alors $g_{i}$, $i=1,2$, s’écrit $z_{i}j_{i}$ où $z_{i}\in F^\x$ et $j_{i}\in \tl J_{1}$ tel que $j_{i}\tau(j_{i})\in J_{1}$. L’égalité précédente donne alors : $$\frac{z_{1}\ol z_{2}}{\ol z_{1}z_{2}}=j_{2}\tau(j_{2})(j_{1}\tau(j_{1}))^{-1} \ssi z_{1}\ol z_{2}\in F_{0}^\x \text{ et } j_{1}\tau(j_{1})=j_{2}\tau(j_{2})$$ quitte à multiplier $z_{1}$ et $j_{1}$ par un élément de $1+{\goth p}$ sans changer leur produit. Encore une fois, $j_{1}$ et $j_{2}$ sont donc $\tau$-$\tl J_{1}$-conjugués. Alors $g_{1}$ et $g_{2}$ sont $\tau$-$\tl J'$-conjugués si et seulement si $z_{1}\ol z_{2}$ est une norme de $F^\x$ dans $F_{0}$. A $\tau$-$\tl J'$-conjugaison près, on a deux $g'\in \tl J'$ de norme cyclique $x'$.
On se place dans le cas 2. Les assertions (i) et (ii) sont immédiates. Soit $x'\in {\scr Cl}^{st}(x)\cap J'$. Alors il existe $j\in \tl J_{1}$, unique à $\tau$-$\tl J_{1}$-conjugaison près, tel que $\N (j)=x'$. Mais pour tout $u\in F_{0}^\x$, $uj$ est de norme cyclique $x'$ et $uj$ est $\tau$-$\tl J'$-conjugué à $j$ si et seulement si $u$ est un carré dans $F_{0}^\x$.\
En effet, si $u$ est un carré, il est clair que $j$ et $uj$ sont $\tau$-$\tl J'$-conjugués. Réciproquement, on peut supposer que $u$ est 1, $\varpi_{0}, \zeta^{q+1}$ ou $\varpi_{0}\zeta^{q+1}$. Si $j$ et $uj$ sont $\tau$-$\tl J'$-conjugués, il existe $v\in F_{0}^\x$ et $j'\in \tl J_{1}$ tels que $uj=v^{-1}({j'}^{-1}j\tau(j'))\tau(v)$. Mais alors, $uv^2\in \tl J_{1}\cap F_{0}^\x=1+{\goth p}_{0}$ donc $u=1$.
{#section-8}
Etablissons l’égalité (\[id\]). Si ${\scr Cl}_{\tau}^{st}(g)\cap \tl J'$ est vide, ${\scr Cl}^{st}(x)$ ne rencontre pas $J'$ (lemme \[cle\], (i)) donc les deux sommes de (\[tr2\]) sont nulles. Supposons maintenant que $g\in \tl J'$ et $x\in J'$ (lemme \[cle\], (i)). Alors l’application $\Nc$ induit une surjection de ${\scr C}_{\tau,\tl J'}^{st}(g)$ sur ${\scr C}_{J'}^{st}(x)$ dont les fibres sont de cardinal 2 dans le cas 1, 4 dans le cas 2 (lemme \[cle\]). De plus, $\tr \tl \Lambda$ est constant sur ces fibres et l’identité s’obtient comme dans [@Bl2 §4.5 et 4.6] en ajoutant le résultat suivant :
Soit $g'\in \tl J'$ tel que $x':=\N (g') \in J'$. Alors $\tl d(g')$ est égal à $d(x')$ dans le cas 1, $2d(x')$ dans le cas 2.
Le résultat est immédiat dans les cas (nr-nr) et (r-r) quand $q\equiv 1 \mod 4$.\
Dans le cas (r-r) quand $q\equiv -1 \mod 4$, $J$ est le produit $\{\pm 1\} J'$ donc $d(x')$ vaut 1. D’autre part, $N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'$ est un sous-groupe d’indice 2 de $\tl J^+$ donc la classe de $\tau$-$\tl J^+$-conjugaison de $g'$ contient au plus deux classes de $\tau$-$N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'$-conjugaison, à savoir celles de $g'$ et de $-\zeta^{-1}g'\tau(\zeta)$. Remarquons qu’il existe un entier $r$ tel que $g'\in \zeta^{(q+1)r}\varpi_{0}^{\Bbb Z}\tl J_{1}$ et que toute la classe de $\tau$-$N_{E_{\vert L}}({\goth o}_{E}^\x)\tl J'$-conjugaison de $g'$ est contenue dans $\displaystyle \bigcup_{s\equiv 0 [4]}\zeta^s\varpi_{0}^{\Bbb Z}\tl J_{1}$. Mais, $-\zeta^{-1}g'\tau(\zeta)$ appartient à $\zeta^s\varpi_{0}^{\Bbb Z}\tl J_{1}$ où $s=\frac{q^2-1}{2}-2+(q+1)r\equiv 2 \mod 4$. Par suite : $\tl d(g')=2=2d(x')$.\
Dans le cas (r-nr), $g'$ s’écrit $zg'_{1}$ avec $z\in (F^\x)^+$ et $g'_{1} \in \tl J_{1}$. Alors $x'=\frac{z}{\ol z}x'_{1}$ où $x'_{1}=\N (g'_{1})\in J_{1}$. Puisque $z$ et $\frac{z}{\ol z}$ sont centraux dans $\tl J^+$ et $J$ respectivement, on a : $\tl d(g')=\tl d(g'_{1})$ et $d(x')=d(x'_{1})$. On peut donc supposer que $g'\in \tl J_{1}$ et $x'\in J_{1}$. Sous cette hypothèse, $\tl J'$ contient toute la classe de $\tau$-$\tl J^+$-conjugaison de $g'$. D’après le lemme \[cle\], $\N$ induit une application de l’ensemble des classes de $\tau$-$\tl J'$-conjugaison contenues dans ${\scr Cl}_{\tau-\tl J^+}(g')$ dans l’ensemble des classes de $J'$-conjugaison contenues dans ${\scr Cl}_{J}(x')$. Cette application est clairement surjective car tout $x''\in {\scr Cl}_{J}(x')$ s’écrit $j^{-1}x'j$, $j\in J$, et est l’image par $\N$ de $g'':=j^{-1}g'\tau(j)$, élément appartenant à ${\scr Cl}_{\tau-\tl J^+}(g')\cap \tl J_{1}$.\
Etudions la fibre en $x''$. D’après le lemme \[cle\], elle contient au plus 2 classes si $q\equiv 1 \mod 4$, 4 si $q\equiv -1 \mod 4$ et, d’après la démonstration de ce même lemme, ces classes de $\tau$-$\tl J'$-conjugaison sont représentées par : $g''$ et $\alpha_{0}g''$ si $q\equiv 1 \mod 4$ ; $ug''$ où $u$ parcourt un ensemble de représentants des classes de $F_{0}^\x$ modulo ses carrés. Mais, une de ces classes est contenue dans ${\scr Cl}_{\tau-\tl J^+}(g')$ si et seulement si son représentant $zg''$ est $\tau$-$\tl J^+$-conjugué à $g''$, c’est-à-dire il existe $h=(\varpi \zeta)^r \zeta^{2s}y$, $r,s\in \Bbb Z$ et $y\in \tl J_{1}$ tel que : $$\begin{aligned}
&zg''=((\varpi \zeta)^r \zeta^{2s}y)^{-1} g'' \tau((\varpi \zeta)^r \zeta^{2s}y) \ssi z=\varpi_{0}^{-r}\zeta^{-(q+1)(r+2s)}\mod \tl J_{1}\\ \ssi &z\!=\varpi_{0}^{-r}\zeta^{-(q+1)(r+2s)}\!\! \!\mod 1\!+\!{\goth p}_{0}\! \ssi
\!\begin{cases}
r=0 \text{ et } z=1\!&\!\text{si }\! q\equiv 1\!\mod 4\\
z=1\! \text{ ou }\! z=\varpi_{0}\zeta^{q+1}\!\! &\!\!\text{si }\! q\!\equiv \!-1\!\!\mod 4\\
\end{cases}
\end{aligned}$$ Ainsi, la fibre en $x''$ est de cardinal 1 si $q\equiv 1 \mod 4$, 2 sinon.
A ce point, sont démontrées les assertions du théorème \[BCdim2\] correspondant aux cas (nr-nr) et (r-r) lorsque $q$ est congru à 1 modulo 4.
{#section-9}
Dans les autres cas de niveau strictement positif, chaque représentation $\tl \pi_{r}$ est l’image par le changement de base stable d’une représentation $\pi_{r}''$ [@Ro §11.4.1] caractérisée par l’identité (\[id\]). D’après ce qui précède, pour tout $x\in G$ régulier, elliptique, de la forme $\N (g)$ pour un $g\in \tl G$, $$\sum _{i=0}^{d-1} \tr \pi ''_i(x)=\sum _{i=0}^{d-1}\tr \pi_i(x).$$ Alors, comme au paragraphe 4.7 de [@Bl2], on obtient que les images des représentations $\pi_{r}, 0\leq r\leq d-1$, par le changement de base stable sont les représentations $\tl \pi_{r}$, $0\leq r\leq d-1$.\
Dans le cas (r-r) où $q$ est congru à -1 modulo 4, les deux représentations $\tl \pi_{0}$ et $\tl \pi_{1}$ ne sont autres que $\tl\pi\cdot \mu^{-1}\circ \Det $ et $\tl \pi'\cdot \mu^{-1}\circ \Det $ et seule $\tl \pi'\cdot \mu^{-1}\circ \Det $ a pour caractère central $\tl \omega_{\pi}$. Elle est donc l’image de $\pi$ par le changement de base stable.\
Dans le cas (r-nr), lorsque $q$ est congru à -1 modulo 4, l’argument du caractère central permet d’affirmer que l’image de $\pi$ par le changement de base stable appartient à $\{ \tl \pi_{r}, r\equiv 0 \mod 2\}$. Sans hypothèse sur $q$, pour montrer que l’image de $\pi$ est bien $\tl \pi_{0}$, il suffit d’évaluer leurs caractères en des éléments bien choisis. Comme dans la démonstration de la proposition \[stable\], on choisit $g=\zeta$ ou $\zeta^2$. Alors, par (\[tr-tlpi(iii)\]) et en poursuivant le raisonnement comme en [@Bl2 §4.7] : $$\tr \tl \pi(g\tau)=\tr \lambda(x)+\tr \lambda(x^{-1}) =\tr \pi(x)\neq 0.$$ Ceci termine la démonstration du théorème dans le cas des représentations de niveau strictement positif.
{#section-10}
Considérons le cas où $\pi$ est de niveau 0. La représentation $\tl \pi$ est une représentation $\tau$-invariante, à caractère central trivial sur $F_{0}$ et de caractère stable : elle appartient donc à l’image du changement de base stable et son antécédent est nécessairement une représentation de niveau 0 décrite par \[donnees2\] (a), conséquence de tout ce qui précède. Ces dernières se distinguent par la restriction de leurs caractères en les éléments de $G$ dont le groupe des points sur $F$ du centralisateur est le groupe multiplicatif d’une extension quadratique de $F$, éléments dits “elliptiques” dans [@Fr] et “très elliptiques” ici pour éviter les confusions.
Soit $g\in \tl G$ tel que $x=g\tau(g)$ est un élément de $G$ régulier et très elliptique. Le groupe $T(F)$ des points sur $F$ du centralisateur de $x$ est le groupe multiplicatif d’une extension quadratique $E$ de $F$, nécessairement non ramifiée sur $F$.
Si ${\scr Cl}_{\tau}^{st}(g)\cap \tl J$ est vide, alors ${\scr Cl}^{st}(x)\cap J$ aussi. En effet, un élément $x'$ de ${\scr Cl}^{st}(x)\cap J$ est de la forme $\N (g')$ où $g'\in {\scr Cl}_{\tau}^{st}(g)$ et commute à $x'$ (puisque $g$ commute à $x$). Or $x'$ est très elliptique donc $g'$ appartient à $F^\x\tl U_{0}(\scr L)=\tl J$ [@Fr th. 1]. Pour un tel $g$, l’identité (\[id\]) est satisfaite.
Si ${\scr Cl}_{\tau}^{st}(g)\cap \tl J$ n’est pas vide, on considère $g'\in {\scr Cl}_{\tau}^{st}(g)\cap \tl J$ et $x'=\N (g')$. Alors $x'\in \tl U_{0}(\scr L)$ et $x'$ est $\tl G$-conjugué à $x$ : $x'=y^{-1}xy, y\in \tl G$.\
Notons que $\tau(x')$ appartient à $\tl U_{0}(\scr L)$ et $\tau(x')=(y^{-1}\tau(y))^{-1}x'y^{-1}\tau(y)$. Ainsi $y^{-1}\tau(y)\in F^\x\tl U_{0}(\scr L)$, autrement dit : $\tau(y)=yj$, $j\in \tl J$. Mais alors la chaîne de réseaux $y\scr L$ est une chaîne autoduale de $F^2$ et $x\in U_{0}(y\scr L)$. De deux choses l’une :\
- la chaîne $y\scr L$ est de même invariant que $\scr L$ auquel cas il existe $y_{0}\in G$ tel que $y\scr L=y_{0}\scr L$ : $y_{0}^{-1}xy_{0}\in U_{0}(\scr L)=J$ ;\
- la chaîne $y\scr L$ n’est pas de même invariant que $\scr L$ auquel cas il existe $y_{0}\in G$ tel que $y\scr L=y_{0}{\scr L}'$ où ${\scr L}'$ est une chaîne de période 1 telle que $U_{0}({\scr L}')$ contient le même sous-groupe d’Iwahori ${\scr I}$ que $U_{0}(\scr L)$ : $y_{0}^{-1}xy_{0}\in U_{0}({\scr L}')$. Mais, ${\scr L}'$ est d’invariant pair et $U_{0}({\scr L}')$ est la réunion disjointe de ${\scr I}$ et de son complémentaire, le premier formé des éléments de déterminant dans $1+{\goth p}$, le second formé des éléments de déterminant dans $-1+{\goth p}$. Or le déterminant de $y_{0}^{-1}xy_{0}$ est égal à $\Det g'/\ol{\Det g'}$ où $\Det g'$ est nécessairement de valuation paire, donc est un élément de $1+{\goth p}$. Ainsi, $y_{0}^{-1}xy_{0}$ appartient à ${\scr I}\subset U_{0}(\scr L)$.\
Dans les deux cas, ${\scr Cl}^{st}(x)\cap J$ n’est pas vide. On peut donc choisir $x\in J$ puis $g\in \tl J$. Par la formule de Mackey et puisque $x$ est très elliptique, on obtient : $\tr \tl \pi (g\tau)=\tr \tl \Lambda (g\tau)\quad$ et $\quad \tr\pi(x)=\tr \lambda(x)$.\
On conclut grâce à (\[tr-lambda-0\]).
Caractères de représentations irréductibles de certaines extensions de groupes finis. {#Traces}
=====================================================================================
On présente dans ce paragraphe un calcul de caractères de représentations admissibles de groupes compacts qui permet de justifier ou de se convaincre de la validité d’affirmations contenues dans le paragraphe \[donnees\] (cas (b’)) et la démonstration du lemme \[construction-choix\].
Le calcul exposé est plus complexe que nécessaire ici. Mais, on rencontre cette même question dans l’étude du transfert de $U(1,1)(F_{0})\x U(1)(F_{0})$ à $U(2,1)(F_{0})$. On a donc choisi un cadre suffisament grand pour réutiliser les résultats lors de cette étude.
Données, hypothèses et objectif. {#Tr1}
--------------------------------
On fixe un nombre premier $p$.
Soient ${\Bbb J}$ un groupe fini, $\J_{1}$ un sous-groupe distingué de ${\Bbb J}$ de centre $Z$ et $\J '$ un sous-groupe de $\J$ tels que $\J=\J' \J_{1}$ et $\J'\cap \J_{1}\subset Z$. On suppose que :
- $\J_{1}$ est un $p$-groupe, extra-spécial de classe 2 ou abélien ;
- $\J'=T\J'_{1}$ où $T$ un sous-goupe abélien de ${\Bbb J}'$ et $\J'_{1}$ un $p$-sous-groupe distingué, ou bien extra-spécial de classe 2 dont le centre $Z'$ contient $\J'_{1}\cap T$, ou bien abélien (dans ce cas on le note aussi $Z'$) ;
- $[\J'_{1}, \J_{1}]=1$ ;
- $\J'/\J'_{1}$ est d’ordre premier à $p$ ;
- $Z'Z$ est contenu dans le centre de $\J$ et $\J'_{1}\J_{1}$ est un $p$-groupe extra-spécial de classe 2, de centre $Z'Z$.
Soient $\theta$ un caractère fidèle de $Z$ et $\theta'$ un caractère fidèle de $Z'$ tels que $\theta_{\vert Z\cap Z'}=\theta'_{\vert Z\cap Z'}$.
Ainsi, le groupe $Z$ est un groupe cyclique d’ordre $p$ et le quotient $V=\J_{1}/Z$ est un ${\Bbb F}_{p}$-espace vectoriel de dimension paire, muni d’une forme bilinéaire alternée non dégénérée $<.,.>$ définie par : $$\begin{matrix} <.,.>&:&V\x V& \fl &\mu_{p}\\ & & (gZ,g'Z)& \mapsto &\theta([g,g']).
\end{matrix}$$ Ceci est également valable pour $Z'$ “en primant” $V$, $\J_{1}$, $Z$ et $\theta$.
On note $\Theta$ le caractère $\theta'\cdot \theta$ de $Z'Z$ et $\eta_{\Theta}$ la représentation de Heisenberg de $\J'_{1}\J_{1}$ de caractère central $\Theta$. Elle est isomorphe au produit tensoriel des représentations de Heisenberg de $\J_{1}$ et $\J'_{1}$ de caractères centraux $\theta$ et $\theta'$ respectivement. On note $\eta_{\theta}$ et $\eta_{\theta'}$ ces deux représentations, $p^a$ et $p^b$ leurs dimensions respectives.
On suppose que les représentations $\eta_{\Theta}$ et $\eta_{\theta'}$ se prolongent en des représentations de $\J$ et $\J'$ respectivement.
Notre but est de définir une bijection entre les prolongements de $\eta_{\Theta}$ à $\J$ et ceux de $\eta_{\theta'}$ à $\J'$ caractérisée par une identité de caractères.
Dans les deux cas, les prolongements se distinguent par les valeurs de leurs caractères sur les éléments de $T$. On cherche donc à calculer la trace des prolongements de $\eta_{\Theta}$ en un élément de $T$ en fonction de celle des prolongements de $\eta_{\theta'}$, au moins dans les situations qui nous sont utiles.
Cinq conséquences des données et hypothèses. {#Tr2}
--------------------------------------------
### {#Tr21}
Tout d’abord, de l’hypothèse (iv), on déduit que : $$\J'\cap \J_{1}\subset \J'_{1}\quad \text{ d'o\`u } \quad \J/\J'_{1}\J_{1}\simeq \J'/\J'_{1}\simeq T/T\cap Z'\quad \text{ et } \quad \J'\cap \J_{1}\subset Z',$$ le dernier isomorphisme et l’inclusion sont alors conséquences de (ii) et (i) respectivement.
### {#Tr22}
A $x\in \J'$, on associe l’automorphisme symplectique de $V$, encore noté $x$, défini par la conjugaison par $x$, le sous-espace $V^x$ de $V$ formé des points fixes sous $x$ et le caractère $\chi_{x}$ de $V^x$ défini par : $$\chi_{x}(v)=\theta([x,v]) \text{ pour tout } v\in V^x.$$ Remarquons d’une part, que $\J'_{1}$ centralisant $\J_{1}$ (iii), $V^x$ et $\chi_{x}$ ne dépendent que de la classe de $x$ modulo $\J'_{1}$ ; d’autre part, que si $x$ n’appartient pas à $\J'_{1}$, son ordre $r$ dans $\J'/\J'_{1}$ est premier à $p$ (par (iv)) et $\chi_{x}$ est trivial. Pour cette dernière assertion, il suffit de remarquer que :
- pour tout $v\in V^x$, tout $s\in {\Bbb N}$, $\chi_{x}(v)^s=1 \ssi s\equiv 0 [p]\, \text{ ou } \chi_{x}(v)=1$;
- pour tout $v\in V^x$, $\chi_{x}(v)^r=\chi_{x^r}(v)=1$.
Par conséquent, puisque $\theta$ est fidèle, la projection naturelle du sous-groupe $\J_{1}^x$ des éléments de $\J_{1}$ invariants par conjugaison par $x$ sur $V^x$ est surjective. En primant les notations, on obtient le résultat analogue pour tout élément de $T$. Ainsi, $$\label{(Tr221)}
\forall x\in \J', \quad \left( \J_{1}/Z\right)^x\simeq\J_{1}^x/Z \quad \text{ et } \quad \forall t\in T, \quad \left( \J'_{1}/Z'\right)^t\simeq{\J'_{1}}^t/Z'$$ Il s’en suit que : $$\forall t\in T, \quad \left( \J'_{1}\J_{1}/Z'Z\right)^t\simeq{\J'_{1}}^t\J_{1}^t/Z'Z.$$
### {#Tr23}
Soit $t\in T$. L’application de $V$ dans $V$ qui à $v\in V$ associe $[t^{-1},v]$ est linéaire de noyau $V^t$. Son image ${\scr I}_{t}=\{ [t^{-1}, v], v\in V\}$ est l’orthogonal de $V^t$. De plus, l’action de $T$ par conjugaison sur $\J_{1}$ fournit une représentation de $T$ dans $GL(V)$ triviale sur $T\cap \J'_{1}$. Puisque $T/T\cap \J'_{1}$ est fini d’ordre premier à $p$ (iv), cette représentation est semi-simple donc somme de caractères. On en déduit que les sous-espaces $V^t$ et ${\scr I}_{t}$ sont supplémentaires. Par conséquent, pour tout $t\in T$ (et par suite pour tout $x\in \J'$), la restriction de $<.,.>$ au sous-espace $V^t$ est non dégénérée et la dimension de $V^t$ est paire, notée $2a_{t}$.
### {#Tr24}
Les paires $(\J, \lambda)$ et $(\J', \lambda')$ sont deux cas de la situation A.1.7 de [@BH2]. En utilisant le corollaire A.1.8 et la surjectivité des projections de $(\J'_{1}\J_{1})^t$ sur $\left( \J'_{1}\J_{1}/Z'Z\right)^t$ et de ${\J'_{1}}^t$ sur $\left( \J'_{1}/Z'\right)^t$ pour tout $t\in T$ (comme dans la démonstration de la proposition 13.1 de [@BH1]), on obtient :
Soit $t\in T$.
- $\tr \lambda(t)\not =0, \quad \tr \lambda'(t)\not =0 \quad$ et $\quad \Vert \tr \lambda(t)\Vert=\vert V^t\vert^{1\over 2}\cdot \vert {V'}^t\vert^{1\over 2},\break \quad \Vert \tr \lambda'(t)\Vert= \vert {V'}^t\vert^{1\over 2}.$
- Soit $y\in \J'_{1}\J_{1}$. Alors $\tr \lambda(ty)$ est non nul si et seulement si $ty$ est $\J'_{1}\J_{1}$-conjugué à un élément de $tZ'Z$.
- Soit $y'\in \J'_{1}$. Alors $\tr \lambda'(ty')$ est non nul si et seulement si $ty'$ est $\J'_{1}$-conjugué à un élément de $tZ'$.
### {#section-11}
Enonçons une dernière conséquence.
\[Tr25\] Deux éléments de $\J'$ sont $\J$-conjugués si et seulement s’ils sont $\J'$-conjugués.
Soient $x, x'$ deux éléments de $\J'$ qui sont $\J$-conjugués. Quitte à conjuguer l’un d’entre eux par un élément de $\J'$, on peut supposer que $x$ et $x'$ sont $\J_{1}$-conjugués. On écrit : $x=ty$ où $t\in T$ et $y\in \J'_{1}$ et $x'=jxj^{-1}$ avec $j\in \J_{1}$. Montrons que $x'$ et $x$ sont égaux.\
Si $x\in \J'_{1}$, il suffit d’utiliser (iii). Si $x\not \in \J'_{1}$, on a grâce à (iii) :
$x'=ty\cdot y^{-1}[t^{-1},j]y\cdot [y^{-1},j]=x\cdot [t^{-1},j]$.\
Ainsi, $[t^{-1},j]=x^{-1}x'$ appartient à $Z$ ($\J_{1}\cap \J' \subset Z$ par hypothèse), c’est-à-dire que $j\in V^t=V^x$. Et puisque $\chi_{x}$ est trivial (§\[Tr22\]) et $\theta$ fidèle, on conclut que $x$ et $x'$ sont égaux.
Par les lemmes \[Tr24\] et \[Tr25\], on obtient :
Soit $x$ un élément de $\J'$. Les quatre propositions suivantes sont équivalentes :
- $\tr \lambda (x)\not =0$ ;
- $\tr \lambda' (x)\not =0$ ;
- $x$ est $\J'$-conjugué à un élément de $TZ'$ ;
- $x$ est $\J$-conjugué à un élément de $TZ'$.
Comparaison de traces. {#Tr3}
----------------------
### {#Tr31}
Soit $\lambda$ un prolongement de $\eta_{\Theta}$ à $\J$. Notons $\J'_{0}$ le centralisateur de $\J_{1}$ dans $\J'$. La restriction de $\lambda$ à $\J'_{0}$ est de la forme $\lambda'_{0}\cdot \eta_{\theta}$ où $\lambda'_{0}$ est un prolongement de $\eta_{\theta'}$ à $\J'_{0}$. Soit $\lambda'$ un prolongement de $\lambda'_{0}$ à $\J'$.
Par la suite, on note ${\scr X}$ l’ensemble des caractères de $\J$ triviaux sur $\J'_{0}\J_{1}$. Il s’identifie via la restriction à l’ensemble des caractères de $\J'$ triviaux sur $\J'_{0}$ ou à ceux de $T$ triviaux sur $T\cap \J'_{0}$. Selon le contexte, on regarde ${\scr X}$ d’une façon ou d’une autre et on note $d$ son cardinal.
Remarquons que lorsque $a=0$, $\lambda$ est isomorphe à $\lambda'\otimes \theta$ pour un prolongement $\lambda'$ de $\eta_{\theta'}$ convenable. Il existe donc une bijection que l’on peut définir “canoniquement” par l’égalité des traces sur les éléments de $T$.
Si $d=1$, $\J$ n’est autre que le produit $\J'_{0}\J_{1}$ et $\J'$ est égal à $\J_{0}'$. La représentation $\lambda$ est de la forme ${\lambda'}\otimes \eta_{\theta}$ pour un unique prolongement $\lambda'$ de $\eta_{\theta'}$. Il existe donc une bijection que l’on peut définir “canoniquement” par l’identité de caractères suivante : $$\forall x\in T,\quad \tr \lambda(x)= p^a\tr\lambda'(x).$$ On suppose donc : $a>0$ et $d>1$.
### {#Tr32}
On désigne par $\K$ le groupe $\J'Z$. Puisque $\lambda$ prolonge $\eta_{\Theta}$, sa restriction à $\K$ est une sous-représentation de la restriction à $\K$ de $\ind_{\J'_{0}\J_{1}}^\J \lambda'_{0}\otimes \eta_{\theta}$ qui est multiple de $\ind_{\J'_{0}Z}^\K \lambda_{0}'\cdot \theta$ d’après la formule de Mackey. Il existe donc $d$ entiers positifs $m_{\xi}$ ($\xi\in {\scr X}$) tels que : $$\label{equ:n321}
\forall k\in \K, \quad \lambda(k)=\bigoplus_{\xi\in {\scr X}} m_{\xi}(\xi \lambda'\cdot \theta)(k)\quad \text{ et } \quad \sum_{\xi\in {\scr X}}m_{\xi}=p^a$$ d’où : $\displaystyle \forall x\in T, \quad \tr\lambda(x)=\tr\lambda'(x)\cdot \sum_{\xi\in {\scr X}}m_{\xi}\xi(x).$
Notre but est d’évaluer $\sum_{\xi\in {\scr X}}m_{\xi}\xi(x)$. De l’assertion [*a*]{} du lemme \[Tr24\], on déduit qu’elle est de module $\vert V^x\vert^{1/2}$ pour tout $x$ de $T$. On obtient alors un système d’équations en les inconnues $m_{\xi}$, à savoir : $$\label{equ:n322}\begin{split}
&\sum_{\xi\in {\scr X}}m_{\xi}=p^a \\
&\sum_{\xi\in {\scr X}}m_{\xi}^2 + \sum_{\xi'\in {\scr X}-\{1\}}(\sum_{\xi\in {\scr X}}m_{\xi}m_{\xi\xi'})\xi'(x)=\vert V^x\vert \text{ pour chaque } \ol x\not =1,
\end{split}$$ où $\ol x$ représente la classe de $x\in T$ modulo $\J'_{0}\cap T$.
### {#Tr33}
On suppose dorénavant que :
[*(H)*]{} [*$T/\J'_{0}\cap T$ est cyclique et que pour tout $\ol x\in T/\J'_{0}\cap T$ non trivial, $\vert V^x\vert =1$.*]{}
Alors les $d$ inconnues $M_{1}=\sum_{\xi\in {\scr X}}m_{\xi}^2 -1$ et $M_{\xi'}=\sum_{\xi\in {\scr X}}m_{\xi}m_{\xi\xi'}$, $\xi'\in {\scr X}$ non trivial, sont solutions du système linéaire $$\left\{ \begin{aligned} &M_{1}+ \sum_{\xi'\in {\scr X}-\{1\}}M_{\xi'}=p^{2a}-1\\
& M_{1}+ \sum_{\xi'\in {\scr X}-\{1\}}\xi'(x)M_{\xi'}=0,\\
\end{aligned}\right.$$ c’est-à-dire : $\forall \xi\in {\scr X}, \quad M_{\xi}={p^{2a}-1\over d}$. En particulier : $\displaystyle \sum_{\xi\in {\scr X}}m_{\xi}^2-1={p^{2a}-1\over d}$.
Soit $\{ m_{\xi}, \xi\in {\scr X}\}$ une solution entière de (\[equ:n322\]). Alors : $$\label{equ:n331}
\sum_{\xi\in {\scr X}}\left({p^a+1\over d}-m_{\xi}\right)^2=\vert {\scr X}\vert\left({p^a+1\over d}\right)^2-2{p^a+1\over d}\sum_{\xi\in {\scr X}}m_{\xi}+\sum_{\xi\in {\scr X}}m_{\xi}^2=1$$ [*i)*]{} Si ${p^a+1\over d}\in {\Bbb Z}$, il existe, à permutation près des $m_{\xi}$, deux solutions entières de (\[equ:n331\]), à savoir :
$m_{1}={p^a+1\over d}\pm 1$ et $m_{\xi}={p^a+1\over d}$ si $\xi\not =1$ ;
et, toujours à permutation près des $m_{\xi}$, il existe une unique solution entière de (\[equ:n322\]), à savoir :
$m_{1}={p^a+1\over d}-1$ et $m_{\xi}={p^a+1\over d}$ si $\xi\not =1$.
[*ii)*]{} Si ${p^a+1\over d}\not\in {\Bbb Z}$, chaque terme de la somme (\[equ:n331\]) appartient à $]0,1[$ donc, pour tout $\xi\in {\scr X}$, $m_{\xi}$ est égal à $u$ ou $u+1$ où $u$ est la partie entière de ${p^a+1\over d}$. Notons $r$ la différence $(p^a+1)-ud$ et $n$ le nombre d’entiers $m_{\xi}$ égaux à $u$. Remarquons que : $0<r<d$.
De la première équation de (\[equ:n322\]), on déduit que $n=d+1-r$, puis de (\[equ:n331\]), on obtient : $$\begin{split}
&\sum_{\xi\in {\scr X}}\left({p^a+1\over d}-m_{\xi}\right)^2=(d+1-r)\left({r\over d}\right)^2+(r-1)\left(1-{r\over d}\right)^2=1\\ \ssi & (r-2)(r-d)=0Ê\impl r=2.
\end{split}$$ Dans ce cas, $d$ divise donc $p^a-1$, l’un des $m_{\xi}$ est égal à ${p^a-1\over d}+1$ et tous les autres à ${p^a-1\over d}$. On obtient ainsi toutes les solutions de (\[equ:n322\]).
En conséquence :
\[(H)\] On suppose l’hypothèse (H) satisfaite. On note $\J'_{0}$ le centralisateur de $\J_{1}$ dans $\J'$, $d$ le cardinal de $ T/\J'_{0}\cap T$ et $p^{2a}$ celui de $\J_{1}/Z$.
Si $d$ divise $p^a+1$, il existe un unique prolongement $\lambda'$ de $\eta_{\theta'}$ tel que : $$\forall x\in T, \quad \tr \lambda(x)=\begin{cases}
p^a\tr\lambda'(x)&\text{ si } x\in \J'_{0}\cap T,\\
-\tr\lambda'(x)\hfill&\text{ sinon.}\end{cases}$$ Sinon, il existe un unique prolongement $\lambda'$ de $\eta_{\theta'}$ tel que : $$\forall x\in T, \quad \tr \lambda(x)=\begin{cases} p^a\tr\lambda'(x)&\text{ si } x\in \J'_{0}\cap T,\\ \tr\lambda'(x)\hfill&\text{ sinon.}\end{cases}$$
### Applications {#precisions}
\
(1) L’affirmation de \[donnees\] (b’) demande une justification lorsque la représentation $\lambda$ n’est pas de dimension 1. D’après [@Bl1 §A.6.2], $\lambda$ est de la forme $\lambda_{\theta}$ pour un caractère $\theta$ d’un sous-groupe de $J$. On applique ce qui précède avec : $$\J=J/\Ker \theta\, , \J_{1}=J\cap U_{1}(\scr L)/\Ker \theta\, , T=H/\Ker \theta\cap H\, , \J'_{1}=Z'=Z.$$ en remarquant que $T/\J'_{0}\cap T\simeq k^1_{\vert k_{0}}$, $V$ est de cardinal $q^2$ et le stabilisateur de chaque élément non nul de $V$ est réduit à $\{1\}$.
\(2) On reprend les notations de \[donnees2\] (b) et on se place dans le cas (r-nr) du paragraphe \[construction\]. Pour calculer $\tr \lambda$ sur ${\goth o}_{E_{\vert L}}^1$, on pose : $$\J= J/\Ker \theta\, , \J_{1}=J_{1}/\Ker \theta\, , T={\goth o}_{E_{\vert L}}^1/{\goth o}_{E_{\vert L}}^1\cap \Ker \theta\, \text{ et } \, \J'_{1}=Z'=Z$$ et on applique le lemme précédent. Remarquons que $\J'_{0}$ est alors égal à $\{\pm 1\}\J'_{1}$ donc $d$ vaut $\frac{q+1}{2}$ tandis que $a$ vaut 1 ou 0 suivant que la dimension de $\lambda$ est $q$ ou $1$. L’hypothèse (H) est bien satisfaite car $T/T\cap \J'_{0}$ est cyclique et tous les éléments de ${\goth o}_{E_{\vert L}}^1$ non centraux sont minimaux.
\(3) On se place dans la même situation que précédemment mais on s’intéresse cette fois à la représentation $\tl \lambda$. On suppose de plus que $\tl \lambda$ n’est pas de dimension 1 (donc $a=1$). On pose : $$\J= \tl J/\Ker\tl \theta\, , \J_{1}=\tl J_{1}/\Ker \tl \theta\, , T={\goth o}_{E}^\x/{\goth o}_{E}^\x\cap \Ker\tl \theta\, \text{ et } \, \J'_{1}=Z'=Z.$$ C’est un cas particulier de \[Tr1\] et toutes les conséquences qui suivent sont valables, en particulier \[Tr32\].
, Depth-zero base change for unramified $U(2,1)$, J. Number Theory, [**114**]{} (2005), 324-360. , Depth-zero base change for ramified $U(2,1)$, arXiv:0807.1528v1, 9 Jul 2008. , Description du dual admissible de $U(2,1)(F)$ par la théorie des types de C. Bushnell et P. Kutzko, Manuscripta Math., [**107**]{} (2002), 151-186. , Types, paquets et changement de base : l’exemple de $U(2,1)(F_0)$, I, Canadian J. Math., [**60**]{} (2008), 790-821. , Local tame lifting for $\GL(N)$ I: simple characters, Publ. Math. IHES, [**83**]{} (1996), 105-233. , Local tame lifting for $\GL(n)$ II: wildly ramified supercuspidals, Astérisque [**254**]{} Soc. Math. France, 1999. , The local Langlands conjecture for $GL(2)$, Grundlerhen der math. Wissenschaften [**335**]{} Springer, 2006. , On the characters of the finite unitary groups, Ann. Acad. Scien. Fenn., [**323**]{} (1963),1-35. , Stable and labile base change for $U(2)$, Duke Math. J., [**49**]{} (1982), 691-729. , On distinguished representations, J.Reine Angew. Math. [**418**]{} (1991), 134-172. , Principal orders and embedding of local fields in algebras, Proc. London Math. Soc., [**54**]{} (1987), 247-266. , Two types of distinguished supercuspidal representations, Int. Math. Res. Not., [**35**]{} (2002), 1857-1889. , Automorphic forms on $GL(2)$, Lectures Notes in Math. [**114**]{}, Springer Verlag,1970. , Rational conjugacy classes on reductive groups, Duke Math. J., [**49**]{} (1982), 785-806. , All supercuspidal representations of $SL_{\ell}$ over a $p$-adic field are induced, Proceedings of the Utah conference on representation Theory, Progress in Math., [**40**]{} (1983), 185-196. , $L$-indistinguishability for $SL(2)$, Can. J. Math., [**31**]{} (1979), 726-785. , Tamely ramified supercuspidal representations, Ann. scient. Ec. Norm. Sup., [**29**]{} (1996), 639-667. , Automorphic representations of unitary groups in three variables, Ann. of Math. Studies [**123**]{} Princeton University Press, 1990. , Characters of special groups, Seminar on Algebraic groups and related finite groups 1968/69, Lectures Notes in Math., [**131**]{} (1970), 121-166. , Double coset decompositions and intertwining, Manuscripta Math., [**106**]{} (2001), 349-364.
|
---
abstract: |
We consider the physical implications of the rapid spindown of Soft Gamma Repeater 1900+14 reported by Woods et al. During an 80 day interval between June 1998 and the large outburst on August 27 1998, the mean spin-down rate increased by a factor 2.3, resulting in a positive period offset of $\Delta P/P = 1\times 10^{-4}$. A radiation-hydrodynamical outflow associated with the August 27th event could impart the required torque, but only if the dipole magnetic field is stronger than $\sim 10^{14}$ G and the outflow lasts longer and/or is more energetic than the observed X-ray flare. A positive period increment is also a natural consequence of a gradual, plastic deformation of the neutron star crust by an intense magnetic field, which forces the neutron superfluid to rotate more slowly than the crust. Sudden unpinning of the neutron vortex lines during the August 27th event would then induce a glitch opposite in sign to those observed in young pulsars, but of a much larger magnitude as a result of the slower rotation.
The change in the persistent X-ray lightcurve following the August 27 event is ascribed to continued particle heating in the active region of that outburst. The enhanced X-ray output can be powered by a steady current flowing through the magnetosphere, induced by the twisting motion of the crust. The long term rate of spindown appears to be accelerated with respect to a simple magnetic dipole torque. Accelerated spindown of a seismically-active magnetar will occur when its persistent output of Alfvén waves and particles exceeds its spindown luminosity. We suggest that SGRs experience some episodes of relative inactivity, with diminished $\dot{P}$, and that such inactive magnetars are observed as Anomalous X-ray Pulsars (AXPs). The reappearence of persistent X-ray emission from SGR 1900+14 within one day of the August 27 event provides strong evidence that the persistent emission is not powered by accretion.
author:
- 'Christopher Thompson, Robert C. Duncan, Peter M. Woods, Chryssa Kouveliotou, Mark H. Finger, and Jan van Paradijs'
title: 'Physical Mechanisms for the Variable Spin-down of SGR $1900+14$'
---
Introduction
============
Woods et al. (1999c; hereafter Paper I) have shown that over the period September 1996 – May 1999, the spin-down history of SGR $1900+14$ is generally smooth, with an average rate of 6 $\times$ 10$^{-11}$ s s$^{-1}$. However, during an 80 day interval starting in June 1998 which contains the extremely energetic August 27 flare (Hurley et al. 1999a; Mazets et al. 1999), the average spindown rate of SGR $1900+14$ increased by a factor $\sim$ 2.3. The sampling of the period history of SGR $1900+14$ is insufficient to distinguish between a long-term (i.e. 80 days) increase of the spin-down rate to an enhanced value and a sudden increase (a ‘braking’ glitch) in the spin period connected with the luminous August 27 flare.
In this paper, we investigate several physical processes that may generate a positive period increment of the observed magnitude ($\Delta P/P
\sim 10^{-4}$) directly associated with the August 27 flare. We focus on two mechanisms: a particle wind coinciding with the period of hyper-Eddington radiative flux; and an exchange of angular momentum between the crustal neutron superfluid and the rest of the neutron star. We show that both models point to the presence of an intense magnetic field. The change in the persistent pulse profile of SGR 1900+14 following the August 27 outburst is considered, and related to continuing particle output in the active region of the burst. We also consider mechanisms that could drive the (nearly) steady spindown observed in both SGRs and AXPs, as well as departures from uniform spindown.
Braking driven by a particle outflow
====================================
The radiative flux during the oscillatory tail of the August 27 event decreased from $1\times 10^{42}\,(D/10~{\rm kpc})^2$ erg/s, with an exponential time constant of $\sim 90$ s (Mazets et al. 1999). The net energy in the tail, radiated in photons of energy $>$ 15 keV, was $\sim 5\times 10^{43}(D/10~{\rm kpc})^2$ erg. The tail was preceded by much harder, narrow pulse of duration $\sim 0.35$ s and energy $> 7\times
10^{43}(D/10~{\rm kpc})^2$ erg (Mazets et al. 1999). The very fast rise time of $\sim 10^{-3}$ s points convincingly to an energy source internal to the neutron star. Just as in the case of the 1979 March 5 event, several arguments indicate the presence of a magnetic field stronger than $10^{14}$ G (Thompson & Duncan 1995; hereafter “TD95"). Not only can such a field spin down the star to its observed 5.16 s period (Hurley et al. 1999c; Kouveliotou et al. 1999), but it can power the burst by inducing a large-scale fracture of the neutron star crust. Indeed, only a fraction $\sim 10^{-2}\,(B_\star/10\,B_{\rm QED})^{-2}$ of the external dipole magnetic energy must be tapped, where $B_{\rm QED} \equiv 4.4\times 10^{13}$ G. This allows for individual SGR sources to emit $\gtrsim 10^2$ such giant flares over their $\sim 10^4$ yr active lifetimes. More generally, [*any*]{} energy source that excites internal seismic modes of the neutron star must be combined with a magnetic field of this strength, if seismic energy is to be transported across the stellar surface at the (minimum) rate observed in the initial spike (cf. Blaes et al. 1989). A field stronger than $1.5\times
10^{14}\,(E/6\times10^{43}~{\rm erg})^{1/2}\, (\Delta R/{\rm
10~km})^{-3/2}[(1+\Delta R/R_\star)/2]^{3}$ G is also required to confine the energy radiated in the oscillatory tail (Hurley et al. 1999a), which maintained a very constant temperature even while the radiative flux declined by an order of magnitude (Mazets et al. 1999).
The radiative flux was high enough throughout the August 27 event to advect outward a large amount of baryonic plasma at relativistic speed. Even though one photon polarization mode (the E-mode) has a suppressed scattering cross-section when $B> B_{\rm QED}$ (Paczyński 1992), splitting of E-mode photons will regenerate the O-mode outside the E-mode scattering photosphere, and ensure than the radiation and matter are hydrodynamically coupled near the stellar surface (TD95). Matter will continue to accumulate further out in the magnetosphere during the burst, but cannot exceed $\tau_{\rm T} \sim 1$ outside a radius where the energy density of the freely streaming photons exceeds the dipole magnetic energy density, \~[B\_\^24]{}([R\_[A]{}R\_]{})\^[-6]{}, or equivalently \[Ralf\] [R\_[A]{}R\_]{} \~([B\_\^2 R\_\^2cL\_[X]{}]{})\^[1/4]{} = 280([B\_10 B\_[QED]{}]{})\^[1/2]{} ([E\_[X]{}10\^[44]{} [erg]{}]{})\^[-1/4]{} ([t\_[burst]{}100 [s]{}]{})\^[1/4]{}. The radiation pressure acting on the suspended matter will overcome the dipole magnetic pressure at a radius $\leq R_{\rm A}$; the same is true for the ram pressure of matter streaming relativistically outward along the dipole field lines.
Photons scattering last at radius $R_{\rm A}$ and polar angle $\theta$ (or relativistic matter escaping the dipole magnetic field from the same position) will carry a specific angular momentum $\sim \Omega R_{\rm A}^2\sin^2\theta$. The net loss of angular momentum corresponding to an energy release $\Delta E$ is \[domega\] I\_-[Ec\^2]{}R\_[A]{}\^2\^2. The period increase accumulated on a timescale $\Delta t_{\rm burst}$ is largest if the outflow is concentrated in the equatorial plane of the star: \[deltap\] [PP]{} (Et\_[burst]{})\^[1/2]{} [B\_R\_\^3I\_c\^[3/2]{}]{} = 810\^[-6]{}([E10\^[44]{} [erg]{}]{})\^[1/2]{} ([t\_[burst]{}100 [s]{}]{})\^[1/2]{} ([B\_10 B\_[QED]{}]{}).
The torque is negligible if the dipole field is in the range $B_\star \sim 0.1 B_{\rm QED}$ typical of ordinary radio pulsars. Even for $B_\star \sim 10~B_{\rm QED}$ this mechanism can induce $\Delta P/P \sim 1\times 10^{-4}$ only if the the outflow lasts longer than the observed duration of the oscillatory tail. Release of $\sim 10^{44}$ erg over $\sim 10^4$ s would suffice; but extending the duration of the outflow to $\sim 10^5$ s would imply $\dot P \sim 1.3\times 10^{-8}$ one day after the August 27 event, in contradiction with the measured value $200$ times smaller. Note also that the short initial spike is expected to impart a negligible torque to the star. This is the basic reason that [*persistent*]{} fluxes of Alfvén waves and particles are more effective at spinning down a magnetar than are sudden, short bursts of equal fluence.
One might consider increasing the torque by increasing the inertia of the outflow, so that it moves subrelativistically at the Alfvén surface, at speed $V$. For a fixed kinetic luminosity, $\dot E = (1/2) \dot{M} V^2$, the Alfvén radius scales in proportion to $(V/c)^{1/4}$, and one finds 1 10\^[-4]{} ([E10\^[44]{} [erg]{}]{})\^[1/2]{} ([t\_[burst]{}100 [s]{}]{})\^[1/2]{} ([B\_10 B\_[QED]{}]{}) ([V0.2 c]{})\^[-3/2]{} . However, the energy needed to lift this material from the surface of the neutron star exceeds $\Delta E = \int \dot E dt$ by a factor $\sim 10\,(V/0.2~c)^{-2}$ (assuming $GM_\star/(R_\star c^2) = 0.2$). This scenario therefore requires some fine-tuning, if the flow is to remain subrelativistic far from the neutron star.
Moreover, such a slow outflow is very thick to Thomson scattering and free-free absorption. The Thomson depth along a radial line through the outflow is \_[T]{}(R\_[A]{}) = 10 ([E10\^[44]{} [erg]{}]{})\^[5/4]{} ([B\_10 B\_[QED]{}]{})\^[-1/2]{}([t\_[burst]{}100 [s]{}]{})\^[-5/4]{}([Vc]{})\^[-13/4]{} at the Alfvén radius. The free-free optical depth is \_[ff]{} ([kTm\_ec\^2]{})\^[-1/2]{} [\_[T]{}\^2 (hc)\^3\_[T]{} R (kT)\^3]{}f([hkT]{}), where f([hkT]{}) ([hkT]{})\^[-3]{} (1-e\^[-h/kT]{}), and $\alpha_{\rm em} = 1/137$ is the fine structure constant. This becomes \_[ff]{}(R) = 310\^[-2]{}([RR\_[A]{}]{})\^[-5]{} ([P/P10\^[-4]{}]{})\^[5/4]{} ([E10\^[44]{} [erg]{}]{})\^[1/2]{} ([B\_10 B\_[QED]{}]{})\^[-16/3]{} ([t\_[burst]{}100 [s]{}]{})\^[-5]{} f([hkT]{}). Here, we have substituted the value of $V/c$ needed to generate the observed $\Delta P/P$. Notice that the magnetic dipole field and burst duration enter into $\tau_{\rm ff}$ with strong negative powers. The optical depth through a flow along rigid dipole magnetic field lines is $\tau_{\rm T}(R) = (R/R_{\rm A})^{-2}\,\tau_{\rm T}(R_{\rm A})$ at constant $V$.
This calculation indicates that the flow will be degraded to a black body temperature corresponding to an emission radius of $\sim 100\,R_\star$ = 1000 km, which is $\sim 1$ keV at a luminosity $\sim 10^4\,L_{\rm edd}$, far below the observed value (Mazets et al. 1999; Feroci et al. 1999). Note, however, that Inan et al. (1999) found evidence for an intense ionizing flux of soft X-rays in the Earth’s ionosphere, coincident with the first second of the August 27th event. They fit this ionization data with an incident spectrum containing two thermal components, of temperatures 200 and 5 keV, and with the soft component carrying $80\%$ of the energy flux at 5 keV. This model contrasts with the initial spectrum of the August 27 event measured by BeppoSAX, which contained a very hard power-law component ($\nu F_\nu \propto \nu^{1/2}$: Feroci et al. 1999). The effects of pair creation on the ionization rate have yet to be quantified.
The four-pronged profile seen within the later pulses of the August 27 event (Feroci et al. 1999; Mazets et al. 1999) has a plausible interpretation in the magnetar model. The radiation-hydrodynamical outflow originates near the surface of the neutron star, where the opacity of X-ray photons moving across the magnetic field lines is smallest (TD95). This is the case even if the trapped $e^\pm$ fireball that powers the burst extends well beyond the stellar surface. In this model, the pattern of the emergent X-ray flux is a convolution of the multipolar structure of the stellar magnetic field, with the orientation of the trapped fireball. The presence of four X-ray ‘jets’ requires that the trapped fireball connect up with four bundles of magnetic field lines extending to at least a few stellar radii.
Braking via the internal exchange of angular momentum
=====================================================
Now let us consider the exchange of angular momentum between the the crustal superfluid neutrons and the rest of the magnetar. Because an SGR or AXP source is slowly rotating, $\Omega_{\rm cr} \sim 1$, the maximum angular velocity difference $\omega =
\Omega_{\rm sf}-\Omega_{\rm cr}$ that can be maintained between superfluid and crust is a much larger fraction of $\Omega_{\rm cr}$ than it is in an ordinary radio pulsar – and may even exceed it. At the same time, these sources are observed to spin down very rapidly, on a timescale comparable to young radio pulsars such as Crab or Vela. If the rotation of the superfluid were to lag behind the crust in the usual manner hypothesized for glitching radio pulsars, the maximum glitch amplitude would increase in proportion to the spin period (Thompson & Duncan 1996, hereafter TD96; Heyl & Hernquist 1999). One deduces $\Delta P/P \simeq -1\times 10^{-5}$ by scaling to the largest glitches of the Crab pulsar, and $\Delta P/P \simeq -1\times 10^{-4}$ by scaling to Vela.
How would a glitch be triggered in a magnetar? A sudden fracture of the crust, driven by a magnetic field stronger than $\sim 10^{14}$ G, induces a horizontal motion at the Alfvén speed $V_{\rm A} = 1.3\times 10^{7}\,(B/10\,B_{\rm QED})\,(\rho/10^{14}\,{\rm g~
cm^{-3}})^{-1/2}$ cm s$^{-1}$, or higher. This exceeds the maximum velocity difference $V_{\rm sf}-V_{\rm cr}$ that can be sustained between superfluid and crust, before the neutron vortex lines unpin (e.g. Link, Epstein, & Baym 1993). The internal heat released in a large flare such as the August 27 event is probably comparable to the external X-ray output, if the flare involves a propagating fracture of the neutron star crust. This heat is $\sim 100$ times the minimum energy of $\sim 10^{42}$ erg that will induce a sudden increase in the rate of thermal vortex creep (Link & Epstein 1993). For both reasons, giant flares from magnetars probably trigger the widespread unpinning of superfluid vortices in the crust and hence large rotational glitches. Magnetically-driven fractures have also been suggested as the trigger for vortex unpinning in ordinary radio pulsars (Thompson & Duncan 1993, hereafter TD93; Ruderman, Zhu, & Chen 1998).
The observation of a period increase associated with the August 27 outburst leads us to re-examine whether the superfluid should, in fact, maintain a faster spin than the crust and charged interior of the star. Transport of superfluid vortices by thermal creep will cause the angular velocity lag $\omega$ to relax to its equilibrium value $\omega_\infty$ on a timescale t\_r\^[-1]{} = |[\_[cr]{}t]{}| ([V\_[cr]{}]{})\_[\_]{}, if the creep is driven primarily by spindown (Alpar, Anderson, Pines, & Shaham 1984; Link, Epstein, & Baym 1993). The partial derivative of the creep velocity $\partial V_{\rm cr}/\partial\omega$ depends mainly on temperature and density. As a result, this relaxation time is expected to be proportional to $t/\Omega_{\rm cr}$ at constant temperature. Comparing with a prompt (intermediate) relaxation time of $\sim 1$ day ($\sim 1$ week) for glitches of the Crab pulsar ($t\simeq 10^3$ yr; Alpar et al. 1996), one infers $t_r \sim 1$ ($10$) [*years*]{} for a magnetar of spin period $6$ s and characteristic age $P/\dot P = 3000$ yr.
The response of the crust to the evolving magnetic field is expected to be a combination of sudden fractures and plastic deformation. When the temperature of the crust exceeds about $\sim 0.1$ of the melt temperature, it will deform plastically (Ruderman 1991). One deduces $T \simeq 2.4\times 10^8\,(B/10^2\,B_{\rm QED})^2$ K for magnetars of age $\sim 10^4$ yr (TD96; Heyl & Kulkarni 1998). Plastic deformation is also expected when $B^2/4\pi > \mu$ in the deep crust (TD96). In a circumstance where the magnetic field is transported through the stellar interior on a timescale shorter than the age of the star, departures from corotation between superfluid and crust are primarily due to advection of the superfluid vortices across the stellar surface by the deforming crust, [*not*]{} due to spindown. (Recall the principal definition of a magnetar: a neutron star in which magnetism, not rotation, is the dominant source of free energy.) If these deformations occur on a timescale much less than the spindown age, they will control the equilibrium lag between the rotation of the superfluid and crust.
Indeed, the SGR bursts provide clear evidence for deformations on short timescales. More precisely, a large burst such as the August 27 event may be preceded (or followed) by an extended period of slow, plastic deformation. If the superfluid starts near corotation with the crust, this process will take angular momentum out of the superfluid, and force its rotation to lag behind the rest of the star. A glitch triggered by a violent disturbance such as the August 27 event will then cause the neutron star crust to [*spin down*]{}.
The angular momentum of the thin shell of crustal superfluid can be expressed simply as J\_[sf]{} = [2]{}M\_[sf]{} R\_\^2 \_[-1]{}\^1 d()\^2n\_V(), when the cylindrical density $n_V(\theta)$ of neutron vortex lines depends only on angle $\theta$ from the axis of rotation. Here $\kappa =
h/2m_n$ is the quantum of circulation, and we neglect that the rotational deformation of the star. One observes from this expression that the outward motion of vortex lines reduces $J_{\rm sf}$, because the weighting factor $\cos^2\theta$ decreases with distance from the axis of rotation.
The simplest deformation of the neutron star crust, which preserves its mass and volume, involves a rotational twist of a circular patch through an angle $\Delta\phi$. Indeed, the stable stratification of the star (Reisenegger & Goldreich 1992) forces the crust to move horizontally, parallel to the local equipotential surfaces. For this reason, one can neglect horizontal displacements of the crustal material that are compressible in the two non-radial dimensions. The patch has radius $a \ll R_\star$ and is centered at an angle $\theta$ from the axis of rotation. The superfluid is assumed initially to corotate with the crust, $\Omega_{\rm sf} = \Omega_{\rm cr}$, everywhere within the patch, so that $n_V(\theta) = 2\Omega_{\rm cr}/\kappa$. As the patch is rotated, the number of vortex lines per unit [*surface area*]{} of crust is conserved. A piece of crust that moves from $\theta_i$ to $\theta_f$ ends up with a vortex density $n_V = (2\Omega_{\rm cr}/\kappa)\cos\theta_i/\cos\theta_f$. The vortex lines are squeezed together in a piece of the crust that moves away from the rotation axis, and are spread apart if the movement is in the opposite direction. If the vortex density is smoothed out in azimuth following this process, the net decrease in the angular momentum of the superfluid is \[deltajsf\] [J\_[sf]{}J\_[sf]{}]{} = -[34]{}([aR\_]{})\^4 (1-)\^2. Here, $J_{\rm sf} = {2\over 3}M_{\rm sf}\Omega_{\rm cr}R_\star^2 \simeq
10^{-2}I_\star\Omega_{\rm cr}$ is the total angular momentum of the crustal superfluid.
A transient, plastic deformation of the crust would induce a measurable spinup of the crust, by forcing the neutron superfluid further from corotation with the crust. Such a gradual glitch would have the same negative sign as in ordinary radio pulsars, but would not necessarily involve any sudden unpinning of the vortex lines. For example, rotation of a patch of radius $a = {1\over 3} R_\star$ through an angle $\Delta\phi \sim 1$ radian would cause a period decrease $\Delta P/P = \Delta J_{\rm sf}/(I_\star-I_{\rm sf})\Omega_\star
= -4\times 10^{-5}$. A transient spinup of this magnitude may have been observed in the AXP source 1E2259+586 (Baykal & Swank 1996). That excursion from a constant, long term spindown trend can be modelled with a glitch of amplitude $\Delta P/P \simeq -3\times 10^{-5}$, although the X-ray period observations are generally too sparse to provide a unique fit.
The long-term spin-down of SGRs and AXPs
========================================
Let us now consider the persistent spindown rate of SGR 1900+14, and its broader implications for the ages and spindown histories of the SGR and AXP sources. Recall that the spindown rate was almost constant at $\dot P \simeq 6.1\times 10^{-11}$ s/s before May 1998, and after August 28 1998 (Paper I). A May 1997 measurement of $P$ revealed a 5% deviation from this trend; and larger variations in the ‘instantaneous’ spindown rate ($\sim 40$%) were found by RXTE in September 1996 and May/June 1998.
Another important constraint comes from the observed angular position of SGR 1900+14. It lies just outside the edge of the $\sim 10^4$ yr-old supernova remnant G42.8+0.6 (Hurley et al. 1994; Vasisht et al. 1994). A strong parallel can be drawn with SGR 0526-66, which also emitted a giant flare (on 5 March 1979) and is projected to lie inside, but near the edge of, SNR N49 in the Large Magellanic Cloud (Cline et al. 1982). The other known SGRs also have positions coincident with supernova remnants of comparable ages (Kulkarni & Frail 1993; Kulkarni et al. 1994; Murakami et al. 1994; Woods et al. 1999b; Smith, Bradt, & Levine 1999; Hurley et al. 1999d). It seems very likely that these physical associations are real; so we will hereafter adopt the hypothesis that SGR 1900+14 formed at the center of SNR G42.8+0.6. The implied transverse velocity is \[vtrans\] V\_ 3400 ([D7 ]{} ) ([t10\^4 ]{} )\^[-1]{} \^[-1]{} (Hurley et al. 1996; Vasisht et al. 1996; Kouveliotou et al. 1999). Several mechanisms may impart large recoil velocities to newborn magnetars (Duncan & Thompson 1992, hereafter “DT92"), but this very high speed indicates that an age much less than $1 \times 10^4$ yrs is unlikely.
In this context, the short charactersitic spindown age $P/2\dot P\sim 1400$ yr of SGR 1900+14 gives evidence that the star is currently in a transient phase of accelerated spindown (Kouveliotou et al. 1999). The almost identical spindown age measured for SGR 1806-20 suggests that a similar effect is being observed in that source (Kouveliotou et al. 1998; Table 1). If each SGR undergoes accelerated spindown during a minor fraction $\epsilon_{\rm active}\sim P/\dot Pt_{\rm SNR}\sim 0.25$ of its life, then its true age increases to \[tactive\] t = \_[active]{}\^[-1]{}([PP]{}).
[cccllcc]{}
SGR $1806-20$ & 7.47 & 8.3 $\times$ 10$^{-11}$ & 1430 & $\sim$ 10$^4$ & $\sim$ 0.1
SGR $1900+14$ & 5.16 & 6.1 $\times$ 10$^{-11}$ & 1340 & $\sim$ 10$^4$ & $\sim$ 0.1
AXP $1E2259+586$ & 6.98 & 5.0 $\times$ 10$^{-13}$ & 220,000 & $\sim$ 13,000 & $\sim$ 10
AXP $1E1841-045$ & 11.8 & 4.13 $\times$ 10$^{-11}$ & 4570 & $\sim$ 2000 & $\sim$ 2.3
Wind-Aided Spindown
-------------------
Seismic activity will accelerate the spindown of an isolated neutron star, if the star is slowly rotating and strongly magnetized (Thompson & Blaes 1998, hereafter “TB98"). Fracturing in the crust generates seismic waves which couple directly to magnetospheric Alfvén modes and to the relativistic particles that support the associated currents. The fractures are frequent and low energy ($\sim 10^{35}$ erg) when the magnetic field is forced across the crust by compressive transport in the core (TD96). When the persistent luminosity $L_{\rm A}$ of waves and particles exceeds the magnetic dipole luminosity $L_{\rm MDR}$ (as calculated from the stellar dipole field and angular velocity), the spindown torque increases by a factor $\sim \sqrt{L_{\rm A}/L_{\rm MDR}}$.
This result follows directly from our treatment of hydrodynamic torques in §2. Magnetic stresses force the relativistic wind to co-rotate with the star out to the Alfvén radius $R_{\rm A}$, which is determined by substituting $L_{\rm A}$ for $L_{\rm X}$ in eq. (\[Ralf\]): \[raqui\] [R\_[A]{}R\_]{} = 1.610\^4L\_[A35]{}\^[-1/4]{} ([B\_10B\_[QED]{}]{})\^[1/2]{}. The torque then has the form $I \dot{\Omega} = - \Lambda (L/c^2) R_{\rm A}^2$, or equivalently \[pdot\] = [B\_R\_\^3 I\_]{} ([L\_[A]{} c\^3]{})\^[1/2]{} P. Here, $\Lambda$ is a numerical factor of order unity that depends on the angle between the angular velocity ${\bf\Omega}$ and the dipole magnetic moment ${\bf m}_\star$. One finds $\Lambda \approx {2\over 3}$ by integrating eq. (\[domega\]) over polar angle, under the assumption that ${\bf\Omega}$ and ${\bf m}_\star$ are aligned, that the ratio of mass flux to magnetic dipole flux is constant, and that the magnetic field is swept into a radial configuration between the Alfvén radius and the light cylinder. This normalization is $\sim 6$ times larger than deduced by TB98 for a rotator with ${\bf m}_\star$ inclined by 45$^\circ$ with respect to ${\bf \Omega}$: they considered the enhanced torque resulting from the sweeping out of magnetic field lines, but not the angular momentum of the outflow itself.
The dipole magnetic field inferred from $P$ and $\dot P$ depends on the persistent wind luminosity. Normalizing $L_{\rm A}$ to the persistent X-ray luminosity, $L_{\rm A} = L_{\rm A\,35}\times 10^{35}$ erg s$^{-1}$, one finds for SGR 1900+14, \[bstar\] B\_= 3 10\^[14]{} L\_[A35]{}\^[-1/2]{} ([2/3]{})\^[-1]{} I\_[45]{} ([610\^[-11]{}]{}) ([P5.16 ]{})\^[-1]{}
The surface dipole field of SGR 1900+14 is inferred to be less than $B_{\rm QED} =
4.4\times 10^{13}$ G only if $L_{\rm A} > 10^{37}$ erg s$^{-1}$. That is, the wind must be $\sim 30-100$ times more luminous than the time-averaged X-ray output of the SGR in either quiescent or bursting modes. Such a large wind luminosity may conflict with observational bounds on the quiescent radio emission of SGR 1900+14 (Vasisht et al. 1994; Frail, Kulkarni, & Bloom 1999). From these considerations alone (which do not involve the additional strong constraints from bursting activity) we find it difficutl to reconcile the observed spindown rate of SGR 1900+14 with dipole fields typical of ordinary radio pulsars (as suggested recently by Marsden, Rothschild, & Lingenfelter 1999).
Note also that the synchrotron nebula surrounding SGR 1806-20 (Frail & Kulkarni 1993), thought until recently to emanate from the SGR itself and to require a particle source of luminosity $\sim 10^{37}$ erg s$^{-1}$ (TD96), appears instead to be associated with a nearby luminous blue variable star discovered by Van Kerkwijk et al. (1995). The new IPN localization of the SGR source (Hurley et al. 1999b) is displaced by 12$''$ from the peak of the radio emission. There is no detected peak in radio emission at the revised location. Since the two SGRs have nearly identical $\dot{P}/P$, we estimate a dipole field $B_\star = 3\times 10^{14} \ L_{\rm A\,35}^{-1/2}$ G for SGR 1806-20.
During episodes of wind-aided spindown, the period grows exponentially: P(t) = [P]{} (t/\_[w]{}), if the luminosity $L_{\rm A}$ in outflowing Alfvén waves and relativistic particles remains constant. In this equation, $\tau_{\rm w} \equiv P/\dot{P} = I_\star c^{3/2}/
(\Lambda B_\star R_\star^3 L_{\rm A}^{1/2})$ is a characteristic braking time, and ${\cal P}$ is the rotation period at the onset of wind-aided spindown. If $L_{\rm A}$ has remained unchanged over the lifetime of the star, then ${\cal P}$ would be set by the condition that the Alfvén radius sit inside the light cylinder, ${\cal P} = 2\pi ( B_\star^2 R_\star^6/c^3L_{\rm A})^{1/4} =
1.9 \, L_{\rm A\,35}^{-1/4} \, (B_{\star\,14}/3)^{1/2}$ s (cf. eq. \[\[Ralf\]\]). (Here, $B_\star = 10^{14} B_{\star\,14}$ G is the polar magnetic field.)
The narrow distribution of spin periods in the SGR/AXP sources ($P = 5$—12 s) would be hard to explain if every source underwent this kind of extended exponential spindown; but the possibility cannot be ruled out in any one source. The total age of such a source would be \[sdage\] t = (P/ ) (P/ [P]{}) + t([P]{}), where $t({\cal P})$ is the time required to spin down to period ${\cal P}$. Notice that $\dot{P}\propto P$ at constant $L_{\rm A}$, as compared with $\dot{P}\propto P^{-1}$ in the case of magnetic dipole radiation (MDR). The net result is to [*lengthen*]{} the spindown age deduced from a given set of $P$ and $\dot{P}$, relative to the usual estimate $t_{\rm MDR} \equiv P/2\dot{P}$ employed for radio pulsars. Note also that $P/\dot P$ remains constant throughout episodes of wind-aided spindown.
Applying these results to SGR 1900+14 (eq. \[\[bstar\]\]), we would infer that wind-aided spindown has been operating for $(P/\dot P)\ln(P/{\cal P}) = 2700$ yrs (assuming a steady wind of luminosity $L_{\rm A\,35} = 1$). Its total age, including the age $t({\cal P})$ at the onset of wind-aided braking, would be $2700 + 1300 = 4000$ yrs. (This number only increases to 5600 yrs if $L_{\rm A}$ increases to $10^{36}$ erg s$^{-1}$.) This age remains uncomfortably short to allow a physical assocation with SNR G42.8+0.6: it would imply a transverse recoil velocity $V_{\perp} \approx 0.03 \,
(D/ 7 \, \hbox{kpc})\, c$ \[eq. (\[vtrans\])\].
The age of SGR 1900+14 can be much longer, and $V_\perp$ much smaller, if the accelerated spindown we now observe occurs only [*intermittently*]{} (eq. \[\[tactive\]\]). In the magnetar model, it is plausible that small-scale seismic activity and Alfvén-driven winds are only vigorous during transient episodes, which overlap periods of bursting activity (§4.4 below).
Connection with Anomalous X-ray Pulsars
---------------------------------------
If each magnetar undergoes accelerated spindown only for a fraction $\epsilon_{\rm active}\sim P/\dot Pt_{\rm SNR}\sim 0.25$ of its life (eq. \[\[tactive\]\]), then [*the observed SGRs should be outnumbered some $\epsilon_{\rm active}^{-1}\sim 4$ times by inactive sources that spin down at a rate $\dot P \leq P/2t_{\rm SNR}$*]{}.
The Anomalous X-ray Pulsars (AXPs) have been identified as such inactive SGRs (Duncan & Thompson 1996; TD96; Vasisht & Gotthelf 1997; Kouveliotou et al. 1998). Although harder to find because they do not emit bright bursts, 6 AXPs are already known in our Galaxy, as compared with 3 Galactic SGRs. Table 1 summarizes the spin behavior and age estimates of the two AXP sources that are presently associated with supernova remnants (1E2259+586 and 1E1841-045). Their characteristic ages are larger than those of SGRs 1900+14 and 1806-20.
The characteristic age of 1E2259+586 also appears to be much longer, by about an order of magnitude, than the age of the associated SNR CTB 109. From Wang et al. (1992), t\_[SNR]{} = 13,000 ([E\_[SN]{}0.4 10\^[51]{} ]{})\^[-1/2]{} ([n0.13 \^[-3]{}]{})\^[1/2]{} where $E_{\rm SN}$ is the supernova energy and $n$ is the ISM particle density into which the remnant has expanded. Such a large characteristic age has a few possible explanations in the magnetar model. First, the source may previously have undergone a period of wind-aided spindown that increased its period to $\sim 4$ times the value that it would have reached by magnetic dipole braking alone. Indeed, there is marginal evidence for an extended X-ray halo surrounding the source, suggesting recent output of energetic particles (Rho & Petre 1997).
Alternatively, the long characteristic age of 1E2259+586 could be caused by significant decay of the dipole field (TD93 §14.3 and 15.2); or by the alignment of a vacuum magnetic dipole with the axis of rotation (Davis & Goldstein 1970; Michel & Goldwire 1970). Episodes of seismic activity can increase the spindown torque in aligned rotators both by driving the conduction current above the displacement current in the outer magnetosphere, and by carrying off angular momentum in particles and waves. Indeed, the outer boundary of the rigidly corotating magnetosphere, calculated by Melatos (1997) to lie at a radius[^1] $R_{\rm mag}/R_\star = 1\times 10^3\,\gamma^{-1/5}
(B_\star/10^{14}~{\rm G})^{2/5}$, is contained well inside the speed of light cylinder, $R_{\rm lc}/R_\star = 3\times 10^4\,(P/6~{\rm s})$. Here, $\gamma$ is the bulk Lorentz factor of the streaming charges. There may be some tendency toward an initial alignment of ${\bf m}_\star$ and ${\bf\Omega}$ in rapidly rotating neutron stars that support a large scale $\alpha$-$\Omega$ dynamo. However, as we argue in §4.3, rapid magnetic field decay will generically force ${\bf m}_\star$ out of alignment with ${\bf\Omega}$ and the principal axes of the star.
The remarkable AXP 1841–045 discovered by Vasisht & Gotthelf (1997) is only $\sim 2000$ yr old, as inferred from the age of the counterpart supernova remnant (Gotthelf & Vasisht 1997). The ratio $t_{\rm MDR}/t_{\rm SNR}$ is consistent with unity, in contrast with all other magnetar candidates that have measured spindown and are associated with supernova remnants (Table 1). Of these sources, AXP 1841–045 is also unique in failing to show measurable variations in its spindown rate, X-ray luminosity, or X-ray pulse shape over 10 years (Gotthelf, Vasisht, & Dotani 1999); nor has it emitted any X-ray bursts, or evinced any evidence for a particle outflow through a radio synchrotron halo. These facts reinforce the hypothesis that [*departures from simple magnetic dipole breaking are correlated with internal activity in a magnetar*]{}, and suggest that inactive phases can occur early in the life of a magnetar.
Free Precession in SGRs and AXPs
--------------------------------
Magnetic stresses will distort the shape of a magnetar (Melatos 1999). The internal magnetic field generated by a post-collapse $\alpha$–$\Omega$ dynamo is probably dominated by a toroidal component (DT92; TD93). A field stronger than $\sim 100 \, B_{\rm QED}$ is transported through the core and deep crust of the neutron star on a timescale short enough for SGR activity (TD96). Such a magnetar is initially [*prolate*]{}, with quadrupole moment $\epsilon=1\times 10^{-5}\,(B_{\rm in}/100~B_{\rm QED})^2$ (Bonazzola & Gourgoulhon 1996). Rapid field decay may cause the magnetic moment ${\bf m}_\star$ to rotate away from the long principal axis $\hat{\bf z}$ of the star, irrespective of any initial tendency for these two axes to align. The distortion of the rotating figure of the star induced by the rigidity of the crust can be neglected when calculating the spin evolution of the star, as long as $B > 10^{12}\,(P/{\rm 1~s})^{-1}$ G (Goldreich 1970).
This hydromagnetic distortion gives rise to free precession on a timescale \[taupr\] \_[pr]{} = [2]{} = 210\^[-2]{} ([B\_[in]{}100 B\_[QED]{}]{})\^[-2]{} ([P6 [s]{}]{}). Even when the magnetosphere is loaded with plasma, the spindown torque will depend on the angle between ${\bf m}_\star$ and the angular velocity ${\bf\Omega}$. Free precession modulates this angle when ${\bf m}_\star$ is canted with respect to the long principal axis $\hat{\bf z}$, and so induces a periodic variation in the spindown torque. [*Observation of free precession in an SGR or AXP source would provide a direct measure of its total magnetic energy.*]{}
How may free precession be excited? In the case of a rigid vacuum dipole, free precession is damped by the radiation torque if the inclination between ${\bf m}_\star$ and $\hat{\bf z}$ is less than $55^\circ$ (Goldreich 1970). At larger inclinations, free precession is excited. In the more realistic case of a plasma-loaded magnetosphere, the rate at which free precession is excited or damped by electromagnetic and particle torques is, unfortunately, not yet known. An additional, [*internal*]{} excitation mechanism, which may be particularly effective in an active SGR, involves rapid transport of the field in short, intense bursts. This is a likely consequence of energetic flares like the March 5 or August 27 events, which probably have occurred $\sim 10^2$ times over the lifetimes of these sources. If the principal axes of the star are rearranged on a timescale less than $\tau_{\rm pr}$, then ${\bf\Omega}$ will not have time to realign with the principal axes and precession is excited. Only if the magnetic field is transported on a timescale longer than $\tau_{\rm pr}$, will ${\bf\Omega}$ adiabatically track the principal axes.
An interesting alternative suggestion (Melatos 1999) is that forced radiative precession in a magnetar drives the bumpy spindown of the AXP sources 1E2259+586 and 1E1048-593 on a timescale of years. When ${\bf m}_\star$ is not aligned with ${\bf\Omega}$, the asymmetric inertia of the corotating magnetic field induces a torque along ${\bf\Omega}\times{\bf m}_\star$ (Davis & Goldstein 1970). This near-field torque acts on a timescale $\tau_{\rm nf}$ that is $(\Omega R/c)$ times the electromagnetic braking time: \[taunf\] \_[nf]{} 0.3 ([B\_10 B\_[QED]{}]{})\^[-2]{}([P6 [s]{}]{}) . As long as $\tau_{\rm nf} < \tau_{\rm pr}$, this near-field torque drives an anharmonic wobble of the neutron star; in particular, Melatos (1999) considers the case where $\tau_{\rm nf} \sim \tau_{\rm pr}$. However, inspection of equations (\[taupr\]) and (\[taunf\]) suggests instead that $\tau_{\rm pr} \ll \tau_{\rm nf}$, because the magnetic energy is dominated by an internal toroidal component. In this case, the near-field torque averages to zero (Goldreich 1970). Note also that this mechanism is predicated on an evacuated inner magnetosphere, although the nonthermal spectra of SGRs and AXPs indicate that this may not be a good approximation (Thompson 1999). The model has the virtue of making clear predictions of the future rotational evolution of the AXPs, which will be tested in coming years.
Almost Constant Long-term Spindown
----------------------------------
We now address the near-uniformity of the long-term spindown rate of SGR 1900+14, before and after the August 27 outburst (Woods et al. 1999a; Marsden, Rothschild & Lingenfelter 1999; Paper I). It provides an important clue to any mechanism causing acceleration of the rate of spindown.
There appears to be no measurable correlation between bursting activity and long-term spindown rate (Paper I). This observation is consistent with the occurence of short, energetic bursts: the period increment caused by the release of a fixed amount of energy is smaller for outbursts of short duration $\Delta t$, scaling as $(\Delta t)^{1/2}$ (eq. \[\[deltap\]\]). The implied constancy of the magnetic dipole moment is also consistent with the energetic output of the August 27 burst: only $\simeq 0.01\,
(E_{\rm Aug~27}/10^{44}~{\rm erg})\,(B_\star/10\,B_{\rm QED})^{-2}$ of the exterior dipole energy need be expended to power the burst. Indeed, if the burst is powered by a large-scale magnetic instability, one infers, from this argument alone, that the dipole field cannot be much smaller than $10\,B_{\rm QED}$.
An additional clue comes from the bursting history of SGR 1806-20. In that source, the cumulative burst fluence grows with time, in a piecewise linear manner (Palmer 1999). This indicates that there exist many quasi-independent active regions in the star, each of which expends a fraction $\sim 10^{-5}$ of the total energy budget. The continuous output of waves and particles from the star is therefore the cumulative effect of many smaller regions. Nonetheless, the long term uniformity of $\dot P$ requires the rate of persistent seismic activity in the crust to remain carefully regulated over a period of years (or longer), even though the bursting activity is much more intermittent.
Persistent seismic activity is excited in a magnetar by the [*compressive*]{} mode of ambipolar diffusion of the magnetic field through the core (TD96). The resulting compressive transport of the magnetic field through the crust requires frequent, low energy ($E \sim 10^{35}$ erg) fractures of the crust induced by the Hall term in the electrical conductivity. The total energy released in magnetospheric particles has the same magnitude as the heat conducted out from the core to the stellar surface. The (orthogonal) rotational model of ambipolar diffusion will shear the crust. It can induce much larger fractures that create optically thick regions of hot $e^\pm$ plasma trapped by the stellar magnetic field (TD95). The strong intermittency of SGR burst activity appears to be closely tied to the energy distribution of SGR bursts, which is weighted toward the largest events (Cheng et al. 1996). This suggests that the rate of low-energy Hall fracturing will more uniform, being modulated by longer term variations in the rate of ambipolar diffusion through the neutron star core.
Nontheless, the modest variability observed in the short term measurements of $\dot P$ (Paper I) must be accounted for. Stochastic fluctuations in the rate of small-scale crustal fractures provide a plausible mechanism. An alternative source of [*periodic*]{}, short-term variability involves free precession in a magnetar whose dipole axis is tilted from the long principal axis (§4.3).
Although angular momentum exchange with the crustal superfluid is a promising mechanism to account for the $\Delta P/P\sim 10^{-4}$ period shift associated with the August 27 event, it is less likely to dominate long-term variations in the spindown rate. An order of magnitude increase in the spindown rate driven such exchange could persist only for a small fraction $\sim 10^{-1} I_{\rm sf}/I_\star \sim 10^{-3}$ of the star’s life. Moreover, a gradual deformation of the neutron star crust by magnetic stresses will remove angular momentum from the superfluid and decrease the rate of spindown.
Changes in the Persistent X-ray Flux and Lightcurve
===================================================
The persistent X-ray lightcurve of SGR 1900+14 measured following the August 27 event (Kouveliotou et al. 1999; Murakami et al. 1999) appears dramatically different from the pulse profile measured earlier: indeed, the profile measured following the burst activity of May/June 1998 (Kouveliotou et al. 1999) is identical to that measured in April 1998 (Hurley et al. 1999c) and September 1996 (Marsden, Rothschild & Lingenfelter 1999). Not only did the pulse-averaged luminosity increase by a factor 2.3 between the 1998 April 30 and 1998 September 17/18 ASCA observations (Hurley et al. 1999c; Murakami et al. 1999), but the lightcurve also simplified into a single prominent pulse, from a multi-pulsed profile before the August 27 flare. The brighter, simplified lightcurve is suggestive of enhanced dissipation in the active region of the outburst (Kouveliotou et al. 1999). We now discuss the implications of this observation for the dissipative mechanism that generates the persistent X-rays, taking into account the additional constraints provided by the period history of SGR 1900+14.
Magnetic Field Decay
--------------------
The X-ray output of a magnetar can be divided into two components (TD96): thermal conduction to the surface, driven by heating in the core and inner crust; and external Comptonization and particle bombardment powered by persistent seismic activity in the star. Both mechanisms naturally generate $\sim 10^{35}$ erg s$^{-1}$ in continuous output. The appearence of a thermal pulse at the surface of the neutron star will be delayed with respect to a deep fracture or plastic rearrangement of the neutron star crust, by the thermal conduction time of $\sim 1$ year (e.g. Van Riper, Epstein, & Miller 1991). By contrast, external heating will vary simultaneously with seismic activity in the star. We have previously argued that if 1E2259+586 is a magnetar, then the coordinated rise and fall of its [*two*]{} X-ray pulses (as observed by Ginga; Iwasawa et al. 1992) requires the thermal component of the X-ray emission to be powered, in part, by particle bombardment of two connected magnetic poles (TD96, §4.2).
Neither internal heating, nor variability in the rate of persistent seismic activity, appears able to provide a consistent explanation for the variable lightcurve of SGR 1900+14. Deposition of $\sim 10^{44}$ erg of thermal energy in the deep crust, of which a fraction $1-\epsilon$ is lost to neutrino radiation, will lead to an increased surface X-ray output of $\sim 3\times
10^{35}\,(\epsilon/0.1)$ erg s$^{-1}$. If, in addition, the heated deposited per unit mass is constant with depth $z$ in the crust, then the heat per unit area scales as $\sim z^4$; whereas the thermal conduction time varies weakly with $z$ at densities above neutron drip (Van Riper et al. 1991). The outward heat flux should, as a result, grow monotonically. This conflicts with the appearance of the new pulse profile of SGR 1900+14 no later than one day after the August 27 event. By the same token, a significant increase in persistent seismic activity – at the rate needed to power the increased persistent luminosity $L_{\rm X} \sim 1.5\times 10^{35}(D/7~{\rm kpc})^2$ erg s$^{-1}$ (Murakami et al. 1999) – would induce a measurable change in the spindown rate that was not observed.
The observations require instead a steady particle source that is confined to the inner magnetosphere. A large-scale deformation of the crust of the neutron star, which likely occured during the August 27 outburst, must involve a horizontal twisting motion (§3). If this motion were driven by [*internal*]{} magnetic stresses,[^2] then the external magnetic field lines connected to the rotating patch would be twisted with respect to their opposite footpoints (which we assume to remain fixed in position). We suppose that the twist angle decreases smoothly from a value $\theta_{\rm max}$ at the center of the patch to its boundary at radius $a$. This means that a component of the twist will remain even after magnetic reconnection eliminates any tangential discontinuities in the external magnetic field resulting from the motion. The current carried by the twisted bundle of magnetic field is I , where $\Phi = \pi a^2 B_\star$ is the magnetic flux carried by the bundle and $L$ is its length.
The surface of an AXP or SGR is hot enough ($T \sim 0.5$ keV) to feed this current via thermionic emission of $Z < 12$ ions from one end of the flux bundle, and electrons from the other end. In magnetic fields stronger than $Z^3\alpha_{\rm em}^2 B_{\rm QED} = 4\times
10^{13}\,(Z/26)^3$ G, even iron is able to form long molecular chains. The cohesion energy per atom is = 1.52([BZ\^3\_[em]{}\^2B\_[QED]{}]{})\^[0.37]{} -[724]{}\^2. In this expression, the first term is the binding energy per atom in the chain (Neuhauser, Koonin, & Langanke 1987; Lai, Salpeter, & Shapiro 1992), from which we subtract the binding energy of an isolated atom (Lieb, Solovej, & Yngvason, 1992). Thermionic emission of ions is effective above a surface temperature T\_[thermionic]{} . Substituting $B = 10\,B_{\rm QED} = 4.4\times 10^{14}$ G, one finds that $T_{\rm thermionic}$ remains well below 0.5 keV for $Z < 12$, but grows rapidly at higher $Z$. Thus, the surface of a magnetar should be an effective thermionic emitter for a wide range of surface compositions.
We can now estimate the energy dissipated by the current flow. The kinetic energy carried by ions of charge $Z$ and mass $A$ is L\_[ion]{} = ([AZ]{}) [Im\_pe]{} = 310\^[35]{}[\_[max]{}AZ]{} ([B\_10B\_[QED]{}]{})([LR\_]{})\^[-1]{} ([a0.5 [km]{}]{})\^2 . Here, $\phi \simeq g_\star R_\star = GM_\star/R_\star$ is the gravitational potential that the charges have to climb along the tube, and we assume $M_\star = 1.4\,M_\odot$, $R_\star = 10$ km. Note that the particle flow estimated here is large enough to break up heavy nuclei even where the outflowing current has a positive sign: electrons returning from the opposite magnetic footpoint are energetic enough for electron-induced spallation to be effective (e.g. Schaeffer, Reeves, & Orland 1982).
On what timescale will this twist decay? Each charge accumulates a potential energy $A m_p g z$ a height $z$ above the surface of the neutron star. Equating this energy with the electrostatic energy released along the magnetic field, one requires a longitudinal electric field $E = Am_p g/Ze$. The corresponding electrical conductivity is = [Ia\^2 E]{} = ([Z\_[max]{}8A]{}) [eBcm\_p g\_L]{}, and the ohmic decay time is t\_[ohmic]{} = [4L\^2c\^2]{} = ([Z\_[max]{}2A]{}) [eB\_Lm\_p g\_c]{} = 300([Z\_[max]{}A]{}) ([B\_10 B\_[QED]{}]{})([L10 [km]{}]{}) . This timescale agrees with that obtained by dividing the persistent luminosity $L_{\rm ion}$ into the available energy of the twisted magnetic field. Further twisting of the field lines would prolong or shorten the lifetime of the current flow.
A static twist in the surface magnetic field will not produce a measurable increase in the torque because the current flow is contained well inside the Alfvén radius (eq. \[\[raqui\]\]). The particles that carry the current lose their energy to Compton scattering and surface impact on a timescale $\sim R_\star/c$ or shorter. By contrast, a persistent flux of low amplitude Alfvén waves into the magnetosphere causes the wave intensity to build up, until the wave luminosity transported beyond the Alfvén radius balances the continuous output of the neutron star (TB98). Thus, the particle flow induced by a localized twist in the magnetic field lines supplements the particle output associated with persistent seismic activity occuring over the larger volume of the star.
Evidence Against Persistent Accretion
-------------------------------------
Direct evidence that the persistent X-ray output of SGR 1900+14 is [*not*]{} powered by accretion comes from measurements one day after the August 27 outburst (Kouveliotou et al. 1999). The increase in persistent $L_{\rm X}$ is not consistent with a constant spindown torque, unless there was a substantial change in the angular pattern of the emergent X-ray flux following the burst. In addition, the radiative momentum deposited by that outburst on a surrounding accretion disk would more than suffice to expel the disk material, out to a considerable distance from the neutron star. In such a circumstance, the time to re-establish the accretion flow onto the neutron star, via inward viscous diffusion from the inner boundary $R_{\rm in}$ of the remnant disk, would greatly exceed one day.[^3] Let us consider this point in more detail.
The accretion rate (assumed steady and independent of radius before the outburst) is related to the surface mass density $\Sigma(R)$ of the hypothetical disk via M = [2R\^2 (R) t\_[visc]{}(R)]{}. The viscous timescale is, as usual, t\_[visc]{}(R) \_[SS]{}\^[-1]{}([H(R)R]{})\^[-2]{} ([R\^3GM\_]{})\^[1/2]{}, where $H(R)$ is the half-thickness of the disk at radius $R$ and $\alpha_{\rm SS} < 1$ is the viscosity coefficient (Shakura & Sunyaev 1973). Balancing the radiative momentum incident on a solid angle $\sim 2\pi (2H/R)$ against the momentum $\sim \pi \Sigma(R) R^2(2GM_\star/R)^{1/2}$ of the disk material moving at the escape speed, and equating the persistent X-ray luminosity $L_{\rm X}$ with $GM_\star\dot M/R_\star$, one finds \[tvisc\] t\_[visc]{} = [E\_[Aug 27]{}L\_[X]{}]{} ([2GM\_R\_c\^2]{})\^[1/2]{} ([R\_[in]{}R\_]{})\^[1/2]{} ([H(R\_[in]{})R\_[in]{}]{} ).
The most important factor in this expression is the ratio of burst energy to persistent X-ray luminosity, $E_{\rm Aug~27}/L_{\rm X} =
30~(E_{\rm Aug~27}/10^{44}~{\rm erg})\,(L_{\rm X}/10^{35}~{\rm erg~s^{-1}})^{-1}$ yr. The timescale is long as the result of the enormous energy of the August 27 flare, and the relatively weak persistent X-ray flux preceding it. It is interesting to compare with Type II X-ray bursts from the Rapid Burster and GRO J1744-28, which are observed to be followed by dips in the persistent emission (Lubin et al. 1992; Kommers et al. 1997). These bursts, which certainly are powered by accretion, involve energies $\sim 10^4$ times smaller and a persistent source luminosity that is $10^2-10^3$ times higher. Indeed, the dips in the persistent emission following the Type II bursts last for only 100-200 s, consistent with the above formula.
Now let us evaluate eq. (\[tvisc\]) in more detail. At a fixed $\dot M$, the surface mass density of the disk increases with decreasing $\alpha_{\rm SS}$, and so a conservative upper bound on $t_{\rm visc}$ is obtained by choosing $\alpha_{\rm SS}$ to be small. (Note that eq. (\[tvisc\]) depends implicity on $\alpha_{\rm SS}$ only through the factor of $R_{\rm in}^{1/2} \propto \alpha_{\rm SS}^{1/2}$.) For the observed parameters $E_{\rm Aug~27} \simeq 10^{44}$ erg (Mazets et al. 1999) and $L_{\rm X} = 10^{35}$ erg s$^{-1}$ (before the August 27 outburst; Hurley et al. 1999a), one finds $R_{\rm in} = 1\times 10^{10}$ cm when $\alpha_{\rm SS} = 0.01$. The corresponding thickness of the gas-pressure dominated disk is (Novikov & Thorne 1973) $H(R_{\rm in})/R_{\rm in}
\simeq 5\times 10^{-3}$. The timescale over which the persistent X-ray flux would be re-established is extremely long, $t_{\rm visc} \simeq 10$ yr.
One final note on disk accretion. There is no observational evidence for a binary companion to any SGR or AXP (Kouveliotou 1999). Because of its large recoil velocity (eq. \[\[vtrans\]\]), SGR 1900+14 almost certainly could not remain bound in a binary system. A similar argument applies to the other giant flare source, SGR 0526–66 (DT92). Thus, any accretion onto SGR 1900 +14 would have to come from a fossil disk. To remain bound, the initial radius of such a disk must be less than $GM_\star/V_{\rm rec}^2 \sim 10^4$ km, for stellar recoil velocity $V_{\rm rec} \sim (3/2)^{1/2} V_\perp$ \[eq. (\[vtrans\])\]. The behavior of a passively spreading remnant disk appears inconsistent with the measured spin evolution of the AXP and SGR sources (Li 1999).
A trigger involving sudden accretion of an unbound planetesimal (Colgate and Petschek 1981) is not consistent with the log-normal distribution of waiting periods between bursts (Hurley et al. 1994) in SGR 1806-20. An internal energy source is also indicated by the power-law distribution of burst energies, with index $dN/dE \sim E^{-1.6}$ similar to the Gutenburg-Richter law for earthquakes (Cheng et al. 1996). In addition, the mass of the accreted planetesimals must exceed $\sim 1/30$ times the mass of the Earth’s Moon in the case of the March 5 and August 27 events. It is very difficult to understand how the accretion of a baryon-rich object could induce a fireball as clean as the initial spike of these giant flares (TD95, §7.3.1). When $B_\star \ll 10^{14}$ G, only a tiny fraction $(B_\star/B_E)^2$ of the hydrostatic released would be converted to magnetic energy; here, $B_E \sim 10^{14}$ G is the minimum field needed to directly power the outburst.
Conclusions
===========
The observation (Paper I) of a rapid spindown associated with the August 27 event, $\Delta P/P = +1\times 10^{-4}$, provides an important clue to the nature of SGR 1900+14. We have described two mechanisms that could induce such a rapid loss of angular momentum from the crust and charged interior of the star. The torque imparted by a relativistic outflow during the August 27 event is proportional to $B_\star$, but falls short by an order of magnitude even if $B_\star \sim 10\,B_{\rm QED} = 4.4\times 10^{14}$ G. Only if SGR 1900+14 released an additional $\sim 10^{44}$ erg for an extended period $\sim 10^4$ s immediately following the August 27 outburst would the loss of angular momentum be sufficient. (The integrated torque increases with the duration $\Delta t$ of the outflow as $(\Delta t)^{1/2}$; eq. \[\[deltap\]\].)
The alternative model, which we favor, involves a glitch driven by the violent disruption of the August 27 event. The unpinned neutron superfluid will absorb angular momentum if it starts out spinning more slowly than the rest of the star – the opposite of the situation encountered in glitching radio pulsars. We have argued that a slowly spinning neutron superfluid is the natural consequence of magnetic stresses acting on the neutron star crust. A gradual, plastic deformation of the crust during the years preceding the recent onset of bursting activity in SGR 1900+14 would move the superfluid out of co-rotation with the rest of the star, and slow its rotation. The magnitude of the August 27 glitch can be crudely estimated by scaling to the largest glitches of young, active pulsars with similar spindown ages and internal temperatures. Depending on the object considered, one deduces $|\Delta P|/P \sim 10^{-5}-10^{-4}$.
This model for the August 27 period increment has interesting implications for the longer-term spindown history of the Soft Gamma Repeaters and Anomalous X-ray Pulsars. It suggests that these objects can potentially glitch, with or without associated bursts, and that $P$ will suddenly shift [*upward,*]{} rather than downward as in radio pulsar glitches. By the same token, an accelerated rate of plastic deformation within a patch of the neutron star crust will force the superfluid further out of co-rotation and induce a transient (but potentially resolvable) [*spin-up*]{} of the crust (TD96). The magnitude of such a ‘plastic spin-up’ event (eq. \[\[deltajsf\]\]), could approach that inferred for the August 27 event, but with the usual (negative) sign observed in radio pulsar glitches. Indeed, RXTE spin measurements provide evidence for a rapid spin-up of the AXP source 1E2259+586 (Baykal & Swank 1996), to the tune of $\Delta P/P = -3\times 10^{-5}$. Transient variations in the persistent X-ray flux of the AXP 1E2259+586, which were not associated with any large outbursts, also require transient plastic deformations of the neutron star crust (TD96).
The rapid spindown rate of SGR 1900+14 during the past few years, $\dot P = 6\times 10^{-11}$ s/s, indicates that this SGR is a transient phase of [*accelerated spindown*]{}, with stronger braking torques than would be produced by simple magnetic dipole radiation (Kouveliotou et al. 1999). Such accelerated spindown can be driven by magnetically-induced seismic activity, with small-scale fractures powering a steady relativistic outflow of magnetic vibrations and particles. This outflow, when channeled by the dipole magnetic field, carries away the star’s angular momentum. A very strong field, $B_\star \gg B_{\rm QED}$, is required to give a sufficiently large “lever arm" to the outflow.
Further evidence for episodic accelerated spindown comes from the two AXPs that are directly associated with supernova remnants: 1E2259+586 and 1E1841-045 (§4.2). The characteristic ages $P/2 \dot P$ of these stars are [*longer*]{} than the the ages of the associated supernova remnant, and also longer than the characteristic ages of the SGRs. This suggests that the AXPs are magnetars observed during phases of seismic inactivity.
The constancy of the long-term spindown rate before and after the bursts and giant flare of 1998 (Woods et al. 1999a; Marsden, Rothschild & Lingenfelter 1999; Paper I) gives evidence that the spindown rate correlates only weakly with bursting activity. It is easy to understand why short, intense bursts are not effective at spinning down a magnetar: the Alfvén radius (the length of the “lever arm") decreases as the flux of Alfvén waves and particles increases.
A persistent output of waves and particles could be driven by the compressive mode of ambipolar diffusion in the liquid neutron star interior (TD96). As the magnetic field is forced through the crust, the Hall term in the electrical conductivity induces many frequent, small fractures ($\Delta E \sim 10^{35}$ erg). By contrast, large fractures of the crust are driven by shear stresses that involve the orthogonal (rotational) mode of ambipolar diffusion. The greater intermittency of bursting activity is a direct consequence of the dominance of the total burst fluence by the largest bursts (Cheng et al. 1996).
Forced radiative precession could cause a short-term modulation of the spindown rate in a magnetar (Melatos 1999), but this requires an evacuated magnetosphere that may not be consistent with the observed non-thermal spectra of the SGR and AXP sources (Thompson 1999). We have argued that transport of the neutron star’s magnetic field will deform the principal axes of the star and induce [*free*]{} precession. The resulting modulation of the spindown torque has an even shorter timescale (eq. \[\[taupr\]\]), and is potentially detectable.
A twist in the exterior magnetic field induced by a large scale fracture of the crust will force a persistent thermionic current through the magnetosphere (§5). The resulting steady output in particles would explain the factor $\sim 2.3$ increase in the persistent X-ray flux of SGR 1900+14 immediately following the August 27 event (Murakami et al. 1999) if $B_\star \sim 10B_{\rm QED}$ and the twist is through $\sim 1$ radian. In this model, the simplification of the lightcurve – into a single large pulse – is due to concentrated particle heating at the site of the August 27 event.
We conclude by emphasizing the diagnostic potential of coordinated measurements of spectrum, flux, bursting behavior and period derivative. When considered together, they constrain not only the internal mechanism driving the accelerated spindown of an SGR source, but also the mechanism powering its persistent X-ray output. For example: an increase in surface X-ray flux will be delayed by $\sim 1$ year with respect to an episode of deep heating (e.g. Van Riper et al. 1991); whereas a shearing and twisting of the external magnetic field of the neutron star will drive a simultaneous increase in the rate of external particle heating (TD96). The magnetar model offers a promising framework in which to interpret these observations.
Alpar, M.A., Anderson, P.W., Pines, D., & Shaham J. 1984, ApJ 276, 325
Alpar, M.A., Chau, H.F., Cheng, K.S., & Pines D. 1996, ApJ, 459, 706
Baykal, A. & Swank, J.H. 1996, ApJ, 460, 470
Blaes, O., Blandford, R., Goldreich, P., & Madau, P. 1989, ApJ, 343, 829
Bonazzola, S. & Gourgoulhon, E. 1996, A& A, 312, 675
Cheng, B., Epstein, R.I., Guyer, R.A., & Young, C. 1996, Nature, 382, 518
Cline, T., et al. 1982, ApJ, 255, L45
Colgate, S.A. & Petchek, A.G. 1981, ApJ, 248, 771
Davis, L. & Goldstein, M. 1970 ApJ, 159, L81
Duncan, R. & Thompson, C. 1992, ApJ, 392, L9 (DT92)
Duncan R.C. & Thompson C. 1996, in AIP Conf. Proc. 366, High Velocity Neutron Stars and Gamma Ray Bursts, ed. R.E. Rothschild & R.E. Lingenfelter (Woodbury: AIP), 111
Feroci, M., Frontera, F., Costa, E., Amati, L., Tavani, M., Rapisarda, M., & Orlandini, M. 1999, ApJ, 515, L9
Frail, D.A., Kulkarni, S.R., & Bloom, J.S. 1999, Nature, 398, 127
Goldreich, P. 1970, ApJ, 160 L11
Gotthelf, E.V. & Vasisht, G. 1997, ApJ, 486, L133
Gotthelf, E.V., Vasisht, G., & Dotani, T. 1999, ApJ Letters, in press ([astro-ph 9906122]{})
Heyl, J.S. & Hernquist, L. 1999, MNRAS, 304, L37
Heyl, J.S. & Kulkarni, S.R. 1998, ApJ, 506, L61
Hurley, K., Sommer, M., Kouveliotou, C., Fishman, G., Meegan, C., Cline, T., Boer, M., & Niel, M. 1994, ApJ, 431, L31
Hurley, K., Kouveliotou, C., Fishman, G., Meegan, C., Briggs, M., Van Paradijs, J., Cline, T., Kommers, J., Lewin, W., Boer, M., & Niel, M. 1996, ApJ, 463, L13
Hurley, K., Cline, T., Mazets, E., Barthelmy, S., Butterworth, P., Marshall, F., Palmer, D., Aptekar, R., Golenetskii, S., Il’inski, V., Frederiks, D., McTiernan, J., Gold, R., & Trombka, J. 1999a, Nature, 397, 41
Hurley, K., Kouveliotou, C., Woods, P., Cline, T., Butterworth, P., Mazets, E., Golenetski, S., & Frederics, D. 1999b, ApJ, 510, L107
Hurley, K., Li, P., Kouveliotou, C., Murakami, T., Ando, M., Strohmayer, T., Van Paradijs, J., Vrba, F., Luginbuhl, C., Yoshida, A., & Smith, I. 1999c, ApJ, 510, L111
Hurley, K., Kouveliotou, C., Woods, P., Mazets, E., Golenetskii, S., Frederiks, D.D., Cline, T., & Van Paradijs J. 1999d, ApJ, 519, L143
Inan, U.S., Lehtinen, N.G., Lev-Tov, S.J., Johnson, M.P., Bell, T.F., & Hurley, K. 1999, Geophys. Res. Lett., in press
Iwasawa, K., Koyama, K., & Halpern, J.P. 1992, PASJ, 44 9
Kommers, J.M., Fox, D.W., Lewin, W.H.G., Rutledge, R.E., Van Paradijs, J., & Kouveliotou, C. 1997, ApJ, 482, L53
Kouveliotou, C., Dieters, S., Strohmayer, T., van Paradijs, J., Fishman, G.J., Meegan, C.A., Hurley, K., Kommers, J., Smith, I., Frail, D. & Murakami, T. 1998, Nature, 393, 235
Kouveliotou, C. 1999, Proc. Natl. Acad. Sci, 96, 5351
Kouveliotou, C., Strohmayer, T., Hurley, K., van Paradijs, J., Finger, M.H., Dieters, S., Woods, P., Thompson, C. & Duncan, R.C. 1999, ApJ, 510, L115
Kulkarni, S., & Frail, D. 1993, Nature, 365, 33
Kulkarni, S., Frail, D.A., Kassim, N.E., Murakami, T., & Vasisht, G. 1994, Nature, 368, 129
Lai, D., Salpeter, E.E., & Shapiro, S.L. 1992, Phys. Rev. A., 45, 4832
Li, X.-D. 1999, ApJ, in press ([astro-ph 9903190]{})
Lieb, E.H., Solovej, J.P., & Yngvason, J. 1992, ApJ, 69, 749
Link, B., Epstein, R.I., & Baym, G. 1993, ApJ, 403, 285
Link, B. & Epstein, R.I. 1996, ApJ, 457, 844
Lubin, L.M., Lewin, W.H.G., Rutledge, R.E., Van Paradijs, J., Van der Klis, M., & Stella, L. 1992, MNRAS, 258, 759
Mazets, E.P., Cline, T., Aptekar, R.L., Butterworth, P., Frederiks, D.D., Golenetskii, S.V., Il’inskii, V.N., & Pal’shin, V.D. 1999, Astronomy Letters, in press
Marsden, D., Rothschild, R.E., & Lingenfelter, R.E. 1999, ApJ, 520 L107
Melatos, A. 1997, MNRAS, 288, 1049
Melatos, A. 1999, ApJ, 519, L77
Mereghetti, S., Israel, G.L., & Stella, L. 1998, MNRAS 296, 689
Michel, F.C. & Goldwire, H.C. 1970, Ap Lett, 5, 21
Murakami, T., Tanaka, Y. Kulkarni, S.R., Ogasaka, Y. Sonobe, T., Ogawara, Y., Aoki, T., & Yoshida, A. 1994, Nature, 368, 127
Murakami, T., Kubo, S., Shibazaki, N., Takeshima, T., Yoshida, A., & Kawai, N. 1999, ApJ, 510, L119
Neuhauser, D., Koonin, S.E., & Langanke, K. 1987, Phys. Rev. A., 36, 4163
Novikov, I.D. & Thorne, K.S. 1973, in Black Holes, C. DeWitt & B. DeWitt, eds., (Gordon and Breach; New York)
Paczyński, B. 1992, Acta Astron. 42, 145
Palmer, D.M. 1999, ApJ, 512, L113
Reisenegger, A. & Goldreich, P. 1992, ApJ, 395, 240
Rho, J. & Petre, R. 1997, ApJ, 484, 828
Ruderman, M. 1991, ApJ, 382, 576
Ruderman, M.A., Zhu, T., & Chen, K. 1998, ApJ, 492, 267
Schaeffer, R., Reeves, H., & Orland, H. 1982, ApJ, 254, 688
Shakura, N.I. & Sunyaev, R.A. 1973, A& A, 24, 337
Smith, D.A., Bradt, H.V., & Levine, A.M. 1999, ApJ, 519, L147
Thompson, C. 1997, in Gamma-Ray Bursts, proceedings of the Fourth Huntsville Symposium, eds. C.A. Meegan, R.D. Preece, and T.M. Koshut (AIP: New York), p. 944
Thompson, C. 1999, in preparation
Thompson, C. & Duncan, R. 1993, ApJ, 408, 194-217 (TD93)
Thompson, C. & Duncan, R. 1995, MNRAS, 275, 255-300 (TD95)
Thompson, C. & Duncan, R. 1996, ApJ, 473, 322-342 (TD96)
Thompson, C. & Blaes, O. 1998, Phys Rev D, 57, 3219 (TB98)
Van Kerkwijk, M.H., Kulkarni, S.R., Matthews, K., & Neugebauer, G. 1995, ApJ, 444, L33
Van Riper, K.A., Epstein, R.I., & Miller, G.S. 1991, ApJ, 381, L47
Vasisht, G. & Gotthelf, E.V. 1997, ApJ, 486, L129
Vasisht, G., Kulkarni, S.R., Frail, D.A., & Greiner, J. 1994, ApJ, 431, L35
Wang, Z., Qu, Q., Luo, D., McCray, R., & MacLow, M. 1992, ApJ, 388, 127
Woods, P.M., Kouveliotou, C., van Paradijs, J., Finger, M.H., & Thompson, C. 1999a, ApJ, 518, L00
Woods, P.M., Kouveliotou, C., van Paradijs, J., Hurley, K., Kippen, R.M., Finger, M.H., Briggs, M.S., Dieters, S. & Fishman, G.J. 1999b, ApJ, 519, L139
Woods, P.M., Kouveliotou, C., van Paradijs, J., Finger, M.H., Thompson, C., Duncan, R.C., Hurley, K., Strohmayer, T., Swank, J., & Murakami, T. 1999c, ApJ, in press ([astro-ph 9907173]{})
[^1]: When the displacement current dominates the conduction current.
[^2]: A sudden unwinding of an external magnetic field could release enough energy to power the March 5 (or August 27) event, but it was argued in TD95 that the timescale $\sim R_\star/c \sim 10^{-4}$ s would be far too short to explain the width of the initial $\sim 0.2$ s hard spike. A pulse broadened by a heavy matter loading would suffer strong adiabatic losses and carry a much greater kinetic energy than is observed in $\gamma$-rays. Shearing of the external magnetic field requires internal motions that will, in themselves, trigger a large outburst by fracturing the crust.
[^3]: This estimate of the viscous timescale is conservative for two reasons: First, if the binding energy of the disk material were balanced with the incident radiative energy, the inner boundary of the remnant disk would like at even larger radius. Second, the central X-ray source may puff up the disk, which increases $\tau_{\rm visc}$ (eq. \[\[tvisc\]\]).
|
---
abstract: 'In the present work we study the Fermi–Pasta–Ulam (FPU) $\beta $–model involving long–range interactions (LRI) in both the quadratic and quartic potentials, by introducing two independent exponents $\alpha_1$ and $\alpha_2$ respectively, which make the [forces decay]{} with distance $r$. Our results demonstrate that weak chaos, in the sense of decreasing Lyapunov exponents, and $q$–Gaussian probability density functions (pdfs) of sums of the momenta, occurs only when long–range interactions are included in the quartic part. More importantly, for $0\leq \alpha_2<1$, we obtain extrapolated values for $q \equiv q_\infty >1$, as $N\rightarrow \infty$, suggesting that these pdfs persist in that limit. On the other hand, when long–range interactions are imposed only on the quadratic part, strong chaos and purely Gaussian pdfs are always obtained for the momenta. We have also focused on similar pdfs for the particle energies and have obtained $q_E$-exponentials (with $q_E>1$) when the quartic-term interactions are long–ranged, otherwise we get the standard Boltzmann-Gibbs weight, with $q=1$. The values of $q_E$ coincide, within small discrepancies, with the values of $q$ obtained by the momentum distributions.'
author:
- |
**Helen Christodoulidi$^{1}$, Tassos Bountis$^{1,2}$, Constantino Tsallis$^{3,4}$ and Lambros Drossos$^{5}$**\
$^{1}$Center for Research and Applications of Nonlinear Systems,\
University of Patras, GR-26500 Patras, Greece.\
$^2$Department of Mathematics, Nazarbayev University,\
Kabanbay-Batyr 53, 010000 Astana, Republic of Kazakhstan\
$^3$Centro Brasileiro de Pesquisas Fisicas and\
National Institute of Science and Technology for Complex Systems,\
Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, Brazil\
$^4$Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA\
$^5$High Performance Computing Systems and Distance Learning Lab,\
Technological Educational Institute of Western Greece,\
GR-26334 Patras, Greece
title: 'Dynamics and Statistics of the Fermi–Pasta–Ulam $\beta $–model with different ranges of particle interactions'
---
Introduction {#intro}
============
In recent years, many authors have examined the effect of long–range interactions on the dynamics of multi–dimensional Hamiltonian systems [@Ruffo95; @Ruffo98; @AnteneodoTsallis1998; @Dauxois2002; @Tarasov06; @Ginelli1; @Ginelli2; @Thanos]. Perhaps the best known example in this class is the so–called Hamiltonian Mean Field model, where the maximal Lyapunov exponent (MLE) was shown numerically to decrease with increasing number of degrees of freedom $N$, according to a specific power law [@Ruffo98; @AnteneodoTsallis1998; @Ginelli1; @Ginelli2]. More recently, another famous example in this category, the FPU $\beta $–Hamiltonian was studied in the presence of long–range interactions. In the complete absence of harmonic terms the MLE appears to vanish in the thermodynamic limit [@Bagchi]. Moreover, when harmonic terms are included in the potential, a similar behavior of the MLE is observed [@CTB], which nevertheless tends to saturate to a non–zero value above a characteristic size $N$.
It is the purpose of the present paper to investigate more thoroughly the FPU $\beta $–model from this point of view, by studying the effect of the interactions through two parameters $\alpha_1$ and $\alpha_2$ introduced in the quadratic and quartic terms of the potential respectively. In so doing, we are able to identify domains of strong and weak chaos, by examining whether probability density functions (pdfs) of sums of the momenta obey Boltzmann Gibbs (BG) statistics or not.
The maximal Lyapunov exponent and other indicators of local dynamics [@GALI] provide useful tools for chaos detection, but are not well suited for distinguishing between different degrees of weak vs. strong chaos [@bountis_book]. For example, if a given orbit is chaotic, its MLE is expected to converge to a positive value. However, if the orbit is trapped for a long time near islands of regular motion, the MLE does not quickly converge and when it does, one cannot tell from its value whether the dynamics can be described as weakly or strongly chaotic.
Now, long–range systems are known to possess long–living quasi–stationary states (QSS) [@Leo1; @Leo2; @Leo3; @PluchinoRapisardaTsallis2007; @PluchinoRapisardaTsallis2007b; @CirtoAssisTsallis2013; @uma1; @uma2; @uma3; @uma4; @uma5; @uma6; @uma7; @TsallisPlastinoAlvarezEstrada2009], whose statistical properties are very different from what is expected within the framework of classical BG thermostatistics [@Gibbs1902]. More specifically, when one studies such QSS in the spirit of the central limit theorem, one finds that the pdfs of sums of their variables are well approximated by $q$–Gaussian functions (with $1<q<3$) or $q$–statistics [@ABB2011; @Tsallis2009; @Tsallis1988; @GellMannTsallis2004; @Tsallis2014]. These pdfs last for very long times beyond which they are expected to tend to the $q=1$ case of pure Gaussians and BG thermal equilibrium. Thus, we will treat the index $q$ as a measure of the “distance” from a Gaussian, and study its time evolution to identify when a “phase transition” will occur from weak chaos and $q>1$–statistics to strong chaos and BG thermostatistics.
In this context, it becomes highly relevant to examine the effect of the range of the interactions on the lifetime of a QSS, and hence the duration of weakly chaotic dynamics. To this end, we recently introduced and studied numerically a generalization of the FPU $\beta$–model, in which we varied the interaction range by multiplying the quartic terms of the potential by coupling constants that decay with distance as $r^{-\alpha}$ [@CTB]. The pdfs of the time–averaged momenta were thus found to be well approximated by $q$–Gaussians with $q>1$, when the range is long enough (i.e. $\alpha<1$). This, however, lasts up to a crossover time $t=t_c$ at which $q$ starts to decrease monotonically to 1, reflecting the transition from $q$–statistics to BG thermostatistics.
In the present paper, we extend our study and investigate additional properties connected with the occurrence of weakly chaotic QSS, taking a closer look at their dynamics as well as associated statistics. In particular, we consider the FPU $\beta$–chain [@FPU] of $N$ particles, whose potential includes harmonic as well as quartic interactions, and employ two different exponents $\alpha_1$ and $\alpha_2$, for the $r^{-\alpha}$ coupling constants of the quadratic and quartic terms respectively. Furthermore, we vary these exponents independently to investigate their effect on the thermostatistics of the orbits at increasingly long times. A recent study on long–range interactions applied only on the harmonic terms can be found in [@Ruffo15].
In Section \[sec2\], we write the Hamiltonian of our model as the sum of its kinetic and potential energies and explain the two exponents that determine the range of the interactions. Next, in Section \[sec3\], we present a detailed study on the behavior of the maximal Lyapunov exponents when long–range is applied either to the quadratic (linear LRI) or the quartic (nonlinear LRI) part of the potential. We choose random initial conditions and compare the dynamics and statistics of these cases computing the pdfs of the sums of their momenta for sufficiently long times.
In Section \[sec4\], we examine the value of $q$, and other parameters on which the pdf depends, focusing especially on the thermodynamic limit, where the total energy $E$ and $N$ tend to infinity at fixed specific energies $\varepsilon=E/N$. We thus discover, for $0\leq \alpha_2<1$, a *linear* relation between $q$ and $1/ \log N$, which allows us to extrapolate the value of $q=q_{\infty }$ in the limit $N\rightarrow\infty$. Since we thus end up with values $q_{\infty }>1$, we conclude that $q$–Gaussian pdfs behave as if they were *attractors* and hence that $q$–statistics prevails over BG thermostatistics in that limit. Finally in Section \[sec5\] we present our conclusions.
The FPU $\beta$–model with different ranges of interaction\[sec2\]
==================================================================
Let us consider the famous Fermi–Pasta–Ulam $\beta $–model of a 1–dimensional lattice of $N$ nonlinearly coupled oscillators governed by the Hamiltonian $$\label{FPU}
{\cal H}_{FPU}= \frac{1}{2}\sum_{n=1}^{N} p_n^2 + \sum_{n=0}^N V_2(x_{n+1}-x_n) + \sum_{n=0}^N V_4(x_{n+1}-x_n) ~~,$$ involving nearest–neighbor interactions, where $V_2$ and $V_4$ represent the quadratic and quartic functions $V_2(u) =a u^2/2$ and $V_4(u)=b u^4 /4$. The $p_n,x_n$ are the canonical conjugate pairs of momentum and position variables assigned to the $nth$ particle, with $n=1,2,...,N$ and fixed boundary conditions, i.e. $x_0=x_{N+1}=p_0=p_{N+1}=0$.
In this paper we modify the above classical form of the FPU $\beta$–model by introducing the parameters $\alpha_1 $ and $\alpha_2$, which enter in the linear and nonlinear parts of the equations of motion, to determine the particle interactions that decay with distance as $1/r^{\alpha _1}$ and $1/r^{\alpha _2}$ respectively. In particular, the modified Hamiltonian function that describes the generalized FPU $\beta$–system has the form $$\begin{aligned}
\label{ham}
{\cal H}_{LRI}=\frac{1}{2}\sum_{n=1}^{N} p_n^2 + \frac{a}{2\widetilde N_1} \sum_{n=0}^{N} \sum_{m=n+1}^{N+1} \frac{(x_n-x_m)^2}{(m-n)^{\alpha_1}} + \frac{b}{4\widetilde N_2} \sum_{n=0}^{N} \sum_{m=n+1}^{N+1} \frac{(x_n-x_m)^4}{(m-n)^{\alpha_2}} ~~,\end{aligned}$$ where $a$ and $b$ are positive constants.
Note that there are three ways to introduce long–range interactions in our model: (a) only in the quadratic potential $V_2$, (b) only in the quartic potential $V_4$ and (c) both in $V_2$ and $V_4$. Case (b) was the one studied in [@CTB] and gave the results mentioned above, where a “phase transition” occurs between $q$–statistics and BG thermostatistics near the value $\alpha_2=1$ that separates the short term $\alpha_2>1$ from the long term $0\leq\alpha_2<1$ interaction range.
As explained above, the critical value $\alpha_1 = \alpha_2= 1$ is expected to determine the crossover between long and short–range interactions. When $\alpha_i <1$, $i=1,2$ the interactions are long range, with the lower bound $\alpha_i =0$ signifying that each particle interacts equally with all others, exactly as in a fully connected network. In contrast, when $\alpha_i > 1$, $i=1,2$, the interactions are short–range and in the limit $\alpha_i \rightarrow \infty $ only the nearest neighbor terms survive in the sums and the classical form of the FPU $\beta $–Hamiltonian is recovered.
The rescaling factors $\widetilde N_i$, $i=1,2$ in (\[ham\]) are given by the expression $$\begin{aligned}
\label{factor}
{\widetilde N_i}(N,\alpha_i) \equiv \frac{1}{N} \sum_{n=0}^{N} \sum_{m=n+1}^{N+1} \frac{1}{(m-n)^{\alpha_i}}, ~~(i=1,2) \end{aligned}$$ and are necessary for making the Hamiltonian extensive. Indeed, without this factor the sums of $V_2$ and $V_4 $ in (\[ham\]) would increase as $O(N^2)$ in the thermodynamic limit, thus rendering the kinetic energy (which grows like $N$) irrelevant [@CTB]. Notice that $\widetilde N_i \simeq 1$ in the limit $\alpha _i\rightarrow \infty $, and thus for large $N$ Hamiltonian (\[ham\]) reduces to Hamiltonian (\[FPU\]).
Conditions for weak chaos and $q$–thermostatistics\[sec3\]
==========================================================
Linear versus nonlinear long–range interactions\[sus\]
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It has been known for some time (see e.g. [@Ruffo98; @AnteneodoTsallis1998]) that Hamiltonian systems possessing LRI display a more organized behavior in the thermodynamic limit. It is also well established that the maximal Lyapunov exponent $\lambda $ of the FPU $\beta$–system converges to a positive constant in the thermodynamic limit, that depends only on the coupling constant $b$ and the system’s specific energy $\varepsilon $. What happens, however, when linear and/or nonlinear interactions between distant particles are taken into account? Choosing to work on the FPU $\beta$–model in the present paper, provides the advantage of making a direct comparison between linear and nonlinear LRI and allows us to examine in detail their effect on the system’s dynamics.
Note that system (\[ham\]) can be studied in the presence of only linear LRI by taking $\alpha _2 \rightarrow \infty $ and letting $\alpha _1$ act as a free parameter that controls the range of interaction. This means that the potential $V_4$ in that case is equivalent to the one used in the classical FPU $\beta$–model. By contrast, if we wish to study the effect of nonlinear LRI alone, we take $\alpha _1 \rightarrow \infty $ and $\alpha _2$ becomes the free parameter.
Let us now display in Fig.\[LE\] the behavior of the maximal Lyapunov exponent $MLE=\lambda $, in the above cases, first in terms of the system size $N$ and then as a function of the specific energy $\varepsilon$ ($a=1$ in every case). These values tend to stabilize and converge to a constant value for times greater than $10^4$, therefore we stopped our simulations at $t=10^6$. Panels (a) and (b) are for $b=1$, $\varepsilon =1$ and $b=10$, $\varepsilon =9$ respectively. Both of them include values $\lambda _{FPU}$ of the classical FPU $\beta$–model as a point of reference, which separate the two LRI cases: Below $\lambda _{FPU}$ we find the maximal Lyapunov exponents $\lambda_{V4}$ of the nonlinear LRI case, while above we encounter the $\lambda_{V2}$ exponents. Note that the longer the range of interaction, the higher the $ \lambda_{V2} $ values. For $\alpha_1=\alpha_2 =10$ the $\lambda_{V2} $ and $\lambda_{V4} $ curves collapse to the $\lambda _{FPU}$ values.
In Fig.\[LE\](a) the MLEs $\lambda_{V2}$ and $\lambda_{V4}$ grow very slowly and even tend to saturate as $N\rightarrow \infty $. By contrast, in Fig.\[LE\](b) for $\varepsilon=9$ and $b=10$, this tendency is reversed in the case of the $\lambda_{V4}$ exponents which are seen to decrease with $N$. The reason this is not observed in Fig.\[LE\](a) is because it requires high $b \varepsilon $ values, as pointed out already in [@CTB]. An additional remark is that, for $\alpha_2=0$ the exponents $\lambda_{V4}$ of Fig.\[LE\](b) seem not to vanish for $N\rightarrow \infty $, but tend to saturate at a positive value. Only when the quadratic part $V_2$ ($a=0$ in (\[ham\])) is completely eliminated from the Hamiltonian, the Lyapunov exponents continue to fall to zero as $N$ keeps increasing (see [@Bagchi] for a detailed numerical study of this issue). So, what are the $b \varepsilon$ values that yield a power–law decrease of $\lambda_{V4}$ vs. $\varepsilon$? When is the system weakly chaotic and why? To find out we have computed the MLEs at various specific energies, keeping the parameter $b=1$ fixed and the number of particles $N=8192$. As Fig.\[LE\](c) clearly shows, linear LRI (represented by the upper curve of squares) make the system much more chaotic than the classical nearest neighbor FPU case (represented by the middle curve of circles). We believe that this is due to the fact that the implementation of LRI on the linear part of the Hamiltonian results in a ‘compression’ of the phonon band $\omega_k = 2 \sin \frac{k \pi }{2(N+1)}$ of the nearest neighbor case ($\alpha_1=\infty$) from the interval $[0,2]$ to a single point with frequency $\Omega =\sqrt{2(N+2)/(N+1)}$, as $\alpha_1$ tends to zero. This suggests that no sizable region of quasiperiodic tori exists to sustain regular motion, while the periodic oscillations of the lattice (with frequency $\Omega$) that become unstable due to the presence of nonlinear terms should have large scale chaotic regions about them that dominate the dynamics in phase space.
On the other hand, when LRI apply only to the nonlinear part of the Hamiltonian and the harmonic terms are of the nearest neighbor type it is interesting to compare the chaotic behavior of the system with that of the classical FPU model. As Fig.\[LE\](c) clearly demonstrates, the corresponding MLE curves are very close to each other when $\varepsilon$ is small, but begin to deviate considerably for $\varepsilon >1$. Indeed, the application of nonlinear LRI is characterized by much weaker chaos in the limit $\varepsilon\rightarrow \infty $ as its MLE behaves like $\lambda_{V4} \sim \varepsilon^{0.05}$, in contrast with the FPU model whose corresponding MLE grows a lot faster, as $\lambda_{FPU} \sim\varepsilon ^{1/4}$ (see [@casetti] for an analytical derivation). It is also interesting to note that in this limit the exponents $\lambda_{V2}$ and $\lambda_{FPU}$ become indistinguishable.
Remarkably, the above picture of the maximal Lyapunov exponent $\lambda_{V4} $ slowing down its increase as the specific energy grows (see the triangles in Fig.\[LE\](c)) is accompanied by the emergence of $q$–Gaussian distributions in the momenta associated with the presence of weakly chaotic behavior. In the next subsection we examine this phenomenon more carefully as we concentrate our study on the statistical aspects of the LRI models.
Emergence of q–Gaussian distributions
-------------------------------------
Besides this striking difference of the level of chaoticity in the above two (short and long–range) situations, there is also a remarkable difference in their statistics, as we now explain.
![Log–log plots of the MLE: (a) For increasing $N$, with $b=1$ and $\varepsilon=1$, (b) as $N$ increases with $b=10$ and $\varepsilon=9$ and (c) as a function of $\varepsilon$ at $N=8192$, $b=1$, for 3 cases: An upper curve of black squares for LRI on $V_2$ only with $(\alpha_1,\alpha_2)=(0,\infty)$, the middle one of red circles for the classical FPU case with $(\alpha_1,\alpha_2)=(\infty,\infty)$ and a lower one of blue triangles for LRI on $V_4$ only for $(\alpha_1,\alpha_2)=(\infty,0)$. []{data-label="LE"}](LE_b1_Energy1.eps "fig:"){width="0.25\linewidth"} ![Log–log plots of the MLE: (a) For increasing $N$, with $b=1$ and $\varepsilon=1$, (b) as $N$ increases with $b=10$ and $\varepsilon=9$ and (c) as a function of $\varepsilon$ at $N=8192$, $b=1$, for 3 cases: An upper curve of black squares for LRI on $V_2$ only with $(\alpha_1,\alpha_2)=(0,\infty)$, the middle one of red circles for the classical FPU case with $(\alpha_1,\alpha_2)=(\infty,\infty)$ and a lower one of blue triangles for LRI on $V_4$ only for $(\alpha_1,\alpha_2)=(\infty,0)$. []{data-label="LE"}](LE_b10_Energy9.eps "fig:"){width="0.25\linewidth"} ![Log–log plots of the MLE: (a) For increasing $N$, with $b=1$ and $\varepsilon=1$, (b) as $N$ increases with $b=10$ and $\varepsilon=9$ and (c) as a function of $\varepsilon$ at $N=8192$, $b=1$, for 3 cases: An upper curve of black squares for LRI on $V_2$ only with $(\alpha_1,\alpha_2)=(0,\infty)$, the middle one of red circles for the classical FPU case with $(\alpha_1,\alpha_2)=(\infty,\infty)$ and a lower one of blue triangles for LRI on $V_4$ only for $(\alpha_1,\alpha_2)=(\infty,0)$. []{data-label="LE"}](LE_energy.eps "fig:"){width="0.5\linewidth"}
The pdfs we study correspond to the momenta $p_1(t),\ldots,p_N(t)$ of orbits starting from a uniform distribution at $t=0$. In particular, these are time averaged pdfs which are evaluated at discrete times $t_j>t_0,j=1,2,\dots$, where $t_0$ is the time at which the kinetic energy has stabilized. The step $\tau =t_{j+1}-t_j$ between these discrete times should be appropriately chosen so as to avoid possible correlations. We then assign to the $i$–th momentum band (i.e. $i$–th column of our histograms) the number of times where the momentum for each single particle falls in. In the four panels of Fig. \[first\] typical momentum histograms are shown, which correspond to the four representative cases we study, as different combinations of short and long–range interactions are applied on $V_2$ and $V_4$. More specifically, in panels (a) and (c) a classical Gaussian shape is observed, either under purely short–range interactions or when LRI apply only to the quadratic part, by setting $\alpha_1=0.7$ and $\alpha_2\rightarrow \infty $ in the Hamiltonian (\[ham\]). Instead in the panels (b) and (d) a clear $q$–Gaussian shape emerges when long–range applies to the quartic interactions, independently of the type of interactions in the quadratic part, i.e. for $\alpha_1\rightarrow \infty $, $\alpha_2=0.7$ and $\alpha_1=0.7$, $\alpha_2=0.7$ in (\[ham\]) respectively. These plots have been evaluated at $\tau =2$ time steps within the interval $[10^5, 5\cdot 10^5]$. Furthermore, as we see from the plots of Fig. \[klk\], the distributions of the time averaged individual particle energies (i.e. $E_n= {1\over 2} p_n^2 + \sum_m V(x_n - x_m)$, which are evaluated after the same time interval as in Fig. \[first\] and for the same initial conditions, show very similar behavior to the momenta distributions: $q$–Exponentials appear when the LRI are nonlinear.
![ The momentum distributions for $N=2048$ particles for the system (\[ham\]). The upper panels show the cases: $\alpha_1 \rightarrow \infty$, $\alpha_2 \rightarrow \infty$, i.e. FPU (left) and $\alpha_1 \rightarrow \infty$, $\alpha_2=0.7$ (right). Lower panels show: $\alpha_1=0.7$, $\alpha_2 \rightarrow \infty$ (right) and $\alpha_1=\alpha_2=0.7$ (right). The yellow lines correspond to the uniform distribution, from which the momenta where randomly extracted.[]{data-label="first"}](h_all.eps){width="0.58\linewidth"}
![ The particle energy distributions for the 4 cases of Fig. \[first\]; [ notice that the fittings (black continuous curves, along the upper bounds of the numerical histograms) exhibit BG distributions ($q_E=1$) when the quartic interactions are short–ranged, and $q_E$-exponential distributions ($q_E>1$) when they are long–ranged]{}. \[klk\]](Energies.eps){width="0.58\linewidth"}
In practice, we have employed an algorithm which uses the least squares method to calculate $q$, determines the intercept and estimates $\beta $ from the slope of the resulting straight line. Dividing then the $q$ interval \[1,3\] into 1000 possible values, we apply the least squares method to all of them. The appropriate $q$ value is chosen as the one corresponding to the minimum standard error and is estimated with an accuracy of at least 3 digits. On the level of $q$-statistics the $\beta$ parameter entering our distributions corresponds to an inverse effective temperature characterizing the width of the distribution. In other words, if this fact is viewed within a thermostatistical framework where an entropy and a partition function can be defined, one expects $\beta=1/kT$, where $T$ corresponds to the temperature and $k$ is Boltzmann constant [@Tsallis2009].
As is evident from these results, the mechanism of LRI drives the system’s behavior away from BG statistics, only if the quartic potential is long–range. Instead, when LRI apply only to the quadratic part, purely Gaussian pdfs are obtained.
In what follows, we examine which of the system’s fundamental parameters affect the shape of the $q$–Gaussian pdfs and how the interaction range of the quadratic part of the potential influences the system’s behavior.
Variation of $q$ for different ranges and system parameters\[sec4\]
===================================================================
As is well–known, $q$–Gaussian distributions are often associated with weak chaos and are linked to QSS which persist for very long times, until the system achieves energy equipartition at complete thermalization. In such cases, there always exists a value of time $t_c$ beyond which the system passes to a strongly chaotic state characterized by $q=1$ and BG thermostatistics[@ABB2011; @bountis_book]. This is also what happens with all weakly chaotic states in the LRI FPU-$\beta$ model and that is why we refer to them as QSS. Thus, our system is different in this regard from the $\alpha$–XY model studied in [@CirtoAssisTsallis2013], for which $q$-Gaussian distributions are found to persist even in the so-called BG regime.
, also evaluated at $\varepsilon=9$. It is not transparent, however, if this is coincidental or due to some deeper explanation related to the fact that the particle energy includes the particle kinetic term. \[qvse\]](q_vs_E_new.eps){width="0.45\linewidth"}
![Momentum distributions for the system with $b=10, \varepsilon=9, \alpha _2=0.7$ and various $N$ values. Note how the pdfs are described by a $q$–Gaussian of higher index $q$ as $N$ grows. More specifically, $q$ ranges from $1.17$ for $N=512$ until $1.25$ for $N=8192$.[]{data-label="Ndep"}](variousN.eps){width="0.55\linewidth"}
![ (a) The linear dependence of $q$ on $1/log N$ for $N=4096, 8192, 16384$ depicted here provides an estimate for $q_{\infty}$ in the thermodynamic limit, [ as $\alpha_2$ changes, with $\alpha_1=\infty$]{}. (b) The values of $q_{\infty }$ are plotted here versus $\alpha _2$. We have not included $\alpha_2>0.8$ in the above results due to the ‘noisy’ behavior of $q$ in the neighborhood $\alpha_2=1$. Nevertheless, for $\alpha_2$ above 1.4, we definitively obtain $q=1$. ($\varepsilon=9$ in both panels.) \[uni\]](q_vs_logN_new.eps "fig:"){width="0.38\linewidth"} ![ (a) The linear dependence of $q$ on $1/log N$ for $N=4096, 8192, 16384$ depicted here provides an estimate for $q_{\infty}$ in the thermodynamic limit, [ as $\alpha_2$ changes, with $\alpha_1=\infty$]{}. (b) The values of $q_{\infty }$ are plotted here versus $\alpha _2$. We have not included $\alpha_2>0.8$ in the above results due to the ‘noisy’ behavior of $q$ in the neighborhood $\alpha_2=1$. Nevertheless, for $\alpha_2$ above 1.4, we definitively obtain $q=1$. ($\varepsilon=9$ in both panels.) \[uni\]](Inter_new.eps "fig:"){width="0.38\linewidth"}
Our main purpose here is to investigate numerically the dependence of $q$ on the system size $N$, the specific energy $\varepsilon$ and the coupling constant $b$ of the Hamiltonian (\[ham\]). Let us mention at the outset that the parameters $\varepsilon$ and $b$ are not independent. Indeed, a simple rescaling of the Hamiltonian shows that the relevant parameter is $b \varepsilon$.
Let us plot in Fig.\[qvse\] the dependence of $q$ on $\varepsilon$ for $ N=2048$ and $\alpha_1\rightarrow\infty$. It turns out that $q$ fluctuates around 1.23 and displays a greater tendency to converge as the specific energy increases. It is important to note that when linear LRI are added to the nonlinear LRI the motion is still weakly chaotic and the index $q$ of the associated pdfs remains unaffected.
On the other hand, when the system size increases, the value of $q$ no longer remains a constant but also increases with $N$. From Fig. \[Ndep\] it becomes evident that the corresponding $q$–Gaussian representing the statistics of the model spreads as $N$ grows. Thus, choosing $\alpha_2 =0.7$, $a=1, b=10$ and $\varepsilon =9$ we find that the momentum histogram for low values of $N$ is very close to a Gaussian, as shown in Fig. \[Ndep\] for $N=256$. It then deviates for $N=512$ to a $q$–Gaussian with $q(N=512)=1.17$, which further increases to $q(N=1024)=1.19$ and so on, as the weakly chaotic properties of the dynamics become more evident.
Asymptotic behavior of $q$ in the limit $N\rightarrow \infty$
-------------------------------------------------------------
Extrapolating the value of $q$ in the limit $N\rightarrow \infty $, we can now estimate the asymptotic value $q=q_{\infty }$ and also vary $\alpha_2$ to determine the dependence of $q_{\infty }$ on the interaction range applied to the quartic part of the potential at the thermodynamic limit. To this end, we consider a given value of $\alpha_2<1$ and systematically calculate the $q$ dependence on $N$. In Fig. \[uni\](a) we plot these $q$ values versus $1/ \log N$ and find that their dependence is accurately described by the following expression: $$\begin{aligned}
\label{qvsN}
q(N,\alpha_2) = q_{\infty } (\alpha_2) - c(\alpha_2)/ \log N ~~ ,\end{aligned}$$ where $c(\alpha_2 )$ is some constant. Each of the data in Fig. \[uni\](a) has been plotted after performing 3 independent realizations of the momentum distributions and taking their average in the time window $[10^5, 5\cdot 10^5]$.
This appears to be an important result because it shows that the $q_{\infty } (\alpha_2)$ obtained from Fig. \[uni\](a) by the intercept of the straight line Eq. (\[qvsN\]) with the vertical axis (as $N\rightarrow\infty$) is larger than 1, which implies that the $q$–Gaussians are attractors in that limit. Next, plotting $q_{\infty } (\alpha_2)$ vs. $\alpha_2$ in Fig. \[uni\](b), we observe that it starts from $5/3$ for $\alpha _2 =0$, and then, after about $\alpha_2 =0.2$, falls linearly towards 1. In particular, for $0.2 \leq \alpha_2 \leq 0.8$ the values of $q_{\infty } (\alpha_2)$ decrease as $q_{\infty } (\alpha_2) = 1.79 -0.475\alpha_2$. Concerning the left limit, we need to recall that $q$-Gaussians have finite variance for $q$ up to $5/3$. Therefore, this value constitutes a natural candidate for quartic infinitely long interactions, i.e. $\alpha_2=0$, as evidenced also by numerical results obtained in [@CTB; @Tsallis2009].
Note that the value of $q$ reaches unity at $\alpha_2 =1.5$ and not at the expected $\alpha_2 =1$ threshold between short and long–range interactions. This is a very interesting phenomenon and may be explained by the fact that $q$ takes a very long time to converge to 1 over the range $1 \leq \alpha_2 \leq 1.4$.
Conclusions\[sec5\]
===================
In the present paper a generalization of the 1-dimensional Fermi-Pasta-Ulam $\beta$–model was studied, where two non–negative exponents $\alpha_1$ and $\alpha_2$ are introduced in the quadratic $V_2$ and quartic $V_4$ part of the potential to control the range of interactions. The role of long–range interactions on the system’s dynamical properties as well as its statistical behavior were examined in detail. In particular we concluded that only when LRI apply on $V_4$ and at high enough $b \varepsilon$ values: (a) the maximal Lyapunov exponent $\lambda_{V4}$ decreases as a power-law with $N$ [@CTB] and increases very slowly, as $\lambda_{V4} \sim \varepsilon^{0.05}$ ($b=1$) with the specific energy, while at the same time (b) $q$–Gaussian pdfs of the momenta appear with $q>1$. Both of these results indicate that LRI on $V_4$ is a necessary condition for what we call weak chaos in the FPU $\beta$–model, especially at high energies.
On the contrary, when LRI are applied to the harmonic part of the potential a much stronger type of chaos is encountered if the nonlinear interactions are short–range. This is especially evident at low energies, indicating that the transition to large scale chaos occurs at much lower levels than in the classical FPU case. The corresponding MLE=$\lambda_{V2}$ tends to saturate as a function of $N$, while, for small $\varepsilon$, $\lambda_{V2}$ is much higher than the MLE=$\lambda_{FPU}$ of the classical FPU model, with $\lambda_{FPU}$ tending to $\lambda_{V2}$ from below as $\varepsilon\rightarrow\infty$. All this is related to momentum pdfs of the purely Gaussian type ($q=1$) and is associated with strong chaos and BG thermostatistics.
We also focused on the value of $q$ in the momentum pdfs, when the main parameters of the problem vary. It turns out that $q$ changes with $N$ and $\alpha _2$ and not with $\alpha _1$. On the other hand, when $0\leq \alpha_2<1$ we find a *linear* relation between $q$ and $1/ \log N$, which allows us to extrapolate the value of $q$ to $q_{\infty}>1$ at $N\rightarrow\infty$. This is important because it suggests that under these conditions BG thermostatistics no longer holds and $q$–Gaussian and $q$–exponential pdfs with $q>1$ describe the true statistics in the thermodynamic limit.
#### Acknowledgments
We are especially indebted to the referees for their many detailed comments and remarks that helped us improve significantly our manuscript. One of us (C.T.) gratefully acknowledges partial financial support by the Brazilian Agencies CNPq and Faperj, and by the John Templeton Foundation (USA). All of us acknowledge that this research has been co-financed by the European Union (European Social Fund–ESF) and Greek national funds through the Operational Program ‘Education and Lifelong Learning’ of the National Strategic Reference Framework (NSRF) - Research Funding Program: [*Thales. Investing in knowledge society through the European Social Fund*]{}.
[99]{}
Antoni M and Ruffo S, 1995 [*Phys. Rev. E*]{} **52** 2361-2374.
Latora V, Rapisarda A and Ruffo S, 1998 [*Phys. Rev. Lett*]{} **80** 692.
Anteneodo C and Tsallis C, [*Phys. Rev. Lett.*]{} [**80**]{} 5313.
Dauxois T, Latora V, Rapisarda A, Ruffo S and Torcini A, 2002 Lecture Notes in Physics, edited by Dauxois T, Ruffo S, Arimondo E, Wilkens M, Vol. 602 p. 458.
Tarasov V E and Zaslavsky G M 2006 [*Commun. Nonlinear Sci. Numer. Simul.*]{} **11** 885-898.
Takeuchi K A, Chat[é]{} H, Ginelli F, Politi A and Torcini A 2011 [*Phys. Rev. Lett.*]{} **107** 124101.
Ginelli F, Takeuchi K A, Chat[é]{} H, Politi A and Torcini A 2011 [*Phys. Rev. E*]{} **84** 066211.
Manos Th and Ruffo S 2011 [*Trans. Theor. and Stat. Phys.* ]{} **40** 360-381.
Bagchi D and Tsallis C [ Sensitivity to initial conditions of $d$-dimensional long–range interacting Fermi-Pasta-Ulam model: Universal scaling]{}, 1509.04697 \[cond-mat.stat-mech\].
Christodoulidi H, Tsallis C and Bountis T 2014 *EPL* [**108**]{} 40006.
Skokos H, Bountis T and Antonopoulos Ch 2007 *Physica D* [**231**]{} 30-54.
Bountis T and Skokos H 2012 [*Complex Hamiltonian Dynamics and Statistics*]{}, Springer Series in Synergetics, Berlin.
Leo M, Leo R A and Tempesta P 2010 [*J. Stat. Mech.*]{} P04021.
Leo M, Leo R A, Tempesta P and Tsallis C 2012 [*Phys. Rev. E*]{} [**85**]{} 031149.
Leo M, Leo R A and Tempesta P 2013 [*Annals Phys.*]{} [**333**]{} 12-18.
Pluchino A, Rapisarda A and Tsallis C 2007 [*Europhys. Lett.*]{} [**80**]{} 26002.
Pluchino A, Rapisarda A and Tsallis C 2008 [*Europhys. Lett.*]{} [**83**]{} 30011.
Cirto L J L, Assis V and Tsallis C [*Physica A*]{} [**393**]{} (2014) 286-296.
Umarov S, Tsallis C and Steinberg S 2008 [*Milan J. Math.*]{} [**76**]{} 307-328.
Umarov S, Tsallis C, Gell-Mann M and Steinberg S 2010 [*J. Math. Phys.*]{} [**51**]{} 033502.
Hahn M G, Jiang X X and Umarov S [*J. Phys. A*]{} [**43**]{}, (16) (2010) 165208. Hilhorst H J 2010 [*J. Stat. Mech.*]{} P10023. Jauregui M, Tsallis C and Curado E M F 2011 [*J. Stat. Mech.*]{} P10016. Jauregui M and Tsallis C, 2011 [*Phys. Lett. A*]{} [**375**]{} 2085-2088. Plastino A and Rocca M C, 2012 [*Milan J. Math.*]{} [**80**]{} 243-249.
Tsallis C, Plastino A and Alvarez-Estrada R F , 2009 [*J. Math. Phys.*]{} [**50**]{} 043303.
Gibbs J W, [*Elementary Principles in Statistical Mechanics – Developed with Especial Reference to the Rational Foundation of Thermodynamics*]{} (C. Scribner’s Sons, New York, 1902; Yale University Press, New Haven, 1948; OX Bow Press, Woodbridge, Connecticut, 1981).
Antonopoulos Ch, Bountis T and Basios V 2011 [*Physica A*]{} [**390**]{} 3290-3307.
Tsallis C, [*Introduction to Nonextensive Statistical Mechanics - Approaching a Complex World*]{} (Springer, New York, 2009).
Tsallis C, 1988 [*Stat. Phys.*]{} [**52**]{} 479 \[First appeared as preprint in 1987: CBPF-NF-062/87, ISSN 0029-3865, Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro\].
Gell-Mann M and Tsallis C, eds., [*Nonextensive Entropy - Interdisciplinary Applications*]{} (Oxford University Press, New York, 2004).
Tsallis C 2014 Contemporary Physics [**55**]{} 179-197.
Fermi E, Pasta J and Ulam S 1955 Los Alamos, Report No. LA-1940. See also: Newell A C, [*Nonlinear Wave Motion*]{}, Lectures in Applied Mathematics [**15**]{} 143-155 (Amer. Math. Soc., Providence, 1974); Berman G P and Izrailev F M, 2005 [*Chaos*]{} **15** 015104.
Miloshevich G, Nguenang J P, Dauxois T, Khomeriki R and Ruffo S 2015 [*Phys. Rev. E*]{} [**91**]{} 032927.
Casetti L, Livi R and Pettini M 1995 [*Phys. Rev. Lett.*]{} **74** 375–378.
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abstract: 'There is the possibility in principle that the noncompensated Casimir force exists in open nanosized metal cavities arranged in the form of periodic structures. It is found that when trapezoid cavities are strictly periodic all the Casimir expulsion forces are completely compensated. However, when the distance of the gap is formed between the cavities, in the periodic configuration a noncompensated expulsion force proportional to the number of cavities appears. There are such effective parameters of the periodic configuration (the angles of the opening of cavities, their lengths and the relationships between them) which lead to the appearance of a maximum of expulsion forces per unit of structure length.'
author:
- 'EvgenyG.Fateev'
title: Casimir expulsion of periodic configurations
---
In Ref. [@Fateev:2012] the possibility in principle is shown that the noncompensated Casimir force can exist in open nanosized metal cavities. The effect is theoretically demonstrated for a single trapezoid configuration. The force manifests itself as the time-constant expulsion of open cavities in the direction of their least opening. The optimal parameters of the angles of opening (broadening) of the cavities’ generetrices and their lengths are found, at which the expulsion force is maximal. It should be noted that the force differs significantly from expulsion forces capable of creating effects of levitation-type over bodies-partners [@Jaffe:2005; @Leonhardt:2007; @Levin:2010; @Rahi:2010; @Rahi:2011]. The question arises if the existence of noncompensated expulsion forces is possible in periodic structures based on trapezoid configurations possessing the effect of expulsion. A particular case of trapezoid configurations is Casimir parallel mirrors [@Casimir:1948; @Casimir:1949] which do not possess an effective expulsion force [@Fateev:2012].
Let us consider a periodic configuration with trapezoid cavities as an illustration of the possibility of the Casimir expulsion force existence. Note that a single cavity is understood as an open thin-walled metal shell with one or several outlets. The inner and outer surfaces of the cavity should have the properties of perfect mirrors. The cavity should entirely be immerged into a material medium or be a part of the medium with the parameters of dielectric permeability being different from those of physical vacuum. In Cartesian coordinates the configuration looks like two thin metal plates with the surface width $L$ (oriented along the $z$-axis) and length R, which are situated at a distance $a$ from one another; the angle $2\varphi $ of the opening of the generating lines of cavities between the plates can be varied (by the same value $\varphi $ imultaneously for both wings of the trapezoid cavity) as it is shown in Fig..
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{width="2.5in" height="1.0in"}
The periodic configuration with trapezoid figures in the Cartesian coordinates looks like it is shown in Fig. . Each figure in the configuration period is similar to a single figure in Fig. . Between the ends of the figures in the period there is a distance of the gap $d$.
For each figure the expulsion force in the $x$-direction can be found in the first approximation in the form [@Fateev:2012] $$\label{eq1}
F_x =\int\limits_0^L {dy} \int\limits_0^R {P_x (\varphi ,\Theta ,r)} \,dr.$$ Here, the local specific force of expulsion at each point $r$ on the cavity wing with the length $R$ and width $L$ is $$\label{eq2}
\begin{gathered}
{P_x}(r) = \frac{{\hbar c{\pi ^2}}}{{{{240}^{}}{s^4}}}\int\limits_{{\Theta _1}}^{{\Theta _2}} {} \sin {(\Theta - 2\varphi )^4}\cos (\Theta - \varphi )d\Theta \\
= - \frac{{\hbar c{\pi ^2}}}{{{{240}^{}}{s^4}}}{A}(\varphi ,{\Theta _1},{\Theta _2}), \\
\end{gathered}$$ where $$\label{eq3}
\begin{gathered}
A (\varphi ,\Theta _1 ,\Theta _2 ) \\
=\frac{1}{240}\Bigl[ 90\sin (\varphi -\Theta _1 )-90\sin (\varphi -\Theta _2 ) \\
+60\sin (3\varphi -\Theta _2 )-60\sin (3\varphi -\Theta _1 ) \\
+20\sin (5\varphi -3\Theta _2 )-20\sin (5\varphi -3\Theta _1 ) \\
+5\sin (7\varphi -3\Theta _1 )-5\sin (7\varphi -3\Theta _2 ) \\
+3\sin (9\varphi -5\Theta _1 )-3\sin (9\varphi -5\Theta _2 )\Bigr].
\end{gathered}$$ In formula (\[eq2\]), $\hbar =h/2\pi$ is reduced Planck constant, c is light velocity, and the functional expressions for limit angles $\Theta _1$, $\Theta _2$ (see Fig. ) in the trapezoid cavity and the parameter $s$ are $$\label{eq4}
\begin{gathered}
\Theta_1 =\mbox{arccos}\Biggl\{-(r+a\sin \varphi - R\cos 2\varphi )\\
\times\Bigl[(a+R\sin \varphi +r\sin \varphi)^2\\
+(r\cos \varphi -R\cos \varphi )^2 \Bigr]^{-\frac{1}{2}}\Biggr\},
\end{gathered}$$ $$\label{eq5}
\Theta _2 =\mbox{arccos}\left[ {-\frac{r+a\sin \varphi }{\sqrt
{a^2+r^2+2ra\sin \varphi } }} \right],$$ and $$\label{eq6}
s=\frac{\sin (2\varphi -\Theta _2 )(a+r\sin \varphi )}{\sin (\varphi -\Theta _2)}.$$
When such cavities are being arranged in the periodic structure, the following should be kept in mind. The periodic arrangement of $n$ trapezoid cavities being at the distance $d$ from one another, which sides with the widest opening are directed against the $x$ axis, leads to the formation of $n$-1 cavities with oppositely directed openings (see Fig. ). In this case, a wing (one of the surfaces of the trapezoid cavity) of each cavity is a wing of the other cavity, the opening of which is oppositely directed. Thus, for $n$ cavities periodically arranged along the $y$ axis we can write the expression for the total force of expulsion along the$ x$ axis $$\label{eq7}
F_{\sum} =nF_x (a)-(n-1)F_x (d).$$ Here, $F_x (a)$ is the force along the $x$ axis for the distance $a$ between the nearest ends of the cavities, and $F_x (d)$ is, respectively, the force for the distance of the gap $d$ between periods instead of $a$ in formulas (\[eq1\]-\[eq6\]). From formula (\[eq7\]) it is clear that for $d = a$ in the strictly periodic configurations the expulsion force is $F_{\sum} \to F_x (a)$ at $n\to \infty $. That is even at $n\to \infty $ the thrust force in a strictly periodic structure always remains at the level of expulsion forces for a single cavity and is directed to the least opening of the cavity wings. It means that in the configuration the thrust force will be created which is directed against the $x$-direction. However, it is clear that at $d\ne a$ the expulsion force of the periodic configuration will not remain at the same level and will depend on the $d/a$ relation according to formula (\[eq7\]) for different angles $\varphi $ of the opening of cavities as it is shown in Fig.. When the number $n$ of trapezoid cavities in the periodic configuration is growing, the character of the curves will be similar to that of the curves presented; however, of course, their level along the coordinate $F_x $ for any angles $\varphi $ and parameters will grow linearly.
{width="1.6in" height="1.6in"} {width="1.6in" height="1.6in"} {width="1.6in" height="1.6in"}
It is possible to determine the effectiveness $Q$ of the expulsion of $n$ cavities as the relation of the total force $F_{\sum}$ to the entire length of the configuration along the $y$ axis $$\label{eq8}
Q=\frac{F_{\sum} }{n\left( {a+2R\tan \varphi} \right)+(n-1)d}.$$ The dependence of $Q$ on the $d/a$ relation is displayed in Fig.. It can be seen, for example, that at $a=4\times 10^{-9}$ m, for any length of the cavity wings $R$ with different angles $\varphi $, there is a maximum of effectiveness $Q$ of expulsion. As is known [@Fateev:2012] there is a maximum of the expulsion forces for each trapezoid figure depending on the angle of the opening of the cavities’ wings \[Fig. \] and their lengths. In the periodic configurations with the distance of the gap $d$, there is a maximum of the expulsion effectiveness $Q$ as well. The maximum of effectiveness, which is common for two parameters $\varphi $ and $d/a$, is shown in Fig. . It was found that for $a=4\times 10^{-9}$ m and $R/a=2.5$, the best angle is $\varphi \approx 5.59\;\mbox{deg}$ at $d/a\approx 1.58$ and $n$ = 2. For the given relation $R/a$ at $n\to \infty $ the angle is $\varphi \to
6.0\;\mbox{deg}$ and $d/a\to 1.7$. At $R/a\to 20$ and $n\to \infty $, the angle $\varphi \to 0.1\;\mbox{deg}$ and $d/a\to 1.85$.
{width="2.4in" height="1.8in"}
Note that when the trapezoid cavities are chequer-wisely arranged in the periodic configuration, i.e. the cavity with the widening of the opening against the $x$ axis is put to the cavity with the wide opening in the $x$-direction (so that the boundary wings become common to both cavities), in the system there will be a torque moment around the centre of mass of the configuration even at $d=a$. Of course, at certain combinations of the arrangement of cavities in the periodic configuration, a much larger torque can be achieved at $d\ne a$ compared to that at $d=a$.
Thus, in the present paper, the possibility in principle is shown that the noncompensated Casimir force exists in open nanosized metal cavities arranged in the form of periodic structures. It is found that in strictly periodic structures based on trapezoid figures all the Casimir expulsion forces are practically completely compensated. However, when the distance of the gap is formed between the cavities, in the periodic configuration a noncompensated expulsion force appears. In this case, at any relations of the configuration parameters (angles of opening and wing length of the cavities, the distance between the cavities, etc.) and at any number of cavities in the period there is an effective maximum of the expulsion forces. In some periodic structures there can be a torque moment around the centre of mass of the configuration.
The author is grateful to T. Bakitskaya for his helpful participation in discussions.
[8]{} E.G. Fateev, <arXiv:1208.0303v1> \[quant-ph\](2012). R.L. Jaffe, A. Scardicchio, J. High Energy Phys. **06**, 006 (2005) <arXiv:hep-th/0501171v2>. U. Leonhardt, New J. Phys. **9**, 254 (2007). M. Levin, A.P. McCauley, A.W. Rodriguez, M.T.Homer Reid, S.G. Johnson, [Phys. Rev. Lett.]{} **105**, 090403 (2010). S. J. Rahi, T. Emig, R. L. Jaffe, Lecture Notes in Physics, **834**, 129 (2011) <arXiv:1007.4355v1> \[quant-ph\]. S. J. Rahi, S. Zaheer, Phys. Rev. Lett. **104**, 070405 (2010). H. B. G. Casimir, Kon. Ned. Akad. Wetensch. Proc. **51**, 793 (1948). H. B. G. Casimir, D. Polder, Phys. Rev. **73**, 360 (1948).
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abstract: 'Optical modes of a multilayered photonic structure with the twisted nematic liquid crystal as a defect layer have been investigated. The electroconvective flow in the nematic makes a spatially periodic structure in the form of abnormal rolls. Non-adiabatic propagation of polarized light in the defect layer causes unique features of the optical modes corresponding to the ordinary $o$-waves. The decay of these modes has been demonstrated with increasing voltage due to the effect of cross-polarization diffraction loss. The modes short-wave shift resulting from the contribution of the non-adiabatic geometric phase to the total phase delay of the wave during a round-trip propagation through the photonic structure has been found [both experimentally and numerically]{}.'
address:
- 'Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Krasnoyarsk, 660036, Russia'
- 'Institute of Engineering Physics and Radio Electronics, Siberian Federal University, Krasnoyarsk, 660041, Russia'
author:
- 'Vladimir A. Gunyakov'
- 'Mikhail N. Krakhalev'
- '[Ivan V. Timofeev]{}'
- 'Victor Ya. Zyryanov'
- 'and Vasily F. Shabanov'
bibliography:
- 'References.bib'
title: Optical modes of multilayered photonic structure containing nematic layer with abnormal electroconvective rolls
---
`Photonic structure` ,Optical modes ,Nematic liquid crystal ,Electroconvective instability ,Cross-polarization diffraction ,Geometric phase
Introduction
============
The Fabry-Pérot cavity-type multilayered photonic structures based on the distributed Bragg mirrors evoke great interest as promising optical materials for nanophotonics and optoelectronics functional elements [@joannopoulos_photonic_2008; @busch_photonic_2006]. An important property of such structures is the presence of photonic bandgap (PBG) with the low density of photonic states and low transmission. The bandgap exhibits specific dispersion properties, which can be used to implement the regimes of light propagation in photonic structures that are interesting for both fundamental research and application [@joannopoulos_photonic_2008; @busch_photonic_2006; @busch_periodic_2007]. In the multilayered photonic structure with broken periodicity, the narrow resonant peaks called defect (localized) modes appear in the bandgap. Using liquid crystals (LCs) as defect layers, one can form photonic structures with controlled spectral characteristics [@busch_periodic_2007; @shabanov_optics_2005; @arkhipkin_electro-_2011]. A thin twisted nematic layer with the strongly violated Mauguin’s waveguide regime [@Mauguin_sur_1911] gives rise to the unique spectral features of multilayered photonic structures. The light waves linearly polarized parallel or perpendicular to the optical axis (director **n**) at the nematic layer input become elliptically polarized when the waveguide propagation regime is violated. The eigenmodes of a twisted nematic are the extraordinary ($te$) and ordinary ($to$) elliptically polarized waves (twist-modes), which adiabatically follow the local director [@yeh_optics_1999]. Then, any polarized beam of light can be decomposed into a linear combination of these two twist-modes.
It was demonstrated using the coupled wave theory that, in the Fabry–Pérot cavity with a twisted nematic layer, the *te*- and *to*-modes at the same frequency couple with each other upon reflection from mirrors and create the cavity eigenmode [@ohtera_analysis_2000]. The state of polarization and, consequently, the cavity eigenmode type (*re* or *ro*) depend on the relative contribution of the twist-modes to each cavity eigenmode. Despite of the ellipticity of the $re$ and $ro$ eigenmodes, their polarizations on the mirrors remain linear in the mutually perpendicular directions [@patel_anticrossing_1991; @yoda_analysis_1997]. In Ref. [@gunyakov_polarization_2017], the electrically-operated polarization exchange of the cavity eigenmodes in the spectral points corresponding to the Gooch–Tarry maxima [@gooch_optical_1975] was demonstrated for a uniformly twisted nematic. With increasing voltage, one can observe series of avoided crossings of the transmission peaks in the spectrum, which cause a bisector (at angles of $\pm45^{\circ}$ to the nematic director) polarization of both cavity modes. The enhancement of the mode coupling leads to the short-wave shift of the *ro*-modes [@gunyakov_electric_2018]. Vice versa, in the vicinity of the spectral points corresponding to the Gooch–Tarry minima, the weak mode coupling does not affect the spectral position of the *ro*-modes. Nevertheless, under certain conditions and at the Gooch–Tarry minima, the shift of the *ro*-modes can be observed. In particular, at the transition from the homeoplanar to twist configuration of the director field, anomalous shifts of the *ro*-modes to the short-wave spectral region were reported [@timofeev_geometric_2015]. It was established that observed shifts are caused by contribution of the non-adiabatic geometric phase [@aharonov_phase_1987] to the total phase delay of the wave during a round-trip propagation.
The twist deformation of the director in the spatially periodic structures, e.g. electroconvective nematic LC structure in the form of abnormal rolls, was predicted theoretically [@plaut_new_1997] and monitored experimentally [@rudroff_relaxation_1999; @dennin_direct_2000]. The wave vector **q** characterizing the periodicity of abnormal rolls is parallel to the orientation of the director **n** on the substrates as in the case of normal rolls (Williams domains [@williams_domains_1963]). While the director projection at the center of abnormal rolls makes an angle with **n** on the substrates and the regions with different twist signs are separated by domain boundaries. The optical contrast between different domains observed using a circular analyzer is resulted from coupling of the extraordinary (*e*) and ordinary (*o*) waves in the twisted nematics. The wave coupling effect in magnetic field twisted nematic director structures (*T*-effect) leading to the ellipticity of the incident linearly polarized light was described previously by Gerber and Schadt [@gerber_measurement_1980]. Under the assumption of smallness of the director gradients, an analytical expression was derived for describing the oscillating behavior of the transverse electrical component $E_{y}$ of the optical eigenmode for LC layer with the anisotropy $\Delta n = n_{e} - n_{o}$ and thickness $d$ as a function of the parameter $w = k_{0}d\Delta n/2\pi$. At the wavelengths $\lambda = 2\pi/k_{0}$ that satisfy the condition $w = 1, 2, 3 .... $ the ellipticity of the modes at the sample output is maximum [@gerber_measurement_1980].
The diffraction in the electroconvective structures is due to periodic optical inhomogeneities in the form of phase gratings [@carroll_liquidcrystal_1972]. The hierarchy of the increasingly complex convective structures in the LCs easily switched by an electric field and their polarization-sensitive optical response make it possible to control the mode amplitudes in the transmission spectrum of photonic structures with a nematic defect layer [@gunyakov_modulation_2016]. At the same time the question related to the transformation of the optical modes of photonic structure under the coupling effect of waves and their diffraction remains understudied. In view of the aforesaid, here we examine the transformation of the polarized transmission spectra of a multilayered photonic structure with a defect layer of the electroconvective nematic with the abnormal roll instability. A homogeneous twist-deformation (torsion effect) of the director field in the abnormal roll structure is controlled by analyzing the states of polarization of the laser radiation diffracted on the LC cell with the electroconvective nematic layer identical to the one in the photonic structure.
Experimental approach
=====================
[Figure \[fig1\] shows two schemes of experimental setups for studying the spectral characteristics of the directly transmitted radiation and the polarization parameters of a diffracted laser beam. Transmission spectra of the photonic structure were measured by setup presented in Fig. \[fig1\]a. The photonic structure with LC defect layer (PS/LC) is composed of two identical Bragg mirrors. Bragg mirrors were made in Technological Design Institute of Applied Microelectronics (Novosibirsk) by electron-beam evaporation technique. Six ZrO$_2$ layers and five SiO$_2$ layers were deposited alternately. It should be noted that available oxide dielectric materials, e.g. TiO$_2$/SiO$_2$ pair has higher refractive indexes difference [@CHIASERA2019107] and, respectively, higher quality factor *Q* of the PS/LC than ZrO$_2$/SiO$_2$ [@Jerman:05]. The fact is that in real photonic structures an increase of the *Q*-factor leads to the decrease of resonant transmission peaks due to scattering and absorption inside the cavity. And strongest decay of these modes occurs at the center of the PBG [@Arkhipkin2008]. That is why we use ZrO$_2$/SiO$_2$ mirrors. This circumstance may be crucial for the determination of the spectral properties of optical modes in photonic structures with electroconvective nematics that are highly scattering media. Transmission spectrum and a schematic view of the Bragg mirror are presented in Fig. \[fig2\]. The multilayer is deposited onto fused quartz substrate with ITO-electrode. The thickness of each layer measured by TEM microscopy on the Bragg mirror was $63~\pm~5$ nm and $82~\pm~5$ nm for the ZrO$_2$ and SiO$_2$, respectively, and a thickness of $168~\pm~5$ nm for the ITO layer (see Supplementary). The first order bandgap ranges from 424 nm to 624 nm. The minimal transmission is about 0.08 and smoothly increases toward the PBG edges. The observed transmittance difference of the short- and long-wave PBG edges is related to optical properties of the ITO thin film that starts to loss transparency for wavelengths shorter than 450 nm [@Krylov2013].]{}
A gap between the mirrors was filled with a 4-methoxybenzylidene-4’-butyl aniline (MBBA) nematic LC with the negative permittivity anisotropy ($\epsilon_a < 0$) and positive conductivity anisotropy ($\sigma_a > 0$). The gap thickness was $7.8~\pm~0.1~\mu$m, [operating area of the PS/LC was $16~\times~22$ mm$^2$]{}. The nematic clearing point is $T_c = 45^\circ$. The Bragg mirrors were coated with rubbed polyvinyl alcohol (PVA) films to specify the planar alignment of the MBBA director **n** $\parallel x$. The uniform alignment of the director ensured the transparent state of the LC under the low-frequency (100 Hz) voltage in the range of 0 $\div$ 7.2 V. With that the spectral properties of the photonic structure remain unchanged. A critical voltage of $U_c = 7.2$ V leads to the convective instabilities in the form of the spatially periodic flows of the MBBA, which significantly change the optical properties of the LC and thereby modify the optical response of the photonic structure.
\[t!\] ![Experimental setups for recording the polarized transmission spectra of the photonic structure with liquid crystal (PS/LC) (a) and diffraction pattern from the LC cell (b) with the convective rolls in nematic layer. Vortex flow of the nematic is indicated by circular arrows.[]{data-label="fig1"}](Figure1-eps-converted-to.pdf "fig:"){width="1.0\linewidth"}
\[t!\] ![[Transmission coefficient of the single Bragg mirror as a function of the light wavelength under normal incidence. Scheme of the mirror is shown on insert.]{}[]{data-label="fig2"}](Figure2-eps-converted-to.pdf "fig:"){width="0.50\linewidth"}
Let an incident beam of light be polarized parallel (perpendicular) to **n** at entrance plane of the twisted nematic layer. This is known as the *e*-mode (*o*-mode) operation [@yeh_optics_1999]. The respective polarized components $T_{e,o}(\lambda)$ of the directed transmission spectra and their field-effect evolution were studied using an Ocean Optics HR4000CG spectrometer equipped with fiber optics (Fig.\[fig1\]a). A collimator of the receiving light guide was used as a diaphragm with an input aperture diameter of 4 mm. A Glan prism (*P*) was used as a polarizing element. Spectra were recorded at a fixed temperature of $25~^\circ$C; the sample thermal stabilization accuracy was not worse than $\pm 0.2~^\circ$C.
To elucidate the effect of diffraction loss on the spectral characteristics of the investigated photonic structure, the electroconvective LC cell with two glass substrates coated with ITO electrodes and a nematic layer identical to the one in photonic structure was assembled. Figure\[fig1\]b shows a scheme of the experimental setup used to observe the diffraction pattern that occurs when the polarized laser beam passes through an electric field-controlled LC cell. We used a Newport R-30972 single mode He-Ne laser with an operation wavelength of $\lambda = 543$ nm. The states of polarization of the diffracted radiation were analyzed using a large-format polaroid (*A*), covering all the observed diffraction orders.
Results and Discussion
======================
Experimental snapshots of the photonic structure with nematic obtained using a polarizing microscope show the evolution of the electroconvective instability in LC with increasing voltage (Fig.\[fig3\]a-d). The highest contrast of convective rolls is reached when the polarizer is parallel to the rubbing direction, i.e. for the $e$-mode operation. Immediately above the critical voltage $U_c$ in the range of 7.2–7.4 V the normal rolls (Williams domains) are observed in the MBBA layer (Fig.\[fig3\]a). As the applied voltage increases, a weak undulating disturbance of normal rolls arises in the sample (Fig.\[fig3\]b). Above a voltage of 7.6 V, the rolls gradually straighten and transform into abnormal ones. Simultaneously, domain boundaries are formed, which now cross straightened abnormal rolls (Fig.\[fig3\]c). [The domain boundaries separating regions of stationary rolls with different twist signs loop close on roll structure defects]{}. [Two types of such defects are observed: topologically stable edge dislocations Fig. \[fig4\]a, non-topological linear defects – a localized screw varicose Fig. \[fig4\]b, as well as their combination Fig. \[fig4\].]{} At the same time both the [roll structure]{} defects and domain boundaries are characterized by the chaotic dynamics at a fixed voltage [@krekhov_spatiotemporal_2015]. [Similar to the normal roll structure defects [@Joets1991], the opposite dislocation can annihilate. At the same time, a non-topological screw varicose may appear. This linear defect either generates a pair of edge dislocations or disappears (see, for example, Supplementary). At a fixed voltage, the length and density of defects in the structure of the rolls remain on average constant.]{} As the applied voltage increases, the density of [the roll structure]{} defects and their travel rate significantly increase (Fig.\[fig3\]d).
\[t!\] ![Snapshots of the photonic structure with convective rolls at applied ac voltages of 7.3 V (a), 7.6 V (b), 8.0 V (c), and 10.0 V (d). Double arrow ($\leftrightarrow$) indicates the polarizer direction, single arrow ($\rightarrow$) indicates the rubbing direction.[]{data-label="fig3"}](Figure3-eps-converted-to.pdf "fig:"){width="1.0\linewidth"}
\[t!\] ![[Snapshots of the rolls structure defects at applied ac voltages of 8.0 V (a), 10.0 V (b) and 10.0 V (c). The pair of opposite edge dislocations (a), screw varicose line (b), combination of the pair of opposite edge dislocations and screw varicose line (c). Dashed single arrows indicate edge defects, double arrow indicates the polarizer direction, solid single arrow indicates the rubbing direction. ]{}[]{data-label="fig4"}](Figure4-eps-converted-to.pdf "fig:"){width="1.0\linewidth"}
A widespread approach to the direct observations of the twist-deformation of the nematic director field in the electroconvective structures like abnormal rolls is based on a polarization-optical technique using a circular analyzer [@rudroff_relaxation_1999; @dennin_direct_2000]. In addition, it is convenient to analyze the states of polarization of the laser radiation diffracted on the roll structure in LC layer. In our case, this method is more relevant, since the decay of the $re$- and $ro$-modes is obviously related to the diffraction loss of light propagating through the roll structure. In particular, under illumination of an LC cell with convective twist domains (Fig.\[fig3\]c) by a polarized laser beam, a pattern of diffraction reflexes is observed in the plane perpendicular to the roll axis (Fig.\[fig1\]b). Moreover, the diffraction arises not only in the trivial case of the $e$-mode operation [@carroll_liquidcrystal_1972], but also at the laser beam polarization parallel to the axis of abnormal rolls, i.e. for the $o$-mode operation. Indeed, in the last case $N^{th}$-order diffraction reflexes ($N = \pm 1, \pm 2, \pm 3, \ldots$) are observed on the left and on the right of directly transmitted beam (Fig.\[fig5\]a). A large-format analyzer oriented parallel to the polarizer decreases slightly the brightness of the zero-order reflex, while the higher-order reflexes become invisible (Fig.\[fig5\]b). Vice versa, when analyzer is oriented perpendicular to the polarizer the brightness of the higher-order reflexes is practically unchanged, while the zero-order reflex brightness decreases significantly (Fig.\[fig5\]c). It means that the deflected beams generating all higher-order reflexes are polarized perpendicular to the polarizer direction (cross-polarization diffraction). Obviously, only twisting the director **n** in the LC layer can lead to the diffraction for the $o$-mode operation and to the polarization change of diffracted light in such a way. It should be recalled, in the case of normal rolls the beams of light produce no deflects for the $o$-mode operation [@carroll_liquidcrystal_1972]. Note the stability of the observed diffraction pattern, in spite of the convective instability which is characterized by the active dynamics of [the roll structure]{} defects and domain boundaries (see Fig.\[fig3\]c). A large number of the twist domains is located in the cross-section of a laser beam with a ($1/e^2$) diameter of 0.83 mm. Nevertheless, the periodicity of abnormal rolls $\Lambda = 7.4~\mu$m is preserved at a fixed voltage $U = 8.0$ V; therefore, the angular distribution of the diffraction maxima remains constant.
\[t!\] ![Snapshots of far-field diffraction patterns from the LC cell with abnormal rolls for the $o$-mode operation without analyzer (a), with analyzer oriented parallel (b) and perpendicular (c) to polarizer. The applied voltage was $U = 8.0$ V.[]{data-label="fig5"}](Figure5-eps-converted-to.pdf "fig:"){width="0.60\linewidth"}
The polarized components of the directed transmission spectra $T_{e,o}(\lambda)$ of the photonic structure at the voltages $U < U_c$ and $U = 10$ V are shown in Figure\[fig6\]. Due to the periodicity of the photonic structure the bandgap is formed with a set of localized modes. Their spectral positions are determined by optical properties of the LC defect. Fig.\[fig6\] shows different response of the $re$- and $ro$-modes of the photonic structure respond differently to the electroconvection process occurring in the nematic layer. By the time when the voltage increases to 10 V, the amplitudes of the $re$-modes damp to the PBG background level, while the signal level at the edges of the bandgap itself decreases by an order of magnitude (Fig.\[fig6\]a). At voltages $U > U_c$ the extraordinary waves diffract on the 1D phase grating of normal and abnormal rolls; therefore, the directed transmission of light decreases due to the spatial redistribution of the wave energy. Thus, the diffraction losses are the main reason for the rapid decay of the $re$-modes [@gunyakov_modulation_2016], starting from a critical voltage of $U_c = 7.2$ V. With voltage increasing, the additional loss comes from [the roll structure]{} defects and domain boundaries. In particular, the density of [the roll structure]{} defects increases from 4 defects per mm$^2$ (Fig.\[fig3\]a) to 35 defects per mm$^2$ (Fig.\[fig3\]d), while their length increases by an order of magnitude on average. Despite these factors, for the $o$-mode operation, at a voltage of 10 V the bandgap is maintained, and the $ro$-modes are still observed (Fig.\[fig6\]b). The amplitudes of these modes decrease by a factor of 2 on average in the long-wave region and by a factor of 3 at the center of the PBG (Fig.\[fig6\]b). It seems that the decay of the $ro$-modes is related to the loss caused by both scattering on [the roll structure]{} defects and the cross-polarized diffraction on abnormal rolls in nematic defect of the photonic structure.
\[t!\] ![Polarized components of the directed transmission spectra $T_{e,o}(\lambda)$ of the multilayered photonic structure for the $e$-mode (a) and $o$-mode (b) operation. Red lines correspond to the voltage range of $U \leq U_c$, blue lines correspond to the voltage $U = 10$ V.[]{data-label="fig6"}](Figure6-eps-converted-to.pdf "fig:"){width="1.0\linewidth"}
Along with a significant decrease in the amplitude, another feature in the behavior of the $ro$-modes at the voltage increasing is their anomalous spectral shift to the short-wave region relative to the initial position (Fig.\[fig6\]b). Field-effect dependencies of the transmittance $T_o(U)$ and spectral positions of the maxima $\lambda(U)$ of the $ro$-modes in the most sensitive central PBG region are shown in Fig.\[fig7\] for the voltage range of $6.8 \leq U \leq 10.0$ V. The measurements were performed with a step of 0.05 V at a fixed temperature. Up to the voltage value of 7.8 V, the amplitudes and spectral positions of the $ro$-mode peaks remain insensitive to the convective instabilities and have the same values as for the initial state of nematic. Above a voltage of 7.8 V, the abnormal rolls are observed. They are characterized by the presence of a homogeneous twist deformation of the director **n**. As the applied voltage increases, the gradually increased twist angle affects the decay of the $ro$-mode amplitudes (Fig.\[fig7\]a) and their smooth spectral shift (Fig.\[fig7\]b). Obviously, the synchronization of both effects is related to the occurrence of the elliptically polarized $te$- and $to$-modes, which redistribute the light fluxes in the photonic structure. The asymptotic behavior of the $\lambda(U)$ curves in Fig.\[fig7\]b evidences for stabilization of the twist-angle value upon approaching a point of $U = 10$ V. Further decay of the $ro$-modes at voltages above 10 V is mainly caused by increasing the density and length of the [roll structure]{} defects (Fig.\[fig3\]d). The maximal shift of the $ro$-modes to the short-wave region amounts to 0.9 nm on average. For the twisted nematic LC layer the probing radiation at some wavelengths has a zero transverse component $E_y~(E_x)$ of the optical eigenmode at the output of the sample [@gerber_measurement_1980]. This means that the elliptical polarization of light inside the LC layer transforms to linear and parallel (perpendicular) to the output director **n**. The $re$- and $ro$-modes of the photonic structure at these wavelengths are polarized in a similar way. At that the effect of coupling the twist-modes upon reflection from mirrors is not revealed. It should be noted that $ro$-modes shown in Fig.\[fig7\]b remain linearly polarized perpendicular to the director **n** on the mirrors. Therefore, the observed anomalous shift of the $ro$-modes to the short-wave region is most likely the spectral manifestation of the non-adiabatic geometric phase. After the voltage switched-off the director field configuration returns to the initial state. In this case, both the amplitudes and spectral positions of the $ro$-modes are recovered.
\[t!\] ![Field-effect dependencies of the transmission spectrum of the photonic structure $T_o(U)$ (a) and spectral positions of the $ro$-modes maxima $\lambda(U)$ (b) at the center of the PBG. The arrow indicates the critical voltage $U_c = 7.2$ V.[]{data-label="fig7"}](Figure7-eps-converted-to.pdf "fig:"){width="1.0\linewidth"}
[The directed transmission spectrum of the *ro*-modes in the multilayered photonic structure has been simulated within one-dimensional approximation of the homogeneous director twist deformation in abnormal rolls [@krekhov_spatiotemporal_2015; @rudroff_relaxation_1999]:]{}
$${\phi = \phi_0 sin(\pi z/d),~~\theta = \theta_0 sin(\pi z/d).}$$
[Here $\phi_0$, $\theta_0$ are maximal azimuthal and polar angles of the director in the midplane of a layer, respectively: $0~\leq~z~\leq~d$. For $U=0$ we take $\theta_0=\phi_0=0$, and for $U=10$ V we assume $\theta_0=24^\circ, \phi_0=60^\circ$ [@krekhov_spatiotemporal_2015]. Then, using the 4$\times$4 transfer matrix method [@Berreman:72], the transmission spectrum for *o*-mode operation in the investigated multilayer structure is simulated with regard to the optical extinction and material dispersion. The following parameters are used. Each of two mirrors is a stack of the SiO$_{2}$ substrate, ITO electrode, ZrO$_{2}$, (SiO$_{2}$, ZrO$_{2})^{5}$, and PVA layer. The thicknesses and refractive indices of the amorphous layers of the dielectric mirrors are 80 nm and 1.45 for SiO$_{2}$, 65 nm and 2.04 for ZrO$_{2}$, 1.515 and 20 nm for the PVA layer. The values for the ITO layer are 160 nm and 1.88858+0.022i with account of the extinction; the substrate refractive index is 1.45 and MBBA refractive indices are $n_{||}$ = 1.765 and $n_{\perp}$ = 1.553, respectively (the wavelength is $\lambda $ = 500 nm and the temperature is $T = 25^\circ$ C). The nematic layer thickness is 7522 nm. Taking into account the dispersion, we use the data and reference from https://refractiveindex.info.The damp of transmission peaks is approximated by increased LC extinction. The imaginary part of MBBA refractive index is taken 4$\cdot $10$^{-4}$i for $U=0$ and 8.5$\cdot $10$^{-4}$i for $U=10$ V.]{} [ Experimental and calculated transmission spectra are presented in Figure \[fig8\] . It can be seen that the experimental and calculated spectral positions of the resonator modes are in good agreement for the $\phi_0$ and $\theta_0$ angles used in simulation.]{}
\[t!\] ![[Spectral position of the modes at the PBG center without (red lines) and under (blue lines) voltage for *o*-mode operation, measured (dashed lines) and simulated using the 4$\times$4 transfer matrix method (solid lines).]{}[]{data-label="fig8"}](Figure8-eps-converted-to.pdf "fig:"){width="0.50\linewidth"}
Conclusion
==========
Thus, we experimentally investigated the transformation of optical modes of the multilayered photonic structure with a defect layer of the electroconvective nematic LC with the abnormal roll instability. In contrast to the well-known Williams domains, this instability is characterized by the homogeneous twist deformation of the director field [@plaut_new_1997]. This leads to the ellipticity of the linearly polarized light travelling through the disturbed nematic. The twist deformation is identified by analyzing the polarization properties of the laser radiation diffracted on the electroconvective LC cell with a nematic layer. In particular, the diffraction for the $o$-mode operation was established. In this case the diffraction pattern is a superposition of zero-order reflex corresponding to the wave linearly polarized perpendicular to the director and zero- and higher-order reflexes corresponding to the wave polarized along the director (cross-polarized diffraction). Under the voltage growth the cross-polarized diffraction results in smooth decay of the $ro$-modes. The anomalous shift of the $ro$-modes to the short-wave range was found, which can be attributed to the contribution of the non-adiabatic geometric phase to the total phase delay experienced by the wave during a round-trip propagation. [The numerical simulation of transmission spectrum of structure with homogeneous twist deformation of the director that is typical for abnormal rolls demonstrates the same effect.]{} The proposed approach based on using a multilayered photonic structure can be used for the study of the features of spatially periodic structures in dissipative liquid-crystalline systems with a complex director field configuration because of high sensitivity of the optical modes to electroconvective processes. This optical material is promising for the applications in various optoelectronic devices: tunable spectral filters, polarizing sensors, etc.
Acknowledgments {#acknowledgments .unnumbered}
===============
[We are grateful to M.N. Volochaev for providing TEM micrograph of the mirror.]{}
|
---
abstract: 'We study the two-time distribution in directed last passage percolation with geometric weights in the first quadrant. We compute the scaling limit and show that it is given by a contour integral of a Fredholm determinant.'
address: ' Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden'
author:
- Kurt Johansson
title: 'The two-time distribution in geometric last-passage percolation'
---
[^1]
Introduction {#secintro}
============
In this paper we will consider the so called two-time distribution in directed last-passage percolation with geometric weights. This last-passage percolation model has several interpretations. It can be related to the Totally Asymmetric Simple Exclusion Process (TASEP) and to local random growth models. It is a basic example of a solvable model in the KPZ universality class. It has been less clear to what extent the two-time problem is also solvable but recently there has been some developments in this direction, [@Dots1], [@JoTt], [@FerSpo], [@NarDou], [@Dous] and [@BaiLiu]. The approach in this paper is different in many ways from that in our previous work [@JoTt]. It is closer to standard computations for determinantal processes, more straightforward and simpler.
To define the model, let $\left(w(i,j)\right)_{i,j\ge 1}$ be independent geometric random variables with parameter $q$, $$\mathbb{P}[w(i,j)=k]=(1-q)q^k,\quad k\ge 0.$$ Consider the last-passage times $$\label{gmn}
G(m,n)=\max_{\pi:(1,1)\nearrow (m,n)} \sum_{(i,j)\in\pi} w(i,j),$$ where the maximum is over all up/right paths from $(1,1)$ to $(m,n)$, see [@JoSh]. We are interested in the correlation between $G(m_1,n_1)$ and $G(m_2,n_2)$, when $(m_1,n_1)$ and $(m_2,n_2)$ are ordered in the time-like direction, i.e. $m_1<m_2$ and $n_1<n_2$. To see why this is called a time-like direction, and give one reason why we are interested in the two-time problem, let us reinterpret the model as a discrete polynuclear growth model. It is clear from (\[gmn\]) that $$\label{growth}
G(m,n)=\max(G(m-1,n), G(m,n-1))+w(m.n).$$ Let $G(m,n)=0$ if $(m,n)\notin\mathbb{Z}_+^2$, and define the height function $h(x,t)$ by $$\label{hxt}
h(x,t)=G\left(\frac{t+x+1}2,\frac{t-x+1}2\right)$$ for $x+t$ odd, and extend it to all $x\in\mathbb{R}$ by linear interpolation. Then (\[growth\]) leads to a growth rule for $h(x,t)$ and this is the discrete time and space polynuclear growth model. We think of $x\mapsto h(x,t)$ as the height above $x$ at time $t$, and we get a random one-dimensional interface. Let the constants $c_i$ be given by (\[scalingconstants\]). It is known, see [@JoDPG], that the rescaled process $$\label{hrescaled}
\mathcal{H}_T(\eta,t)=\frac{h(2c_1\eta(tT)^{2/3},2tT)-c_2tT}{c_3(tT)^{1/3}},$$ as a process in $\eta\in\mathbb{R}$ for a fixed $t>0$, converges as $T\to\infty$ to $\mathcal{A}_2(\eta)-\eta^2$, where $\mathcal{A}_2(\eta)$ is the Airy-2-process, [@PrSp]. In particular, for any fixed $\eta,t$, $$\lim_{T\to\infty}\mathbb{P}[\mathcal{H}_T(\eta,t)\le\xi-\eta^2]=F_2(\xi)=\det(I-K_{{\text{Ai\,}}})_{L^2(\xi,\infty)},$$ where $F_2$ is the Tracy-Widom distribution, and $$K_{\text{Ai}}(x,y)=\int_0^\infty {\text{Ai\,}}(x+s){\text{Ai\,}}(y+s)\,ds,$$ is the Airy kernel. The two-time problem is concerned with the question of the correlation between heights at different times. What is the limiting joint distribution of $\mathcal{H}_T(\eta_1,t_1)$ and $\mathcal{H}_T(\eta_2,t_2)$ for $t_1<t_2$, as $T\to\infty$? From (\[hxt\]), we see that this is related to understanding the correlation between last-passage times in the time-like direction. That a time separation of order $T$ is the correct order to get non-trivial correlations is quite clear if we think about how much random environment e.g. $G(n,n)$ and $G(N,N)$, $n<N$, share. It can also be seen from the slow de-correlation phenomenon, see [@Fer], [@CorFerPec]. Looking at (\[hrescaled\]) we see that we have the fluctuation exponent $1/3$ (fluctuations have order $T^{1/3}$), the spatial correlation exponent $2/3$, and we also have the time correlation exponent $1=3/3$ as explained. This is the KPZ 1:2:3 scaling. For further references and more on random growth models in the KPZ-universality class and related interacting particle systems, we refer to the survey papers [@BorPet], [@Corw] and [@Quas].
The main result of the present paper is a limit theorem for the following two-time probability. Fix $m,M,n,N$ with $1\le m<M$ and $1\le n<N$. For $a,A\in\mathbb{Z}$, we will consider the probability $$\label{paA}
P(a,A)=\mathbb{P}[G(m,n)< a,\,G(M,N)<A],$$ in the appropriate scaling limit. The result is formulated in Theorem \[ThMain\] below.
The first studies of the two-time problem, using a non-rigorous based on the replica method, was given by Dotsenko in [@Dots1], [@Dots2], see also [@Dots3]. However, the formulas are believed not to be correct, [@NarDou]. The replica method has also been used by De Nardis and Le Doussal, [@NarDou], to derive very interesting results in the limit $t_1/t_2\to 1$ and, for arbitrary $t_1/t_2$, in the partial tail of the joint law of $\mathcal{H}_T(\eta_1,t_1)$ and $\mathcal{H}_T(\eta_2,t_2)$ when $\mathcal{H}_T(\eta_1,t_1)$ is large positive. In [@Dous], Le Doussal gives a conjecturally exact formula for the limit $t_1/t_2\to 0$. See also [@FerSpo] for some rigorous work on this with quantitative results for the height correlation in the stationary case, which is not investigated here. We will not discuss these limits although to do so would be interesting. There are very interesting experimental and numerical results on the two-time problem by K. A. Takeuchi and collaborators, see [@Take], [@TaSa] and [@NaDoTa].
Recently there has been a striking new development on the two-time problem, and more generally the multi-time problem, by J. Baik and Z. Liu, [@BaiLiu]. They consider the totally asymmetric simple exclusion process (TASEP) in a circular geometry, the periodic TASEP. Baik and Liu are able to give formulas for the multi-time distribution as contour integrals of Fredholm determinants, and take the scaling limit in the so-called relaxation time scale, $T=O(L^{3/2})$, where $L$ is the period. In principle their formulas include the problem studied here, but they are not able to take the scaling limit that we study in this paper. It would be interesting to understand the relation between the two approaches. For some comments on the multi-time problem in the setting used here see Remark \[remmultitime\]. A related problem is to understand the Markovian time evolution of the whole limiting process with some fixed initial condition, the so called KPZ-fixed point. There has recently been very interesting progress on this problem by Matetski, Quastel and Remenik, see [@MaQuRe] and [@MaQu].
An outline of the paper is as follows. In section \[secresults\] we give the formula for the two-time distribution using an integral of a Fredholm determinant and state the main theorem. The main theorem is proved in section \[secproofmain\] using a sequence of lemmas proved in sections \[secprooflemmas\] and \[secasymptotics\]. In section \[secoldformula\], we briefly discuss the relation to the result in our previous work [@JoTt].
[**Notation**]{} Throughout the paper $1(\cdot)$ denotes an indicator function, $\gamma_r(a)$ is a positively oriented circle of radius $r$ around the point $a$, and $\gamma_r=\gamma_r(0)$. Also, $\Gamma_c$ is the upward oriented straight line through the point $c$, $t\mapsto c+it$, $t\in\mathbb{R}$.
Results {#secresults}
=======
Let $0<t_1<t_2$, $\eta_1,\eta_2\in\mathbb{R}$ and $\xi_1,\xi_2\in\mathbb{R}$ be given. Furthermore $T$ is a parameter that will tend to infinity. To formulate the scaling limit we need the constants, $$\label{scalingconstants}
c_0=q^{-1/3}(1+\sqrt{q})^{1/3},\quad c_1=q^{-1/6}(1+\sqrt{q})^{2/3},\quad
c_2=\frac{2\sqrt{q}}{1-\sqrt{q}},\quad c_3=\frac{q^{1/6}(1+\sqrt{q})^{1/3}}{1-\sqrt{q}}$$ We will investigate the asymptotics of the probability distribution defined by (\[paA\]). The appropriate scaling is then $$\begin{aligned}
\label{scaling}
n&=t_1T-c_1\eta_1(t_1T)^{2/3},\quad m=t_1T+c_1\eta_1(t_1T)^{2/3}\\
N&=t_2T-c_1\eta_2(t_2T)^{2/3},\quad M=t_2T+c_1\eta_2(t_2T)^{2/3}\notag\\
a&=c_2t_1T+c_3\xi_1(t_1T)^{1/3}, \quad A=c_2t_2T+c_3\xi_2(t_2T)^{1/3}.\notag\end{aligned}$$ Let $\Delta t=t_2-t_1$, and write $$\label{alpha}
\alpha=\left(\frac{t_1}{\Delta t}\right)^{1/3}.$$ Introduce the notation $$\label{deltaeta}
\Delta\eta=\eta_2\left(\frac{t_2}{\Delta t}\right)^{2/3}-\eta_1\left(\frac{t_1}{\Delta t}\right)^{2/3},\quad \Delta\xi=\xi_2\left(\frac{t_2}{\Delta t}\right)^{1/3}-\xi_1\left(\frac{t_1}{\Delta t}\right)^{1/3}.$$
We will now define the limiting probability function. Before we can do that we need to define some functions. Fix $\delta$ such that $$\label{deltacondition}
\delta>\max(\eta_1,\alpha\Delta\eta),$$ and define $$\begin{aligned}
\label{S1}
S_1(x,y)&=-\alpha e^{(\eta_1-\delta)x+(\delta-\alpha\Delta\eta)y}\int_0^\infty e^{(\alpha\Delta\eta-\eta_1)s}
K_{\text{Ai}}(\xi_1+\eta_1^2-s,\xi_1+\eta_1^2-x)\notag\\
&\times K_{\text{Ai}}(\Delta\xi+\Delta\eta^2+\alpha s,\Delta\xi+\Delta\eta^2+\alpha y)\,ds,\end{aligned}$$ $$\begin{aligned}
\label{T1}
T_1(x,y)&=\alpha e^{(\eta_1-\delta)x+(\delta-\alpha\Delta\eta)y}
\int_{-\infty}^0 e^{(\alpha\Delta\eta-\eta_1)s}
K_{\text{Ai}}(\xi_1+\eta_1^2-s,\xi_1+\eta_1^2-x)\notag
\\&\times K_{\text{Ai}}(\Delta\xi+\Delta\eta^2+\alpha s,\Delta\xi+\Delta\eta^2+\alpha y)\,ds,\end{aligned}$$ $$\label{S2}
S_2(x,y)=\alpha e^{(\delta-\alpha\Delta\eta)(y-x)}K_{\text{Ai}}(\Delta\xi+\Delta\eta^2+\alpha x,\Delta\xi+\Delta\eta^2+\alpha y),$$ and $$\label{S3}
S_3(x,y)=e^{(\delta-\eta_1)(y-x)}K_{\text{Ai}}(\xi_1+\eta_1^2-x,\xi_1+\eta_1^2-y).$$ Using these, we can define the functions $$\label{Sxy}
S(x,y)=S_1(x,y)+1(x> 0)S_2(x,y)-S_3(x,y)1(y<0),$$ $$\label{Txy}
T(x,y)=-T_1(x,y)-1(x>0)S_2(x,y)+S_3(x,y)1(y< 0).$$
Let $u$ be a complex parameter and set $$\label{Ru}
R(u)(x,y)=S(x,y)+u^{-1}T(x,y).$$ Consider the space $$\label{Xspace}
X=L^2(\mathbb{R}_-,dx)\oplus L^2(\mathbb{R}_+,dx),$$ and define the following matrix kernel on $X$, $$\label{Kuv}
K(u)(x,y)=\begin{pmatrix} R_u(x,y) & R_u(x,y) \\
uR_u(x,y) & uR_u(x,y)
\end{pmatrix}.$$ $K(u)$ defines a trace-class operator on $X$, which we also denote by $K(u)$. Let $\gamma_r$ denote a circle around the origin of radius $r$ with positive orientation. We define the two-time probability distribution by $$\label{ftt}
F_{\text{two-time}}(\xi_1,\eta_1;\xi_2,\eta_2;\alpha)=\frac{1}{{2\pi\mathrm{i}}}\int_{\gamma_r}\frac 1{u-1}\det(I+K(u))_X\,du,$$ where $r>1$.
We can now formulate our main theorem.
\[ThMain\] Let $P(a,A)$ be defined as in (\[paA\]) and consider the scaling (\[scaling\]). Then, $$\label{limit}
\lim_{T\to\infty} P(a,A)=F_{\text{two-time}}(\xi_1,\eta_1;\xi_2,\eta_2;\alpha).$$
The theorem will be proved in section \[secproofmain\]. The fact that $K(u)$ is a trace-class operator is Lemma \[lemtraceclass\] below.
The formula for the two-time distribution can be written in different ways. In section \[SecFormulas\], we will give formulas suitable for studying the limits $\alpha\to 0$, $\alpha\to \infty$ and expansions in $\alpha$ and $1/\alpha$ respectively. We will not discuss these expansions here, but refer to [@DJL] for more on this and comparison with the results in [@Dous].
For comments on the relation between this formula and the formula derived in [@JoTt], see the discussion in section \[secoldformula\].
\[remmultitime\] [It would be interesting to be able to prove the same type of scaling limit for the multi-time case, i.e. to consider the probability function $$P(a_1,\dots,a_{L})=\mathbb{P}\left[G(m_1,n_1)<a_1,\dots,G(m_{L},n_{L})<a_{L}\right],$$ where $m_1< m_2<\dots< m_L$, and $n_1< n_2<\dots< n_L$. It is possible to write a formula analogous to (\[paAformula4\]) below but with $L-1$ contour integrals. This can be proved in a very similar way as the proof of (\[paAformula4\]). We hope to say more on this problem in future work.]{}
Proof of the Main theorem {#secproofmain}
=========================
In this section we will prove the main theorem. Along the way we will use several lemmas that will be proved in sections \[secprooflemmas\] and \[secasymptotics\].
Write $$\label{Gm}
\mathbf{G}(m)=(G(m,1),\dots,G(m,N)),$$ for $m\ge 0$, and a fixed $N\ge 1$. Let $\mathbf{G}(0)=0$. By $\Delta$ we denote the finite difference operator defined on functions $f:\mathbb{Z}\mapsto\mathbb{C}$ by $\Delta f(x)=f(x+1)-f(x)$, which has the inverse $$\Delta^{-1} f(x)=\sum_{y=-\infty}^{x-1} f(y),$$ for all functions $f$ for which the series converges. The negative binomial weight is $$\label{negbinom}
w_m(x)=(1-q)^m\binom{x+m-1}{x} q^x 1(x\ge 0),$$ for $m\ge 1$, $x\in\mathbb{Z}$. Write $$\label{WN}
W_N=\{\mathbf{x}=(x_1,\dots,x_N)\in\mathbb{Z}^N\,,\,x_1\le\dots\le x_N\}.$$ Note that $\mathbf{G}(m)\in W_N$.
The following proposition is the starting point for the proof. It is proved in [@JoMar] following the paper [@Warr] by J. Warren, see also [@DiWa] for a more systematic treatment.
\[PropTransition\] The vectors $(\mathbf{G}(m))_{m\ge 0}$ form a Markov chain with transition function $$\label{Trans}
\mathbb{P}[\mathbf{G}(m)=\mathbf{y}\,|\,\mathbf{G}(\ell)=\mathbf{x}]=\det(\Delta^{j-i}w_{m-\ell}(y_j-x_i))_{1\le i,j\le N},$$ for any $\mathbf{x},\mathbf{y}\in W_N$, $m>\ell\ge 0$.
Write $$\label{deltam}
\Delta m=M-m,\quad\Delta N=N-n,\quad \Delta a=A-a,$$ and $$W_{N,n}(a)=\{\mathbf{x}\in W_N;\,x_n< a\}.$$ We can the write $$\label{paAformula}
P(a,A)=\sum_{\mathbf{x}\in W_{N,n}(a)}\sum_{\mathbf{y}\in W_{N,N}(A)}\det(\Delta^{j-i}w_m(x_j))_{1\le i,j\le N}\det(\Delta^{j-i}w_{\Delta m}(y_j-x_i))_{1\le i,j\le N}.$$ Here we would like to perform the sum over $\mathbf{y}$, which is straightforward, and then the sum over $\mathbf{x}$, which is tricky since we cannot use the Cauchy-Binet identity directly. An important step is part a) of the following lemma, which is proved in section \[secprooflemmas\]. The proof of (\[Partsum\]) uses successive summations by parts and generalizes the proof of Lemma 3.2 in [@JoMar].
\[LemSumparts\] Let $f,g:\mathbb{Z}\mapsto\mathbb{R}$ be given functions and assume that there is an $L\in\mathbb{Z}$ such that $f(x)=g(x)=0$ if $x<L$.
\(a) Let $a_i,d_i\in\mathbb{Z}$, $1\le i\le N$ and fix $k$, $1\le k\le N$. Then, $$\begin{aligned}
\label{Partsum}
&\sum_{\mathbf{x}\in W_{N,k}(a)}\det\big(\Delta^{j-a_i}f(x_j-y_i)\big)_{1\le i,j\le N}\det\big(\Delta^{d_i-j}g(z_i-x_j)\big)_{1\le i,j\le N}\\
=&\sum_{\mathbf{x}\in W_{N,k}(a)}\det\big(\Delta^{k-a_i}f(x_j-y_i)\big)_{1\le i,j\le N}\det\big(\Delta^{d_i-k}g(z_i-x_j)\big)_{1\le i,j\le N}.\notag\end{aligned}$$
\(b) For $1\le n\le N$, we have the identity $$\label{secondsum}
\sum_{\mathbf{x}\in W_{N,N}(A)}\det\big(\Delta^{i-n}w_m(x_i-y_j)\big)_{1\le i,j\le N}=\det\big(\Delta^{i-n-1}w_m(A-y_j)\big)_{1\le i,j\le N}.$$
If we use (\[Partsum\]) and (\[secondsum\]) in (\[paAformula\]), we find $$\label{paAformula2}
P(a,A)=\sum_{\mathbf{x}\in W_{N,n}(a)}\det\big(\Delta^{n-i}w_m(x_j)\big)_{1\le i,j\le N}\det\big(\Delta^{j-n-1}w_{\Delta m}(A-x_i)\big)_{1\le i,j\le N}.$$
Before we show how we can use the Cauchy-Binet identity to do the summation in (\[paAformula2\]), we will modify it somewhat. Below, this modification will be a kind of orthogonalization procedure, and will be important for obtaining a Fredholm determinant. Let $A=(a_{ij})$ and $B=(b_{ij})$ be two $N\times N$-matrices that satisfy $a_{ij}=0$ if $j>i$ and $b_{ij}=0$ if $j<i$, so that $A$ is lower- and $B$ upper-triangular. Assume that $$\label{detAB}
\det AB=\prod_{i=1}^Na_{ii}b_{ii}=1.$$ For $x\in\mathbb{Z}$, $1\le i,j\le N$, we define $$\label{f01}
f_{0,1}(i,x)=\sum_{k=1}^Na_{ik}(-1)^n\Delta^{n-k}w_m(x+a),$$ and $$\label{f12}
f_{1,2}(x,j)=\sum_{k=1}^N(-1)^n\Delta^{k-1-n}w_{\Delta m}(\Delta a-x)b_{kj},$$ where $w_m$ is the negative binomial weight (\[negbinom\]). If we shift $x_i\to x_i+a$, $1\le i\le N$, in (\[paAformula2\]), and use (\[detAB\]), (\[f01\]) and (\[f12\]), we get $$\label{paAformula3}
P(a,A)=\sum_{\mathbf{x}\in W_{N,n}(0)}\det\big(f_{0,1}(i,x_j)\big)_{1\le i,j\le N}\det\big(f_{1,2}(x_i,j)\big)_{1\le i,j\le N}.$$ This formula is the basis for the next lemma, the proof of which is based on the Cauchy-Binet identity. However, because of the restriction $x_n< 0$ in the summation in (\[paAformula3\]), we cannot apply the identity directly. In order to state the result we need some further notation. Define $$\label{L1}
L_1(i,j)=\sum_{x=-\infty}^{-1}f_{0,1}(i,x)f_{1,2}(x,j),$$ $$\label{L2}
L_2(i,j)=\sum_{x=0}^{\infty}f_{0,1}(i,x)f_{1,2}(x,j).$$ Let $u$ be a complex parameter and set $$\label{Lijuv}
L(i,j;u)=u^{1(i>n)}L_1(i,j)+u^{-1(i\le n)}L_2(i,j).$$
\[LemCB\] We have the formula, $$\label{paAformula4}
P(a,A)=\frac 1{{2\pi\mathrm{i}}}\int_{\gamma_r}\frac 1{u-1}\det\big(L(i,j;u)\big)_{1\le i,j\le N}du,$$ for any $r>1$.
The lemma is proved in section \[secprooflemmas\]. The contour integral come from the need to capture the restriction $x_n<0$ and still use the Cauchy-Binet identity.
We now come to the choice of the matrices $A$ and $B$. The aim is to get a good formula for $f_{0,1}$ and $f_{1,2}$ and make it possible to write the determinant in (\[paAformula4\]) as a Fredholm determinant suitable for asymptotic analysis. Define $$\label{Hnmx}
H_{n,m,x}(w)=\frac{w^n(1-w)^{x+m}}{\big(1-\frac{w}{1-q}\big)^m}.$$ Using a generating function for the negative binomial weight (\[negbinom\]), it is straightforward to show that for all $m\ge 1$, $k,x\in\mathbb{Z}$, $$\label{Deltakwm}
\Delta^n w_m(x)=\frac{(-1)^{k-1}}{{2\pi\mathrm{i}}}\int_{\gamma_r}H_{n,m,x}(z)\frac{dz}{1-z},$$ if $r>1$. For $k,x\in\mathbb{Z}$, $m\ge 1$, $\epsilon\in\{0,1\}$ and $0<\tau<1$, we define $$\label{beta}
\beta_k^{\epsilon}(m,a)=\frac 1{{2\pi\mathrm{i}}}\int_{\gamma_\tau}\zeta^{k-1}\frac{\left(1-\frac{\zeta}{1-q}\right)^m}{(1-\zeta)^{a+m-\epsilon}}d\zeta.$$ Note that $\beta_0^{\epsilon}=1$ and $\beta_k^{\epsilon}=0$ if $k\ge 1$. By expanding $(z-\zeta)^{-1}$ in powers of $\zeta/z$, we see that $$\label{betaformula}
\sum_{k=1}^N\frac{\beta_{k-i}^{\epsilon}(m,a)}{z^k}=\frac 1{{2\pi\mathrm{i}}}\int_{\gamma_\tau}\frac{(1-\zeta)^\epsilon}{H_{i,m,a}(\zeta)(z-\zeta)}\,d\zeta,$$ provided $|z|>\tau$.
We now define the matrices $A$ and $B$. Let $c(i)$ be a conjugation factor defined below in (\[conjfactor\]) which we need to make the asymptotic analysis work. Set $$\label{ABdef}
a_{ik}=c(i)(-1)^{-k}\beta_{k-i}^1(m,a),\quad b_{kj}=c(j)^{-1}(-1)^k\beta_{j-k}^0(\Delta m,\Delta a).$$ From the properties of $\beta_k^\epsilon$, we see that $(a_{ik})$ is lower- and $(b_{kj})$ upper-triangular, and that the condition (\[detAB\]) is satisfied.
\[Lemf01f12\] If $f_{0,1}$ and $f_{1,2}$ are defined by (\[f01\]) and (\[f12\]) respectively, and $a_{ik}$ and $b_{kj}$ by (\[ABdef\]), then $$\label{f01formula}
f_{0,1}(i,x)=-\frac {c(i)}{({2\pi\mathrm{i}})^2}\int_{\gamma_r}dz\int_{\gamma_\tau}d\zeta\frac{H_{n,m,a+x}(z)(1-\zeta)}{H_{i,m,a}(\zeta)(z-\zeta)(1-z)},$$ $$\label{f12formula}
f_{1,2}(x,j)=\frac {c(j)^{-1}}{({2\pi\mathrm{i}})^2}\int_{\gamma_r}dw\int_{\gamma_\tau}d\omega\frac{H_{\Delta n,\Delta m,\Delta a-x}(w)}{H_{N+1-j,\Delta m,\Delta a}(\omega)(w-\omega)(1-w)},$$ where $0<\tau<1<r$.
The proof of the lemma, which will be given in section \[secprooflemmas\], is a straightforward computation using the definitions and (\[betaformula\]).
We now turn to rewriting the determinant in (\[paAformula4\]) as a Fredholm determinant and performing the asymptotic analysis. The conjugation factor $c(i)$ in (\[ABdef\]) is given by $$\label{conjfactor}
c(i)=(1-\sqrt{q})^i e^{-\delta i/c_0(t_1T)^{1/3}},$$ where $\delta>0$ is fixed, and satisfies (\[deltacondition\]), and $c_1$ is given by (\[scalingconstants\]). Let $\tau_1, \tau_2,\rho_1,\rho_2$ and $\rho_3$ be radii such that $$\label{radii}
0<\tau_1,\tau_2<1-\rho_1<1-\rho_2<1-\rho_3<1-q.$$ We denote by $\gamma_\rho(1)$ a positively oriented circle around the point $1$ with radius $\rho$. For $\epsilon\in\{0,1\}$ and $1\le i,j\le N$, we define $$\label{A1}
A_1(i,j)=\frac{c(i)}{c(j)({2\pi\mathrm{i}})^4}\int_{\gamma_{\rho_1}(1)}dz\int_{\gamma_{\rho_2}(1)}dw\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{\tau_2}}d\omega
\frac{H_{n,m,a}(z)H_{\Delta n,\Delta m,\Delta a}(w)(1-\zeta)(1-z)^{-1}}{H_{i,m,a}(\zeta)H_{N+1-j,\Delta m,\Delta a}(\omega)(z-\zeta)(w-\omega)(z-w)},$$ $$\label{B1}
B_1(i,j)=\frac{c(i)}{c(j)({2\pi\mathrm{i}})^4}\int_{\gamma_{\rho_3}(1)}dz\int_{\gamma_{\rho_2}(1)}dw\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{\tau_2}}d\omega
\frac{H_{n,m,a}(z)H_{\Delta n,\Delta m,\Delta a}(w)(1-\zeta)^(1-z)^{-1}}{H_{i,m,a}(\zeta)H_{N+1-j,\Delta m,\Delta a}(\omega)(z-\zeta)(w-\omega)(z-w)},$$ $$\label{A2}
A_2(i,j)=\frac{c(i)}{c(j)({2\pi\mathrm{i}})^2}\int_{\gamma_{\rho_2}(1)}dw\int_{\gamma_{\tau_2}}d\omega
\frac{H_{N-i,\Delta m,\Delta a}(w)}{H_{N+1-j,\Delta m,\Delta a}(\omega)(w-\omega)},$$ and $$\label{A3}
A_3(i,j)=\frac{c(i)}{c(j)({2\pi\mathrm{i}})^2}\int_{\gamma_{\rho_1}(1)}dz\int_{\gamma_{\tau_1}}d\zeta
\frac{H_{j-1,m,a}(z)(1-\zeta)}{H_{i,m,a}(\zeta)(z-\zeta)(1-z)}.$$ We also define, for $\epsilon\in\{0,1\}$ and $1\le i,j\le N$, $$\label{Cepsilon}
C(i,j)=A_1(i,j)-1(i>n)A_2(i,j)+A_3(i,j)1(j\le n),$$ $$\label{Depsilon}
D(i,j)=-B_1(i,j)+1(i>n)A_2(i,j)-A_3(i,j)1(j\le n),$$ compare with (\[Sxy\]) and (\[Txy\]).
We can now express $L_p$, $p=1,2$, in terms of these objects.
\[LemLformulas\] We have the formulas $$\label{L1delta}
L_1(i,j)=1(i\le n)\delta_{ij}+C(i,j),$$ and $$\label{L2delta}
L_2(i,j)=1(i>n)\delta_{ij}+D(i,j).$$
The proof is based on (\[L1\]), (\[L2\]), and Lemma \[Lemf01f12\], and suitable contour deformations in order to get the contours into positions that can be used in the asymptotic analysis, see section \[secprooflemmas\].
Combining (\[Lijuv\]) with Lemma \[LemLformulas\] we obtain $$\label{Liju}
L(i,j;u)=\delta_{ij}+M_u(i,j),$$ where $$\label{Muij}
M_u(i,j)=u^{-1(i\le n)}\left(uC(i,j)+D(i,j)\right),$$ and we also set $M_{u}(i,j)=0$ if $i,j\notin\{1,\dots,N\}$. Thus we have the formula $$\label{paAformula5}
P(a,A)=\frac 1{{2\pi\mathrm{i}}}\int_{\gamma_r}\frac 1{u-1}\det\big(\delta_{ij}+M_u(i,j)\big)_{1\le i,j\le N}du.$$
Next, we want to rewrite the determinant in (\[paAformula5\]) in a block determinant form, corresponding to $i\le n$ and $i>n$, and similarly for $j$. For $r,s\in\{1,2\}$, and $x,y\in\mathbb{R}$, we define $$\label{Fuv}
F_{u}(r,x;s,y)=M_{u}(n+[x]+1,n+[y]+1),$$ where $[\cdot]$ denotes the integer part. The right side of (\[Fuv\]) does not depend on $r$ or $s$ explicitely but we have $x<0$ for $r=1$ and $x\ge 0$ for $r=2$, and correspondingly for $y$ depending on $s$. Let $\Lambda=\{1,2\}\times\mathbb{R}$ and define the measures $$d\nu_1(x)=1(x<0)dx,\quad d\nu_2(x)=1(x\ge 0)(x)dx.$$ On $\Lambda$ we define a measure $\rho$ by $$\label{rhomeasure}
\int_{\Lambda}f(\lambda)d\rho(\lambda)=\sum_{r=1}^2\int_{\mathbb{R}}f(r,x)\,d\nu_r(x),$$ for every integrable function $f:\Lambda\mapsto\mathbb{R}$. $F_{u}$ defines an integral operator $F_{u}$ on $L^2(\Lambda,\rho)$ with kernel $F_{u}(r,x;s,y)$. Note that the space $L^2(\Lambda,\rho)$ is isomorphic to the space $X$ defined in (\[Xspace\]), and we can also think of $F_{u}$ as a matrix operator.
\[LemBlockdet\] We have the identity, $$\label{detidentity}
\det(\delta_{ij}+M_{u}(i,j))_{1\le i,j\le N}=\det(I+F_{u})_{L^2(\Lambda,\rho)}.$$
This is straightforward, using Fredholm expansions, and the lemma will be proved in section \[secprooflemmas\].
We can now insert the formula (\[detidentity\]) into (\[paAformula5\]). This leads to a formula that can be used for taking a limit, but before considering the limit, we have to introduce the appropriate scalings. For $s=1,2$, we define $$\label{Fuvtilde}
\tilde{F}_{u,T}(r,x;s,y)=
c_0(t_1T)^{1/3}F_{u}(r,c_0(t_1T)^{1/3}x;s,c_0(t_1T)^{1/3}y)$$ where $c_0$ is given by (\[scalingconstants\]). The next lemma follows from (\[paAformula5\]), Lemma \[LemBlockdet\], and (\[Fuvtilde\]), see section \[secprooflemmas\].
\[LemRescaled\] We have the formula, $$\label{paAformula6}
P(a,A)=\frac 1{{2\pi\mathrm{i}}}\int_{\gamma_r}\frac 1{u-1}\det\big(I+\tilde{F}_{u,T}\big)_{L^2(\Lambda,\rho)}du.$$
Theorem \[ThMain\] now follows by combining this lemma with the next lemma which will be proved in section \[secasymptotics\].
\[LemScalinglimit\] Consider the scaling (\[scaling\]) and let $K(u)$ be the matrix kernel defined by (\[Kuv\]). Then, $$\label{deterlimit}
\lim_{T\to\infty}\det\big(I+\tilde{F}_{u,T}\big)_{L^2(\Lambda,\rho)}=
\det\big(I+K(u)\big)_X,$$ uniformly for $u$ in a compact set.
Proof of Lemmas {#secprooflemmas}
===============
In this section we will prove the lemmas that were used in section \[secproofmain\]. Some results related to the asymptotic analysis will be proved in section \[secasymptotics\].
Write $$W^*_{N,k}(a)=\{\mathbf{x}\in W_N\,;\,x_k=a\}$$ so that $$W_{N,k}(a)=\bigcup_{t=-\infty}^a W^*_{N,k}(t)$$ Hence, it is enough to prove the statement with $W_{N,k}(a)$ replaced by $W^*_{N,k}(t)$. Let $a_i, b_i, c_i, d_i\in\mathbb{Z}$, $1\le i,j\le N$, and let $k<\ell\le N$. Assume that $b_{\ell-1}=b_\ell-1$, and $c_{\ell}=c_{\ell+1}$ if $\ell<N$. Set $$b_j'=\begin{cases}b_j &\text{if $j\neq \ell$} \\ b_\ell-1 &\text{if $j=\ell$} \end{cases}, \quad
c_j'=\begin{cases}c_j &\text{if $j\neq \ell$} \\ c_\ell-1 &\text{if $j=\ell$} \end{cases}.$$ Then, $$\begin{aligned}
\label{sum1}
&\sum_{\mathbf{x}\in W^*_{N,k}(t)}\det\left(\Delta^{b_j-a_i}f(x_j-y_i)\right)_{1\le i,j\le N}\det\left(\Delta^{d_i-c_j}g(z_i-x_j)\right)_{1\le i,j\le N}\\
=&\sum_{\mathbf{x}\in W^*_{N,k}(t)}\det\left(\Delta^{b'_j-a_i}f(x_j-y_i)\right)_{1\le i,j\le N}\det\left(\Delta^{d_i-c'_j}g(z_i-x_j)\right)_{1\le i,j\le N}.\notag\end{aligned}$$
To prove (\[sum1\]), we use the summation by parts identity, $$\label{sumparts}
\sum_{y=a}^b\Delta u(y-x)c(z-y)=\sum_{y=a}^b u(y-x)\Delta c(z-y) + u(b+1-x)v(z-b)-u(a-x)v(z+1-a).$$ Consider the $x_\ell$-summation in the left side of (\[sum1\]) with all the other variables fixed. Let $x_{\ell+1}=\infty$ if $\ell=N$ and let $\Delta_x$ denote the finite difference with respect to the variable $x$. Using (\[sumparts\]) in the second inequality we get $$\begin{aligned}
\label{sumpartsdet}
&\sum_{x_\ell=x_{\ell-1}}^{x_{\ell+1}}\det\left(\Delta^{b_j-a_i}f(x_j-y_i)\right)_{1\le i,j\le N}\det\left(\Delta^{d_i-c_j}g(z_i-x_j)\right)_{1\le i,j\le N}\\
=&\sum_{x_\ell=x_{\ell-1}}^{x_{\ell+1}}\Delta_{x_{\ell}}\det\left(\Delta^{b'_j-a_i}f(x_j-y_i)\right)_{1\le i,j\le N}\det\left(\Delta^{d_i-c_j}g(z_i-x_j)\right)_{1\le i,j\le N}\notag\\
=&\sum_{x_\ell=x_{\ell-1}}^{x_{\ell+1}}\det\left(\Delta^{b'_j-a_i}f(x_j-y_i)\right)_{1\le i,j\le N}\det\left(\Delta^{d_i-c'_j}g(z_i-x_j)\right)_{1\le i,j\le N}\notag\\
+&\left.\det\left(\Delta^{b_j'-a_i}f(x_j-y_i)\right)_{1\le i,j\le N}\right|_{x_\ell\to x_{\ell+1}+1}\left.\det\left(\Delta^{d_i-c_j}g(z_i-x_j)\right)_{1\le i,j\le N}\right|_{x_\ell\to x_{\ell+1}}\notag\\
-&\left.\det\left(\Delta^{b_j'-a_i}f(x_j-y_i)\right)_{1\le i,j\le N}\right|_{x_\ell\to x_{\ell-1}}\left.\det\left(\Delta^{d_i-c_j}g(z_i-x_j)\right)_{1\le i,j\le N}\right|_{x_\ell\to x_{\ell-1}-1}.\notag\end{aligned}$$ If $\ell=N$, then the first boundary term in (\[sumpartsdet\]) is $=0$. This follows since $\Delta^{d_i-c_\ell}g(z_i-\infty)=0$ (assumption that all series are convergents, expressions well-defined), so one column in the second determinant the first boundary term in (\[sumpartsdet\]) is $=0$. If $\ell<N$, then the first boundary term in (\[sumpartsdet\]) is $=0$ because $c_{\ell}=c_{\ell+1}$, and $x_\ell\to x_{\ell+1}$ means that columns $\ell$ and $\ell+1$ will be identical in the second determinant. Since $b'_{\ell}=b_\ell-1=b_{\ell-1}$, we see that columns $\ell$ and $\ell-1$ in the first determinant in the second boundary term in (\[sumpartsdet\]) will be identical.
Similarly, if $1\le \ell<k$, and $c_{\ell+1}=c_\ell+1$, $b_\ell=b_{\ell-1}$, then $$\begin{aligned}
\label{sum2}
&\sum_{\mathbf{x}\in W^*_{N,k}(t)}\det\left(\Delta^{b_j-a_i}f(x_j-y_i)\right)_{1\le i,j\le N}\det\left(\Delta^{d_i-c_j}g(z_i-x_j)\right)_{1\le i,j\le N}\\
=&\sum_{\mathbf{x}\in W^*_{N,k}(t)}\det\left(\Delta^{b''_j-a_i}f(x_j-y_i)\right)_{1\le i,j\le N}\det\left(\Delta^{d_i-c''_j}g(z_i-x_j)\right)_{1\le i,j\le N},\notag\end{aligned}$$ where $$b_j''=\begin{cases}b_j &\text{if $j\neq \ell$} \\ b_\ell+1 &\text{if $j=\ell$} \end{cases}, \quad
c_j''=\begin{cases}c_j &\text{if $j\neq \ell$} \\ c_\ell+1 &\text{if $j=\ell$} \end{cases}.$$ The proof of (\[sum2\]) is analogous to the proof of (\[sum1\]).
To prove lemma \[LemSumparts\], we apply (\[sum1\]) successively to $x_N,x_{N-1},\dots,x_{k+1}$, and then to $x_N,x_{N-1},\dots,x_{k+2}$ etc., and then finally just to $x_N$. Similarly, we apply (\[sum2\]) to $x_1,x_2,\dots, x_{k-1}$, then to $x_1,x_2,\dots, x_{k-2}$, and finally just to $x_1$. This proofs part a) of the lemma.
Part b) of the lemma follows from the identity $$\label{sum3}
\sum_{\mathbf{x}\in W_{N,N}(a)}\det\left(\Delta^{i-n}f_j(x_i)\right)_{1\le i,j\le N}=\det\left(\Delta^{i-1-n}f_j(a+1)\right)_{1\le i,j\le N}.$$ To prove (\[sum3\]), first sum over $x_N$ from $x_{N-1}$ to $a$ in the last row. This gives $\Delta^{N-1-n}f_j(a+1)-\Delta^{N-1}f_j(x_{N-1})$. The last term does not contribute since it is the same as in row $N-1$. We can now sum over $x_{N-1}$ from $x_{N-2}$ to $a$ in row $N-1$ etc. In this way we obtain (\[sum3\]).
We see that $$\begin{aligned}
\label{QaAformula}
P(a,A)&=\sum_{\mathbf{x}\in W_N\,;\,x_n<0}\det\big(f_{0,1}(i,x_j)\big)_{1\le i,j\le N}\det\big(f_{1,2}(x_i,j)\big)_{1\le i,j\le N}\\
&=\sum_{\mathbf{x}\in W_N}\det\big(f_{0,1}(i,x_j)\big)_{1\le i,j\le N}\det\big(f_{1,2}(x_i,j)\big)_{1\le i,j\le N} 1\left(\sum_{j=1}^N1(x_j<0)\ge n\right)\notag\end{aligned}$$ Now, for any $r>0$, $$\frac 1{{2\pi\mathrm{i}}}\int_{\gamma_r}\frac{u^{\sum_{j=1}^N1(x_j<0)}}{u^{\ell+1}}du=1\left(\sum_{j=1}^N1(x_j<0)=\ell\right).$$ Summing over $\ell\ge n$ and assuming that $r>1$, we get $$\label{contourformula}
\frac 1{{2\pi\mathrm{i}}}\int_{\gamma_r}\frac{u^{\sum_{j=1}^N1(x_j<0)}}{u^n(u-1)}du=1\left(\sum_{j=1}^N1(x_j<0)\ge n\right).$$ Since, $$u^{\sum_{j=1}^N1(x_j<0)}=\prod_{j=1}^N\left(u1(x_j<0)+1(x_j\ge 0)\right),$$ it follows from (\[QaAformula\]), (\[contourformula\]), and the Cauchy-Binet identity that $$\begin{aligned}
P(a,A)&=\frac 1{{2\pi\mathrm{i}}}\int_{\gamma_r}\frac{du}{u^n(u-1)}
\sum_{\mathbf{x}\in W_N}\det\big(f_{0,1}(i,x_j)\big)_{1\le i,j\le N}\det\big(f_{1,2}(x_i,j)\big)_{1\le i,j\le N}\\
&\times \prod_{j=1}^N\left(u1(x_j<0)+1(x_j\ge 0)\right)\notag\\
&=\frac 1{{2\pi\mathrm{i}}}\int_{\gamma_r}\frac{du}{u^n(u-1)}\det\left(\sum_{z\in\mathbb{Z}}f_{0,1}(i,x)f_{1,2}(x,j)(u1(x<0)+1(x\ge 0))\right)_{1\le i,j\le N}\notag\\
&=\frac 1{{2\pi\mathrm{i}}}\int_{\gamma_r}\frac{du}{u-1}\det\left(u^{-1(i\le n)}(uL_1(i,j)+L_2(i,j))\right)_{1\le i,j\le N}\notag\\
&=\frac 1{{2\pi\mathrm{i}}}\int_{\gamma_r}\frac{du}{u-1}\det\left(L(i,j;u)\right)_{1\le i,j\le N}.\notag\end{aligned}$$
It follows from (\[f01\]), (\[betaformula\]), and (\[ABdef\]), that $$\begin{aligned}
f_{0,1}(i,x)&=c(i)\sum_{k=1}^N\beta_{k-i}^1(m,a)(-1)^{n-k}\Delta^{n-k}w_m(a+x)\\&=
-\frac {c(i)}{{2\pi\mathrm{i}}}\int_{\gamma_r}\left(\sum_{k=1}^N\frac{\beta_{k-i}^1(m,a)}{z^k}\right)H_{n,m,a+x}(z)\frac{dz}{1-z}\\
&=-\frac {c(i)}{({2\pi\mathrm{i}})^2}\int_{\gamma_r}dz\int_{\gamma_\tau}d\zeta\frac{H_{n,m,a+x}(z)(1-\zeta)}{H_{i,m,a}(\zeta)(z-\zeta)(1-z)}.\end{aligned}$$ Similarly, by (\[f12\]), (\[betaformula\]) and (\[ABdef\]), $$\begin{aligned}
f_{1,2}(i,x)&=c(j)^{-1}\sum_{k=1}^N(-1)^{k-n}\Delta^{k-n-1}w_{\Delta m}(\Delta a-x)\beta_{j-k}^0(\Delta m,\Delta a)\\&=
\frac {c(j)^{-1}}{{2\pi\mathrm{i}}}\int_{\gamma_r}\left(\sum_{k=1}^N\frac{\beta_{j-k}^0(\Delta m,\Delta a)}{w^{N+1-k}}\right)H_{\Delta n,\Delta m,\Delta a-x}(w)\frac{dw}{1-w}\\
&=\frac {c(j)^{-1}}{{2\pi\mathrm{i}}}\int_{\gamma_r}\left(\sum_{k=1}^N\frac{\beta_{j-(N+1-k)}^0(\Delta m,\Delta a)}{w^{k}}\right)H_{\Delta n,\Delta m,\Delta a-x}(w)\frac{dw}{1-w}\\
&=\frac {c(j)^{-1}}{({2\pi\mathrm{i}})^2}\int_{\gamma_r}dw\int_{\gamma_\tau}d\omega\frac{H_{\Delta n,\Delta m,\Delta a-x}(z)}{H_{N+1-j,\Delta m,\Delta a}(\omega)(w-\omega)(1-w)}.\end{aligned}$$ This proves the lemma.
Recall the condition (\[radii\]) and choose $r_1, r_2$ so that $r_1>r_2>1+\max(\rho_1,\rho_2)$, which means that $\gamma_{r_i}(1)$ surrounds $\gamma_{\rho_i}$ and $\gamma_{\tau_i}$, $i=1,2$. It follows from (\[f01formula\]) and (\[f12formula\]), that $$\begin{aligned}
L_1(i,j)&=-\frac{c(i)c(j)^{-1}}{({2\pi\mathrm{i}})^4}\sum_{x=-\infty}^{-1}\left(\int_{\gamma_{r_1}(1)}dz\int_{\gamma_{\tau_1}}d\zeta\frac{H_{n,m,a+x}(z)(1-\zeta)}{H_{i,m,a}(\zeta)(z-\zeta)(1-z)}\right)
\\
&\times\left(\int_{\gamma_{r_2}(1)}dw\int_{\gamma_{\tau_2}}d\omega\frac{H_{\Delta n,\Delta m,\Delta a-x}(w)}{H_{N+1-\Delta m,\Delta a}(\omega)(w-\omega)(1-w)}\right)\\
&=-\frac{c(i)c(j)^{-1}}{({2\pi\mathrm{i}})^4}\int_{\gamma_{r_1}(1)}dz\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{r_2}(1)}dw\int_{\gamma_{\tau_2}}d\omega\left(\sum_{x=-\infty}^{-1}
\left(\frac{1-z}{1-w}\right)^x\right)\\
&\times\frac{H_{n,m,a}(z)H_{\Delta n,\Delta m,\Delta a}(w)(1-\zeta)}{H_{i,m,a}(\zeta)H_{N+1-j,\Delta m,\Delta a}(\omega)(z-\zeta)(w-\omega)
(1-z)(1-w)}.\end{aligned}$$ Since $r_1>r_2$, $$\sum_{x=-\infty}^{-1}\left(\frac{1-z}{1-w}\right)^x=-\frac{1-w}{z-w},$$ and we obtain $$\begin{aligned}
L_1(i,j)&=\frac{c(i)c(j)^{-1}}{({2\pi\mathrm{i}})^4}\int_{\gamma_{r_1}(1)}dz\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{r_2}(1)}dw\int_{\gamma_{\tau_2}}d\omega
\\&\times \frac{H_{n,m,a}(z)H_{\Delta n,\Delta m,\Delta a}(w)(1-\zeta)}{H_{i,m,a}(\zeta)H_{N+1-j,\Delta m,\Delta a}(\omega)(z-\zeta)
(w-\omega)(z-w)(1-z)}.\end{aligned}$$ We now deform $\gamma_{r_2}(1)$ to $\gamma_{\rho_2}(1)$. Doing so, we cross the pole at $w=\omega$, and hence $$\begin{aligned}
\label{L1step}
L_1(i,j)&=\frac{c(i)c(j)^{-1}}{({2\pi\mathrm{i}})^4}\int_{\gamma_{r_1}(1)}dz\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{\rho_2}(1)}dw\int_{\gamma_{\tau_2}}d\omega
\\&\times \frac{H_{n,m,a}(z)H_{\Delta n,\Delta m,\Delta a}(w)(1-\zeta)}{H_{i,m,a}(\zeta)H_{N+1-j,\Delta m,\Delta a}(\omega)(z-\zeta)
(w-\omega)(z-w)(1-z)}\notag\\
&+\frac{c(i)c(j)^{-1}}{({2\pi\mathrm{i}})^3}\int_{\gamma_{r_1}(1)}dz\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{\tau_2}}d\omega
\frac{H_{n,m,a}(z)(1-\zeta)}{H_{i,m,a}(\zeta)\omega^{n+1-j}
(z-\omega)(z-\zeta)(1-z)}:=I_1+I_2.\notag\end{aligned}$$ In $I_1$ we can shrink $\gamma_{r_1}(1)$ to $\gamma_{\rho_1}(1)$. We then cross the pole at $z=\zeta$ (but not $z=w$ since $\rho_2<\rho_1$). Thus, by (\[A1\]), $$\begin{aligned}
\label{I1}
I_1&=A_1(i,j)+\frac{c(i)c(j)^{-1}}{({2\pi\mathrm{i}})^3}\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{\rho_2}(1)}dw\int_{\gamma_{\tau_2}}d\omega
\frac{\zeta^{n-i}H_{\Delta n,\Delta m,\Delta a}(w)}{H_{N+1-j,\Delta m,\Delta a}(\omega)(w-\omega)(\zeta-w)}\\
:&=A_1(i,j)+I_3.\notag\end{aligned}$$ We note that $$\label{Int1}
\frac 1{{2\pi\mathrm{i}}}\int_{\gamma_{\tau_1}}\frac{d\zeta}{\zeta^{i-n}(\zeta-w)}=-\frac{1(i>n)}{w^{i-n}},$$ since $|w|>|\zeta|$, and hence by (\[A2\]), $$\label{I3}
I_3=-1(i>n)A_2(i,j).$$ Also $$\label{Int2}
\frac 1{{2\pi\mathrm{i}}}\int_{\gamma_{\tau_2}}\frac{d\omega}{\omega^{n+1-j}(z-\omega)}=\frac{1(j\le n)}{z^{n+1-j}},$$ and we obtain $$I_2=\frac{1(j\le n)c(i)c(j)^{-1}}{({2\pi\mathrm{i}})^2}\int_{\gamma_{r_1}(1)}dz\int_{\gamma_{\tau_1}}d\zeta\frac{H_{j-1,m.a}(z)(1-\zeta)}{H_{i,m,a}(\zeta)(z-\zeta)(1-z)}.$$ Deform $\gamma_{r_1}(1)$ to $\gamma_{\rho_1}(1)$. We then cross the pole at $z=\zeta$ and we obtain, using (\[A3\]), $$\label{I2}
I_2=1(j\le n)A_3(i,j)+\frac{1(j\le n)c(i)c(j)^{-1}}{{2\pi\mathrm{i}}}\int_{\gamma_{\tau_1}}\zeta^{i-j-1}\,d\zeta=1(j\le n)A_3(i,j)+1(i\le n)\delta_{ij}.$$ Combining (\[L1step\]), (\[I1\]), (\[I3\]), (\[I2\]) and (\[Cepsilon\]), we get (\[L1delta\]).
Consider next, $$\begin{aligned}
L_2(i,j)&=-\frac{c(i)c(j)^{-1}}{({2\pi\mathrm{i}})^4}\int_{\gamma_{r_3}(1)}dz\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{r_2}(1)}dw\int_{\gamma_{\tau_2}}d\omega\left(\sum_{x=0}^{\infty}
\left(\frac{1-z}{1-w}\right)^x\right)\\
&\times\frac{H_{n,m,a}(z)H_{\Delta n,\Delta m,\Delta a}(w)(1-\zeta)}{H_{i,m,a}(\zeta)H_{N+1-j,\Delta m,\Delta a}(\omega)(z-\zeta)(w-\omega)
(1-z)(1-w)},\end{aligned}$$ where now $r_2>r_3>1+\max(\rho_1,\rho_2)$. Thus, $$\sum_{x=0}^{\infty}\left(\frac{1-z}{1-w}\right)^x=\frac{1-w}{z-w},$$ and consequently $$\begin{aligned}
L_2(i,j)&=-\frac{c(i)c(j)^{-1}}{({2\pi\mathrm{i}})^4}\int_{\gamma_{r_3}(1)}dz\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{r_2}(1)}dw\int_{\gamma_{\tau_2}}d\omega
\\&\times \frac{H_{n,m,a}(z)H_{\Delta n,\Delta m,\Delta a}(w)(1-\zeta)}{H_{i,m,a}(\zeta)H_{N+1-j,\Delta m,\Delta a}(\omega)(z-\zeta)
(w-\omega)(z-w)(1-z)}.\end{aligned}$$ We now deform $\gamma_{r_3}(1)$ to $\gamma_{\rho_3}(1)$, and doing so we pass the pole at $z=\zeta$, and find $$\begin{aligned}
L_2(i,j)&=-\frac{c(i)c(j)^{-1}}{({2\pi\mathrm{i}})^4}\int_{\gamma_{\rho_3}(1)}dz\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{r_2}(1)}dw\int_{\gamma_{\tau_2}}d\omega
\\&\times \frac{H_{n,m,a}(z)H_{\Delta n,\Delta m,\Delta a}(w)(1-\zeta)}{H_{i,m,a}(\zeta)H_{N+1-j,\Delta m,\Delta a}(\omega)(z-\zeta)
(w-\omega)(z-w)(1-z)}\\
&-\frac{c(i)c(j)^{-1}}{({2\pi\mathrm{i}})^3}\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{r_2}(1)}dw\int_{\gamma_{\tau_2}}d\omega
\frac{H_{\Delta n,\Delta m,\Delta a}(w)}{\zeta^{i-n}H_{N+1-j,\Delta m,\Delta a}(\omega)
(w-\omega)(\zeta-w)}:=J_1+J_2.\end{aligned}$$ In $J_1$ we deform $\gamma_{r_2}(1)$ to $\gamma_{\rho_2}(1)$. Since $\rho_2>\rho_3$, we only cross the pole at $w=\omega$, and we get $$\begin{aligned}
J_1&=-B_1(i,j)-\frac{c(i)c(j)^{-1}}{({2\pi\mathrm{i}})^3}\int_{\gamma_{\rho_3}(1)}dz\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{\tau_2}}d\omega
\frac{H_{n,m,a}(z)(1-\zeta)}{\omega^{n+1-j}H_{i,m,a}(\zeta)(z-\zeta)(z-\omega)(1-z)}\\:&=-B_1(i,j)+J_3.\end{aligned}$$ Using (\[Int1\]), we find $$\begin{aligned}
J_2&=\frac{1(i>n)c(i)c(j)^{-1}}{({2\pi\mathrm{i}})^2}\int_{\gamma_{r_2}(1)}dw\int_{\gamma_{\tau_2}}d\omega\frac{H_{N-i,\Delta m,\Delta a}(w)}{H_{N+1-j,\Delta m,\Delta a}(\omega)(w-\omega)}\\
&=1(i>n)A_2(i,j)+\frac{1(i>n)c(i)c(j)^{-1}}{{2\pi\mathrm{i}}}\int_{\gamma_{\tau_2}} \omega^{j-i-1}d\omega=1(i>n)(A_2(i,j)+\delta_{ij}),\end{aligned}$$ which gives (\[L2delta\]) and the lemma is proved.
We start with the right side of (\[detidentity\]), $$\begin{aligned}
&\det(I+F_{u})_{L^2(\Lambda,\rho)}=\sum_{k=0}^{\infty}\frac 1{k!}\int_{\Lambda^k}d\rho^k(\lambda)\det(F_{u}(\lambda_p,\lambda_q))_{1\le p,q\le k}\\
&=\sum_{k=0}^{\infty}\frac 1{k!}\sum_{r_1,\dots,r_k=1}^2\int_{\mathbb{R}^k}d\nu_{r_1}(x_1)\dots d\nu_{r_k}(x_k)
\det\left(M_{u}(n+[x_p]+1,n+[x_q]+1)\right)_{1\le p,q\le k}\\
&=\sum_{k=0}^{\infty}\frac 1{k!}\sum_{i_1=-n}^{N-n-1}\dots\sum_{i_k=-n}^{N-n-1}\det\left(M_{u,v}(n+i_p+1,n+i_q+1)\right)_{1\le p,q\le k}\\
&=\det(\delta_{ij}+M_{u})(i,j))_{1\le i,j\le N},\end{aligned}$$ where we recall that $M_{u}(i,j)=0$ if $i,j\notin\{1,\dots,N\}$.
By the formula (\[paAformula5\]) for $P(a;A)$ and Lemma \[LemBlockdet\], we see that $$\begin{aligned}
\label{paArescaled}
P(a;A)&=\frac{c_3(t_1T)^{1/3}}{{2\pi\mathrm{i}}}\int_{\gamma_r}\frac{1}{u-1}\det(I+F_{u})_{L^2(\lambda,\rho)}du\\
&=\frac{1}{{2\pi\mathrm{i}}}\int_{\gamma_r}\frac{du}{u-1}\det(I+F_{u})_{L^2(\lambda,\rho)}du.\notag\end{aligned}$$ We have the Fredholm expansion, $$\label{Fredexp}
\det\left(I+F_{u}\right)_{L^2(\lambda,\rho)}=
\sum_{k=0}^\infty\frac 1{k!}\sum_{r_1,\dots,r_k=1}^2\int_{\mathbb{R}^k}d\nu_{r_1}(x_1)\dots d\nu_{r_k}(x_k)\det\left(F_{u}(r_p,x_p;r_q,x_q)\right)_{1\le p,q\le k}.$$ The change of variables $x_p\to c_0(t_1T)^{1/3}x_p$ gives $$d\nu_{r_p}(c_0(t_1T)^{1/3}x_p)=c_0(t_1T)^{1/3}d\nu_{r_p}(x_p).$$ Take the factor $c_0(t_1T)^{1/3}$ into row $p$. We see then that the right side of (\[Fredexp\]) equals, $$\sum_{k=0}^\infty\frac 1{k!}\sum_{r_1,\dots,r_k=1}^2\int_{\mathbb{R}^k}d\nu_{r_1}(x_1)\dots d\nu_{r_k}(x_k)\det\left(\tilde{F}_{u}(r_p,x_p;r_q,x_q)\right)_{1\le p,q\le k}
=\det\left(I+\tilde{F}_{u}\right)_{L^2(\lambda,\rho)}.$$ Combining this with (\[paArescaled\]) we have proved the lemma.
We want to prove that the operator $K(u)$ in the definition of the two-time distribution is a trace-class operator.
\[lemtraceclass\] The operator $K(u)$ defined by (\[Kuv\]) is a trace-class operator on the space $X$ given by (\[Xspace\]).
Write $$S_2^*(x,y)=1(x>0)S_2(x,y),\quad S_3^*(x,y)=S_3(x,y)1(y<0)$$ so that $$S=S_1-S_2^*+S_3^{*},\quad T=-T_1+S_2^{*}-S_3^*.$$ By splitting $K(u)$ into several parts and factoring out multiplicative constants, we see that it is enough to prove that $$\begin{pmatrix} A & A \\A & A \end{pmatrix}$$ is a trace-class operator on $X$ for $A=S_1, T_1, S_2^*,S_3^*$. We can think of $A$ as an operator on $L^2(\Lambda,\rho)$ instead, where $\Lambda=\{1,2\}\times\mathbb{R}$ and $\rho$ is given by (\[rhomeasure\]).
Define the kernels $$\begin{aligned}
\label{HSfunctions}
&a_1(x,s)=S_3(x,s)e^{-\delta s},\quad a_2(s,y)=e^{\delta s}S_2(s,y),\\
&b_1(x,s)=\alpha 1(x>0)e^{-(\delta-\alpha\Delta\eta)x}{\text{Ai\,}}(\Delta\xi+\Delta\eta^2+\alpha x+s),\notag\\
&b_2(x,s)=e^{(\delta-\alpha\Delta\eta)y}{\text{Ai\,}}(\Delta\xi+\Delta\eta^2+\alpha y+s),\notag\\
&c_1(x,s)=e^{-(\delta-\eta_1)x}{\text{Ai\,}}(\xi_1+\eta_1^2-x+s),\quad
c_2(x,s)=e^{(\delta-\eta_1)y}{\text{Ai\,}}(\xi_1+\eta_1^2-y+s)1(y<0).\notag\end{aligned}$$ Using the definitions, we see that $$\begin{aligned}
\label{STformulas}
S_1(x,y)&=\int_0^\infty(-a_1(x,s))a_2(s,y)\,ds,\quad T_1(x,y)=\int_{-\infty}^0a_1(x,s)a_2(s,y)\,ds\\
S_2^*(x,y)&=\int_0^\infty b_1(x,s))b_2(s,y)\,ds,\quad S_3^*(x,y)=\int_0^\infty c_1(x,s)c_2(s,y)\,ds,\notag\end{aligned}$$ To get kernels on $L^2(\Lambda,\rho)$, we define $$\begin{aligned}
a_1(r_1,x;1,s)&=b_1(r_1,x;1,s)=c_1(r_1,x;1,s)=0\\
a_2(1,s;r_3,y)&=b_2(1,s;r_3,y)=c_2(1,s;r_3,y)=0.\end{aligned}$$ for $r_1=1,2$, and $$\tilde{a}_1(r_1,x;2,s)=\tilde{a}_2(2,s;r_3,y)=0$$ for $r_1=1,2$. Furthermore, we define $$\begin{aligned}
-a_1(r_1,x;2,s)&=\tilde{a}_1(r_1,x;1,s)=a_1(x,s)\\
a_2(2,s;r_3,y)&=\tilde{a}_2(2,s;r_3,y)=a_2(s,y)\\
b_1(r_1,x;2,s)&=b_1(x,s),\quad b_2(2,s;r_3,y)=b_2(s,y)\\
c_1(r_1,x;2,s)&=c_1(x,s),\quad c_2(2,s;r_3,y)=c_2(s,y).\end{aligned}$$ Then, by (\[STformulas\]) and (\[rhomeasure\]), $$\int_{\Lambda}a_1(r_1,x;r_2,z)a_2(r_2,z;r_3,y)\,d\rho(r_2,z)=S_1(r_1,x;r_2,y),$$ so $S_1=a_1a_2$. Similarly, we see that $T_1=\tilde{a}_1\tilde{a}_2$, $S_2^*=b_1b_2$ and $S_3^*=c_1c_2$. Using (\[deltacondition\]) and asymptotic properties of the Airyfunction, we see that $a_1,a_2,b_1,b_2, c_1, c_2$ are square integrable over $\mathbb{R}^2$, and also over $\mathbb{R}$ if we fix one of the variables to be zero. It follows from this that $a_1, a_2, \tilde{a}_1,\dots, c_2$ are Hilbert-Schmidt operators on $L^2(\Lambda,\rho)$. Since the composition of two Hilbert-Schmidt operators is a trace-class operator, we have that $S_1, T_1, S_2^*$ and $S_3^{*}$ are trace-class operators on $L^2(\Lambda,\rho)$, and hence $K(u)$ is a trace-class operator also.
Asymptotic analysis {#secasymptotics}
===================
In this section we will prove Lemma \[LemScalinglimit\]. The proof has several steps and we will split it into a sequence of lemmas. The proofs of these lemmas will appear later in the section.
For $k=1,2,3$, we define the rescaled kernels $$\begin{aligned}
\label{Atilde}
\tilde{A}_{1,T}(x,y)&=c_0(t_1T)^{1/3}A_1(n+[c_0(t_1T)^{1/3}x]+1,n+[c_0(t_1T)^{1/3}y]+1),\\
\tilde{A}_{2,T}(x,y)&=1(x\ge 0)c_0(t_1T)^{1/3}A_2(n+[c_0(t_1T)^{1/3}x]+1,n+[c_0(t_1T)^{1/3}y]+1),\notag\\
\tilde{A}_{3,T}(x,y)&=1(y<0)c_0(t_1T)^{1/3}A_3(n+[c_0(t_1T)^{1/3}x]+1,n+[c_0(t_1T)^{1/3}y]+1),\notag\\
\tilde{B}_{1,T}(x,y)&=c_0(t_1T)^{1/3}B_1(n+[c_0(t_1T)^{1/3}x]+1,n+[c_0(t_1T)^{1/3}y]+1).\notag\end{aligned}$$
\[LemA1\] Uniformly, for $x,y$ in a compact subset of $\mathbb{R}$, we have the limits $$\begin{aligned}
\label{Sklimit}
\lim_{T\to\infty}\tilde{A}_{1,T}(x,y)&=S_1(x,y),\\
\lim_{T\to\infty}\tilde{A}_{2,T}(x,y)&=1(x\ge 0)S_2(x,y)\notag\\
\lim_{T\to\infty}\tilde{A}_{3,T}(x,y)&=S_3(x,y)1(y<0),\notag\end{aligned}$$ and $$\label{T1limit}
\lim_{T\to\infty}\tilde{B}_{1,T}(x,y)=T_1(x,y).$$
The lemma is proved below. In order to prove the convergence of the Fredholm determinant we also need some estimates.
\[LemKernelestimates\] Assume that $|\xi|,|\eta|\le L$ for some fixed $L$. If we choose $\delta$ in (\[conjfactor\]) sufficiently large, depending on $q$ and $L$, there are positive constants $C_0, C_1, C_2$ that only depend on $q$ and $L$, so that for all $x,y$ satisfying $$\label{xycondition}
0\le n+[c_0(t_1T)^{1/3}x]<N,\quad 0\le n+[c_0(t_1T)^{1/3}y]<N,$$ we have the estimates $$\begin{aligned}
\label{xyest}
\left|\tilde{A}_{1,T}(x,y)\right|&\le C_0 e^{-C_1(-x)_+^{3/2}-C_2(x)_+-C_1(y)_+^{3/2}-C_2(-y)_+},\\
\left|\tilde{B}_{1,T}(x,y)\right|&\le C_0 e^{-C_1(-x)_+^{3/2}-C_2(x)_+-C_1(y)_+^{3/2}-C_2(-y)_+},\notag\\
\left|\tilde{A}_{2,T}(x,y)\right|&\le C_0 1(x\ge 0)e^{-C_1(x)_+^{3/2}-C_1(y)_+^{3/2}-C_2(-y)_+},\notag\\
\left|\tilde{A}_{3,T}(x,y)\right|&\le C_0 1(y<0)e^{-C_1(-x)_+^{3/2}-C_2(x)_+ -C_1(-y)_+^{3/2}}.\notag\end{aligned}$$ Here $(x)_+=\max(0,x)$.
The proof is given below. We now have the estimates that we need to prove Lemma \[LemScalinglimit\]
Recall from (\[Ru\]) and (\[Kuv\]) that $$K_u(1,x;s,y)=S(x,y)+u^{-1}T(x,y),\quad K_u(2,x;s,y)=uS(x,y)+T(x,y),$$ $s=1,2$. It follows from Lemma \[LemA1\] that $$\label{Ftildelimit}
\lim_{T\to\infty}\tilde{F}_{u,T}(r,x;s,y)=K_{u}(r,x;s,y),$$ for $r,s\in\{1,2\}$, uniformly for $u,x,y$ in compact sets. From (\[xyest\]) we see that for all $\xi,\eta,u$ in compact sets there are positive constants $C_0,C_1$ so that $$\label{Ftildeest}
\left|\tilde{F}_{u,T}(r,x;s,y)\right|\le C_0e^{-C_1(|x|+|y|)},$$ for $r,s\in\{1,2\}$ and all $x,y\in\mathbb{R}$. Note that, by definition $\tilde{F}_{u,T}$ is zero if $x,y$ do not satisfy (\[xycondition\]). We can expand the Fredholm determinant, $$\label{Fredexpt}
\det(I+\tilde{F}_{u,T})_{L^2(\Lambda,\rho)}=\sum_{k=0}^\infty\frac 1{k!}\int_{\Lambda^k}\det(\tilde{F}_{u,T}(\lambda_i, \lambda_j))_{1\le i,j\le k}d^k\rho(\lambda)$$ in its Fredholm expansion. It follows from (\[Ftildelimit\]), (\[Ftildeest\]) and Hadamard’s inequality that we can take the limit $T\to\infty$ in (\[Fredexp\]) and get $$\sum_{k=0}^\infty\frac 1{k!}\int_{\Lambda^k}\det(K_u(\lambda_i,\lambda_j))_{1\le i,j\le k}d^k\rho(\lambda)=\det(I+K_u)_X.$$ This completes the proof.
Consider $$H_{k,\ell,b}(w)=\frac{w^k(1-w)^{b+\ell}}{\left(1-\frac w{1-q}\right)^\ell}$$ with the scalings ($K\to\infty$, $\eta,\xi,v$ fixed), $$\begin{aligned}
\label{klbscaling}
k&=K-c_1\eta K^{2/3}+c_0vK^{1/3},\\
\ell&=K+c_1\eta K^{2/3},\notag\\
b&=c_2K+c_3\xi K^{1/3}.\notag\end{aligned}$$ Here the constants $c_i$ are given by (\[scalingconstants\]). Write $$\label{fw}
f(w)=\log H_{k,\ell,b}(w)=k\log w+(b+\ell)\log(1-w)-\ell\log(1-\frac{w}{1-q}).$$ If $\eta=\xi=v=0$, then $f(w)$ has a double critical point at $$\label{wc}
w_c=1-\sqrt{q}.$$ Define $$\label{Hstar}
H^*_{k,\ell,b}(w)=\frac{H_{k,\ell,b}(w)}{H_{k,\ell,b}(w_c)}.$$ The local asymptotics around the critical point is given by the next lemma.
\[LemHstarlimit\] Fix $L>0$ and assume that $|\xi|,|\eta|, |v|\le L$. Furthermore, assume that we have the scaling (\[klbscaling\]). Then, uniformly for $w'$ in a compact set in $\mathbb{C}$ $$\label{Hstarlimit}
\lim_{K\to\infty} H^*_{k,\ell,b}\left(w_c+\frac{c_4}{K^{1/3}}w'\right)=\exp(\frac 13w'^3+\eta w'^2-(\xi-v)w'),$$ where $$\label{c4}
c_4=\frac{q^{1/3}(1-\sqrt{q})}{(1+\sqrt{q})^{1/3}}.$$
Let $$\begin{aligned}
f_1(w)&=\log w+(c_2+1)\log(1-w)-\log\left(1-\frac w{1-q}\right),\\
f_2(w)&=-\log w+\log(1-w)-\log\left(1-\frac w{1-q}\right),\\
f_3(w)&=c_0x\log w+c_3\xi\log(1-w),\end{aligned}$$ so that $$\label{Hnmx2}
f(w)=Kf_1(w)+c_1\eta K^{2/3}f_2(w)+K^{1/3}f_3(w).$$ Then $f_1'(w)$ has a double zero at $w_c$ only if the constant $c_2=2\sqrt{q}/(1-\sqrt{q})$. A computation gives $$f_1^{(3)}(w_c)=\frac{2(1+\sqrt{q})}{q(1-\sqrt{q})^3},$$ and we find $$\label{ftaylor}
K\left(f_1\left(w_c+\frac{c_4}{K^{1/3}}w'\right)-f_1(w_c)\right)=\frac 13 w'^3+O\left(\frac{|w'|^4}{K^{1/3}}\right).$$ Also, $$\label{g1taylor}
c_1\eta K^{2/3}\left(f_2\left(w_c+\frac{c_4}{K^{1/3}}w'\right)-f_2(w_c)\right)=\eta w'^2+O\left(\frac{|w'|^3}{K^{1/3}}\right),$$ and $$\label{g2taylor}
K^{2/3}\left(f_3\left(w_c+\frac{c_4}{K^{1/3}}w'\right)-f_3(w_c)\right)=-(\xi-v)w'+O\left(\frac{|w'|^2}{K^{1/3}}\right).$$ Using (\[ftaylor\]), (\[g1taylor\]) and (\[g2taylor\]) in (\[Hnmx2\]), we obtain $$\label{Hstarnmx}
H^*_{k,\ell,b}\left(w_c+\frac{c_4}{K^{1/3}}w'\right)=
\exp\left(\frac 13w'^3+\eta w'^2-(\xi-x)w'+O(|w'|^4/K^{1/3})\right)\notag$$ as $K\to\infty$.
To prove the estimates that we need, we use some explicit contours in (\[A1\]) to (\[A3\]). Let $d>0$ and define $$\label{w1}
w_1(\sigma)=w_1(\sigma;d)=w_c(1-\frac d{K^{1/3}})e^{\mathrm{i}\sigma/K^{1/3}},$$ and $$\label{w2}
w_2(\sigma)=w_2(\sigma;d)=1- \sqrt{q}(1-\frac d{K^{1/3}})e^{\mathrm{i}\sigma/K^{1/3}},$$ for $|\sigma|\le\pi K^{1/3}$, where $K$ is as in (\[klbscaling\]). Thus, $w_1$ gives a circle around the origin of radius $w_c(1-\frac d{K^{1/3}})$, and $w_2$ gives a circle of radius $ \sqrt{q}(1-\frac d{K^{1/3}})$ around $1$.
\[LemHstarest1\] Fix $L>0$. Assume that we have the scaling (\[klbscaling\]) and that $|\xi|,|\eta|, |v|\le L$. Then, there are positive constants $C_j$, $1\le j\le 4$ that only depend on $q$ and $L$, so that if $C_1\le d\le C_2$, then $$\label{Hstarest1}
\left|H^*_{k,\ell,b}(w_1(\sigma;d))\right|^{-1}\le C_3e^{-C_4\sigma^2},$$ and $$\label{Hstarest2}
\left|H^*_{k,\ell,b}(w_2(\sigma;d))\right|\le C_3e^{-C_4\sigma^2},$$ for $|\sigma|\le \pi K^{1/3}$.
We will also need estimates that work for large $v$.
\[LemHstarest2\] Assume that $|\xi|,|\eta|\le L$ for some fixed $L>0$, and assume that we have the scaling (\[klbscaling\]) and $v$ is such that $k\ge 0$. Then, we can choose $d=d(v)\ge C_0$, so that $$\label{Hstarest3}
\left|H^*_{k,\ell,b}(w_1(\sigma;d(v)))\right|^{-1}\le C_1e^{-C_2\sigma^2-\mu_1(-v)_+^{3/2}+\mu_2(v)_+},$$ for $|\sigma|\le \pi K^{1/3}$, where $C_0,C_1,C_2,\mu_1,\mu_2$ are positive constants that only depend on $q$ and $L$. Similarly, there is a choice of $d=d(v)$ so that $$\label{Hstarest4}
\left|H^*_{k,\ell,b}(w_2(\sigma;d(v)))\right|\le C_1e^{-C_2\sigma^2-\mu_1(-v)_+^{3/2}+\mu_2(v)_+}.$$
These two Lemmas will be proved below. We can use Lemma \[LemHstarlimit\] and Lemma \[LemHstarest1\] to prove Lemma \[LemA1\].
It follows from (\[conjfactor\]), (\[A1\]) and (\[Hstar\]) that $$\begin{aligned}
\label{Atildestar}
\tilde{A}_{1,T}(i,j)&=\frac{c_0(t_1T)^{1/3}e^{-\delta(x-y)}}{({2\pi\mathrm{i}})^4}\int_{\gamma_{\rho_1}(1)}dz\int_{\gamma_{\rho_2}(1)}dw\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{\tau_2}}d\omega
\\&\times\frac{H^*_{n,m,a}(z)H^*_{\Delta n,\Delta m,\Delta a}(w)(1-\zeta)(1-\sqrt{q})^{-1}}{H^*_{n+[c_0(t_1T)^{1/3}x]+1,m,a}(\zeta)H^*_{\Delta n-[c_0(t_1T)^{1/3}y],\Delta m,\Delta a}(\omega)(z-\zeta)(w-\omega)(z-w)(1-z)}.\notag\end{aligned}$$
Let $\Gamma_D$ denote the vertical line through $D$ oriented upwards, $\mathbb{R}\ni t\mapsto D+\mathrm{i}t$. Let $D_1>D_2>0$, $d_1,d_2>0$ be such that $$C_1\le\frac{c_4}{\sqrt{q}}D_r\le C_2,\quad C_1\le\frac{c_4}{\sqrt{q}}d_r\le C_2,$$ $r=1,2$, where $C_1,C_2$ are the constants in Lemma \[LemHstarest1\] with some fixed $L$ arbitrarily large. We choose the following parametrizations in (\[Atildestar\]), $$\label{paramet1}
z(\sigma_1)=w_2\left(\frac{c_4\sigma_1}{\sqrt{q}},\frac{c_4D_1}{\sqrt{q}}\right),\quad \zeta(\sigma_3)=w_1\left(\frac{c_4\sigma_3}{\sqrt{q}},\frac{c_4d_1}{\sqrt{q}}\right),$$ where $K=K_1=(t_1T)^{1/3}$ in (\[w1\]), (\[w1\]), and $$\label{paramet2}
w(\sigma_2)=w_2\left(\frac{c_4\sigma_2}{\sqrt{q}},\frac{c_4D_2}{\sqrt{q}}\right),\quad \omega(\sigma_4)=w_1\left(\frac{c_4\sigma_4}{\sqrt{q}},\frac{c_4d_2}{\sqrt{q}}\right),$$ where $K=K_2=(\Delta tT)^{1/3}$, $$\label{sigmai}
|\sigma_i|\le\pi K_1^{1/3}, \text{ for $i=1,3$}, \quad |\sigma_i|\le\pi K_2^{1/3}, \text{ for $i=2,4$}.$$ Recall the condition (\[radii\]) on the radii. Let $$\begin{aligned}
\label{hfunctions}
h_1(\sigma_1)=H^*_{n,m,a}(z(\sigma_1))&,\quad h_2(\sigma_2)=H*_{\Delta n,\Delta m,\Delta a}(w(\sigma_2)),\\
h_3(\sigma_3)=H^*_{n+[c_0(t_1T)^{1/3}x]+1,m,a}(\zeta(\sigma_3))&,\quad h_4(\sigma_4)=H^*_{\Delta n-[c_0(t_1T)^{1/3}y],\Delta m,\Delta a}(\omega(\sigma_4)).
\notag\end{aligned}$$ Now, a computation shows that, for some constant $C$, $$\label{estimate}
\left|\frac{c_0K_1^{1/3}}{(z(\sigma_1)-\zeta(\sigma_3))(w(\sigma_2)-\omega(\sigma_4))(z(\sigma_1)-w(\sigma_2))}\frac{dz}{d\sigma_1}\frac{dw}{d\sigma_2}
\frac{d\zeta}{d\sigma_3}\frac{d\omega}{d\sigma_4}\right|\le C$$ for all $\sigma_i$ satisfying (\[sigmai\]). Thus, for $x,y$ in a compact set, we have the following bound on the integrand in (\[Atildestar\]), $$\begin{aligned}
\label{intbound}
&\left|\frac{c_0K_1^{1/3}h_1(\sigma_1)h_2(\sigma_2)(1-\zeta(\sigma_3))(1-z(\sigma_1))^{-1}}{h_3(\sigma_3)h_4(\sigma_4)
(z(\sigma_1)-\zeta(\sigma_3))(w(\sigma_2)-\omega(\sigma_4))(z(\sigma_1)-w(\sigma_2))}\frac{dz}{d\sigma_1}\frac{dw}{d\sigma_2}
\frac{d\zeta}{d\sigma_3}\frac{d\omega}{d\sigma_4}\right|\\
&\le C\left|\frac{h_1(\sigma_1)h_2(\sigma_2)}{h_3(\sigma_3)h_4(\sigma_4)}\right|\le C_3'e^{-C_4'(\sigma_1^2+\sigma_2^2+\sigma_3^2+\sigma_4^2)},\notag\end{aligned}$$ where the last inequality follows from Lemma \[LemHstarest1\].
For $\sigma_i$ in a bounded set, we see that $$\begin{aligned}
\label{approxparamet}
z(\sigma_1)&=w_c+\frac{c_4}{K_1^{1/3}}(-\mathrm{i}\sigma_1+D_1)+O(K_1^{-2/3}),\\
w(\sigma_2)&=w_c+\frac{c_4}{K_2^{1/3}}(-\mathrm{i}\sigma_2+D_2)+O(K_2^{-2/3}),\notag\\
\zeta(\sigma_3)&=w_c+\frac{c_4}{K_1^{1/3}}(\mathrm{i}\sigma_3+d_1)+O(K_1^{-2/3}),\notag\\
\omega(\sigma_4)&=w_c+\frac{c_4}{K_2^{1/3}}(\mathrm{i}\sigma_4+d_2)+O(K_2^{-2/3}),\notag\end{aligned}$$ It follows from (\[scaling\]) that $$\begin{aligned}
n&=K_1-c_1\eta_1K_1^{2/3},\quad \Delta n=K_2-c_1\Delta\eta K_2^{2/3}\\
m&=K_1-c_1\eta_2K_1^{2/3},\quad \Delta m=K_2+c_1\Delta\eta K_2^{2/3}\notag\\
a&=c_2K_1+c_3\xi_1K_1^{1/3}, \quad \Delta a=c_2K_2+c_3\Delta\xi K_2^{1/3},\notag\end{aligned}$$ and hence $$\begin{aligned}
n+c_0x(t_1T)^{1/3}&=K_1-c_1\eta_1 K_1^{2/3}+c_0xK_1^{1/3},\\
\Delta n-c_0y(t_1T)^{1/3}&=K_2-c_1\Delta\eta K_1^{2/3}-c_0\alpha yK_2^{1/3}.\notag\end{aligned}$$ Write $z'=-\mathrm{i}\sigma_1+D_1$, $w'=-\mathrm{i}\sigma_2+D_2$, $\zeta'=\mathrm{i}\sigma_3+d_1$, $\omega'=\mathrm{i}\sigma_4+d_2$. Note that $$\begin{aligned}
\label{differentials}
&c_0(t_1T)^{1/3}\frac{dzdwd\zeta d\omega}{(z-\zeta)(w-\omega)(z-w)}=\alpha (1-\sqrt{q})\frac{dz'dw'd\zeta' d\omega'}{(z'-\zeta')(w'-\omega')(z'-\alpha w')},
\\
&c_0(t_1T)^{1/3}\frac{dzd\zeta}{z-\zeta}=(1-\sqrt{q})\frac{dz'd\zeta'}{z'-\zeta'},\quad
c_0(t_1T)^{1/3}\frac{dwd\omega}{w-\omega}=\alpha (1-\sqrt{q})\frac{dw'd\omega'}{w'-\omega'}.\notag\end{aligned}$$
It follows from Lemma \[LemHstarlimit\], (\[Atildestar\]), (\[approxparamet\]), (\[intbound\]) and the dominated convergence theorem that $$\begin{aligned}
\label{A1limit}
\lim_{T\to\infty}\tilde{A}_{1,T}(x,y)&=\frac{\alpha e^{\delta(y-x)}}{({2\pi\mathrm{i}})^4}\int_{\Gamma_{D_1}}dz'\int_{\Gamma_{D_2}}dw'
\int_{\Gamma_{-d_1}}d\zeta'\int_{\Gamma_{-d_2}}d\omega'\\
&\times\frac{e^{\frac 13z'^3+\eta_1z'^2-\xi_1z'+\frac 13w'^3+\Delta\eta w'^2-\Delta\xi w'}}{e^{\frac 13\zeta'^3+\eta_1\zeta'^2-(\xi_1-x)\zeta'
+\frac 13\omega'^3+\Delta\eta \omega'^2-(\Delta\xi+\alpha y)\omega'}
(z'-\zeta')(w'-\omega')(z'-\alpha w')},\notag\end{aligned}$$ and we have the condition $$\label{Gammacondition}
d_1,d_2>0,\quad 0<D_1<\alpha D_2<D_3.$$
Define $$\label{Gxieta}
G_{\xi,\eta}(z)=e^{\frac 13 z^3+\eta z^2-\xi z},$$ and let $$\label{S1intformula}
S_1(x,y)=\frac{\alpha e^{\delta(y-x)}}{({2\pi\mathrm{i}})^4}\int_{\Gamma_{D_1}}dz\int_{\Gamma_{D_2}}dw
\int_{\Gamma_{-d_1}}d\zeta\int_{\Gamma_{-d_2}}d\omega\frac{G_{\xi_1,\eta_1}(z)G_{\Delta\xi,\Delta\eta}(w)}
{G_{\xi_1-x,\eta_1}(\zeta)G_{\Delta\xi+\alpha y,\Delta\eta}(\omega)(z-\zeta)(w-\omega)(z-\alpha w)}.$$ If $d,D>0$, we have the formulas, $$\begin{aligned}
\label{Gint}
\frac 1{{2\pi\mathrm{i}}}\int_{\Gamma_D}G_{\xi,\eta}(z)\,dz&={\text{Ai\,}}(\xi+\eta^2)e^{\xi\eta+\frac 23\eta^3},\\
\frac 1{{2\pi\mathrm{i}}}\int_{\Gamma_{-d}}\frac{d\zeta}{G_{\xi,\eta}(\zeta)}&={\text{Ai\,}}(\xi+\eta^2)e^{-\xi\eta-\frac 23\eta^3},
\notag\end{aligned}$$ with absolutely convergent integrals. Using (\[Gammacondition\]), we see that $$\frac 1{z-\zeta}=\int_0^\infty e^{-s_1(z-\zeta)}ds_1,\quad \frac 1{w-\omega}=\int_0^\infty e^{-s_2(w-\omega)}ds_2,
\quad \frac 1{z-\alpha w}=-\int_0^\infty e^{s_3(z-\alpha w)}ds_3.$$ It follows from these formulas, (\[S1intformula\]) and (\[Gint\]) that $S_1$ is also given by (\[S1\]).
The proof of (\[T1limit\]) is identical with $D_1$ replaced by $D_3$ satisfying (\[Gammacondition\]). The integral formula for $T_1$ reads $$\label{T1intformula}
T_1(x,y)=\frac{\alpha e^{\delta(y-x)}}{({2\pi\mathrm{i}})^4}\int_{\Gamma_{D_3}}dz\int_{\Gamma_{D_2}}dw
\int_{\Gamma_{-d_1}}d\zeta\int_{\Gamma_{-d_2}}d\omega\frac{G_{\xi_1,\eta_1}(z)G_{\Delta\xi,\Delta\eta}(w)}
{G_{\xi_1-x,\eta_1}(\zeta)G_{\Delta\xi+\alpha y,\Delta\eta}(\omega)(z-\zeta)(w-\omega)(z-\alpha w)}.$$ The other cases are treated similarly. For $S_2$ and $S_3$ we get the formulas $$\label{S2intformula}
S_2(x,y)=\frac{\alpha e^{\delta(y-x)}}{({2\pi\mathrm{i}})^2}\int_{\Gamma_{D_2}}dw\int_{\Gamma_{-d_2}}d\omega\frac{G_{\Delta\xi+\alpha x,\Delta\eta}(w)}
{G_{\Delta\xi+\alpha y,\Delta\eta}(\omega)(w-\omega)},$$ and $$\label{S3intformula}
S_3(x,y)=\frac{e^{\delta(y-x)}}{({2\pi\mathrm{i}})^2}\int_{\Gamma_{D_1}}dz\int_{\Gamma_{-d_1}}d\zeta\frac{G_{\xi_1-y,\eta_1}(\zeta)}{G_{\xi_1-x,\eta_1}(\zeta)(z-\zeta)}.$$
This proves Lemma \[LemA1\].
Consider first $\tilde{A}_{1,T}$. By Lemma (\[LemHstarest1\]), we can choose $d_1$ and $d_2$, with $d_1<\alpha d_2$, so that $$\begin{aligned}
\label{Est1}
|H^*_{n,m,a}(w_2(\sigma_1,d_1))|&\le C_3e^{-C_4\sigma_1^2}, \quad |\sigma_1|\le\pi K_1^{1/3},\\
|H^*_{\Delta n,\Delta m,\Delta a}(w_2(\sigma_2,d_1))|&\le C_3e^{-C_4\sigma_1^2}, \quad |\sigma_2|\le\pi K_2^{1/3},\notag\end{aligned}$$ where $C_3,C_4$ are some positive constants independent of $\sigma_1$ and $\sigma_2$. By Lemma \[LemHstarest2\], we can choose $d=d_3(x)\ge C_0$, and $d=d_4(y)\ge C_0$, so that $$\begin{aligned}
\label{Est2}
|H^*_{n+[c_0x(t_1T)^{1/3}]+1,m,a}(w_1(\sigma_3,d_3(x)))|^{-1}&\le C_1e^{-C_2\sigma_3^2-\mu_1(-x)_+^{3/2}+\mu_2(x)_+},\\
|H^*_{\Delta n-[c_0y(t_1T)^{1/3}],\Delta m,\Delta a}(w_1(\sigma_4,d_4(y)))|^{-1}&\le C_1e^{-C_2\sigma_3^2-\mu_1(y)_+^{3/2}+\mu_2(-y)_+},\notag\end{aligned}$$ It is not difficult to check that if $z=w_2(\sigma_1,d_1)$, $w=w_2(\sigma_2,d_2)$, $\zeta=w_1(\sigma_3,d_3(x))$ and $\omega=w_1(\sigma_4,d_4(y))$, then there is a constant $C_5$ so that $$|z-\zeta|\ge C_5K_1^{-1/3}, \quad |w-\omega|\ge C_5K_2^{-1/3},$$ and $$|z-w|\ge \sqrt{q}|d_1-\alpha d_2|K_1^{-1/3}\ge C_5K_1^{-1/3}.$$ Introducing these parametrizations into (\[Atildestar\]) and using the estimates above, we find $$\begin{aligned}
|\tilde{A}_{1,T}(x,y)|&\le Ce^{-\delta(x-y)-\mu_1(-x)_+^{3/2}+\mu_2(x)_+-\mu_1(y)_+^{3/2}+\mu_2(-y)_+}\int_{\mathbb{R}^4}
e^{-C_4(\sigma_1^2+\sigma_2^2+\sigma_3^2+\sigma_4^2)}d^4\sigma\\
&\le Ce^{-\delta(x-y)-\mu_1(-x)_+^{3/2}+\mu_2(x)_+-\mu_1(y)_+^{3/2}+\mu_2(-y)_+}.\notag\end{aligned}$$ We see that for large enough $|x|$, we can choose $\delta$ so large that $$-\mu_1(-x)_+^{3/2}+\mu_2(x)_+-\delta x\le -C_1(-x)_+^{3/2}-C_2(x)_+$$ for some positive constants $C_1,C_2$. This proves the estimate for $\tilde{A}_{1,T}$. The proof for $\tilde{B}_{1,T}$ is completely analogous.
Consider now $\tilde{A}_{3,T}$, $$\label{A3tildestar}
\tilde{A}_{3,T}(x,y)=\frac{c_0(t_1T)^{1/3}e^{-\delta(x-y)}1(y<0)}{(2{2\pi\mathrm{i}})^2}\int_{\gamma_{\rho_1}(1)}dz\int_{\gamma_{\tau_1}}d\zeta
\frac{H^*_{n+[c_0y(t_1T)^{1/3}],m,a}(z)(1-\zeta)}{H^*_{n+[c_0x(t_1T)^{1/3}]+1,m,a}(z)(1-z)(z-\zeta)}.$$ Using Lemma \[LemHstarest2\], we see that, just as for $\tilde{A}_{1,T}$, we can choose $d_1(y)$ and $d_2(x)$ so that $$\begin{aligned}
|H^*_{n+[c_0y(t_1T)^{1/3}],m,a}(w_2(\sigma_1,d_1(y)))|&\le C_1e^{-C_2\sigma_1^2-\mu_1(-y)_+^{3/2}+\mu_2(y)_+},\\
|H^*_{n+[c_0x(t_1T)^{1/3}],m,a}(w_1(\sigma_2,d_2(x)))^{-1}|&\le C_1e^{-C_2\sigma_1^2-\mu_1(-x)_+^{3/2}+\mu_2(x)_+},\end{aligned}$$ and we get the estimate $$|\tilde{A}_{3,T}(x,y)|\le Ce^{-\mu_1(-x)_+^{3/2}+\mu_2(x)_+-\delta x-\mu_1(-y)_+^{3/2}+\delta y}1(y<0).$$ This gives us the estimate we want by choosing $\delta$ large enough. The proof for $\tilde{A}_{2,T}$ is analogous.
The statements in Lemma \[LemHstarest1\] and in Lemma \[LemHstarest2\] are consequences of two other lemmas that we will now state and prove. The first lemma is concerned with the decay along the paths given by $w_1(\sigma)$ and $w_2(\sigma)$.
\[LemFirstest\] Assume that we have the scaling (\[klbscaling\]) and let $|\xi|,|\eta|\le L$ for some fixed $L>0$. There are positive constants $C_1,C_2,C_3,C_4$ that only depend on $q$ and $L$, so that if $$\label{dcondition}
C_1\le d\le C_2K^{1/3}$$ then for $|\sigma|\le\pi K^{1/3}$, $$\label{Hklbest1}
\left|\frac{H_{k,\ell,b}(w_1(\sigma;d))}{H_{k,\ell,b}(w_1(0;d))}\right|^{-1}\le C_3 e^{-C_4d\sigma^2},$$ for all $v\in\mathbb{R}$. Furthermore, for $|\sigma|\le\pi K^{1/3}$, $$\label{Hklbest2}
\left|\frac{H_{k,\ell,b}(w_2(\sigma;d))}{H_{k,\ell,b}(w_1(0;d))}\right|\le C_3 e^{-C_4d\sigma^2},$$ for all $v\le 0$ such that $k\ge 0$, and all $v$ such that $|v|\le L$.
Recall the definition of $f(w)$ in (\[fw\]) and the parametrizations (\[w1\]) and (\[w2\]). Define $$\label{gr}
g_r(\sigma)={\text{Re\,}}f(w_r(\sigma))=
k\log|w_r(\sigma)|+(b+\ell)\log|1-w_r(\sigma)|-\ell\log\left|1-\frac{w_r(\sigma)}{1-q}\right|,$$ $r=1,2$, $|\sigma|\le\pi K^{1/3}$. Note that for any real numbers $\alpha,\beta$, $$\label{dsigma}
\frac{d}{d\sigma}\log|1-\alpha e^{\mathrm{i}\beta\sigma}|=\frac{\alpha\beta\sin{\beta\sigma}}{(1-\alpha)^2+4\alpha\sin^2(\beta\sigma/2)}.$$ Let $\beta=K^{-1/3}$, $\alpha_1=w_c(1-dK^{-1/3})$, $\alpha_2=\alpha_1/(1-q)$. Then a computation using (\[gr\]) and (\[dsigma\]) gives $$\label{g1prime}
g_1'(\sigma)=\frac{(b+\ell)\alpha_1(1-\alpha_2)^2-\ell\alpha_2(1-\alpha_1)^2+4b\alpha_1\alpha_2\sin^2\frac{\beta\sigma}{2}}
{((1-\alpha_1)^2+4\alpha_1\sin^2\frac{\beta\sigma}{2})((1-\alpha_2)^2+4\alpha_2\sin^2\frac{\beta\sigma}{2})}\beta\sin\beta\sigma.$$ By symmetry it is enough to consider $0\le \sigma\le\pi K^{1/3}$. We have to compute $$\label{alphaexpr}
(b+\ell)\alpha_1(1-\alpha_2)^2-\ell\alpha_2(1-\alpha_1)^2=\frac{\alpha_1}{1-q}[(1-q)(b+\ell)(1-\alpha_2)^2-\ell(1-\alpha_1)^2].$$ Now, $$1-\alpha_1=\sqrt{q}+(1-\sqrt{q})d\beta,\quad 1-\alpha_2=\frac 1{1+\sqrt{q}}(\sqrt{q}+d\beta),$$ and using (\[klbscaling\]) a computation gives $$\begin{aligned}
&(1-q)(b+\ell)(1-\alpha_2)^2-\ell(1-\alpha_1)^2\\&=\left(2qd-\frac{2c_1q^{3/2}}{1+\sqrt{q}}\eta\right)K^{2/3}+\left(\sqrt{q}d^2-\frac{2c_1q(1-\sqrt{q})}{1+\sqrt{q}}\eta d+
\frac{c_3(1-\sqrt{q})q}{1+\sqrt{q}}\xi\right)K^{1/3}\\
&-\frac{c_1\sqrt{q}(1-\sqrt{q})}{1+\sqrt{q}}\eta d^2+\frac{2c_3\sqrt{q}(1-\sqrt{q})}{1+\sqrt{q}}\xi d+\frac{c_3(1-\sqrt{q})}{1+\sqrt{q}}\xi d^2K^{-1/3}.\end{aligned}$$ Since $|\xi|,|\eta|\le L$, we see that $$\label{alphaest}
(1-q)(b+\ell)(1-\alpha_2)^2-\ell(1-\alpha_1)^2\ge qdK^{2/3}+\Delta_1K^{2/3}+\Delta_2K^{1/3},$$ where $$\begin{aligned}
\Delta_1&=qd-\frac{2c_1q^{3/2}}{1+\sqrt{q}}L,\\
\Delta_2&=\sqrt{q}d^2-\frac{2c_1q(1-\sqrt{q})}{1+\sqrt{q}}Ld-\frac{c_3(1-\sqrt{q})q}{1+\sqrt{q}}L-\frac{c_1\sqrt{q}(1-\sqrt{q})}{1+\sqrt{q}}Ld^2K^{-1/3}\\
&-\frac{2c_3\sqrt{q}(1-\sqrt{q})}{1+\sqrt{q}}LdK^{-1/3}-\frac{c_3(1-\sqrt{q})}{1+\sqrt{q}}L d^2K^{-2/3}.\end{aligned}$$ We note that we can choose $C_1$ and $C_2$, depending only on $q$ and $L$, so that if $C_1\le d\le C_2K^{1/3}$, then $\Delta_1\ge 0$ and $\Delta_2\ge 0$, and also $$\frac{\alpha_1}{1-q}\ge\frac{w_c}{2(1-q)}=\frac 1{2(1+\sqrt{q})}.$$ Thus, we see from (\[alphaexpr\]) and (\[alphaest\]) that $$(b+\ell)\alpha_1(1-\alpha_2)^2-\ell\alpha_2(1-\alpha_1)^2\ge\frac{q}{2(1+\sqrt{q})}dK^{2/3}$$ provided that $C_1\le d\le C_2K^{1/3}$. Consequently, by (\[g1prime\]), $$\label{g1primeestt}
g_1'(\sigma)\ge\frac{qdK^{2/3}\sin K^{-2/3}\sigma}{2(1+\sqrt{q})(1+\alpha_1)^2(1+\alpha_2)^2}\ge \frac{q}{8(1+\sqrt{q})}dK^{2/3}\sin K^{-2/3}\sigma$$ since $$(1+\alpha_1)^2(1+\alpha_2)^2\le 4.$$ It follows, by integration, that, for $0\le\sigma\le\pi K^{1/3}$, $$g_1(\sigma)-g_1(0)\ge \frac{q}{4(1+\sqrt{q})}dK^{4/3}\sin^2\left(\frac{\sigma}{2K^{2/3}}\right)\ge\frac{q}{4(1+\sqrt{q})}dK^{4/3}\left(\frac{2\sigma}{2\pi K^{2/3}}\right)^2
=\frac{q}{4\pi^2(1+\sqrt{q})}d\sigma^2,$$ since by convexity $\sin t\ge 2t/\pi$ for $0\le t\le \pi/2$. This proves the estimate (\[Hklbest1\]).
Next, we turn to the proof of (\[Hklbest2\]) which is similar. In this case we get $$g_2'(\sigma)=\frac{d}{d\sigma}\left(k\log|1-\sqrt{q}(1-d\beta)e^{\mathrm{i}\beta\sigma}|-\ell\log|1-\frac 1{\sqrt{q}}(1-d\beta)e^{\mathrm{i}\beta\sigma}|\right),$$ where $\beta=K^{-1/3}$. Let $\alpha_1=\sqrt{q}(1-d\beta)$, $\alpha_2=\frac 1q \alpha$. Then, using (\[dsigma\]), we obtain $$\label{g2prime}
g_2'(\sigma)=\frac{k\alpha_1(1-\alpha_2)^2-\ell\alpha_2(1-\alpha_1)^2+4(k-\ell)\alpha_1\alpha_2\sin^2\frac{\beta\sigma}{2}}
{((1-\alpha_1)^2+4\alpha_1\sin^2\frac{\beta\sigma}{2})((1-\alpha_2)^2+4\alpha_2\sin^2\frac{\beta\sigma}{2})}\beta\sin\beta\sigma.$$ Now, $$\label{alphaexpr2}
k\alpha_1(1-\alpha_2)^2-\ell\alpha_2(1-\alpha_1)^2=\frac{\alpha_1}q\left[kq(1-\alpha_2)^2-\ell(1-\alpha_1)^2\right],$$ and a computation gives $$kq(1-\alpha_2)^2-\ell(1-\alpha_1)^2=-3(1-\sqrt{q})dK^{2/3}-\Delta K^{2/3},$$ where $$\begin{aligned}
\label{Delta}
\Delta&=(1-\sqrt{q})d+2c_1(1-\sqrt{q})^2\eta-(1-q)dK^{-1/3}-c_0(1-\sqrt{q})^2vK^{-1/3}\\
&+2c_1(1+q)\eta dK^{-2/3}+2c_0(1-\sqrt{q})vdK^{-2/3}-c_0vd^2K^{-1}.\notag\end{aligned}$$ If $|\xi|,|\eta|,|v|\le L$, we see that we can choose $C_1, C_2$, depending only on $q,L$, so that if $C_1\le d\le C_2K^{1/3}$, the $\Delta\ge 0$, and we obtain $$\label{alphaest2}
kq(1-\alpha_2)^2-\ell(1-\alpha_1)^2\le-3(1-\sqrt{q})dK^{2/3}.$$ If $|\xi|,|\eta|\le L$ and $v\le 0$, we can also choose $C_1, C_2$ so that $\Delta\ge 0$ if $C_1\le d\le C_2K^{1/3}$. Also, we see that $$\begin{aligned}
\label{alphaest3}
4(k-\ell)\alpha_1\alpha_2\sin^2\frac{\beta\sigma}{2}&=\left(-2c_1\eta K^{2/3}+c_0vK^{1/3}\right)\alpha_1\alpha_2\sin^2\frac{\sigma}{2K^{1/3}}\\
&\le 8(c_0+c_1)L\alpha_1\alpha_2K^{2/3}\le 8(c_0+c_1)LK^{2/3}.\notag\end{aligned}$$ if $v\le 0$ or $|v|\le L$. Assume that $C_2$ is such that $\alpha_1\ge\sqrt{q}/2$. Then (\[alphaexpr2\]), (\[alphaest2\]) and (\[alphaest3\]) give $$\begin{aligned}
&k\alpha_1(1-\alpha_2)^2-\ell\alpha_2(1-\alpha_1)^2+4(k-\ell)\alpha_1\alpha_2\sin^2\frac{\beta\sigma}{2}\\&\le
-\frac 1{\sqrt{q}}(1-\sqrt{q})dK^{2/3}+\left(-\frac{1-\sqrt{q}}{2\sqrt{q}}d+8(c_0+c_1)L\right)K^{2/3}\\
&\le -\frac 1{\sqrt{q}}(1-\sqrt{q})dK^{2/3},\end{aligned}$$ if we choose $C_1$ so that $$-\frac{1-\sqrt{q}}{2\sqrt{q}}d+8(c_0+c_1)L\le 0$$ for $d\ge C_1$. Since $\alpha_1\le\sqrt{q}$, $\alpha_1\le 1/\sqrt{q}$, $$\frac 1{(1+\alpha_1)^2(1+\alpha_2)^2}\ge \frac 1{(2+\sqrt{q}+1/\sqrt{q})^2},$$ and (\[g2prime\]) gives $$g_2'(\sigma)ß\le -\frac{1-\sqrt{q}}{\sqrt{q}(2+\sqrt{q}+1/\sqrt{q})^2}dK^{2/3}.$$ We can now proceed, as for $g_1$, to prove that $$g_2(\sigma)-g_2(0)\le -\frac{1-\sqrt{q}}{\pi^2\sqrt{q}(2+\sqrt{q}+1/\sqrt{q})^2}d\sigma^2.$$ This completes the proof of the Lemma.
The next Lemma is concerned with the decay for large $|v|$.
\[LemSecest\] Assume that we have the scaling (\[klbscaling\]) and that $v$ is such that $k\ge 0$, which will always be the case. Also, assume that $|\xi|,|\eta|\le L$ for some $L>0$. There are positive constants $\mu_1,\mu_2,\mu_3$ that only depend on $q,L$, and a choice $d=d(v)$ satisfying (\[dcondition\]) so that $$\label{Hklbest3}
\left|\frac{H_{k,\ell,b}(w_c)}{H_{k,\ell,b}(w_1(0;d(v)))}\right|\le\mu_3e^{-\mu_1(-v)_+^{3/2}+\mu_2(v)_+}.$$ There is also a choice $d=d(v)$ satisfying (\[dcondition\]) so that $$\label{Hklbest4}
\left|\frac{H_{k,\ell,b}(w_2(0;d(v)))}{H_{k,\ell,b}(w_c)}\right|\le\mu_3e^{-\mu_1(-v)_+^{3/2}+\mu_2(v)_+}.$$ If we assume that $|v|\le L$, we can choose $d$ independent of $v$ in some interval so that (\[Hklbest3\]) and (\[Hklbest4\]) hold.
Using (\[gr\]) we see that $$\left|\frac{H_{k,\ell,b}(w_1(0;d(v)))}{H_{k,\ell,b}(w_c)}\right|=e^{g_1(0)-\log f(w_c)},$$ so we want to estimate $g_1(0)-\log f(w_c)$ from below, and then make a good choice of $d$. We see that $$\label{g10}
g_1(0)-\log f(w_c)=k\log(1-dK^{-1/3})+(b+\ell)\log\left(1+\frac{1-\sqrt{q}}{\sqrt{q}}dK^{-1/3}\right)-\ell\log\left(1+\frac 1{\sqrt{q}}dK^{-1/3}\right).$$ To estimate this expression, we will use the inequalities $$\label{logineq}
-x-\frac{x^2}2-\frac{2x^3}{3}\le\log(1-x)\le -x-\frac{x^2}2-\frac{x^3}3,$$ for $1/2\le x\le 1$, and $$\label{logineq2}
x-\frac{x^2}2\le\log(1+x)\le x-\frac{x^2}2+\frac{x^3}3,$$ for $x\ge 0$. It follows from (\[g10\]) and these inequalities that $$\begin{aligned}
g_1(0)-\log f(w_c)&\ge k\left(-dK^{-1/3}-\frac 12d^2K^{-2/3}-\frac 23d^3K^{-1}\right)\\&+(b+\ell)\left(\frac{1-\sqrt{q}}{\sqrt{q}}dK^{-1/3}-\frac 12\left(\frac{1-\sqrt{q}}{\sqrt{q}}\right)^2d^2K^{-2/3}
\right)\\
&+\ell\left(-\frac 1{\sqrt{q}}dK^{-1/3}+\frac 1{2q}d^2K^{-2/3}-\frac 1{3q^{3/2}}d^3K^{-1}\right)\end{aligned}$$ Substitute the expressions in (\[klbscaling\]). After some manipulation this gives $$\begin{aligned}
\label{g10est}
g_1(0)-\log f(w_c)&\ge\left(-c_0v+\frac{1-\sqrt{q}}{\sqrt{q}}c_3\xi\right)d+
\left(\frac 1{\sqrt{q}}c_1\eta-\frac12c_0vK^{-1/3}-\frac{(1-\sqrt{q})^2}{2q}c_3\xi K^{-1/3}\right)d^2\\
&+\left(-\frac 23-\frac 1{3q^{3/2}}+\left(\frac 23-\frac 1{3q^{3/2}}\right)c_1\eta K^{-1/3}-\frac 23c_0vK^{-2/3}\right)d^3\notag\\
&\ge\left(-c_0v-\frac{1-\sqrt{q}}{\sqrt{q}}c_3L\right)d+\left(-\frac 1{\sqrt{q}}c_1L-\frac12c_0vK^{-1/3}-\frac{(1-\sqrt{q})^2}{2q}c_3LK^{-1/3}\right)d^2\notag\\
&+\left(-\frac 23-\frac 1{3q^{3/2}}-\left|\frac 23-\frac 1{3q^{3/2}}\right|c_1LK^{-1/3}-\frac 23c_0vK^{-2/3}\right)d^3.\notag\end{aligned}$$ If $|v|\le L$, we see that if we choose $d$ so that $C_1'\le d\le C_2'$, then $$g_1(0)-\log f(w_c)\ge -C_3'.$$ Here $C_1',C_2', C_3'$ only depend on $q,L$. If $v\le 0$, then it follows from (\[g10est\]) that $$\begin{aligned}
\label{g10est2}
g_1(0)-\log f(w_c)&\ge
\left(-c_0v-\frac{1-\sqrt{q}}{\sqrt{q}}c_3L\right)d+\left(-\frac 1{\sqrt{q}}c_1L-\frac{(1-\sqrt{q})^2}{2q}c_3LK^{-1/3}\right)d^2\\
&+\left(-\frac 23-\frac 1{3q^{3/2}}-\left|\frac 23-\frac 1{3q^{3/2}}\right|c_1LK^{-1/3}\right)d^3.\notag\end{aligned}$$ Choose $d=\epsilon\sqrt{-v}$. Then, by (\[g10est2\]), $$\begin{aligned}
\label{g10est3}
g_1(0)-\log f(w_c)&\ge
c_0\epsilon(-v)^{3/2}\left[1-\left(\frac{1-\sqrt{q}}{\sqrt{q}}\right)\frac{c_3L}{-v}-\left(\frac 1{\sqrt{q}}c_1L+\frac{(1-\sqrt{q})^2}{2q}\frac{c_3L}{K^{1/3}}\right)\epsilon^2\frac 1{\sqrt{-v}}\right.\\
&-\left.\left(\frac 23+\frac 1{3q^{3/2}}+\left|\frac 23-\frac 1{3q^{3/2}}\right|\frac{c_1L}{K^{1/3}}\right)\epsilon^2\right] \notag\end{aligned}$$ Choose $D_1$ large, depending on only $q,L$, so that $$\left(\frac{1-\sqrt{q}}{\sqrt{q}}\right)\frac{c_3L}{-v}\le \frac14,\quad \left(\frac 1{\sqrt{q}}c_1L+\frac{(1-\sqrt{q})^2}{2q}\frac{c_3L}{K^{1/3}}\right)\frac 1{\sqrt{-v}}\le 1,$$ if $\sqrt{-v}\ge D_1$. Since $k\ge 0$, there is a constant $D_2$ so that $\sqrt{-v}\le D_2K^{1/3}$. The condition (\[dcondition\]) becomes $$\frac{C_1}{\sqrt{-v}}\le\epsilon\le\frac{C_2K^{1/3}}{\sqrt{-v}},$$ which is satisfied if $$\label{epsiloncondition}
\frac{C_1}{D_1}\le\epsilon\le\frac{C_2}{D_2}.$$ We can choose $D_1$ so large that $C_1/D_1$ is as small as we want, and hence we can choose $\epsilon$ so small that $$\left(1+\frac 23+\frac 1{3q^{3/2}}+\left|\frac 23-\frac 1{3q^{3/2}}\right|\frac{c_1L}{K^{1/3}}\right)\epsilon^2\le \frac14.$$ It then follows from (\[g10est3\]) that $$g_1(0)-\log f(w_c)\ge\frac 12c_0\epsilon(-v)^{3/2}$$ for $\sqrt{-v}\ge D_1$. By adjusting $\mu_3$, we see that (\[Hklbest3\]) holds if $v\le 0$.
If $v\ge 0$, we choose a $d$ satistying (\[dcondition\]) depending on $q,L$, but not on $v$ or $K$. It follows from (\[g10est2\]) that there are constants $\mu_1$ and $\mu_3'$, so that $$g_1(0)-\log f(w_c)\ge -\mu_1(v)_+-\mu_3'.$$ Hence (\[Hklbest3\]) holds also when $v\ge 0$.
To prove (\[Hklbest4\]) we consider instead $$\begin{aligned}
g_2(0)-\log f(w_c)&=k\log\left(1+\frac{\sqrt{q}}{1-\sqrt{q}}dK^{-1/3}\right)+(b+\ell)\log(1-dK^{-1/3})-\ell\log\left(1-\frac 1{1-\sqrt{q}}dK^{-1/3}\right)\\
&\le k\left(\frac{\sqrt{q}}{1-\sqrt{q}}dK^{-1/3}-\frac{q}{2(1-\sqrt{q})^2}d^2K^{-2/3}+\frac{q^{3/2}}{(1-\sqrt{q})^3}d^3K^{-1}\right)\\&+
(b+\ell)\left(-dK^{-1/3}-\frac 12d^2K^{-2/3}-\frac 13d^3K^{-1}\right)\\
&+\ell\left(\frac{1}{1-\sqrt{q}}dK^{-1/3}+\frac 1{2(1-\sqrt{q})^2}d^2K^{-2/3}+\frac{2}{(1-\sqrt{q})^3}d^3K^{-1}\right),\end{aligned}$$ by (\[logineq\]) and (\[logineq2\]). Into this estimate we insert the expressions in (\[klbscaling\]), and after some computation we get $$\begin{aligned}
&g_2(0)-\log f(w_c)\le\left(\frac{\sqrt{q}}{1-\sqrt{q}}c_0v-c_3\xi\right)d\\
&+\frac 1{2(1-\sqrt{q})^2}\left(2\sqrt{q}c_1\eta-qc_0vK^{-1/3}+c_3(1-\sqrt{q})^2\xi K^{-1/3}\right)d^2\\
&+\frac 1{3(1-\sqrt{q})^3}\left(1+\sqrt{q}+q+(1+3\sqrt{q}-3q)c_1\eta K^{-1/3}+q^{3/2}c_0vK^{-2/3}-c_3(1-\sqrt{q})^3\xi K^{-2/3}\right)d^3.\end{aligned}$$ We can now proceed in analogy with the previous case to show (\[Hklbest4\]).
More formulas for the two-time distribution {#SecFormulas}
===========================================
In this section we give an alternative formula for the two-time distribution, see Proposition \[propQ\] below.
Recall the notation (\[Gxieta\]), $$\label{Gxieta2}
G_{\xi,\eta}(z)=e^{\frac 13 z^3+\eta z^2-\xi z}.$$ Looking at (\[Gint\]), we see that it is natural to write $$\label{Aixieta}
{\text{Ai\,}}_{\xi,\eta}(x,y)={\text{Ai\,}}(\xi+\eta^2+x+y)e^{(\xi+x+y)\eta+\frac 23\eta^3},$$ since we then get the formulas $$\begin{aligned}
\label{GAiformulas}
\frac 1{{2\pi\mathrm{i}}}\int_{\Gamma_D}G_{\xi+x+y,\eta}(z)\,dz&={\text{Ai\,}}_{\xi,\eta}(x,y),\\
\frac 1{{2\pi\mathrm{i}}}\int_{\Gamma_{-d}}\frac{d\zeta}{G_{\xi+x+y,\eta}(\zeta)}&={\text{Ai\,}}_{\xi,-\eta}(x,y),\end{aligned}$$ for any $d,D>0$. We can think of (\[Aixieta\]) as the kernel of an integral operator on $L^2(\mathbb{R}_+)$.
In order to give a different formula for the two-time distribution, we need to define several kernels. We will write $$\label{alphaprime}
\alpha'=(1+\alpha^3)^{1/3}=\left(\frac{t_2}{\Delta t}\right)^{1/3}.$$ Let $$\begin{aligned}
\label{M1}
M_1(v_1,v_2)&=\frac{e^{\delta(v_1-v_2)}}{({2\pi\mathrm{i}})^2}\int_{\Gamma_D}dz\int_{\Gamma_{-d}}d\zeta\frac{G_{\xi_1+v_1,\eta_1}(z)}{G_{\xi_1+v_2,\eta_1}(\zeta)(z-\zeta)}\\
&=e^{\delta(v_1-v_2)}\int_0^\infty{\text{Ai\,}}_{\xi_1,\eta_1}(v_1,\lambda){\text{Ai\,}}_{\xi_1,-\eta_1}(\lambda,v_2)\,d\lambda,\notag\end{aligned}$$ $$\begin{aligned}
\label{M2}
M_2(v_1,v_2)&=\frac{1}{({2\pi\mathrm{i}})^2\alpha'}\int_{\Gamma_D}dz\int_{\Gamma_{-d}}d\zeta\frac{G_{\xi_2+v_2/\alpha',\eta_2}(z)}{G_{\xi_2+v_1/\alpha',\eta_2}(\zeta)(z-\zeta)}\\
&=\frac 1{\alpha'}\int_0^\infty{\text{Ai\,}}_{\xi_2,-\eta_2}(v_1/\alpha',\lambda){\text{Ai\,}}_{\xi_2,\eta_2}(\lambda,v_2/\alpha')\,d\lambda,\notag\end{aligned}$$ and $$\begin{aligned}
\label{M3}
M_3(v_1,v_2)&=\frac{1}{({2\pi\mathrm{i}})^2}\int_{\Gamma_D}dz\int_{\Gamma_{-d}}d\zeta\frac{G_{\Delta\xi+v_2,\Delta\eta}(z)}{G_{\Delta\xi+v_1,\Delta\eta}(\zeta)(z-\zeta)}\\
&=\int_0^\infty{\text{Ai\,}}_{\Delta\xi,-\Delta\eta}(v_1,\lambda){\text{Ai\,}}_{\Delta\xi,\Delta\eta_1}(\lambda,v_2)\,d\lambda,\notag\end{aligned}$$
We will also need the following kernels. Let $$\label{dDconditions}
0<d_1<\alpha d_2<d_3,\quad 0<D_1<\alpha D_2<D_3.$$ Define $$\begin{aligned}
\label{k1def}
&k_1(v_1,v_2)\\&=\frac{\alpha}{({2\pi\mathrm{i}})^4}\int_{\Gamma_{D_3}}dz\int_{\Gamma_{D_2}}dw\int_{\Gamma_{-d_3}}d\zeta\int_{\Gamma_{-d_2}}d\omega
\frac{G_{\xi_1,\eta_1}(z)G_{\Delta\xi+v_2,\Delta\eta}(w)}{G_{\xi_1,\eta_1}(\zeta)G_{\Delta\xi+v_1,\Delta\eta}(\omega)(z-\zeta)(z-\alpha w)(\alpha\omega-\zeta)}\notag\\
&=\alpha\int_{\mathbb{R}_+^3}{\text{Ai\,}}_{\Delta\xi,-\Delta\eta}(v_1,-\alpha\lambda_1){\text{Ai\,}}_{\xi_1,-\eta_1}(\lambda_1,\lambda_2){\text{Ai\,}}_{\xi_1,\eta_1}(\lambda_2,\lambda_3)
{\text{Ai\,}}_{\Delta\xi,\Delta\eta}(-\alpha\lambda_3,v_2)\,d^3\lambda,\notag\end{aligned}$$ $$\begin{aligned}
\label{k2def}
&k_2(v_1,v_2)\\&=\frac{\alpha}{({2\pi\mathrm{i}})^3}\int_{\Gamma_{D_3}}dz\int_{\Gamma_{D_2}}dw\int_{\Gamma_{-d_2}}d\omega
\frac{G_{\xi_1,\eta_1}(z)G_{\Delta\xi+v_2,\Delta\eta}(w)}{G_{\xi_2+v_1/\alpha',\eta_2}(\omega)(\alpha'z-\alpha\omega)(z-\alpha w)}\notag\\
&=\alpha\int_{\mathbb{R}_+^2}{\text{Ai\,}}_{\xi_2,-\eta_2}(\frac{v_1}{\alpha'},\alpha\lambda_1){\text{Ai\,}}_{\xi_1,\eta_1}(\alpha'\lambda_1,\lambda_2)
{\text{Ai\,}}_{\Delta\xi,\Delta\eta}(-\alpha\lambda_2,v_2)\,d^2\lambda,\notag\end{aligned}$$ $$\begin{aligned}
\label{k3def}
k_3(v_1,v_2)&=\frac{\alpha e^{-\delta v_2}}{({2\pi\mathrm{i}})^2}\int_{\Gamma_{-d_3}}d\zeta\int_{\Gamma_{-d_2}}d\omega
\frac 1{G_{\xi_1+v_2,\eta_1}(\zeta)G_{\Delta\xi+v_1,\Delta\eta}(\omega)(\alpha\omega-\zeta)}\\
&=\alpha e^{-\delta v_2}\int_{\mathbb{R}_+}{\text{Ai\,}}_{\Delta\xi,-\Delta\eta}(v_1,-\alpha\lambda){\text{Ai\,}}_{\xi_1,-\eta_1}(\lambda,v_2)\,d\lambda,\notag\end{aligned}$$ $$\begin{aligned}
\label{k4def}
k_4(v_1,v_2)&=\frac{\alpha e^{-\delta v_2}}{\alpha'{2\pi\mathrm{i}}}\int_{\Gamma_{-d_2}}\frac{d\omega}{G_{\xi_2+(v_1+\alpha v_2)/\alpha',\eta_2}(\omega)}\\
&=e^{-\delta v_2}\frac {\alpha}{\alpha'}{\text{Ai\,}}_{\xi_2,-\eta_2}\left(\frac{v_1}{\alpha'},\frac{\alpha v_2}{\alpha'}\right),\notag\end{aligned}$$ $$\begin{aligned}
\label{k5def}
&k_5(v_1,v_2)\\&=\frac{\alpha}{({2\pi\mathrm{i}})^3}\int_{\Gamma_{D_2}}dw\int_{\Gamma_{-d_3}}d\zeta\int_{\Gamma_{-d_2}}d\omega
\frac{G_{\xi_2+v_2/\alpha',\eta_2}(w)}{G_{\xi_1,\eta_1}(\zeta)G_{\Delta\xi+v_1,\Delta\eta}(\omega)(\alpha w-\alpha'\zeta)(\alpha\omega-\zeta)}\notag\\
&=\alpha\int_{\mathbb{R}_+^2}{\text{Ai\,}}_{\Delta\xi,-\Delta\eta}(v_1,-\alpha\lambda_1){\text{Ai\,}}_{\xi_1,-\eta_1}(\lambda_1,\alpha'\lambda_2)
{\text{Ai\,}}_{\xi_2,\eta_2}(\alpha\lambda_2,\frac{v_2}{\alpha'})\,d^2\lambda,\notag\end{aligned}$$ $$\begin{aligned}
\label{k6def}
&k_6(v_1,v_2)\\&=\frac{e^{\delta v_1}}{({2\pi\mathrm{i}})^4}\int_{\Gamma_{D_3}}dz_1\int_{\Gamma_{D_1}}dz_2\int_{\Gamma_{D_2}}dw\int_{\Gamma_{-d_1}}d\zeta
\frac{G_{\xi_1,\eta_1}(z_1)G_{\xi_1+v_1,\eta_1}(z_2)G_{\Delta\xi+v_2,\Delta\eta}(w)}{G_{\xi_1,\eta_1}(\zeta)(z_1-\zeta)(z_2-\zeta)(z_1-\alpha w)}\notag\\
&=e^{\delta v_1}\int_{\mathbb{R}_+^3}{\text{Ai\,}}_{\xi_1,\eta_1}(v_1,\lambda_1){\text{Ai\,}}_{\xi_1,-\eta_1}(\lambda_1,\lambda_2){\text{Ai\,}}_{\xi_1,\eta_1}(\lambda_2,\lambda_3)
{\text{Ai\,}}_{\Delta\xi,\Delta\eta}(-\alpha\lambda_3,v_2)\,d^3\lambda,\notag\end{aligned}$$ and $$\begin{aligned}
\label{k7def}
&k_7(v_1,v_2)\\&=\frac{e^{\delta v_1}}{({2\pi\mathrm{i}})^3}\int_{\Gamma_{D_1}}dz\int_{\Gamma_{D_2}}dw\int_{\Gamma_{-d_1}}d\zeta
\frac{G_{\xi_1+v_1,\eta_1}(z)G_{\xi_2+v_2/\alpha',\eta_2}(w)}{G_{\xi_1,\eta_1}(\zeta)(\alpha w-\alpha'\zeta)(z-\zeta)}\notag\\
&=e^{\delta v_1}\int_{\mathbb{R}_+^2}{\text{Ai\,}}_{\xi_1,\eta_1}(v_1,\alpha'\lambda_1){\text{Ai\,}}_{\xi_1,-\eta_1}(\lambda_1,\alpha'\lambda_2)
{\text{Ai\,}}_{\xi_2,\eta_2}(\alpha\lambda_2,\frac{v_2}{\alpha'})\,d^2\lambda.\notag\end{aligned}$$ The kernels $M_i$ and $k_i$ depend on the parameters $\alpha, \xi_1, \Delta\xi, \eta_1, \Delta\eta$ and $\delta$. When we need to indicate this dependence we write $M_i(\alpha, \xi_1, \Delta\xi, \eta_1, \Delta\eta, \delta)$ and $k_i(\alpha, \xi_1, \Delta\xi, \eta_1, \Delta\eta, \delta)$. We then think of $\xi_2$ and $\eta_2$ as functions of $\alpha$, $\xi_1$ and $\Delta\xi$, and $\alpha$, $\eta_1$ and $\Delta\eta$ respectively. Explicitly, $$\label{xi2}
\xi_2=\xi_2(\alpha, \xi_1, \Delta\xi)=\frac 1{\alpha'}(\alpha\xi_1+\Delta\xi),$$ $$\label{eta2}
\eta_2=\eta_2(\alpha, \eta_1, \Delta\eta)=\frac 1{\alpha'^2}(\alpha^2\eta_1+\Delta\eta).$$
Let $$\label{Yspace}
Y=L^2(\mathbb{R}_+)\oplus L^2(\mathbb{R}_+)$$ On $Y$, we define a matrix operator kernel $Q(u)$ by $$\label{Qu}
Q(u)=\begin{pmatrix} Q_{11}(u) & Q_{12}(u)\\
Q_{21}(u) & Q_{22}(u)
\end{pmatrix},$$ where $$\begin{aligned}
\label{Qdef}
Q_{11}(u)&=(2-u-u^{-1})k_1+(u-1)(k_2+k_5)+(u-1)M_3-uM_2\\
Q_{12}(u)&=(u+u^{-1}-2)k_3+(1-u)k_4\notag\\
Q_{21}(u)&=(1-u^{-1})k_6-k_7\notag\\
Q_{22}(u)&=(u^{-1}-1)M_1.\notag
\end{aligned}$$ We will write $Q(u,\alpha, \xi_1, \Delta\xi, \eta_1, \Delta\eta, \delta)$ to indicate the dependence on all parameters.
\[propQ\] The two-time distribution (\[ftt\]) is given by $$\label{Fnew1}
F_{\text{two-time}}(\xi_1,\eta_1;\xi_2,\eta_2;\alpha)=\frac{1}{{2\pi\mathrm{i}}}\int_{\gamma_r}\frac 1{u-1}\det(I+Q(u))_Y\,du,$$ where $r>1$.
We will give the proof below. The formula (\[Fnew1\]) is suitable for investigating the limit $\alpha\to 0$ (long time separation). For more on this limit see [@DJL]. To study the limit $\alpha\to\infty$ (short time separation), we can use (\[Fnew1\]) and the next Proposition which gives an $\alpha$ and $1/\alpha$ relation. Let $$\label{betadef}
\beta=\frac 1{\alpha},\quad \beta'=(1+\beta^3)^{1/3}=\frac{\alpha'}{\alpha}.$$ To indicate the dependence of the kernel $K(u)$ on all parameters we write $K(u,\alpha, \xi_1, \Delta\xi, \eta_1, \Delta\eta, \delta)$.
\[propshortlongrelation\] We have the formula $$\label{Fshortlong}
F_{\text{two-time}}(\xi_1,\eta_1;\xi_2,\eta_2;\alpha)=\frac{1}{{2\pi\mathrm{i}}}\int_{\gamma_r}\frac 1{u-1}\det(I+K(u^{-1},\beta,\Delta\xi,\xi_1,\Delta\eta,\eta_1,\delta))_X\,du,$$ where $r>1$.
The proof is given below. Recall that $$\label{Deltaxieta2}
\Delta\xi=\alpha'\xi_2-\alpha\xi_1,\quad \Delta\eta=\alpha'^2\eta_2-\alpha^2\eta_1.$$ Combining the two Propositions above we see that $$\label{Fnew2}
F_{\text{two-time}}(\xi_1,\eta_1;\xi_2,\eta_2;\alpha)=\frac{1}{{2\pi\mathrm{i}}}\int_{\gamma_r}\frac 1{u-1}\det(I+Q(u^{-1},\beta,\Delta\xi,\xi_1,\Delta\eta,\eta_1,\delta))_Y\,du.$$ Note that $\alpha$ is replaced by $\beta=1/\alpha$, $\xi_1$ and $\Delta\xi$, as well as $\eta_1$ and $\Delta\eta$, are interchanged, and $u$ is replaced by $u^{-1}$. This formula is suitable for studying the limit $\alpha\to\infty$ since this corresponds to $\beta\to 0$, see [@DJL]. Note that combining (\[xi2\]), (\[eta2\]) and (\[Deltaxieta2\]), we get $$\label{xi2eta2}
\xi_2=\xi_2(\beta,\Delta\xi,\xi_1),\quad \eta_2=\eta_2(\beta,\Delta\eta,\eta_1).$$ We now turn to the proofs of the Propositions.
Define the kernels $$\begin{aligned}
\label{pqkernels}
p_1(x,v)&=-\frac{e^{-\delta x}}{({2\pi\mathrm{i}})^3}\int_{\Gamma_{D_3}}dz\int_{\Gamma_{D_2}}dw\int_{\Gamma_{-d_1}}d\zeta
\frac{G_{\xi_1,\eta_1}(z)G_{\Delta\xi+v,\Delta\eta}(w)}{G_{\xi_1-x,\eta_1}(\zeta)(z-\zeta)(z-\alpha w)},\\
p_2(x,v)&=-\frac{e^{-\delta x}1(x>0)}{{2\pi\mathrm{i}}}\int_{\Gamma_{D_2}}G_{\Delta\xi+\alpha x+v,\Delta\eta}(w)\,dw,\notag\\
p_3(x,v)&=\frac{e^{-\delta (x+v)}}{{2\pi\mathrm{i}}}\int_{\Gamma_{-d_1}}\frac{d\zeta}{G_{\xi_1+v-x,\eta_1}(\zeta)},\notag\\
p_4(x,v)&=-\frac{e^{-\delta x}}{({2\pi\mathrm{i}})^2}\int_{\Gamma_{D_2}}dw\int_{\Gamma_{-d_1}}d\zeta
\frac{G_{\xi_2+v/\alpha',\eta_2}(\alpha' w)}{G_{\xi_1-x,\eta_1}(\zeta)(\alpha w-\zeta)},\notag\\
q_1(v,y)&=\frac{\alpha e^{\delta y}}{{2\pi\mathrm{i}}}\int_{\Gamma_{-d_2}}\frac{d\omega}{G_{\Delta\xi+\alpha y+v,\Delta\eta}(\omega)},\notag\\
q_2(v,y)&=\frac{e^{\delta (y+v)}1(y<0)}{{2\pi\mathrm{i}}}\int_{\Gamma_{D_1}}G_{\xi_1+v-y,\eta_1}(z)\,dz.\notag\end{aligned}$$ The factors involving $\delta v$ have been introduced in order to get well-defined operators. We also define $$\label{S4intformula}
S_4(x,y)=-\frac{\alpha e^{\delta(y-x)}}{({2\pi\mathrm{i}})^3}\int_{\Gamma_{D_2}}dw\int_{\Gamma_{-d_1}}d\zeta\int_{\Gamma_{-d_2}}d\omega
\frac{G_{\xi_2,\eta_2}(\alpha'w)}{G_{\xi_1-x,\eta_1}(\zeta)G_{\Delta\xi+\alpha y,\Delta\eta}(\omega)(\alpha w-\zeta)(w-\omega)}.$$ From (\[S1intformula\]) and (\[T1intformula\]), we see that $$\label{S4S1T1}
S_4=S_1-T_1,$$ by moving the $z$-integration contour. We then pick up a contribution from the pole at $z=\alpha w$, which gives $S_4$. It follows from (\[T1intformula\]), (\[S2intformula\]), (\[S3intformula\]), (\[S4intformula\]) and (\[pqkernels\]) that $$\begin{aligned}
\label{STfactor}
T_1(x,y)&=-\int_{\mathbb{R}_+}p_1(x,v)q_1(v,y)\,dv,\\
1(x>0)S_2(x,y)&=-\int_{\mathbb{R}_+}p_2(x,v)q_1(v,y)\,dv,\notag\\
S_3(x,y)1(y<0)&=\int_{\mathbb{R}_+}p_3(x,v)q_2(v,y)\,dv,\notag\\
S_4(x,y)&=\int_{\mathbb{R}_+}p_4(x,v)q_1(v,y)\,dv.\notag\end{aligned}$$ From the definition of $R(u)$, (\[S4S1T1\]) and (\[STfactor\]), we see that $$\label{RuFormula}
R(u)(x,y)=(u^{-1}-1)\int_{\mathbb{R}_+}p_1(x,v)q_1(v,y)+p_2(x,v)q_1(v,y)+p_3(x,v)q_2(v,y)\,dv+\int_{\mathbb{R}_+}p_4(x,v)q_1(v,y)\,dv.$$ Let $p_i^{\pm}$ be the operator from $L^2(\mathbb{R}_+)$ to $L^2(\mathbb{R}_\pm)$ with kernel $p_i(x,v)$, and $q_i^{\pm}$ be the operator from $L^2(\mathbb{R}_\pm)$ to $L^2(\mathbb{R}_+)$ with kernel $q_i(v,y)$. From the definition of $K(u)$ and (\[RuFormula\]) it follows that $$K(u)=pq,$$ where $$p= \begin{pmatrix} &(u^{-1}-1)p_1^-+p_4^- &(u^{-1}-1)p_3^-\\
&(1-u)p_1^++(1-u)p_2^++up_4^+ &(1-u)p_3^+
\end{pmatrix}$$ and $$q= \begin{pmatrix} &q_1^- &q_1^+\\ &q_2^- &0
\end{pmatrix},$$ are matrix operators $p:Y\mapsto X$ and $q:X\mapsto Y$. Note that $p_2^-=q_2^+=0$. Let $$Q(u)=qp$$ which gives an operator from $Y$ to itself. A straightforward computation using (\[pqkernels\]), (\[k1def\]) - (\[k7def\]) and (\[M1\]) - (\[M3\]) shows that $$\begin{matrix}
&q_1^-p_1^-=-k_1(\alpha), &q_1^+p_1^+=k_1(\alpha)-k_2(\alpha), &q_1^+p_2^+=-M_3, \\
&q_1^-p_3^-=k_3(\alpha), &q_1^+p_3^+=-k_3(\alpha)+k_4(\alpha), &q_1^-p_4^-=-k_5(\alpha), \\
&q_1^+p_4^+=k_5(\alpha)-M_2(\alpha), &q_2^-p_1^-=-k_6(\alpha), &q_2^-p_4^-=-k_7(\alpha),\\
&q_2^-p_3^-=M_1. & &
\end{matrix}$$
From this we see that $Q(u)$ is given by (\[Qdef\]). In these computations we use (\[xi2\]) and (\[eta2\]) to get $\xi_2,\eta_2$ from $\xi_1,\Delta\xi, \eta_1,\Delta\eta$. The Proposition now follows from $$\det(I+K(u))_X= \det(I+pq)_X=\det(I+qp)_Y=\det(I+Q(u))_Y.$$
To indicate the dependence of $S$, $T$ and $R(u)$ on all parameters we write $S(\alpha,\xi_1,\Delta\xi,\eta_1,\Delta\eta,\delta)$ etc. It is straightforward to check from the definitions that $$\frac 1{\alpha}S(\alpha,\xi_1,\Delta\xi, \eta_1,\Delta\eta,\delta)(\frac x{\alpha},\frac y{\alpha})=T(\beta,\Delta\xi,\xi_1,\Delta\eta,\eta_1,\beta\delta)(-y,-x),$$ and $$\frac 1{\alpha}T(\alpha,\xi_1,\Delta\xi, \eta_1,\Delta\eta,\delta)(\frac x{\alpha},\frac y{\alpha})=S(\beta,\Delta\xi,\xi_1,\Delta\eta,\eta_1,\beta\delta)(-y,-x).$$ It follows that $$\frac 1{\alpha}R(u,\alpha,\xi_1,\Delta\xi, \eta_1,\Delta\eta,\delta)(\frac x{\alpha},\frac y{\alpha})=u^{-1}R(u^{-1},\beta,\Delta\xi,\xi_1,\Delta\eta,\eta_1,\beta\delta)(-y,-x).$$ If we write $$\tilde{R}(u)(x,y)=R(u^{-1},\beta,\Delta\xi,\xi_1,\Delta\eta,\eta_1,\beta\delta)(x,y),$$ we see that $$\frac 1{\alpha}R(u)(-\frac y{\alpha},-\frac x{\alpha})=u^{-1}\tilde{R}(u^{-1}(x,y).$$ Let $K^*_{\alpha}(u)(x,y)=\alpha^{-1}K(\alpha^{-1}y,\alpha^{-1}x)$, and define $V:X\mapsto X$ by $$V\begin{pmatrix} f_1(x) \\ f_2(x) \end{pmatrix}
=\begin{pmatrix} f_2(-x) \\ f_1(-x) \end{pmatrix}.$$ Note that $V^2=I$. Since taking the adjoint and rescaling the kernel does not change the Fredholm determinant, we see that $$\det(I+K(u))_{X}=\det(I+K^*_{\alpha}(u))_X=\det(I+VK^*_{\alpha}(u)V)_X,$$ Using these definitions a computation shows that $$VK^*_{\alpha}(u)V=\begin{pmatrix} &\tilde{R}(u^{-1}(x,y) &\tilde{R}(u^{-1}(x,y) \\ &\tilde{R}(u^{-1}(x,y) &\tilde{R}(u^{-1}(x,y)
\end{pmatrix}
\begin{pmatrix} &I &0\\&0 &u^{-1}I \end{pmatrix}$$ This operator has the same determinant as $$\begin{aligned}
&\begin{pmatrix} &I &0\\&0 &u^{-1}I \end{pmatrix}
\begin{pmatrix} &\tilde{R}(u^{-1}(x,y) &\tilde{R}(u^{-1}(x,y) \\ &\tilde{R}(u^{-1}(x,y) &\tilde{R}(u^{-1}(x,y)
\end{pmatrix}=
\begin{pmatrix} &\tilde{R}(u^{-1}(x,y) &\tilde{R}(u^{-1}(x,y)\\ &u^{-1}\tilde{R}(u^{-1}(x,y) &u^{-1}\tilde{R}(u^{-1}(x,y)
\end{pmatrix}\\
&=K(u^{-1},\beta,\Delta\xi,\xi_1,\Delta\eta,\eta_1,\beta\delta)(x,y).\end{aligned}$$ Thus, $$\begin{aligned}
&\det(I+K(u,\alpha,\xi_1,\Delta\xi,\eta_1,\Delta\eta,\delta))_X=\det(I+K(u^{-1},\beta,\Delta\xi,\xi_1,\Delta\eta,\eta_1,\beta\delta))_X\\&=
\det(I+K(u^{-1},\beta,\Delta\xi,\xi_1,\Delta\eta,\eta_1,\delta)_X\end{aligned}$$ since the Fredholm determinant is independent of the value of $\delta$ as long as the condition (\[deltacondition\]) is satisfied. Note that this condition is $\delta>\max(\eta_1,\alpha\Delta\eta)$ so $\beta\delta>\max(\Delta\eta,\beta\eta_1)$ and we can replace $\beta\delta$ with $\delta$ as long as $\delta>\max(\Delta\eta,\beta\eta_1)$.
Relation to the previous two-time formula {#secoldformula}
=========================================
The approach in the present paper can be modified to study the probability $$\label{paA2}
p(a;A)=\mathbb{P}[G(m,n)=a,\,G(M,N)<A],$$ under the same scaling (\[scaling\]).
Let $$X'=L^2(\mathbb{R}_-,dx)\oplus L^2(\mathbb{R}_+,dx)\oplus L^2(\{0\},\delta_0)$$ and modify the definition of $S$ and $T$ into $$\label{Sxy2}
S(x,y)=S_1(x,y)+1(x\ge 0)S_2(x,y)-S_3(x,y)1(y<0),$$ $$\label{Txy2}
T(x,y)=-T_1(x,y)-1(x>0)S_2(x,y)+S_3(x,y)1(y\le 0).$$ Define the matrix kernel $$\label{Kuv2}
K_{uv}(x,y)=\begin{pmatrix} R_u(x,y) & R_u(x,y) & R_u(x,y) \\
uR_u(x,y) & uR_u(x,y) & uR_u(x,y)\\
vR_u(x,y) & vR_u(x,y) & vR_u(x,y)
\end{pmatrix},$$ where $R_u$ is defined as in (\[Ru\]) but with $S$ and $T$ given by (\[Sxy2\]) and (\[Txy2\]) instead. Then, under (\[scaling\]), $$\label{limit2}
\lim_{T\to\infty} c_3(t_1T)^{1/3}p(a;A)=\frac{1}{({2\pi\mathrm{i}})^2}\int_{\gamma_r}du\int_{\gamma_r}\frac{dv}{v^2}\det(I+K_{uv})_{X'},$$ for any $r>0$. From this formula, it is possible to derive the formula for the two-time distribution given in [@JoTt]. It should be possible to get the formula in [@JoTt] also by taking the partial derivative with respect to $\xi_1$ in (\[ftt\]). We have not been able to carry out that computation.
[**Acknowledgement**]{}: I thank Jinho Baik for an interesting discussion and correspondence. Also, thanks to Mustazee Rahman for helpful comments on the paper.
[99]{}
=
J. Baik, Z. Liu, *Multi-point distribution of periodic TASEP*, arXiv:1710.03284
A. Borodin and L. Petrov, *Integrable probability: from representation theory to MacDonald processes*, Probab. Surv. [**11**]{} (2014), 1–58
I. Corwin, *The Kardar-Parisi-Zhang equation and universality class*, Random Matrices Theory Appl. [**1**]{} (2012), no. 1, 1130001
I. Corwin, P.L. Ferrari, S. Péché, *Universality of slow de-correlation in KPZ growth*, Ann. Inst. Henri Poincaré Probab. Stat. [**48**]{} (2012), 134–150
I. Corwin and J. Quastel, *Renormalization fixed point of the KPZ universality class*, arXiv:1103.3422
J. De Nardis and P. Le Doussal, *Tail of the two-time height distribution for KPZ growth in one dimension*, J. Stat. Mech. Theory Exp., (5):053212, 2017
J. De Nardis, P. Le Doussal, K. A. Takeuchi, *Memory and universality in interface growth*, Phys. Rev. Lett. [**118**]{} (2017), 125701
J. De Nardis, K. Johansson and P. Le Doussal, in preparation.
A.B. Dieker, J. Warren, *Determinantal transition kernels for some interacting particles on the line*, Ann. Inst. Henri Poincaré Probab. Stat. [**44**]{} (2008), no. 6, 1162–1172
V. Dotsenko, *Two-time free energy distribution function in $(1+1)$ directed polymers*, J. Stat. Mech. Theory Exp. 2013, no. 6, P06017
V. Dotsenko, *Two-point free energy distribution function in (1+1) directed polymers*, J. Phys. A [**46**]{} (2013), no. 35, 355001
V. Dotsenko., *On two-time distribution functions in (1 + 1) random directed polymers*, J. Phys. A, [**49**]{} (2016), no:27, 27LT01
P. Le Doussal, *Maximum of an Airy process plus Brownian motion and memory in KPZ growth*, Phys. Rev. E [**96**]{} (2017), 060101
P. L. Ferrari, *Slow decorrelations in KPZ growth*, J. Stat. Mech. Theory Exp. 2008, P07022
P.L. Ferrari, H. Spohn, *On time correlations for KPZ growth in one dimension* SIGMA Symmetry Integrability Geom. Methods Appl. [**12**]{} (2016), Paper No. 074, 23 pp
I. Gohberg, S. Goldberg and N. Krupnik, *Traces and determinants of linear operators*, Opeartor Theory Advances and Applications Vol. 116, Birkhäuser Verlag, Basel, 2000
K. Johansson, [*Shape fluctuations and random matrices,*]{} Commun. Math. Phys., [**209**]{}, (2000), 437–476
K. Johansson, *Discrete polynuclear growth and determinantal processes*, Comm. Math. Phys. [**242**]{} (2003), 277–329
K. Johansson, [*A multi-dimensional Markov chain and the Meixner ensemble*]{}, Ark. Mat. [**48**]{} (2010), 437–476
K. Johansson, [*Two time distribution function in Brownian directed percolation*]{}, Comm. Math. Phys. [**351**]{} (2017), 441–492
K. Matetski, J. Quastel and D. Remenik, *The KPZ fixed point*, arXiv:1701.00018
K. Matetski and J. Quastel, *From the totally asymmetric simple exclusion process to the KPZ fixed point*, arXiv:1710.02635
M. Prähofer, H. Spohn, *Scale invariance of the PNG droplet and the Airy process*, J. Statist. Phys. [**108**]{} (2002), 1071–1106
J. Quastel, *Introduction to KPZ. Current developments in mathematics*, (2011), 125–194, Int. Press, Somerville, MA, 2012
K. A. Takeuchi, *Statistics of circular interface fluctuations in an off-lattice Eden model*, J. Stat. Mech. 2012, P05007
K. A. Takeuchi, M. Sano, *Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence*, J. Stat. Phys. [**147**]{} (2012), , 853–890
J. Warren, *Dyson’s Brownian motions, intertwining and interlacing*, Electron. J. Probab. 12 (2007), no. 19, 573–590
[^1]: Supported by the grant KAW 2015.0270 from the Knut and Alice Wallenberg Foundation
|
---
abstract: 'Community question answering (CQA) gains increasing popularity in both academy and industry recently. However, the redundancy and lengthiness issues of crowdsourced answers limit the performance of answer selection and lead to reading difficulties and misunderstandings for community users. To solve these problems, we tackle the tasks of answer selection and answer summary generation in CQA with a novel joint learning model. Specifically, we design a question-driven pointer-generator network, which exploits the correlation information between question-answer pairs to aid in attending the essential information when generating answer summaries. Meanwhile, we leverage the answer summaries to alleviate noise in original lengthy answers when ranking the relevancy degrees of question-answer pairs. In addition, we construct a new large-scale CQA corpus, WikiHowQA, which contains long answers for answer selection as well as reference summaries for answer summarization. The experimental results show that the joint learning method can effectively address the answer redundancy issue in CQA and achieves state-of-the-art results on both answer selection and text summarization tasks. Furthermore, the proposed model is shown to be of great transferring ability and applicability for resource-poor CQA tasks, which lack of reference answer summaries.'
author:
- |
Yang Deng^1^, Wai Lam^1^, Yuexiang Xie^3^, Daoyuan Chen^4^, Yaliang Li^4^, Min Yang^5^, Ying Shen^23$\dagger$^\
^$\dagger$^Corresponding Author\
^1^The Chinese University of Hong Kong, ^2^South China University of Technology,\
^3^Peking University Shenzhen Graduate School, ^4^Alibaba Group, ^5^Chinese Academy of Sciences,\
[{ydeng,wlam}@se.cuhk.edu.hk, {xieyx,shenying}@pku.edu.cn,]{}\
[{daoyuanchen.cdy,yaliang.li}@alibaba-inc.com, min.yang@siat.ac.cn]{}
bibliography:
- 'AAAI-DengY.5372.bib'
title: 'Joint Learning of Answer Selection and Answer Summary Generation in Community Question Answering[^1]'
---
Introduction
============
Recent years have witnessed a spectacular increase in real-world applications of community question answering (CQA), such as Yahoo! Answer[^2] and StackExchange[^3]. Many studies have been made on different tasks in CQA, such as answer selection, question-question relatedness, and comment classification [@DBLP:conf/eacl/MoschittiBU17; @DBLP:conf/emnlp/JotyMN18; @DBLP:conf/semeval/NakovHMMMBV17]. However, due to the length and redundancy of answers in CQA scenario, there are several challenges that need to be tackled in real-world applications. (i) The noise introduced by the redundancy of answers makes it difficult for answer selection model to pick out correct answers from a set of candidates. (ii) Compared with other QA systems (e.g., factoid question answering), answers in CQA are often too long for community users to read and comprehend.
Current state-of-the-art answer selection models [@Tan2016Improved; @DBLP:conf/acl/WuWS18] employ the attention mechanism to attend the important correlated information between question-answer pairs. These methods perform well when ranking short answers, while the accuracy goes down with the increase in the length of answers [@dos2016attentive; @COALA]. Recent studies on coarse-to-fine question answering for long documents, such as Reading Comprehension (RC) [@DBLP:conf/acl/ChoiHUPLB17; @DBLP:conf/aaai/WangYGWKZCTZJ18; @DBLP:conf/acl/XiaoWLWL18], focus on the answer span extraction in factoid QA, in which those factoid questions can be answered by a certain word or a short phrase. Conversely, in non-factoid CQA, discrete and complex information from multiple sentences makes up the answers together. Besides, generative RC methods [@DBLP:conf/acl/NishidaSNSOAT19] only give one certain answer, while there are often multiple useful answers in CQA. Thus, these approaches are not suitable for addressing the redundancy issue of answers in CQA.
On the other hand, text summarization provides an effective approach to alleviating the aforementioned issue. Text summarization methods can generally be divided into two categories: extractive summarization [@DBLP:conf/acl/0001L16; @DBLP:conf/aaai/NallapatiZZ17] and abstractive summarization [@DBLP:conf/acl/SeeLM17; @DBLP:conf/conll/NallapatiZSGX16]. The aim is to assemble or generate summaries from the source article or external vocabulary, based on the information from the source text. In the existing studies, answer summarization in CQA is mainly explored by extractive summarization models [@DBLP:conf/acl/TomasoniH10; @DBLP:conf/wsdm/SongRLLMR17]. However, due to the length of answers, extractive methods sometimes fall short of generalization of all the important information in the whole answer and consistency of the core idea. Besides, the correlation information between question and answer, which plays a crucial role in human comprehension, is underutilized by current query-based summarization studies [@DBLP:conf/acl/NemaKLR17; @DBLP:conf/ecir/SinghMOBK18]. Therefore, we intend to take advantage of both the contextual information from the source text and the relationship between the question-answer pair to generate abstractive answer summaries in CQA.
We aim to simultaneously tackle the above issues in CQA, including (i) improving the performance of non-factoid answer selection with long answers, (ii) generating abstractive summaries of the answers. We jointly learn answer selection and abstractive summarization to generate answer summaries for CQA. First, we exploit the correlated information between question-answer pairs to improve abstractive answer summarization, which enables the summarizer to generate abstractive summaries related to questions. Then, we measure the relevancy degrees between questions and answer summaries to alleviate the impact of noise from original answers. Besides, since obtaining reference summaries is usually labor-intensive and time-consuming in a new domain, a transfer learning strategy is designed to improve resource-poor CQA tasks with large-scale supervision data.
We summarize our contributions as follows:
1\. We jointly learn answer selection and answer summary generation to tackle the lengthiness and redundancy issues of the answer in CQA with a unified model. A novel joint learning framework of answer selection and abstractive summarization (ASAS) is proposed to employ the question information to guide the abstractive summarization, and meanwhile leverage the summaries to reduce noise in answers for precisely measuring the correlation degrees of QA pairs.
2\. We construct a new dataset, WikiHowQA, for the task of answer summary generation in CQA, which can be adapted to both answer selection and summarization tasks. Experimental results on WikiHowQA show that the proposed joint learning method outperforms SOTA answer selection methods and meanwhile generates more precise answer summaries than existing summarization methods.
3\. To handle resource-poor CQA tasks, we design a transfer learning strategy, which enable those tasks without reference answer summaries to conduct the joint learning with impressive experimental results.
Related Work
============
**Community Question Answering.** Answer selection is the core and the most widely-studied problem in community question answering. Recent studies have evolved from feature-based methods [@DBLP:conf/sigir/WangMC09; @DBLP:conf/coling/WangM10a] into deep learning models, such as convolutional neural network (CNN) [@Severyn2015Learning] and recurrent neural network (RNN) [@DBLP:conf/acl/WangN15]. In order to capture the interactive information in QA sentences, various attention mechanisms [@Tan2016Improved; @dos2016attentive] are developed to align the related words between questions and answers. However, the lengthy and redundant answers in CQA scenario may introduce much noise and scatter important information, which causes difficulties in answer selection. Some studies leverage additional information to compensate the imbalance of information between questions and answers, such as user model [@DBLP:conf/aaai/WenMFZ18; @DBLP:conf/wsdm/LiDLXFLGS17], latent topic [@DBLP:conf/naacl/YoonSJ18], external knowledge [@DBLP:conf/sigir/ShenDYLD0L18] or question subject [@DBLP:conf/acl/WuWS18]. Some existing transfer learning studies on CQA focus on cross-domain adaptation [@DBLP:conf/coling/DengSYLDFL18; @DBLP:conf/wsdm/YuQJHSCC18]. In this work, we employ summarization method to reduce noise in the original lengthy answers to improve the answer selection performance in CQA.
**Text Summarization.** Text summarization techniques are mainly classified into two categories: extractive and abstractive summarization. Extractive approaches regard summarization as a sentence classification [@DBLP:conf/aaai/NallapatiZZ17] or a sequence labeling task [@DBLP:conf/acl/0001L16] to select sentences from the article to form the summary, while abstractive approaches usually employ attention-based encoder-decoder models [@DBLP:conf/conll/NallapatiZSGX16; @DBLP:conf/acl/SeeLM17] to generate abstractive summaries. Answer summarization in CQA was first introduced by @DBLP:conf/lrec/ZhouLH06 (2006) as an application of extractive summarization. After that, studies on answer summarization are still regarded as a separate extractive summarization module in QA pipeline [@DBLP:conf/acl/TomasoniH10; @DBLP:conf/wsdm/SongRLLMR17]. Besides, query-based summarization methods [@DBLP:conf/acl/NemaKLR17; @DBLP:conf/ecir/SinghMOBK18] also can be a good solution for this task, however, these approaches are reported to perform worse than answer selection methods on question answering scenario [@DBLP:conf/acl/KhapraSSA18].
**Multi-task Learning.** Inspired by the success of multi-task learning in other NLP tasks, several attempts have been made to solve answer selection with different tasks. @DBLP:conf/eacl/MoschittiBU17 (2017) and @DBLP:conf/emnlp/JotyMN18 (2018) enhance answer selection in CQA via multi-task learning with the auxiliary tasks of question-question relatedness and question-comment relatedness. @DBLP:conf/ijcai/0007CCWZS19 (2019) leverage the question categorization to enhance the question representation learning for CQA. @DBLP:conf/aaai/DengXLYDFLS19 (2019) propose a multi-view attention based multi-task learning model to jointly tackle answer selection and knowledge base question answering tasks. In this work, we jointly learn answer selection and abstractive summarization to select and generate precise answers in CQA.
Method
======
Problem Definition
------------------
We aim to jointly conduct two tasks, answer selection and abstractive summarization, to select and generate concise answers for CQA. Given a question $q_i$, the goal is to simultaneously select the set of correct answers from a set of candidates $A_i=\{a^{(1)}_i,...,a^{(j)}_i\}$ and generate an abstractive summary $\beta^{(*)}_i$ for each selected answer $a^{(*)}_i$.
The dataset $D$ for learning typically contains a set of questions $Q$ with the number of $N$. For each question $q_i \in Q$, there are $M_i$ candidate answers $A_i$ with the corresponding reference summary $\beta^{(j)}_i$ written by human and the label $y^{(j)}_i$ determining whether $a^{(j)}_i$ can answer $q_i$. $$D = \{(q_i,\{(a^{(j)}_i,\beta^{(j)}_i,y^{(j)}_i)\}^{M_i}_{j=1})\}^{N}_{i=1}.$$
Model
-----
![The Joint Learning Framework of Answer Selection and Abstractive Summarization (ASAS).[]{data-label="method"}](5372_method.pdf){width="50.00000%"}
We introduce the proposed joint learning model for answer selection and abstractive summarization (ASAS). As is depicted in Fig. \[method\], The overall framework of ASAS consists of four components: (i) Shared Compare-Aggregate Bi-LSTM Encoder, (ii) Sequence-to-sequence Model with Question-aware Attention, (iii) Question Answer Alignment with Summary Representations, (iv) Question-driven Pointer-generator Network.
### Shared Compare-Aggregate Bi-LSTM Encoder.
The word embeddings of the question and the original answer, $W_q$ and $W_a$, are fed into a compare layer as @compare-aggregate (2017) to generate the model input $\hat{W_q}$ and $\hat{W_a}$. Then, a pair of Bi-LSTM encoders are adopted to aggregate the context information. We encode a pair of word sequences of the question $q$ and the answer $a$ into sentence representations $H\in\mathbb{R}^{L\times{d_h}}$, where $L$ and $d_h$ are the length of sentences and the size of hidden states: $$\begin{gathered}
H_{q} = \textbf{Bi-LSTM}(\hat{W_q}), \quad H_{a} = \textbf{Bi-LSTM}(\hat{W_a}).\end{gathered}$$
### Seq2Seq Model with Question-aware Attention.
With the intuition that the information in the question is supposed to be helpful in attending the important elements in the original answer sentence, we propose a question-aware attention based seq2seq model to decode the encoded sentence representation of the answer. We adopt a unidirectional LSTM as the decoder. On each step $t$, the decoder produces the hidden state $s_t$ with the input of the previous word $w_{t-1}$. The question-aware attention $\alpha^t$ is generated by: $$\begin{gathered}
s_t= \textbf{LSTM}(s_{t-1},w_{t-1}),\\
o_q = Average(H_q);\\
e_i^t = m^t \text{tanh} (W_h h^a_i+W_s s_t+W_q o_q+b), \\
\alpha^t = \text{softmax}(e^t),\end{gathered}$$ where $m$, $W_h$, $W_s$, $W_q$ are attention parameter matrices to be learned. The question-aware attention weight $\alpha^t$ is used to generate context vector $\hat{h}_t$ as a probability distribution over the source words: $$\hat{h}_t = \sum\nolimits_i \alpha^t h^a_i.$$
The context vector aggregates the information from the source text and the question for the current step. We concatenate it with the decoder state $s_t$ and pass through a linear layer to generate the summary representation $h^s_t$: $$h^s_t=W_1[s_t:\hat{h}_t]+b_1,$$ where $W_1$ and $b_1$ are parameters to be learned.
### Question Answer Alignment with Summary Representations.
We apply a two-way attention mechanism to generate the co-attention between the encoded question representation $H_q$ and the decoded summary representation $H_s$: $$\begin{gathered}
M_{qa}=\text{tanh}\left(H_q^TUH_s\right),\\
\alpha_q = \text{softmax}(\text{Max}(M_{qa})),\\
\alpha_a =\text{softmax}(\text{Max}({M_{qa}}^T)),\end{gathered}$$ where $U\in\mathbb{R}^{d_{s}\times{d_{s}}}$ is the attention parameter matrix to be learned; $d_{s}$ is the dimension of QA representations; $\alpha_q$ and $\alpha_a$ are the co-attention weights for the question and the answer summary respectively.
We conduct dot product between the attention vectors and the question and summary representations to generate the final attentive sentence representations for answer selection: $$\begin{aligned}
r_q = H_q^T\alpha_q, \quad r_a = H_s^T\alpha_a.\end{aligned}$$
Compared with encoded answer representations, decoded summary representations are more concise and compressive, which enable answer selection model to precisely capture the interactive information between questions and answers.
### Question-driven Pointer-generator Network.
First, the probability distribution $P_{vocab}$ over the fixed vocabulary is obtained by passing the summary representation $h^s_t$ through a softmax layer: $$P_{vocab} = \text{softmax}(W_2 h^s_t + b_2),$$ where $W_2$ and $b_2$ are parameters to be learned. Then, a question-aware pointer network is proposed to copy words from the source article with the guidance of the question information. The question-aware generation probability $p_{gen}\in [0,1]$ takes into account the decoded summary representation $h^s_t$, the decoder input $x_t$ and the question representation $o_q$: $$p_{gen} = \sigma (w_h^T h^s_t + w_x^T x_t + w_q^T o_q + b_p),$$ where $w_h$, $w_x$, $w_q$ and $b_p$ are parameters to be learned, and $\sigma$ is the sigmoid function. Following the basic pointer-generator network (PGN) [@DBLP:conf/acl/SeeLM17], we obtain the final probability distribution over both the fixed vocabulary and words from the source article: $$P = p_{gen}P_{vocab} + (1-p_{gen})\sum\nolimits_{i:w_i=w} \alpha^t_i.$$
To be specific, the question information is involved in not only the generating process, but also the copying process in the question-driven PGN. (i) The question information directs the calculation of the generation probability to decide whether generating a word from the vocabulary or copying from the source text. (ii) The question-aware attention weights integrate the question information to attend the important words in the source text for copying. (iii) The probability distribution over the vocabulary is learned from the question-aware attentive summary representations.
Joint Training Procedure
------------------------
### Answer Selection Loss.
The attentive representations of questions and summaries go through a softmax layer for binary classification: $$y(q,a)=\text{softmax}\left(W_s[r_q:r_a]+b_s\right),$$ where $W_s\in\mathbb{R}^{d_x\times{2}}$ and $b_s\in\mathbb{R}^2$ are parameters to be learned. The answer selection task is trained to minimize the cross-entropy loss function: $$L_{qa}=-\sum_{i=1}^N\left[y_i\log{p_i}+\left(1-y_i\right)\log{\left(1-p_i\right)}\right],$$ where $p$ is the output of the softmax layer and $y$ is the binary classification label of the QA pair.
### Summarization Loss.
The summarization task is trained to minimize the negative log likelihood: $$L_{sum} = - \frac{1}{T}\sum^T_{t=0}\text{log}P(w_t^*).$$
### Coverage Loss.
Coverage loss [@DBLP:conf/acl/SeeLM17] was proposed to discourage the repetition in abstractive summarization. In each decoder timestep $t$, the coverage vector $c^t=\sum^{t-1}_{t'=0}a^{t'}$ is used to represent the degree of coverage so far. The coverage vector $c^t$ will be applied to compute the attention weight $\alpha^t$. The coverage loss is trained to penalize the repetition in updated attention weight $\alpha^t$: $$L_{cov}=\frac{1}{T}\sum^T_{t=1}\sum\nolimits_i \text{min}(\alpha^t_i,c^t_i).$$
### Overall Loss Function.
For joint training, the final objective function is to minimize above three loss functions: $$L=\lambda_1 L_{qa}+\lambda_2 L_{sum}+\lambda_3 L_{cov},$$ where $\lambda_1$, $\lambda_2$, $\lambda_3$ are hyper-parameters to balance losses.
Handling Resource-poor Datasets
-------------------------------
Since annotating gold answer summaries is a labor-intensive work, we intend to leverage the knowledge learned from the joint learning of answer selection and answer summary generation on a large-scale supervision dataset and apply it to resource-poor datasets without reference answer summaries. The goal can be achieved by a transfer learning strategy involving two steps: (i) initialize the the parameters of model pre-trained on the source dataset, (ii) further fine-tune on the target dataset. A straightforward way is to fine-tune all the parameters learned from the source data on the target training dataset. Another fashion is to fine-tune a certain part of parameters and keep the remaining part of model fixed during fine-tuning. In this case, we first pre-train the whole joint learning model on the source dataset, and then only fine-tune the answer selection modules (including Shared Compare-Aggregate Bi-LSTM Encoder & Question Answer Alignment). On one hand, fixing the summarization part can not only reduce the demand for annotating summary data, but also prevent model over-fitting. On the other hand, questioning styles and answer contents vary from CQA tasks in different domains, thus, the answer selection part is supposed to benefit from fine-tuning in target domains.
Datasets and Experimental Setting
=================================
train / dev / test
------------- ----------------------------
\#Questions 76,687 / 8,000 / 22,354
\#QA Pairs 904,460 / 72,474 / 211,255
\#Summaries 142,063 / 18,909 / 42,624
Avg QLen 7.20 / 6.84 / 6.69
Avg ALen 520.87 / 548.26 / 554.66
Avg SLen 67.38 / 61.84 / 74.42
Avg \#CandA 11.79 / 9.06 / 9.45
: Statistic of WikiHowQA Dataset[]{data-label="stat"}
Datasets
--------
Most of the widely-adopted answer selection benchmark datasets are composed of short sentences, such as WikiQA [@Yang2015WikiQA], SemEval [@DBLP:conf/semeval/NakovHMMMBV17]. WikiPassageQA [@DBLP:conf/sigir/CohenYC18] and StackExchange [@COALA], two latest non-factoid answer selection datasets with long passages (about 150 words) as candidate answers, lack of the reference summary for answer summarization evaluation in our defined answer summary generation task.
We present a new CQA corpus, WikiHowQA, for answer summary generation, which contains labels for the answer selection task as well as reference summaries for the text summarization task. To prepare this dataset, we modify a latest text summarization dataset, WikiHow [@DBLP:journals/corr/abs-1810-09305], which was obtained from *WikiHow*[^4] knowledge base. The WikiHow dataset contains detailed answers written by community users for non-factoid questions starting with “**How to**". The original answers are composed by multiple steps of different methods for the question, and the description in each step is associated with an abstractive summary. The WikiHow dataset only contains the selected ground-truth answers and the reference summaries for each answer, while the whole candidate answer set is required when we wish to conduct answer selection experiments on this dataset. Therefore, we construct a new CQA dataset based on the WikiHow dataset.
We first clean up the WikiHow dataset by filtering out those questions without answers or summaries and those answers with punctuation only. After that, the dataset size is reduced from 230,843 to 203,596, including 107,041 unique questions. The clean WikiHow dataset is split into 142,063 / 18,909 / 42,624 as train / dev / test sets. In order to retrieve the candidate answer pool for all the questions, we write a crawler to collect the relevant questions for each question from the *WikiHow* website. The answers of the relevant questions posted on *WikiHow* are labeled as negative answers for the given question. Finally, we obtain 1,188,189 question-answer pairs with corresponding answer summaries and matching labels as the WikiHowQA dataset. In accordance with the clean WikiHow dataset, we split the WikiHowQA dataset into 904,460 / 72,474 / 211,255 as train / dev / test sets, which implies that there is no overlapping of samples among the three split sets. The statistics of the WikiHowQA[^5] dataset are shown in Table \[stat\].
---------- ----------------------- -----
\#Questions
(train/dev/test)
Travel 3,572 / 765 / 766 214
Cooking 3,692 / 791 / 792 189
Academia 2,856 / 612 / 612 229
Apple 5,831 / 1,249 / 1,250 114
Aviation 3,035 / 650 / 652 281
---------- ----------------------- -----
: Statistic of StackExchange CQA Dataset[]{data-label="stat3"}
In addition, we evaluate the proposed method on a resource-poor CQA dataset, StackExchange [@COALA], which lacks of reference answer summaries. The statistics of the StackExchange dataset are presented in Table \[stat3\], which is a real-life CQA dataset containing data with long answers from different domains, including travel, cooking, academia, apple, and aviation. We adopt WikiHowQA as the source dataset for transfer learning due to its high quality and large quantity, while StackExchange are used as the target dataset.
Implementation Details
----------------------
We train all the implemented models with pre-trained GloVE embeddings[^6] of 100 dimensions as word embeddings and set the vocabulary size to 50k for both source and target text. During training and testing procedure, we truncate the article to 400 words and restrict the length of generated summaries within 100 words. We apply early stopping based on the answer selection evaluation result on the validation set. We train our model and implement answer selection models for 5 epochs, while we implement summarization models for 20 epochs for fair comparisons, since the answers may repetitively occur in the candidates for different questions in the WikiHowQA dataset.
In our model, we train with a learning rate of 0.15 and an initial accumulator value of 0.1. The dropout rate is set to 0.5. The hidden unit sizes of the BiLSTM encoder and the LSTM decoder are all set to 150. We train our models with the batch size of 32. All other parameters are randomly initialized from \[-0.05, 0.05\]. $\lambda_1$, $\lambda_2$, $\lambda_3$ are all set to 1.
Experimental Result
===================
Answer Selection Result
-----------------------
We first compare the proposed method with several state-of-the-art methods on the answer selection task, including Siamese BiLSTM [@DBLP:conf/aaai/MuellerT16], Att-BiLSTM [@Tan2016Improved], AP-LSTM [@dos2016attentive], CA (Compare-Aggregate) [@compare-aggregate] and COALA [@COALA]. Besides, we perform several **Two-Stage** methods, which first summarize the original answers and then conduct answer selection. To validate the effectiveness of different components of ASAS, we also conduct ablation tests. MAP and MRR are adopted as evaluation metrics.
**Models** **<span style="font-variant:small-caps;">MAP</span>** **<span style="font-variant:small-caps;">MRR</span>**
-------------------------------------------------------------- ------------------------------------------------------- -------------------------------------------------------
<span style="font-variant:small-caps;">Random Guess</span> 0.4088 0.4319
<span style="font-variant:small-caps;">BM25</span> 0.4212 0.4377
<span style="font-variant:small-caps;">Siamese BiLSTM</span> 0.4604 0.4734
<span style="font-variant:small-caps;">Att-BiLSTM</span> 0.4573 0.4721
<span style="font-variant:small-caps;">AP-BiLSTM</span> 0.4896 0.5058
<span style="font-variant:small-caps;">CA</span>
<span style="font-variant:small-caps;">COALA</span> 0.5003 0.5196
GOLD + AP-BiLSTM 0.5261 0.5377
PGN + AP-BiLSTM 0.4992 0.5078
QPGN + AP-BiLSTM 0.5237 0.5343
QPGN + CA 0.5246 0.5373
QPGN + COALA 0.5197 0.5302
**Joint Learning (ASAS)** **0.5522** **0.5686**
w/o two-way attention 0.5208 0.5311
w/o pointer network 0.5341 0.5483
: Evaluation on Answer Selection[]{data-label="asresult"}
Answer selection results on WikiHowQA are summarized in Table \[asresult\]. We show that the joint learning model (ASAS) achieves state-of-the-art performance. There are several notable observations in the results. (i) BM25 model and even the basic deep learning model slightly improve the performance compared to random guessing, which signifies that the testing set is indeed a difficult one. (ii) The Compare-Aggregate methods (including CA and COALA) and AP-BiLSTM, which have been proven to be relatively effective in long-sentence answer selection [@COALA; @dos2016attentive], outperforms other strong baseline methods. (iii) Although **Two-Stage** methods actually improve the final answer selection result, it is time-consuming and inconvenient to train two separate models. In specific, using gold summary (GOLD) achieves the best performance, and Question-driven PGN (QPGN) performs better than original PGN. With the same summarization method, different answer selection models achieve similar results. (iv) Finally, the proposed joint learning model (ASAS) decently and substantially enhances the performance, which not only achieves the SOTA result, but also is easily trained by end-to-end fashion. By doing so, we precisely pick out the correct answers from candidate answers with long sentences, and meanwhile generate abstractive summaries for the convenience of community users. (v) The ablation study shows both the two-way attention mechanism and the pointer network contribute to the final result. The two-way attention mechanism enhances the interaction between questions and decoded answer summaries, while the pointer network aids in generating a better summary.
Answer Summary Generation Result
--------------------------------
To evaluate the generated answer summary, we also compare the proposed method with the following state-of-the-art baseline methods on text summarization subtask, including four extractive methods (Lead3, TextRank [@DBLP:conf/emnlp/MihalceaT04], NeuralSum [@DBLP:conf/acl/0001L16], NeuSum [@DBLP:conf/acl/ZhaoZWYHZ18]), two abstractive methods (Seq2Seq [@DBLP:conf/conll/NallapatiZSGX16], PGN [@DBLP:conf/acl/SeeLM17]) and two query-based methods ($\text{SD}_2$ [@DBLP:conf/acl/NemaKLR17], biASBLSTM [@DBLP:conf/ecir/SinghMOBK18]). ROUGE F1 scores are used to evaluate the summarization methods.
**Models** **<span style="font-variant:small-caps;">ROUGE 1</span>** **<span style="font-variant:small-caps;">ROUGE 2</span>** **<span style="font-variant:small-caps;">ROUGE L</span>**
-------------------------------------------------------------------- ----------------------------------------------------------- ----------------------------------------------------------- -----------------------------------------------------------
<span style="font-variant:small-caps;">Lead3</span> 24.66 5.56 22.67
<span style="font-variant:small-caps;">TextRank</span> 26.42 7.12 23.79
<span style="font-variant:small-caps;">NeuralSum</span> 6.78 25.10
<span style="font-variant:small-caps;">NeuSum</span> 26.78 6.88 25.14
<span style="font-variant:small-caps;">Seq2Seq w/ Attention</span> 20.31 5.53 19.75
<span style="font-variant:small-caps;">PGN w/ coverage</span> 26.83
<span style="font-variant:small-caps;">$\text{SD}_2$</span> 26.65 6.92 24.77
<span style="font-variant:small-caps;">biASBLSTM</span> 24.74 6.02 22.75
**Question-driven PGN** 27.32 7.98 25.46
**Joint Learning (ASAS)** **27.78** **8.16** **25.86**
: Evaluation on Text Summarization[]{data-label="comparisons"}
Text summarization results on WikiHowQA are summarized in Table \[comparisons\]. The experimental results show that the question-driven PGN outperforms all the state-of-the-art methods of both extractive and abstractive summarization, which demonstrates the effectiveness of incorporating question information to generate summaries for answers. The question information directly involves in the calculation of the generation probability to determine the next word whether generated from the vocabulary or copied from the source text. In addition, jointly learning with answer selection, ASAS further improves the result with a noticeable margin. The correlation information between question-answer pairs also aids in attending important words in the original answer, which are related to the question. These results show that ASAS can effectively generate high-quality summaries for the selected answers.
Analysis of The Length of Answers
---------------------------------
In order to validate the effectiveness of the proposed method on long-sentence answer selection, we split the test set in terms of the length of the answer. As shown in Fig. \[length\], we compare ASAS with two baseline methods, AP-LSTM and Compare-Aggregate Model (CA), by measuring the accuracy, which is the ratio of correctly selected answers. We observe that ASAS performs better especially for long answers. For answers that are shorter than 100 words, CA and AP-LSTM is slightly better than ASAS, which indicates that the summary may have lost some information for short answers. However, the performance of these two methods goes down with the increase in the answer length, while ASAS maintains a great stability.
![Model Accuracy in terms of Answer Length[]{data-label="length"}](5372_example1.pdf){width="35.00000%"}
Method Info Conc Read Corr
--------------------------------------------------------------- ---------- ---------- ---------- ----------
<span style="font-variant:small-caps;">NeuralSum</span> 3.60 2.70 3.22 3.24
<span style="font-variant:small-caps;">PGN w/ coverage</span> 2.90 3.51 3.09 3.04
**<span style="font-variant:small-caps;">ASAS</span>** **3.67** **3.88** **3.59** **3.71**
: Human Evaluation Results[]{data-label="human_eval"}
Human Evaluation on Summarization
---------------------------------
We conduct human evaluation on a sample of test set to evaluate the generated answer summaries from four aspects: (1) Informativity: how well does the summary capture the key information from the original answer? (2) Conciseness: how concise the summary is? (3) Readability: how fluent and coherent the summary is? (4) Correlatedness: how correlated the summary and the given question are? We randomly sample 50 answers and generate their summaries by three methods, including NeuralSum, PGN w/ coverage and the proposed ASAS. Three data annotators are asked to score each generated summary with 1 to 5 (higher the better).
![Case Study. ASAS generates the answer summary highly related to the question (Underlined), while PGN may misunderstand the core idea of the answer (Wavy-lined).[]{data-label="case"}](5372_acl_case.pdf){width="48.00000%"}
Table \[human\_eval\] shows the human evaluation results. The results show that ASAS consistently outperforms other methods in all aspects. Noticeably, the proposed method learns well to generate answer summaries that are highly related to the given questions so there is a substantial margin on Correlatedness. In order to intuitively observe the advantage of the proposed method, we randomly choose one example to show the answer summary generation results. As shown in the Fig. \[case\], the extractive method (e.g., NeuralSum) selects important sentences from the original answer to form the answer summary, which still contains many insignificant or redundant information. The abstractive method (e.g., PGN) generates the answer summary from the vocabulary and the original answer, which may miss some key words and essential information. Upon these defects, the proposed joint learning method (ASAS) takes into account the information provided by the question to capture the core idea of the original answer and generate a precise summary. More importantly, unlike other methods, answer summaries are generated at the same time that the answers are selected.
Resource-poor CQA Results
-------------------------
To evaluate the transferring ability and applicability of the proposed method, we conduct experiments on the resource-poor CQA task with transfer learning. We also conduct several ablations that use no pre-training or no fine-tuning, including (i) *Finetune/-* is the baseline without pre-training, (ii) *Finetune/No* is trained with the training set of source data without fine-tuning on the target training data, (iii) *Finetune/Yes* is to first pre-train a model on the source data, and then use the learned parameters to initialize the model parameters for only fine-tuning the answer selection part on the target data. Following previous studies [@COALA], we adopt the ratio of correctly selected answers as the evaluation metrics. Note that we use an unsupervised summarization method, TextRank [@DBLP:conf/emnlp/MihalceaT04], to generate reference summaries roughly for *Finetune/-* settings with ASAS, since there is no reference summary in the original StackExchange dataset.
**Models** Finetune Travel Cooking Academia Apple Aviation
---------------------------------------------------------- ---------- ---------- ---------- ---------- ---------- ----------
<span style="font-variant:small-caps;">BM25</span> - 38.1 30.9 29.2 21.8 37.0
<span style="font-variant:small-caps;">BiLSTM</span> - 45.3 35.2 31.5 27.2 37.3
<span style="font-variant:small-caps;">Att-BiLSTM</span> - 43.0 36.2 31.2 24.7 33.9
<span style="font-variant:small-caps;">AP-BiLSTM</span> - 38.8 32.2 27.3 22.9 34.5
<span style="font-variant:small-caps;">CA</span> - 46.5 39.4 36.1 29.2 46.5
<span style="font-variant:small-caps;">COALA</span> - 47.3 32.0 48.4
AP-BiLSTM No 39.7 34.4 30.6 25.7 34.8
CA No 33.4 28.1 21.4 21.2 31.5
COALA No 35.6 32.2 24.5 22.8 37.2
AP-BiLSTM Yes 44.9 38.1 36.7 29.1 46.3
CA Yes 46.2 39.9 36.6 29.5 45.2
COALA Yes 52.7 41.5
**ASAS** - 54.8 48.1 42.8 32.6 50.1
**ASAS** No 52.3 45.8 39.9 30.9 48.2
**ASAS** Yes **56.5** **52.8** **44.4** **35.1** **52.9**
: Evaluation on Resource-poor Answer Selection[]{data-label="transfer"}
The experimental results show that even with the coarse reference summaries, ASAS (Finetune/-) achieves the best performance in 4 out of 5 domains, which demonstrates the applicability of the proposed joint learning framework. Under the zero-shot setting, ASAS (Finetune/No) also achieves competitive results as those strong baseline methods, which shows the strong transferring ability of the proposed method and the value of the large-scale source dataset, WikiHowQA. Fine-tuning the answer selection part further outperforms all the baselines by about 4%. This result indicates that there are actually some gaps between different CQA datasets and the fine-tune strategy effectively overcomes these domain differences. Compared with ASAS and AP-BiLSTM, CA and COALA hardly benefit from pre-training due to their reliance on unsupervised embedding matching features.
![Generated Summaries for Resource-poor CQA[]{data-label="case2"}](5372_tl_case.pdf){width="48.00000%"}
In addition, Fig. \[case2\] presents examples of answer summary generation results from target datasets. For those resource-poor CQA tasks without reference answer summaries, ASAS can not only achieve state-of-the-art results on answer selection, but also automatically generate decent and concise summaries via a simple transfer learning strategy with a resource-rich dataset.
Conclusion
==========
We study the joint learning of answer selection and answer summary generation in CQA. We propose a novel model to employ the question information to improve the summarization result, and meanwhile leverage the summaries to reduce noise in answers for a better performance on long-sentence answer selection. In order to evaluate the answer generation task in CQA, we construct a new large-scale CQA dataset, WikiHowQA, which contains both labels for answer selection task and reference summaries for text summarization task. The experimental results show that the proposed joint learning method outperforms the state-of-the-art methods on both answer selection and summarization tasks, and processes robust applicability and transferring ability for resource-poor CQA tasks.
[^1]: This work was financially supported by the National Natural Science Foundation of China (No.61602013) and a grant from the Research Grant Council of the Hong Kong Special Administrative Region, China (Project Code: 14204418).
[^2]: https://answers.yahoo.com/
[^3]: https://stackexchange.com/
[^4]: http://www.wikihow.com/
[^5]: https://github.com/dengyang17/wikihowQA
[^6]: http://nlp.stanford.edu/data/glove.6B.zip
|
---
abstract: 'The image of the branch set of a PL branched cover between PL $n$-manifolds is a simplicial $(n-2)$-complex. We demonstrate that the reverse implication also holds: a branched cover $f \colon {\mathbb S}^n \to {\mathbb S}^n$ with the image of the branch set contained in a simplicial $(n-2)$-complex is equivalent up to homeomorphism to a PL mapping.'
address:
- 'Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 00 Prague 8, Czech Republic Department of Mathematics and Statistics, P.O. Box 35, FI-40014 University of Jyväskylä, Finland'
- 'UCLA, 520 Portola Plaza, Los Angeles, CA 90095, USA '
author:
- Rami Luisto
- Eden Prywes
title: Characterization of branched covers with simplicial branch sets
---
[^1]
Introduction {#sec:intro}
============
A mapping between topological spaces is said to be *open* if the image of every open set is open and *discrete* if the preimages of points are discrete sets in the domain. A continuous, discrete and open mapping is called a *branched cover*. The canonical example is the winding map in the plane $w_p(z) = \frac{z^p}{|z|^{p-1}}$, $p \in {\mathbb Z}$, and the higher dimensional analogues, $w_p \times {\operatorname{id}}_{{\mathbb R}^k} \colon {\mathbb R}^{k+2} \to {\mathbb R}^{k+2}$. An important subclass of branched covers is that of quasiregular mappings. A mapping $f \colon {\mathbb R}^n \to {\mathbb R}^n$ is *$K$-quasiregular* for some $K \geq 1$ if $f \in W^{1,n}_{\text{loc}}({\mathbb R}^n)$ and for almost every $x \in {\mathbb R}^n$, $$\|Df\| \le K\det(Df),$$ where $\|Df\|$ is the norm of the weak differential of $f$ and $J_f$ is the Jacobian determinant of $f$ (see [@Rickman-book]). By the Reshetnyak theorem quasiregular mappings are branched covers ([@Reshetnyak67] or [@Rickman-book Section IV.5, p. 145]) and so branched coverings can be seen as generalizations of quasiregular mappings, see e.g. [@LuistoPankka-Stoilow] for some further discussion.
We denote by $B_f$ the *branch set* of $f$. This is the set of points where $f$ fails to be a local homeomorphism. In dimension two the branch set of branched covers is well understood. By the classical Stoïlow theorem (see e.g. [@Stoilow] or [@LuistoPankka-Stoilow]) the branch set of a branched cover between planar domains is a discrete set. In higher dimensions the Černavskii-Väisälä theorem [@Vaisala] states that the branch set of a branched cover between two $n$-manifolds has topological dimension of at most $n-2$. Note that the aforementioned winding map $w_p \colon {\mathbb R}^n \to {\mathbb R}^n$ gives an extremal example as the branch set of $w_p$ is the $(n-2)$-dimensional subspace $$\begin{aligned}
\{ (0,0,x_3, \dots, x_n):(x_3,\dots,x_n) \in {\mathbb R}^{n-2} \}.\end{aligned}$$ On the other hand the Černavskii-Väisälä result is not strict in all dimensions. In Section \[sec:Nemesis\] we describe a classical example by Church and Timourian of a branched cover ${\mathbb S}^5 \to {\mathbb S}^5$ with $\dim_{\mathcal{T}}(B_f) = 1$. It is currently not known if such examples exist in lower dimensions. For example, the Church-Hemmingsen conjecture asks if there exists a branched cover in three dimensions with a branch set homeomorphic to a Cantor set (see [@ChurchHemmingsen1] and [@AaltonenPankka]). In general the structure of the branch set of a branched cover, or even a quasiregular mapping, is not well understood but the topic garners great interest. In Heinonen’s ICM address, [@HeinonenICM Section 3], he asked the following:
*Can we describe the geometry and the topology of the allowable branch sets of quasiregular mappings between metric $n$-manifolds?*
In the setting of piecewise linear (PL) branched covers between PL manifolds the Černavskii-Väisälä result is exact in the sense that the branch set is $(n-2)$-dimensional. Furthermore, it is a simplicial subcomplex of the underlying PL structure and the branched cover is locally a composition of winding maps. Even without an underlying PL structure of the mapping, we can in some situations identify that a branched cover between Euclidean domains is a winding map. Indeed, by the classical results of Church and Hemmingsen [@ChurchHemmingsen1] and Martio, Rickman and Väisälä [@MRV1971], if the image of the branch set of a branched cover $f\colon \Omega \to {\mathbb R}^n$ is contained in an $(n-2)$-dimensional affine subset, then the mapping is locally topologically equivalent to a winding map. Winding maps, in turn, admit locally a canonical PL-structure.
The following is the main theorem of this paper. For terminology on simplicial complexes and cones we refer to Section \[sec:Preli\].
\[thm:maintheorem\] Let $\Omega \subset {\mathbb R}^n$ be a domain and $f \colon \Omega \to {\mathbb R}^n$ be a branched cover. Suppose that $f(B_f)$ is contained in a simplicial $(n-2)$-complex. Then $f$ is locally topologically equivalent to a piecewise linear map which is a cone of a lower-dimensional PL-mapping $g \colon {\mathbb S}^{n-1} \to {\mathbb S}^{n-1}$.
We formulate and prove our results in the topological setting, but a quasiregular version of the theorem can be acquired using similar methods (see Section \[sec:EdenMap\]). Theorem \[thm:maintheorem\] also yields the following corollary.
\[coro:GlobalMainTheorem\] Let $f \colon {\mathbb S}^n \to {\mathbb S}^n$ be a branched cover such that $f(B_f)$ is contained in a simplicial $(n-2)$-complex. Then $f$ is topologically equivalent to a PL mapping.
This gives a partial answer to a question posed by Heinonen and Semmes in [@HeinonenSemmes Question 28] – see Remark \[remark:HeinonenRickman\]. The previous two statements assume that $f(B_f)$ is contained in a simplicial $(n-2)$-complex. Since the results are stated up to topological equivalence, Theorem \[thm:maintheorem\] still applies when $f(B_f)$ is contained in a set $X$ such that for each point in $X$ there exists a neighborhood $U$ and a homeomorphism $\phi\colon U \to B(0,1)$ that sends $X \cap U$ to an $(n-2)$-simplicial complex. Theorem \[thm:maintheorem\] was shown by Martio and Srebro [@MartioSrebro] in dimension three.
A straightforward consequence of Theorem \[thm:maintheorem\] is that when $f(B_f)$ is contained in a codimension two simplicial complex, the topological dimension of $f(B_f)$ must be exactly $(n-2)$. However, there are many branched covers for which the image of the branch set is complicated. Indeed, Heinonen and Rickman construct a quasiregular branched cover $f \colon {\mathbb S}^3 \to {\mathbb S}^3$ containing a wild Cantor set in the branch set. The set ${\mathbb S}^3 \setminus f(B_f)$ is not simply connected. So, as a Cantor set with a topologically nontrivial complement, the set $f(B_f)$ cannot be contained in a codimension 2 simplicial complex (see [@HeinonenRickman] and [@HeinonenRickman2]). Here a wild Cantor set refers to any Cantor set $C$ in ${\mathbb R}^n$ such that there is no homeomorphism $h \colon {\mathbb R}^n \to {\mathbb R}^n$ for which $h(C) \subset {\mathbb R}\times \{ 0 \}^{n-1}$.
We also note that the hypothesis of the PL structure must be made on the image of the branch set and not on the branch set itself. In Section \[sec:Nemesis\] we present a classical example due to Church and Timourian [@ChurchTimourian] of a branched cover whose branch set is a simplicial complex, but whose image is not.
A crucial step in the proof of Theorem \[thm:maintheorem\] is showing that the boundaries of so-called *normal domains* of the mapping $f$ are $(n-1)$-manifolds. In dimensions above three we need to study, not only the boundary of a normal domain $U$, but also the boundaries of the $(n-1)$-dimensional normal domains of the restriction $f|_{\partial U} \colon \partial U \to f \partial U$, and so forth continuing these restrictions to boundaries of normal domains all the way down to dimension 1.
This added level of detail turns out to be natural in higher dimensions. The complexity of a mapping is reflected in how many levels of normal domain boundaries are manifolds. In general, the boundaries of normal domains can be manifolds when the branched cover in question is not locally a cone of a lower dimensional map. In dimension three, Martio and Srebro proved that the boundaries of normal domains are manifolds exactly when the branched cover in question is locally a *path* of lower dimensional branched covers (see Sections \[sec:Preli\] and \[sec:ReverseImplication\] for the terminology).
We extend this result to higher dimensions in Section \[sec:ReverseImplication\]. A crucial step in this proof is showing that, in Euclidean spaces, codimension one manifold foliations of punctured domains are necessarily spherical under certain conditions (see Section \[sec:HomotopiesOfHomotopies\]). In dimensions four and above, the theorem also generalizes naturally to state that the more levels of boundaries of lower dimensional normal domains are manifolds, the more the mapping displays path-like properties. A path of branched covers with nonempty branch sets increases the topological dimension of the resulting branch set by one. So this result gives lower bounds on the topological dimension of the branch set of a branched cover (see Section \[sec:ReverseImplication\]).
The underlying motivation of this paper is to better understand the connections between the behavior of a branched cover $f$ and the sets $B_f$ and $f(B_f)$. From this point of view, we find the apparent ‘duality’ between the structure of the branch set and the properties of lower dimensional normal domains very promising.
Finally, as an application of our results, we construct examples of quasiregular mappings in Section \[sec:EdenMap\] in the form of the following proposition.
\[prop:projective\] For each $n \in {\mathbb N}$ there exists a non-constant quasiregular mapping $f \colon {\mathbb R}^{2n} \to \mathbb{CP}^n$.
As mentioned above, a large motivation for the contemporary study of branched covers comes from their subclass of quasiregular mappings. Often quasiregular mappings in dimensions larger than $2$ are difficult to construct, but it can oftentimes be easier to construct branched covers. Thus Proposition \[prop:projective\] demonstrates that Theorem \[thm:maintheorem\] can be applied in some cases to enhance a branched cover into a quasiregular mapping.
Preliminaries {#sec:Preli}
=============
We follow the conventions of [@Rickman-book] and say that $U \subset X$ is a *normal domain* for $f \colon X \to Y$ if $U$ is a precompact domain such that $$\begin{aligned}
\partial f (U) = f (\partial U).\end{aligned}$$ A normal domain $U$ is *a normal neighborhood* of $x \in U$ if $$\begin{aligned}
\overline{U} \cap f {^{-1}}(\{ f(x) \}) = \{ x \}.\end{aligned}$$ By $U(x,f,r)$, we denote the component of the open set $f {^{-1}}(B_Y(f(x),r))$ containing $x$. The existence of arbitrarily small normal neighborhoods is essential for the theory of branched covers. The following lemma guarantees the existence of normal domains, the proof can be found in [@Rickman-book Lemma I.4.9, p.19] (see also [@Vaisala Lemma 5.1.]).
\[lemma:TopologicalNormalDomainLemma\] Let $X$ and $Y$ be locally compact complete path-metric spaces and $f \colon X \to Y$ a branched cover. Then for every point $x \in X$ there exists a radius $r_0 > 0$ such that $U(x,f,r)$ is a normal neighborhood of $x$ for any $r \in (0,r_0)$. Furthermore, $$\begin{aligned}
\lim_{r\to 0}{\operatorname{diam}}U(x,f,r) = 0.
\end{aligned}$$
The following Černavskii-Väisälä theorem (see [@Vaisala]) is fundamental in the study of branched covers.
\[thm:CernavskiiVaisala\] Let $X$ and $Y$ be $n$-dimensional manifolds. If $f\colon X \to Y$ is a branched cover, then the topological dimension of $B_f, f(B_f)$ and $f^{-1}(f(B_f))$ is bounded above by $n-2$. In particular, $B_f$, $f(B_f)$ and $f{^{-1}}(f (B_f))$ have no interior points and do not locally separate the spaces $X$ nor $Y$.
Another concept that we will use below is that of a *cone*.
Let $X$ be a topological space.
1. [ The *cone* of $X$ is the set $(X\times [0,1])/(X\times \{0\}) \equalscolon {\operatorname{cone}}(X)$. ]{}
2. [ The *suspension* of $X$, denoted $S(X)$, is the disjoint union of two copies of ${\operatorname{cone}}(X)$ glued together by the identity at $X\times \{1\}$. ]{}
3. If $Y$ is another topological space, a *cone map* $f\colon {\operatorname{cone}}(X) \to {\operatorname{cone}}(Y)$ is a continuous map such that $f(x,t) = (h(x),t)$ for some $h \colon X \to Y$ and for all $t \in [0,1]$. Note that a mapping $g \colon X \to Y$ induces a canonical cone map ${\operatorname{cone}}(X) \to {\operatorname{cone}}(Y)$, $(x,t) \mapsto (g(x),t)$ which we will denote by ${\operatorname{cone}}(g)$.
The *suspension map of $f$*, denoted $S(f) \colon S(X) \to S(Y)$, is defined in an identical manner.
Note that ${\operatorname{cone}}({\mathbb S}^k)$ is homeomorphic to the closed $(k+1)$-ball, and $S({\mathbb S}^k)$ is homeomorphic to ${\mathbb S}^{k+1}$.
A mapping $f\colon X \to Y$ is *topologically equivalent* to $g \colon X' \to Y'$ if there exists homeomorphisms $\phi$ and $\psi$ such that $$f = \psi^{-1} \circ g \circ \phi.$$ In other words the following diagram commutes: $$\begin{tikzcd}
X \arrow{r}{f} \arrow[swap]{d}{\phi} & Y \arrow{d}{\psi} \\
X' \arrow{r}{g} & Y'
\end{tikzcd}.$$
Simplicial complexes and PL-structures
--------------------------------------
We largely follow [@rourkesanderson] in our notation and terminology. We list some of the basic definitions and concepts in this section for the sake of completeness.
Let $\{v_0,\dots,v_k\} \subset {\mathbb R}^n$ be a finite set of points not contained in any $(k-1)$-dimensional affine subset. The $k$-*simplex* $D$ is defined as $$D = \left\{ \sum_{i=1}^k \lambda_iv_i : \sum_{i=1}^k\lambda_i = 1, \lambda_i \ge 0\right\}.$$ We say $D$ is *spanned* by $\{v_1,\dots,v_k\}$.
A *face* of a simplex $D$ is a simplex spanned by a subset of the vertices that span $D$.
A *simplicial complex* $X$ is a finite collection of simplices such that
1. if $D_1 \in X$ and $D_2 $ is a face of $D_1$, then $D_2 \in X$, and
2. if $D_1,D_2 \in X$, then $D_1\cap D_2$ is a face of both $D_1$ and $D_2$.
The simplicial complex $X$ is *$k$-dimensional* if the highest degree simplex in $X$ is a $k$-simplex.
We will often consider $X$ as a subset of ${\mathbb R}^n$. In this case we tacitly identify $X$ with the union of the simplices contained in $X$.
Let $\Omega \subset {\mathbb R}^n$ be a domain. A mapping $f \colon \Omega \to {\mathbb R}^n$ is piecewise linear if there exists a simplicial complex $X = \Omega$ such that $f$ is linear on each $n$-simplex in $X$.
Algebraic topology
------------------
We refer to [@Hatcher] for basic definitions and theory of homotopy and homology. We denote the homotopy groups and the singular homology groups of a space $X$ by $\pi_k(X)$ and $H_k(X)$, respectively, for $k \in {\mathbb N}$. A closed $n$-manifold $M$ is said to be a *homology sphere* if $H_0(M) = H_n(M) = {\mathbb Z}$ and $H_k(M) = 0$ for all $k \neq 0,n$.
A homology sphere need not be a sphere. The canonical example of a nontrivial homology sphere is the so-called *Poincaré homology sphere*, defined by gluing the opposing edges of a solid dodecahedron together with a twist (see e.g. [@Cannon] and [@KirbyScharlemann]). We will denote the Poincaré homology sphere by $P$ and note that even though the suspension $S(P)$ of $P$ is not a manifold, the double suspension $S^2(P)$ of $P$ is homeomorphic to ${\mathbb S}^5$ (see again e.g. [@Cannon] and [@KirbyScharlemann]).
An important result for us is the following theorem that is an immediate corollary of the Hurewicz isomorphism theorem [@Hatcher Theorem 4.32] combined with the generalized Poincaré conjecture (see [@smale], [@freedman], and [@kleinerlott]).
\[prop:WhiteheadHomology\] If $M$ is a simply connected homology sphere, then $M$ is homeomorphic to the $n$-dimensional sphere ${\mathbb S}^n$.
The double suspension of the cover ${\mathbb S}^3 \to P$. {#sec:Nemesis}
---------------------------------------------------------
To contrast our results and underline the necessity of the more technical arguments we recall in this section a classical branched cover ${\mathbb S}^5 \to {\mathbb S}^5$ constructed by Church and Timourian [@ChurchTimourian] with complicated branch behavior. This example shares many of the properties of branched covers with $f(B_f)$ contained in an $(n-2)$-simplicial complex, but it is not a PL mapping. For further discussion on this map see e.g.[@AaltonenPankka].
We note first that the Poincaré homology sphere can be equivalently defined as a quotient of ${\mathbb S}^3$ under a group action of order 120 (see [@KirbyScharlemann]). The mapping $f \colon {\mathbb S}^3 \to P$ induced by the group action is a covering map, and since ${\mathbb S}^3$ is simply connected we see that ${\mathbb S}^3$ is the universal cover of the Poincaré homology sphere $P$. As a covering map $f$ has an empty branch set but the suspension of $f$, $S(f) \colon S({\mathbb S}^3) \to S(P)$, has a branch set equal to the two suspension points. By definition of the cone of a map, the preimage of either suspension point $P \times\{0\}$ or $P \times \{1\}$ is a point and the preimage of any other point is a discrete set of $120$ points. Thus the double suspension of $f$, $$\begin{aligned}
S^2(f) \colon S^2({\mathbb S}^3) \simeq {\mathbb S}^5 \to S^2(P) \simeq {\mathbb S}^5\end{aligned}$$ is a branched cover between $5$-spheres and has a branch set equal to the suspension of the two branch points of $S(f)$. Thus the branch set $B_{S^2(f)}$ is PL-equivalent to ${\mathbb S}^1$ and so we see that $S^2(f)$ is a branched cover between two spheres with a branch set of codimension four.
The image of the branch set $B_{S^2(f)}$ is complicated since its complement has a fundamental group of 120 elements. Furthermore even though the branch set is PL-equivalent to ${\mathbb S}^1$, the image of the branch set is not PL-equivalent to a simplicial complex even though it is a Jordan curve in ${\mathbb S}^5$. Thus the map $S^2(f)$ does not satisfy the hypothesis of our main theorem.
We also remark for future comparison that for $S^2(f)$ the boundaries of normal neighborhoods $U(x_0,f,r)$, where $x_0$ is one of the two suspension points of the second suspension, are homeomorphic to $S(P)$. This means that the suspension of the Poincaré homology sphere foliates a punctured neighborhood of a point in ${\mathbb R}^5$, but the simply connected space $S(P)$ with homology groups of a sphere is not a manifold.
Boundary of a normal domain {#sec:Bdry}
===========================
In this section we show that for a branched cover $f \colon \Omega \to {\mathbb R}^n$ with $f(B_f)$ contained in a simplicial $(n-2)$-complex, the boundaries of sufficiently small normal domains are homeomorphic to a sphere. The main step of the proof takes the form of an inductive argument where in the inductive step we restrict a branched cover to the boundary of a small normal domain and study the new branched cover between the lower dimensional spaces. Since we do not a priori know that the boundary of a normal domain is a manifold, many of the results in this section are proved in a more general setting where the domain of the mapping is not assumed to be a manifold.
We begin with a few preliminary results on the behavior of $f$ on the boundary of a normal domain. The following Lemma \[lemma:MappingAtTheBoundary\] is known to the experts in the field (see e.g. [@MartioSrebro]) but we give a short proof for the convenience of the reader.
\[lemma:MappingAtTheBoundary\] Let $X$ be a locally compact and complete metric space and $f \colon X \to {\mathbb R}^n$ a branched cover. Fix $x_0 \in X$ and let $r_0>0$ be such that $U_r \colonequals U(x_0,f,r)$ is a normal neighborhood of $x_0$ for all $r \leq r_0$. Then the restriction $$\begin{aligned}
f|_{\partial U_r} \colon \partial U_r \to \partial B(f(x_0),r)
\end{aligned}$$ is a branched cover for all $r<r_0$.
The restriction is clearly continuous and discrete, so it suffices to show that it is an open map. Let $V \subset \partial U_r$ be a relatively open set and suppose $y = f(x_1) \in f(V)$, where $x_1 \in V$. Additionally, suppose that $\{x_1,\dots,x_k\} = f^{-1}(y)$. For $\delta > 0$ let $N_\delta (y) = B(y,\delta)\cap \partial B(x_0,r)$ and for $\epsilon > 0$ let $N_\epsilon(x_i) = B(x_i,\epsilon) \cap \partial U_r$.
Fix $\epsilon > 0$ so that $N_\epsilon(x_i) \cap N_\epsilon(x_j) = \emptyset$ for $i \ne j$ and $N_\epsilon(x_i) \subset V$ for $1\le i\le k$. By [@bonkmeyer Lemma 5.15], there exists a $\delta >0$ such that $$\begin{aligned}
f^{-1}(B(y,\delta)) \subset \cup_{i=1}^k B(x_i,\epsilon).
\end{aligned}$$ Let $y' \in N_\delta(y)$. There exists a path $\gamma$ connecting $y$ to $y'$ in $N_\delta(y)$. By the path-lifting properties of branched covers, (see e.g. [@Rickman-book Chapter II.3] for the Euclidean setting or [@Luisto-Characterization] for a general case), $\gamma$ can be lifted to paths $\gamma_1,\dots,\gamma_k$ each contained in $N_\epsilon(x_i)$. The end point of each lift $x_i'$ maps to $y'$. So $N_\delta(y) \subset f(V)$, which means that $f(V)$ is open.
We will repeatedly choose suitably small normal neighborhoods for points in the domain. For clarity we formulate this selection as the following lemma.
\[lemma:LemmaX\] Let $X$ be a locally connected, locally compact and complete metric space and $f \colon X \to {\mathbb R}^n$ a branched cover. Then for every $x \in X$ there exists a radius $r(x,f) > 0$ such that for all $r < r(x,f)$, $U(x,f,r)$ is a normal neighborhood of $x$.
Furthermore if $f(B_f)$ is contained in an $(n-2)$-simplicial complex we may assume that $f(B_f) \cap f (\partial U(x,f,r)) = f(B_f) \cap \partial B(f(x),r)$ is contained in an $(n-3)$-simplicial complex (up to a global homeomorphism) for all $r < r(x,f)$.
Radial properties of the mapping $f$
------------------------------------
In the following arguments we need a consistent way of describing boundaries of normal domains of mappings which are themselves restrictions of ambient mappings to boundaries of normal domains. To this end we define nested collections of lower dimensional normal domains.
Let $\Omega \subset {\mathbb R}^n$ be a domain and $f \colon \Omega \to {\mathbb R}^n$ a branched cover. Denote by $\mathcal{U}_{n-1}$ the collection of boundaries of normal domains $U(x,f,r) \subset \Omega$ with $r < r(x,f)$ as in Lemma \[lemma:LemmaX\]. For $k = n-1, \ldots, 2$ we similarly define $\mathcal{U}_{k-1}$ to be the collection of boundaries of normal domains $U(x,f|_{V},r) \subset V$, $V \in \mathcal{U}_{k}$, with $r < r(x,f|_{V})$ as in Lemma \[lemma:LemmaX\]. We call these collections as *lower dimensional normal domains*.
By Lemma \[lemma:LemmaX\], in the case where $f(B_f)$ is contained in an $(n-2)$-simplicial complex we may assume that for given $1 \le k \le n-1$ and $V\in \mathcal{U}_k$ that the set $f(B_f) \cap f(\partial U(x,f|_{V},r))$ is contained, up to a homeomorphism, in an $(n-3)$-simplicial for $r < r(x,f|_{V})$.
\[lemma:ImageIsASphere\] Let $\Omega \subset {\mathbb R}^n$ be a domain and $f \colon \Omega \to {\mathbb R}^n$ a branched cover with $f(B_f)$ contained in an $(n-2)$-simplicial complex. Then for any $k = n-1,\ldots,1$ and $V \in \mathcal{U}_k$, $fV$ is homeomorphic to a sphere.
By using an inductive argument we see that it suffices to study the case where $f(V) \subset f(U)$ with $U \in \mathcal{U}_{k+1}$ and $f(U)$ is a $(k+1)$-sphere. The proof in this setting identical to the proof of [@Rickman-book Lemma I.4.9].
We now prove the main proposition needed for the proof of Theorem \[thm:maintheorem\]. It captures the fact that for branched covers with $f(B_f)$ contained in an $(n-2)$-simplicial complex, the branching should occur ‘tangentially’, i.e., inside the boundaries of normal domains. Some of the steps of the proof are described in Figure \[fig:RadialLifts\].
\[prop:UniqueLiftsTopDim\] Let $f \colon \Omega \to {\mathbb R}^n$ be a branched cover such that $f(B_f)$ is contained in an $(n-2)$-simplicial complex. Then for any $x_0 \in \Omega$, there exists a sufficiently small $r < r(x_0,f)$ so that for $v \in {\mathbb S}^{n-1}$, the path $$\begin{aligned}
\beta \colon [0,r] \to \overline{B}(f(x_0),r),
\quad
\beta(t) = (r-t) v + f(x_0)
\end{aligned}$$ has a unique lift starting from any point $z_0 \in \overline{U}(x_0,f,r) \cap f {^{-1}}\{ \beta (0) \}$.
Choose $r$ small enough so that $f(B_f) \cap B(f(x_0,r))$ is contained in a codimension-2 radial set. That is, there exists an $(n-2)$-simplicial complex $D$ such that $D \cap B(x_0,r_1) = \frac{r_1}{r_2} (D \cap B(x_0,r_2))$.
Suppose towards contradiction that the claim is false. Then there exists two different lifts of $\beta$, say $\alpha_1, \alpha_2 \colon [0,r] \to \overline{U}(x_0,f,r)$ satisfying, $$\begin{aligned}
\alpha_1(0) = \alpha_2(0) = z_0 \quad \text{and} \quad \alpha_1(s_0) \neq \alpha_2(s_0),
\end{aligned}$$ for some $s_0 \in (0,r)$. Set $$\begin{aligned}
t_0 = \inf \{ t \in [0,r] \mid \alpha_1(t) \ne \alpha_2(t) \}.
\end{aligned}$$ So $\alpha_1(t) = \alpha(t)$ for all $t \in [0,t_0]$, but for $s \in (t_0,t_0 + \epsilon)$ for small $\epsilon$, $\alpha_1(s) \ne \alpha_2(s)$. Without loss of generality we may assume that $t_0 = 0$ and that $$\begin{aligned}
\alpha_1(t_0) = \alpha_2(t_0) = z_0.
\end{aligned}$$ (see top part of Figure \[fig:RadialLifts\]).
Fix a radius $R < r(z_0,f)$ such that $\overline{B}(f(z_0),R) \subset B(f(x_0),r(x_0,f))$ (see middle part of Figure \[fig:RadialLifts\]). Let $s_0 \in (t_0, t_0 + \epsilon)$, we may assume that $s_0$ is sufficiently small so that $\beta(s_0) \in B(f(x_0),R)$. We now let $U(\alpha_1(s_0))$ and $U(\alpha_2(s_0))$ be normal neighborhoods of $\alpha_1(s_0)$ and $\alpha_2(s_0)$ respectively. Let $\zeta$ be a line segment that has one endpoint at $\beta(s_0)$ and intersects $f(B_f)$ only at $\beta(s_0)$. Additionally, suppose that $\zeta$ is small so that $$\begin{aligned}
\zeta \subset fU(\alpha_1(s_0))\cap fU(\alpha_2(s_0)).
\end{aligned}$$ Since everything is contained in the image of normal neighborhoods we can lift $\zeta$ to $\gamma_1 \subset U(\alpha_1(s_0))$ and $\gamma_2 \subset U(\alpha_2(s_0))$ from the points $\alpha_1(s_0)$ and $\alpha_2(s_0)$, respectively – note though that these lifts might not be unique. Let $\gamma_3$ be a path connecting $\gamma_1$ and $\gamma_2$ that lies outside of $f^{-1}(f(B_f))$. The path $f(\gamma_1\cup\gamma_2\cup\gamma_3)$ will be a loop based at $\beta(s_0)$ that consists of a line segment and a loop. The loop will lie outside of $f(B_f)$ (see bottom part of Figure \[fig:RadialLifts\]).
The image of the branch set is contained in a simplicial complex so if the normal neighborhood around $x_0$ is chosen to be sufficiently small the image of the branch set will be *radial in the normal neighborhood*. By this we mean that it is contained in the union of $(n-2)$-dimensional planes whose intersection contains $f(x_0)$. We may choose the normal neighborhood $V$ of $z_0$ to be so small that the image of the branch set is also radial with respect to $z_0$ in this normal neighborhood $V$. The point $\beta(s_0)$ lies on a path between $f(z_0)$ and $f(x_0)$ so the branch set will be radial at $\beta(s_0)$ with respect to small enough normal domains as well; indeed, for any $w \in B(f(z_0),R) \cap f(B_f)$ the line segment $[w,f(z_0)]$ belongs to the branch, and so will the line $[w,f(x_0)]$. Additionally, for each $w' \in [w,f(z_0)]$, the line $[w',f(x_0)]$ will be in $f(B_f)$ and so we conclude that $f(B_f)$ contains the segment $[w,\beta(s_0)]$.
Define a homotopy that consists of the straight line from each point in $f(\gamma_1\cup\gamma_2\cup\gamma_3)$ to $\beta(s_0)$. Due to the local radial structure of $f(B_f)$ at $\beta(s_0)$ we see that the homotopy will take $f(\gamma_1\cup\gamma_2\cup\gamma_3)$ to an arbitrarily small neighborhood of $\beta(s_0)$ without intersecting $f(B_f)$. Additionally, the end loop will be contained in the image of the normal neighborhoods, $U(\alpha_1(s_0))$ and $U(\alpha_2(s_0))$.
The homotopy will always preserve a small straight line in $\zeta$ and so we can lift the homotopy uniquely. The end loop of the homotopy will be lifted separately to the normal neighborhood of $\alpha_1(s_0)$ and $\alpha_2(s_0)$. This gives a homotopy from a connected curve to two disconnected loops, which is a contradiction.
![Showing that radial lifts are unique.[]{data-label="fig:RadialLifts"}](RadialPicture.pdf){width="100.00000%"}
The previous proposition allows us to uniquely lift radial paths in normal neighborhoods. We would also like to be able to lift radial paths uniquely in lower dimensional normal neighborhoods $U \subset V$ with $V \in \mathcal{U}_k$ for any $k$.
Let $f \colon \Omega \to {\mathbb R}^n$ be a branched cover such that $f(B_f)$ is contained in an $(n-2)$-simplicial complex. Let $x_0 \in V \in \mathcal{U}_k$. By Lemma \[lemma:ImageIsASphere\], $f(V) \simeq {\mathbb S}^k$. Up to homeomorphism we can assume that $f(V)$ minus a point maps to a $k$-dimensional plane. In this case $f|_V$ will have a branch set contained in a $(k-2)$-simplicial complex.
\[prop:UniqueLifts\] If $r < r(x_0,f|_V)$ and $v \in {\mathbb S}^{k-1}$, the path $$\begin{aligned}
\beta \colon [0,r] \to \overline{B}(f(x_0),r),
\quad
\beta(t) = (r-t) v
\end{aligned}$$ has a unique lift starting from any point $z_0 \in \overline{U}(x_0,f|_V,r) \cap f|_V^{-1} \{ \beta (0) \}$.
By Proposition \[prop:UniqueLiftsTopDim\] we know that $\beta$ has a unique lift in $\Omega$ starting from any preimage of $\beta(0)$. Thus we only need to show that such a lift is contained $V$. But this is clear since $f(V)$ maps surjectively onto a $k$-dimensional plane containing $\beta$ and thus there will be a preimage of $\beta(0)$ in $V$ and the lift of $\beta$ under $f|_V$ starting from this preimage is contained in $V$.
\[prop:HomeomorphismFoliation\] Let $\Omega \subset {\mathbb R}^n$ be a domain and $f \colon \Omega \to {\mathbb R}^n$ a branched cover with $f(B_f)$ contained in an $(n-2)$-simplicial complex. Suppose $k = n-1, \ldots, 2$ and $W \in \mathcal{U}_k$. Then for any $x_0 \in W$ and all normal domains $U(x_0,f|_W,r)$ with $r < r(x,f) \equalscolon r_0$ (as in Lemma \[lemma:LemmaX\]) there exists a parameterized collection of homeomorphisms $$\begin{aligned}
h_t \colon \partial U(x_0,f|_W,r_0) \to \partial U(x_0,f|_W,t),
\end{aligned}$$ $t \in (0,r_0)$ such that the mapping $$\begin{aligned}
H \colon (0, r_0) \times \partial U(x_0,f|_W,r_0) \to U(x_0, f|_W, r_0) \setminus \{ x_0 \},\end{aligned}$$ $$\begin{aligned}
H(t,x) = h_t(x)
\end{aligned}$$ is also a homeomorphism and $U(x_0,f|_W,r_0) \simeq {\operatorname{cone}}(\partial U(x_0,f|_W,r_0))$.
For $t \in (0,r_0)$ and any given point $x \in \partial U(x_0,f|_W,t)$ we define the homeomorphism $h_{t}$ to map $x$ to the endpoint of the unique lift, guaranteed by Proposition \[prop:UniqueLifts\], of the straight line connecting $f(x)$ and $f(x_0)$. Since these lifts are unique, there exists a canonical inverse map for $h_{t}$. Since these two maps are defined symmetrically it suffices to show that $h_t$ is continuous to prove the claim.
Suppose that there exists a sequence $\{a_j\}_{j \in \mathbb N}$ such that $a_j \in U(x_0,f|_W,r_0)$ and $\lim_{j \to \infty} a_j = a \in U(x_0,f|_W,r_0)$. This would imply that there is a radial line segment $I$ together with a sequence $(I_j)$ of line segments converging to $I$. We must show that the unique lifts $\alpha_j$ of $I_j$ converge to the unique lift $\alpha$ of $I$. By compactness of the Hausdorff metric (see e.g. [@BridsonHaefliger pp.70–77]) $\{\alpha_j\}_{j \in \mathbb N}$ must have a converging subsequence. So by taking a subsequence suppose that $\lim_{j \to \infty} \alpha_j = \beta$. Additionally, $\beta$ will be connected since for each $j \in \mathbb N$, $\alpha_j$ is connected.
We can parametrize the $I_j$ by a time parameter $t$ in the obvious way. Similarly, we can parametrize $\alpha_j$ so that $f\circ \alpha_j(t) = I_j(t)$. By [@BridsonHaefliger Lemma 5.32], for every $x \in \beta$, there exists a sequence $\{\alpha_j(t)\}_{j\in\mathbb N}$ so that $\lim_{j \to \infty} \alpha_j(t_j) = x$. Since $f$ is continuous and $f(\alpha_j(t_j)) \in I_j$, we have that $f(x) \in I$. So $\beta \subset \alpha$. Note that $\beta$ cannot be contained in a different preimage of $I$ by $f$ since $\lim_{j \to \infty} a_j = a \in \alpha$ and $\beta$ is connected.
If $x \in \alpha$, then there exists a sequence of points $y_j \in I_j$ such that $\lim_{j \to \infty} y_j = f(x)$. The point $y_j$ has a unique preimage $x_j \in \alpha_j$ for all $j \in \mathbb N$. By [@BridsonHaefliger Lemma 5.32] there exists a subsequence $\{x_{j_k}\}_{k \in \mathbb N}$ such that $\lim_{k \to \infty} x_{j_k} = x' \in \beta \subset \alpha$. So $f(x') \in I$ and by uniqueness of lifts we have that $x = x'$. This gives that $\beta = \alpha$. The argument shows that every subsequence of $\{\alpha_j\}_{j \in \mathbb N}$ must limit to $\alpha$ and so $\lim_{j \to \infty} h_t(a_j) = h_t(a)$, which gives that $h_t$ is continuous.
Finally, it is straightforward to check that $H$ is also a homeomorphism, which implies that $U(x_0,f|_W,r_0) = {\operatorname{cone}}(\partial U(x_0,f|_W,r_0))$.
The previous Proposition \[prop:HomeomorphismFoliation\] shows that we can foliate the small punctured lower dimensional normal domains with their boundaries. Note that this does not a priori imply that the boundaries are spheres, see again the example in Section \[sec:Nemesis\].
Boundaries of normal domains are homeomorphic to spheres
--------------------------------------------------------
We wish to show that the boundary of a normal domain is homeomorphic to a sphere for a branched cover $f$ with $f(B_f)$ contained in an $(n-2)$-simplicial complex. The proof is based on an inductive argument on the dimension of the lower dimensional normal domains. Most of the complications in the statements and proofs of the following proposition arise from the fact that we need to study the restriction of $f$ to the boundary of a normal domain before showing that the boundary is a manifold.
We first compute the homology groups of the boundary of a lower dimensional normal neighborhood.
\[lemma:homologyequality\] Fix $k \in \{2,\dots,n-1\}$. Let $U$ be a normal neighborhood of dimension $k$ centered at a point $x \in {\mathbb R}^n$. Let also $\partial U = V \in \mathcal{U}_k$. If $U$ is sufficiently small, then $$\begin{aligned}
H_l(V) = H_l({\mathbb S}^k)
\end{aligned}$$ for $0 \le l \leq k$, where $H_l$ is the simplicial homology group.
By Proposition \[prop:HomeomorphismFoliation\], $U \simeq {\operatorname{cone}}(V)$ and therefore $U\setminus\{x\} \simeq V\times(0,1)$.
Since $V \in \mathcal{U}_k$ we know that $U$ is a normal neighborhood contained in some $W \in \mathcal{U}_{k+1}$. By Proposition \[prop:HomeomorphismFoliation\] there exists an open set containing $U$ in $W$ that is homeomorphic to $U \times (0,1)$. Removing the point $x \in U$ thus gives rise to a neighborhood of $U\setminus\{x\}$ homeomorphic to $U \setminus \{x\} \times (0,1) \simeq V \times (0,1)^2$.
We can continue inductively to find an open set containing $U$ in the top level normal neighborhood (which is a domain in ${\mathbb R}^n$) that is homeomorphic to $U \times (0,1)^{n-k-1}$. Furthermore, $U\setminus\{x\}$ is contained in an open set that is homeomorphic to $V \times (0,1)^{n-k}$. These are now open sets in ${\mathbb R}^n$ and are therefore manifolds. Recall that $U \simeq {\operatorname{cone}}(V)$ and therefore $U$ is contractible. So $U\times (0,1)^{n-k-1}$ is also contractible.
By extending $U$ to an open domain in ${\mathbb R}^n$ the point $x \in U$ is extended radially. Therefore $\{x\}\times (0,1)^{n-k-1} \subset U\times (0,1)^{n-k-1}$ is an $(n-k-1)$-submanifold. For $1 \le l < k$, consider a map $$\begin{aligned}
\gamma \colon {\mathbb S}^l \to (U\times (0,1)^{n-k-1})\setminus (\{x\} \times(0,1)^{n-k-1}).
\end{aligned}$$ Since $U \times (0,1)^{n-k-1}$ is contractible, there is a homotopy $H$ that takes $\gamma$ to a point $x' \ne x$. The dimension of ${\mathbb S}^l \times (0,1)$ is $l+1$ and $$\begin{aligned}
(l+1) +(n-k-1) < n,
\end{aligned}$$ since $l < k \le n-1$. So we claim that the image of $H$ can be guaranteed to avoid $\{x\} \times (0,1)^{n-k-1}$. To prove this claim note that $H$ can be assumed to be smooth since $U\times (0,1)^{n-k-1})\setminus (\{x\} \times(0,1)^{n-k-1})$ is an open set and hence a smooth manifold. By the compactness of the image of $H$, there exists an $\epsilon > 0$ so that the $\epsilon$-neighborhood of $H$ lies in $U\times (0,1)^{n-k-1})\setminus (\{x\} \times(0,1)^{n-k-1})$. Smooth functions are dense in the uniform topology. Therefore there exists a smooth function $\tilde H \colon {\mathbb S}^l \times [0,1] \to U\times (0,1)^{n-k-1})\setminus (\{x\} \times(0,1)^{n-k-1})$ such that $\|H - \tilde H\| < \epsilon$. A straight-line homotopy takes $H$ to $\tilde H$ and hence the claim is shown.
We can also assume that the image of $H$ is transverse to the submanifold $\{x\} \times(0,1)^{n-k-1}$ (see [@guilleminpollack Chapter 2]). Since the dimensions add up to a number less than $n$, the transversality condition implies that they are actually disjoint. The entire homotopy is disjoint from the removed set and thus $$\begin{aligned}
\pi_l((U\times (0,1)^{n-k-1})\setminus (\{x\} \times(0,1)^{n-k-1})) = 0
\end{aligned}$$ for $1 \le l < k$. By the above argument, $\pi_l(V) = 0$ for $1\le l < k$. The lemma now follows by the Hurewicz theorem [@Hatcher p. 366] for this index range.
It remains to show that $H_k(V) = H_k(S^k)$. We show this case by the use of the Mayer-Vietoris theorem. Let $M = U\times (0,1)^{n-k-1}$ and let $L =\{x\} \times\mathbb R^{n-k-1}$. Note that $$\begin{aligned}
M \setminus L = (U\times (0,1)^{n-k-1})\setminus (\{x\} \times(0,1)^{n-k-1}).
\end{aligned}$$ The Mayer-Vietoris theorem implies that $$\begin{aligned}
\cdots \to H_{k+1}(M \cup (\mathbb R^n \setminus L) ) &\to H_k(M \cap (\mathbb R^n\setminus L) ) \to H_k(M)\oplus H_k(\mathbb R^n \setminus L)\\
&\to H_k(M \cup (\mathbb R^n \setminus L) ) \to \cdots
\end{aligned}$$ is an exact sequence. We have that $M$ and $ M \cup (\mathbb R^n \setminus L)$ are contractible. Additionally, $$\begin{aligned}
M \cap (\mathbb R^n\setminus L) = M \setminus (\{x\} \times(0,1)^{n-k-1}).
\end{aligned}$$ So $$\begin{aligned}
0 \to H_k(M\setminus (\{x\} \times(0,1)^{n-k-1})) \to H_k(\mathbb R^n \setminus L) \to 0.\end{aligned}$$ This implies that $$\begin{aligned}
H_k(V) &= H_k((U\times (0,1)^{n-k-1})\setminus (\{x\} \times(0,1)^{n-k-1})\\
&\cong H_k(\mathbb R^n \setminus L) \cong H_k(S^k). \end{aligned}$$
We next show that the boundary of normal domains are homeomorphic to spheres.
\[prop:boundaryspheres\] Let $k \in \{2,\dots,n-1\}$. If $V \in \mathcal{U}_k$, then $V \simeq {\mathbb S}^k$.
We begin by noting that by the proof in Lemma \[lemma:homologyequality\], $$\begin{aligned}
V\times (0,1)^{n-k} \simeq U\times (0,1)^{n-k-1} \setminus(\{x\} \times (0,1)^{n-k-1}),
\end{aligned}$$ where $U$ is a normal neighborhood on the $(k+1)$-level and $\partial U = V$. Normal neighborhoods are connected and removing a set of dimension ${n-k-1}$ does not disconnect the set for $k \ge 1$. So $V\times (0,1)^{n-k}$ is connected and therefore $V$ is connected.
We now continue to prove the main claim in the proposition. Suppose first that $k=1$ and fix $V \in \mathcal{U}_1$. We denote the restriction $f|_V \colon V \to fV$ by $g$. By Lemma \[lemma:ImageIsASphere\], $gV$ is homeomorphic to a circle. At this level the intersection of $f(B_f)$ with the $g(V)$ is at most a finite set of points. Since $V$ is the boundary of a normal neighborhood we can therefore conclude that actually $$\begin{aligned}
g(V) \cap f(B_f) = \emptyset.
\end{aligned}$$ This implies that $V \cap B_f = \emptyset$ and $$\begin{aligned}
g \colon V \to g(V) \simeq {\mathbb S}^1
\end{aligned}$$ is a covering map. Since the branched cover $f$ is finite-to-one in any normal domain, we see that $g$ is a finite-to-one cover of ${\mathbb S}^1$. This implies that $V$ is homeomorphic to ${\mathbb S}^1$.
Suppose next that the claim holds true for some $k < n-1$ and $V \in \mathcal{U}_{k+1}$. Fix a point $x \in V$ and take a normal neighborhood $W$ of $x$ such that $\partial W \in \mathcal{U}_{k}$. By the inductive assumption $\partial W$ is homeomorphic to ${\mathbb S}^k$. By Proposition \[prop:HomeomorphismFoliation\], $$\begin{aligned}
W
\simeq {\operatorname{cone}}{\partial W}
\simeq {\operatorname{cone}}{{\mathbb S}^k}
\simeq B^n.
\end{aligned}$$ The point $x$ has a neighborhood in $V$ homeomorphic to a ball and therefore $V$ is a closed $k$-manifold. By Lemma \[lemma:homologyequality\], $$\begin{aligned}
H_l(V) \cong H_l({\mathbb S}^k)\end{aligned}$$ for $0 \le l \leq k$. In the proof of Lemma \[lemma:homologyequality\], we also showed that $\pi_l(V) = 0$ for $1 \le l \le k-1$. Combining these we see that $V$ is a simply connected homology $k$-sphere and so $V \simeq {\mathbb S}^k$ by Proposition \[prop:WhiteheadHomology\].
PL cone mappings {#sec:PL}
================
In this section we prove Theorem \[thm:maintheorem\]. We divide the proof into a local and global part. A branched cover $f \colon {\mathbb S}^n \to {\mathbb S}^n$ is called *locally PL with respect to a simplicial decomposition $A$ of ${\mathbb S}^n$* if, for all $x \in {\mathbb S}^n$, there exists an open set $U \subset \overline U \subset {\mathbb S}^n$ containing $x$ and a homeomorphism $$\begin{aligned}
\phi \colon \overline{U'} \to \overline{U} \subset {\mathbb S}^n
\end{aligned}$$ such that $f \circ \phi$ is a PL mapping. Additionally, if $\overline{U}$ is given a simplicial decomposition defined by $f \circ \phi$, the $k$-simplices in $\overline{U}$ are mapped to $k$-simplices in a subdivision of $A$.
\[lemma:localtoglobal\] Let $g \colon {\mathbb S}^n\to {\mathbb S}^n$ be a branched cover whose branch set is contained in an $(n-2)$-simplicial complex. Let $A$ be a simplicial decomposition of ${\mathbb S}^n$ that contains $g(B_g)$ in its $(n-2)$-skeleton. Additionally, suppose that $g$ is locally $PL$ with respect to $A$. Under these conditions, there exists a homeomorphism $$\begin{aligned}
\Phi \colon {\mathbb S}^n \to {\mathbb S}^n
\end{aligned}$$ such that $g \circ \Phi$ is a PL mapping and the $k$-simplices defined by $g \circ \Phi$ are mapped to $k$-simplices in $A$.
We remark that the proof here uses ideas from the proof in [@bonkmeyer Lemma 5.12].
The strategy of the proof will be to pull back the simplicial structure $A$ by $g$. The set ${\mathbb S}^n$ can be covered by finitely many open sets $U$ that satisfy the conditions in the definition of the local PL property of $g$. We refine $A$ so that $g$ maps the simplices in $\overline{U'}$ to simplices in $A$, where $\phi \colon \overline{U'} \to \overline{U}$ is a homeomorphism as described above.
In the spirit of pulling back $A$ by $g$, let $B$ be the set of simplices $\sigma$ such that $g(\sigma) \in A$ and $g|_\sigma$ is a homeomorphism onto its image. To show that this is a simplicial structure for ${\mathbb S}^n$ it suffices to show that every point lies in the interior of a unique simplex and that the intersection of two simplices is a face of those simplices.
We first show that every point lies in the interior of a unique simplex. Let $x \in {\mathbb S}^n$ and $y=g(x) \in \Delta_k^o$, where $\Delta_k$ is a $k$-simplex in $B$ and $\Delta_k^o$ is the interior of $\Delta_k$. Let $U$ be an open set containing $x$ such that there exists a homeomorphism $\phi \colon U' \to U$ satisfying that $g \circ \phi$ is a PL mapping. By our assumption, $x' = \phi^{-1}(x)$ is contained in a simplex $D$ that is mapped by $g \circ \phi$ onto a simplex in $A$. Since $g\circ \phi(x') = y \in \Delta_k^o$, the simplex $D$ must be a degree $k$ simplex and $x' \in D^o$. Additionally, $D^o = (g\circ \phi)^{-1}(\Delta_k^o)\cap U'$.
The map $g \circ \phi$ is a PL branched cover. Therefore it is locally injective on $D^o$. So $g$ defines a covering map from the component $\tau$ of $g^{-1}(\Delta_k^o)$ containing $x$ to $\Delta_k^o$. Since $\Delta_k$ is simply connected, $g$ is actually a homeomorphism from $\tau$ to $\Delta_k^o$.
We claim that $g$ extends to a homeomorphism from $\sigma = \overline{\tau}$ to $\Delta_k$. It suffices to show that $g^{-1}\colon \Delta_k^o \to \sigma$ extends continuously to the boundary. Let $\{y_n\}_{n \in \mathbb N}$ be a sequence of points such that $y_n \to y \in \partial \Delta_n$. Then there exists a sequence of points $\{x_n\}_{n\in \mathbb N}$ such that $g(x_n)= y_n$. Let $a$ and $b$ be accumulation points of $\{x_n\}_{n\in\mathbb N}$. Let $a_n$ be a subsequence that converges to $a$ and $b_n$ a subsequence that converges to $b$.
By [@bonkmeyer Lemma 5.15], for all $\epsilon > 0$, there exists $\delta > 0$ so that $$\begin{aligned}
g^{-1}(B(y,\delta)) \subset \cup_{z \in g^{-1}(y)} B(z,\epsilon).\end{aligned}$$ By choosing $\epsilon$ sufficiently small, the sets $B(z,\epsilon)$ will be pairwise disjoint for $z \in g^{-1}(y)$. However, for large $n$, $g(a_n)$ and $g(b_n)$ will be in $B(y,\delta)$. If $a_n$ and $b_n$ are connected by a path $\gamma$, then $g^{-1}(\gamma)$ must be a path connecting $a_n \in B(a,\epsilon)$ and $b_n \in B(b,\epsilon)$. So $g^{-1}(\gamma)$ lies outside $\cup_{z\in g^{-1}(y)}B(z,\epsilon)$, which gives a contradiction if $a \ne b$. Thus $g^{-1}$ extends continuously to $\partial \Delta_k$ and $g$ defines a homeomorphism from $ \sigma$ to $\Delta_k$. This shows that $ \sigma$ defines a $k$-simplex in $B$ and that $x \in \sigma$. This shows that every $x$ is in a simplex defined by $B$.
Let $\sigma_1$ and $\sigma_2$ be simplices in $B$ and suppose $\sigma_1\cap \sigma_2 \ne \emptyset$. If $\sigma_1^o\cap \sigma_2^o \ne \emptyset$, then they must both be $k$-simplices and by construction must be mapped homeomorphically onto the same $k$-simplex $\Delta_k \in A$. This is not possible since $\Delta_k$ is simply connected.
If $\sigma_1^o\cap \sigma_2^o = \emptyset$, then suppose $\tau$ is a simplex such that $\tau^o \cap \sigma_1\cap \sigma_2 \ne \emptyset$. It follows that $g(\tau) \subset g(\sigma_1)\cap g(\sigma_2)$. The map $g$ defines an inverse on $g(\tau)^o$, which must agree with the inverses that it defines on $g(\sigma_1)^o$ and $g(\sigma_2)^o$. So the entirety of $\tau$ must be contained in $\sigma_1\cap \sigma_2$. This implies that $\sigma_1\cap \sigma_2$ is comprised of the union of finitely many simplices.
Finally, we claim that $A$ and $B$ can be refined so that the intersection of two simplices is a face. Let $\sigma_1$ and $\sigma_2$ be $k$-simplices. Suppose that $\sigma_1\cap \sigma_2 \ne \emptyset$ and that there are two $(k-1)$-simplices whose interiors are in $\sigma_1\cap \sigma_2$. We apply a barycentric subdivision $A$. Then $g$ pulls back this decomposition to a refinement of $B$ and the new simplices in $\sigma_1$ and $\sigma_2$ cannot share more than one $(k-1)$-simplex. We may now proceed by repeated barycentric subdivision to rule out the cases when $\sigma_1\cap\sigma_2$ contain more than one interior of lower degree shared simplices. At the end of this process, the refined $B$ must be a simplicial decomposition of ${\mathbb S}^n$.
The construction implies that $g$ is a simplicial map from $S_B^n$ to $S_A^n$. Thus there exists a PL map from $S_B^n$ to $S_A^n$ that is topologically equivalent to $g$ with respect to $A$.
\[lemma:LocallyPL\] Let $f \colon {\mathbb S}^n\to {\mathbb S}^n$ be a branched cover with $f(B_f)$ contained in a simplicial $(n-2)$-complex. Let $A$ be a simplicial decomposition of ${\mathbb S}^n$ that contains $f(B_f)$ in its $(n-2)$-skeleton. Then $f$ is locally PL with respect to $A$.
We proceed by induction on $n$. The base case, $n = 2$, follows from Stoïlow’s theorem (see [@Stoilow] or [@LuistoPankka-Stoilow]) as $f$ is topologically equivalent to a rational map $S^2 \to S^2$ and rational maps are topologically equivalent to PL mappings.
We now suppose that $f\colon {\mathbb S}^n \to {\mathbb S}^n$ is defined as in the statement of the lemma. Then there exists a Euclidean simplicial decomposition $A$ (when ${\mathbb S}^n$ is viewed as $\mathbb R^n \cup \{\infty\}$) such that $f(B_f)$ is contained in the $(n-2)$-skeleton of $A$.
Fix $x \in {\mathbb S}^n$. For a small radius $r_0$, there exists a ball $B(f(x),r_0)$ that is radially symmetric with respect to the simplicial decomposition $A$. More precisely, for any simplex $\Delta \in A$, $$\begin{aligned}
\Delta \cap \partial B(f(x),r) = \frac{r}{s} (\Delta \cap \partial B(f(x),s))
\end{aligned}$$ for $0<r,s \le r_0$, where $r/s$ is the dilation mapping the $s$-sphere at $f(x_0)$ to the $r$-sphere.
By Proposition \[prop:HomeomorphismFoliation\] and Proposition \[prop:boundaryspheres\], for sufficiently small $r_0$, the normal neighborhood $U(x,f,r_0) \simeq {\operatorname{cone}}(V)$, where $V = \partial U(x,f,r_0)$, is homeomorphic to $S^{n-1}$. Let $g = f|_V$. By the construction of the homeomorphism in Proposition \[prop:HomeomorphismFoliation\], $f$ is topologically equivalent to ${\operatorname{cone}}(g) \colon {\operatorname{cone}}(V) \to B(f(x),r_0)$. By the choice of $B(f(x),r_0)$, the map $g \colon V \to \partial B(f(x),r_0)$ sends its branch set into the $(n-3)$-skeleton of $B(f(x),r_0)$ induced by $A$. The induction hypothesis gives that $g$ is locally a PL mapping which respects the simplicial decomposition $A$. By Lemma \[lemma:localtoglobal\] it is globally a PL mapping, which respects the simplicial decomposition $A$.
The set $B(f(x),r_0)$ was chosen to be radially symmetric. Therefore, the map ${\operatorname{cone}}(g)$ also respects the simplicial decomposition $A$ on ${\mathbb S}^n$. Thus $f$ satisfies the conclusion of the lemma.
Theorem \[thm:maintheorem\] follows immediately from Lemma \[lemma:LocallyPL\]. The combination of Lemmas \[lemma:localtoglobal\] and \[lemma:LocallyPL\] proves Corollary \[coro:GlobalMainTheorem\].
Homotopic properties of foliations {#sec:HomotopiesOfHomotopies}
==================================
In Section \[sec:Bdry\] we showed that the boundaries of normal neighborhoods locally foliate a punctured normal neighborhood. Furthermore, when the image of the branch is a simplicial $(n-2)$-complex, this foliation is the trivial one, i.e., it consists only of spheres. The proof in Section \[sec:Bdry\] relied strongly on the fact that by Proposition \[prop:HomeomorphismFoliation\] the boundaries are homeomorphic. This enabled us to show that the boundaries are not only manifolds but even spheres.
In this section we show that the existence of the homeomorphisms given by Proposition \[prop:HomeomorphismFoliation\] is not needed if we a priori assume the normal domains to be manifolds with boundaries, in which case the boundaries are naturally also manifolds.. Compare these results to the example in Section \[sec:Nemesis\], where we noted that the boundaries of normal neighborhoods of the double suspension map are not manifolds but do foliate a punctured domain in ${\mathbb S}^5$. For clarity we state the results here as concerning codimension 1 closed submanifolds in ${\mathbb R}^n$ instead of focusing on boundaries of normal neighborhoods. We prove that the only topological codimension 1 manifold foliations of punctured domains in euclidean spaces are the trivial spherical ones. We have not been able to find this statement recorded in the literature in this generality, but we do not assume it to be unknown to the specialists in the field. See however [@MartioSrebro Theorem 3.7 and Lemma 6.2] for the three dimensional case and compare to the Reeb Stability Theorem [@Foliations-book Theorem 2.4.1, p. 67] for a related claim in the smooth setting. Compare also to the Perelman stability theorem in [@Perelman-Stability] (see also [@Kapovitch-PerelmanStability]) from which a similar result could be deduced in the smooth setting.
We do not assume that leaves in foliations are homeomorphic, which arises from unique path lifts in the setting of Section \[sec:Bdry\]. Rather, we rely here on the fact that in manifolds with positive injectivity radius homotopy arguments can be essentially reduced to discrete homotopy. Note that the above-mentioned Perelman stability theorem, [@Perelman-Stability], requires the assumption of a lower bound to the Ricci curvature, and such a lower bound also gives rise to a lower bound for the injectivity radius of a closed Riemannian manifold.
Let $\{U_t\}$, $t \in (0,1)$, be a family of $n$-dimensional connected manifolds with boundary contained in ${\mathbb R}^n$. If $\partial U_t = X_t$, then $\mathcal{X} \colonequals \{ X_t \}, t \in (0,1)$ is a *topological foliation* if the following conditions hold.
1. $X_t \cap X_s = \emptyset$ when $t \neq s$.
2. $0 \in U_t \subset U_s$ for all $t \leq s$.
3. $0 \notin X_t$ for all $t \in (0,1)$ and ${\operatorname{diam}}(X_t) \to 0$ when $t \to 0$.
4. $U \colonequals \{ 0 \} \cup \bigcup_{t} X_t$ is an open neighborhood of the origin.
By the above properties, there exists a parameter $t_0 \in (0,1)$ and a radius $r_0 > 0$ such that $U_{t_0} \subset B(0,r_0) \subset U$. We call the pair $(t_0,r_0)$ a *break point* of the foliation.
We remark that the assumption that $U_t$ is a manifold with boundary seems unnecessary to us. We believe that instead it should suffice to assume that $X_t$ are closed, connected $(n-1)$-submanifolds in ${\mathbb R}^n$ along with (F1)-(F4) should suffice for the results that follow.
The aim of this section is to show that the definition above always leads to a trivial foliation.
\[thm:SphericalityOfFoliation\] For any topological foliation $\mathcal{X} = \{ X_t \}$ there exists a break point $(t_0,r_0)$ with$t_0 \in (0,1), r_0 > 0$ such that $X_t$ is a topological sphere for all $t \leq t_0$.
To prove this claim we will require several auxiliary results. For the purposes of the upcoming proofs we denote $$\begin{aligned}
A^\mathcal{X}(a,b)
= \bigcup_{t \in (a,b)} X_t\end{aligned}$$ and call such sets *foliation annuli*. For subintervals $I$ of $(0,1)$ we similarly use the notation $A^\mathcal{X}(I)$.
\[lemma:FoliatonAnnuliIsAManifold\] Let $\mathcal{X} = \{ X_t \}_{t \in (0,1)}$ be a topological foliation. Then for any $a,b \in (0,1)$, $a<b$, the foliation annulus $A^\mathcal{X}(a,b)$ is an $n$-manifold.
We show that $A^\mathcal{X}(a,b)$ is an open subset of ${\mathbb R}^n$. To this end, fix a point $x_0 \in A^\mathcal{X}(a,b)$, and let $t_0 \in (0,1)$ be such that $x_0 \in X_{t_0}$. Since the sets $\{ x_0 \}$, $X_a$ and $X_b$ are all compact, they have positive distances. Call the smallest one of these ${\varepsilon}$. Since the domains $U_t$ form a monotone sequence with disjoint boundaries $X_t$, this implies that $B(x_0,{\varepsilon}) \subset A^\mathcal{X}(a,b)$.
\[lemma:HomotopyLemma\] Let $X$ be a metric space and let $M$ be a topological manifold embedded in $\mathbb R^n$. There exists a $\delta > 0$ such that for any continuous maps $f,g\colon X \to M $, if $\sup_{x \in X} \|f(x) - g(x)\| < \delta$, then $f$ and $g$ are homotopic.
By [@Hatcher Theorem A.7], there exists an open neighborhood $N \subset \mathbb R^n$ of $M$ such that $M$ is a retract of $N$. If $\delta$ is sufficiently small, then for all $p \in M$, $B_\delta(p) \subset N$. So the image of the homotopy $H(t,x) = (1-t)f(x) + tg(x)$, for $t \in (0,1)$, is contained in $N$. Let $r \colon N \to M$ be the retraction map. Then $r \circ H$ is a homotopy from $f$ to $g$ lying in $M$.
Motivated by this lemma we fix some terminology on discrete approximations of homotopies in the setting of manifolds.
Let $M$ be an $n$-manifold and let $\delta$ be the parameter given in Lemma \[lemma:HomotopyLemma\]. By Lemma \[lemma:HomotopyLemma\], if $X$ is a metric space, then any two mappings $f,g \colon X \to M$ with $\sup_{x \in X} \|f(x) - g(x)\| < \delta$ are homotopic. In such a setting we say that *a discrete homotopy approximation of a continuous map* $f \colon [0,1]^k \to M$ is a discrete collection of points $D \subset [0,1]^k$ together with a mapping $g \colon D \to M$ such that there is a continuation $\tilde g$ of $g$ that satisfies $\sup_{x \in [0,1]^k} \|f(x) - \tilde g(x)\| < \delta$.
We next show that any homotopy performed in the union of a foliation $\mathcal{X}$ can be ‘pulled’ within one of the leaves. For the sake of clarity and readability we state the main proposition for a general mapping instead of a homotopy. Note that with minor modifications this argument could be used with more general topological foliations that are not converging to a point.
\[prop:HomotopyOfHomotopies\] Let $\mathcal{X} = \{ X_t \}$ be a topological foliation. Then for any $k \in {\mathbb N}$ and any continuous mapping $$\begin{aligned}
f \colon {\mathbb S}^k \times [0,1] \to A^\mathcal{X}(0,1)
\end{aligned}$$ there exists a continuous mapping $$\begin{aligned}
g \colon {\mathbb S}^k \times [0,1] \to X_{t_0}
\end{aligned}$$ such that $f$ and $g$ are homotopic in $A^\mathcal{X}(0,1)$.
Fix $k \in {\mathbb N}$ and note that for any mapping $$\begin{aligned}
f \colon {\mathbb S}^k \times [0,1] \to A^{\mathcal{X}}(0,1)
\end{aligned}$$ there is a positive distance from the image of $f$ to the boundary of $A^{\mathcal{X}}(0,1)$.
We define $I$ to be a maximal, possibly trivial, subinterval of $(0,1)$ such that for any mapping $$\begin{aligned}
f \colon {\mathbb S}^k \times [0,1] \to A^{\mathcal{X}}(I)
\end{aligned}$$ there exists a mapping $$\begin{aligned}
g \colon {\mathbb S}^k \times [0,1] \to X_{t_0}
\end{aligned}$$ such that $f$ and $g$ are homotopic. We wish to show that in fact $I = (0,1)$, which will prove the claim. In order to show this we demonstrate that $I$ is non-empty and both open and closed.
Since $\{ t_0 \} \subset I$, the interval is clearly non-empty. To show that $I$ is open, we fix $s_0 \in I$ and cover the $(n-1)$-dimensional manifold $X_{s_0}$ with open sets that form an atlas. Let $\delta_{s_0}'$ be the Lebesgue number of this open cover. Next we cover $X_{s_0}$ with charts of the $n$-manifold $A^{\mathcal{X}}(0,1) \supset X_{s_0}$ and denote by $\delta_{s_0}''$ the Lebesgue number of this open cover. We set $\delta_{s_0} = 8{^{-1}}\min\{\delta_{s_0}', \delta_{s_0}''\}$ and fix ${\varepsilon}> 0$ such that $A^{\mathcal{X}}(s_0-{\varepsilon}, s_0+{\varepsilon}) \subset B^n(X_{s_0}, \delta_{s_0})$.
Fix now a mapping $$\begin{aligned}
f \colon {\mathbb S}^k \times [0,1] \to A^{\mathcal{X}}(I \cup (s_0-{\varepsilon}, s_0 + {\varepsilon})).
\end{aligned}$$ Since the domain of $f$ is compact, $f$ is uniformly continuous, so there exists $\delta > 0$ such that $\| f(x) - f(y) \| < {\varepsilon}/4$ for all $x,y$ with $d(x,y) < \delta$. Let $D \subset {\mathbb S}^k \times [0,1]$ be a discrete set such that $B(D,\delta/4) = {\mathbb S}^k \times [0,1]$ and denote $\hat f \colonequals f|_D$. By our selection of ${\varepsilon}$ we can now define a mapping $$\begin{aligned}
\hat g \colon D \to X_{s_0}
\end{aligned}$$ such that $d_\infty(\hat f, \hat g) < {\varepsilon}/4$, where $d_\infty$ denotes the supremum norm.
Now, since ${\varepsilon}< \delta_{s_0}'/8$, we see that the mapping $\hat g$ can be extended through affine continuations in the charts of $X_{t_0}$ into a continuous map $$\begin{aligned}
g \colon {\mathbb S}^k \times [0,1] \to X_{s_0}.
\end{aligned}$$ We immediately see that we also have $d_\infty(f,g) < {\varepsilon}/2$ and so by the definition of ${\varepsilon}$ we see that $f$ and $g$ are homotopic as we can use the affine line homotopy within the charts of $A^{\mathcal{X}}(0,1) \supset X_{s_0}$. Thus we conclude that $$(s_0 - {\varepsilon}, s_0 + {\varepsilon})
\subset I$$ and so $I$ is open.
Finally we need to show that $I$ is closed. We may suppose that $b \in (0,1)$ is such that $(b-\epsilon,b) \subset I$ for some $\epsilon > 0$. Suppose $f\colon {\mathbb S}^k \times [0,1] \to A^{\mathcal{X}}([b-\epsilon,b])$ is a continuous map. Then $f$ maps into $U_b \cup X_b$, where $X_b = \partial U_b$. Since $U_b$ is a manifold with boundary, there exists a number $\delta' > 0$ such that we can form an atlas for $U_b$ with Lebesgue number $\delta'$. The same argument as above gives a discrete set $D\subset S^k \times [0,1]$, with $B(D, \delta) = S^k\times [0,1]$, and a map $$\begin{aligned}
\hat g \colon D \to A^{\mathcal{X}}(b-\epsilon,\epsilon)\end{aligned}$$ such that $\hat g$ is uniformly close to $\hat f := f|_D$. That is, if $d(x,y) < \delta/4$, then $\|\hat g(x) - \hat g(y)\| < \delta'/2$. Since $U_b$ is a manifold with boundary we can extend $\hat g$ to a map $$\begin{aligned}
g \colon S^k\times [0,1] \to A^{\mathcal{X}}(b-\epsilon,b).\end{aligned}$$ Note that if $U_b$ were just an open set and not a manifold with boundary, it would not be clear that the image of $g$ would not intersect $X_b$. Since $U_b$ is a manifold with boundary, the extension can be done affinely in the charts near the boundary and therefore will not intersect the boundary.
We still have that $f$ and $g$ are uniformly close and so $f$ is homotopic to $g$. Thus we can conclude that $b \in I$, and the proof is complete.
By replacing $f$ and $g$ with homotopies we can deduce the following corollary.
\[coro:HomotopyOfHomotopies\] Let $\mathcal{X} = \{ X_t \}$ be a topological foliation. Then for any $k \in {\mathbb N}$, any $t_0 \in (0,1)$ and any homotopy $$\begin{aligned}
H \colonequals {\mathbb S}^k \times [0,1] \to A^\mathcal{X}(0,1)
\end{aligned}$$ there exists a homotopy $$\begin{aligned}
\tilde H \colonequals {\mathbb S}^k \times [0,1] \to X_{t_0}
\end{aligned}$$ such that $H$ and $\tilde H$ are homotopic in $A^\mathcal{X}(0,1)$.
\[coro:Homotopyspheres\] Let $\mathcal{X} = \{ X_t \}$ be a topological foliation with $(t_0,r_0)$ its break point. Then for any $k = 1, \ldots, n-2$ and $t \leq t_0$, $\pi_k(X_{t}) = 0$.
Fix $t < t_0$ and let $\alpha \colon {\mathbb S}^k \to X_t$, $k \in 1, \ldots, n-2$. Denote by $\iota \colon X_t \to B(0,r_0)$ the inclusion map. Since $\pi_k(B(0,r_0) \setminus \{ 0 \}) = 0$, there exists a homotopy $$\begin{aligned}
H \colonequals {\mathbb S}^k \times [0,1] \to B(0,r_0) \setminus \{ 0 \}
\end{aligned}$$ taking $\alpha$ to to the constant path $\alpha(0)$. By Proposition \[coro:HomotopyOfHomotopies\] there exists a homotopy $$\begin{aligned}
\tilde H \colonequals {\mathbb S}^k \times [0,1] \to X_{t}
\end{aligned}$$ such that $H$ and $\tilde H$ are homotopic; especially the homotopy $\tilde H$ takes $\alpha$ to the constant path as a homotopy in $X_t$. Thus $\pi_k(X_t) = 0$.
We are finally ready to prove the main result of this section.
As closed $(n-1)$-submanifolds of ${\mathbb R}^n$, the spaces $X_t$ are all orientable (see [@Hatcher Theorem 3.26]) so $H_{n-1}(X_t) = {\mathbb Z}$ for all $t \in (0,1)$. On the other hand by Corollary \[coro:Homotopyspheres\], if $t \le t_0$ for $(t_0,r_0)$ a break point of $\mathcal{X}$, then $\pi_k(X_t) = \pi_k({\mathbb S}^{n-1})$ for $k = 0, \ldots , n-2$. This, combined with the Hurewicz isomorphism theorem implies that the spaces $X_t$ with $t \leq t_0$ are homotopy spheres, and thus topological spheres by Proposition \[prop:WhiteheadHomology\].
Reverse implication {#sec:ReverseImplication}
===================
A crucial step in the proof of our main result, Theorem \[thm:maintheorem\] was to detect that the boundaries of sufficiently small normal domains are manifolds when the image of the branch has a PL-structure. Conversely the regularity of the boundaries of normal domains is strongly connected to the structure of both the branch set and the mapping in general. This was noted already by Martio and Srebro in dimension three.
We begin with a simple example demonstrating that we cannot hope the PL property of $f(B_f)$ to be equivalent to the property of boundaries of normal domains being manifolds.
\[example:CW-complex\] Let $w \colon {\mathbb R}^3 \to {\mathbb R}^3$ be the standard 2-to-1 winding around the $z$ axis. Denote by $h \colon {\mathbb R}^3 \to {\mathbb R}^3$ a homeomorphism that takes the $z$-axis to the graph of the function $t \mapsto (0,t, t^2 \cos(t{^{-1}}))$. Also define $f \colonequals (w \circ h) \circ w$. The image of the branch set of $f$ is equivalent to two copies of infinitely many connected circles converging to the origin and in particular it is not an $(n-2)$-dimensional simplicial complex. However, the mapping $f$ has the property that the boundaries of sufficiently small normal neighborhoods are manifolds.
The branch set and its image in the example above do have some regularity – even though $f(B_f)$ does not have a PL structure, it is a CW-complex. We remark that the regularity of $f(B_f)$ being a CW-complex is not enough for our main results. The quasiregular mappings constructed by Heinonen and Rickman in [@HeinonenRickman] and [@HeinonenRickman2] also have CW-complex branch sets but otherwise behave pathologically. In particular, the boundaries of normal domains are not manifolds in those examples.
Example \[example:CW-complex\] demonstrates that with just the assumption that the boundaries of normal domains are manifolds we cannot deduce that the mapping is locally a cone-type map. Instead we need to study a weaker notion of the mapping being locally a *path-type map*.
\[def:Path-TypeMap\] Let $\Omega \subset {\mathbb R}^n$ be a domain and let $f \colon \Omega \to {\mathbb R}^n$ be a branched cover. We say that $f$ is a *path-type mapping* at $x_0 \in \Omega$ or that $f$ *is a path of branched covers at $x_0 \in \Omega$* if there exists a radius $r_0 > 0$ and a path $t \mapsto f_t$ of branched covers $f_t \colon {\mathbb S}^{n-1} \to {\mathbb S}^{n-1}$ such that $$\begin{aligned}
f(x)
= \| x_0 - x \| f_{\| x_0 - x \|}\left(\frac{x}{\| x \|}\right)
\end{aligned}$$ for all $x \in B(x_0,r)$.
We use similar terminology also when $f$ and the mappings in the path are quasiregular mappings.
With the aid of the results in Section \[sec:HomotopiesOfHomotopies\] we can now prove the following proposition.
\[prop:ReverseTopDim\] Let $\Omega \subset {\mathbb R}^n$ be a domain and let $f \colon \Omega \to {\mathbb R}^n$ be a branched cover (or a quasiregular mapping.) Suppose that for any $x \in \Omega$ and for all $r < r_x$ small enough $\overline{U}(x,f,r)$ is a manifold with boundary. Then for every $x_0 \in \Omega$, $f$ is a path of branched covers (or quasiregular mappings) at $x_0$.
For any fixed $x_0 \in \Omega$ it is immediate to see that for small enough $r_0 > 0$ the boundaries $\partial U(x_0,f,r)$ with $r < r_0$ form a topological foliation since we assumed them to be manifolds. Thus by Theorem \[thm:SphericalityOfFoliation\] each $\partial U(x_0,f,r)$ is a topological sphere and we may set $f_t = f|_{\partial U(x_0,f,r)}$. After conjugating by homeomorphisms $(f_t)$ becomes a path of branched covers between spheres and thus a path-type map at $x_0$.
In higher dimensions it is again natural to ask about the structure and behavior of the boundaries of lower dimensional normal domains.
\[lemma:ReverseMidlevelLemma\] Let $\Omega \subset {\mathbb R}^n$ be a domain and let $f \colon \Omega \to {\mathbb R}^n$ be a branched cover. Suppose that for some $k\in \{2, \ldots, n-2\}$ all the $k+1$-dimensional normal domains are manifolds with boundary. Then for any $V \in \mathcal{U}_{k+1}$ the restriction $f|_{\partial V}$ is locally a path-type map.
The proof is identical to the proof of Proposition \[prop:ReverseTopDim\].
The above lemma has a natural corollary when more ‘levels’ of lower dimensional normal domains are manifolds. To state the corollary we define that a mapping $f \colon \Omega \to {\mathbb R}^n$ is a *$2$-repeated path* at $x_0 \in \Omega$ if $f$ is a at $x_0$ a path of path-type branched covers. Likewise a mapping $f$ is a *$k$-repeated path* if it is locally a path of $(k-1)$-repeated paths.
\[coro:RepeatedPath\] Let $\Omega \subset {\mathbb R}^n$ be a domain and let $f \colon \Omega \to {\mathbb R}^n$ be a branched cover. Suppose that for $k$ consecutive integers all the normal domains of those dimensions are manifolds with boundary. Then $f$ is an $(n-k)$-repeated path at $x_0$.
Note that for path-type maps, since branched covers are locally uniformly continuous, we necessarily have for any $t_0$ that $f_t \to f_{t_0}$ uniformly when $t \to t_0$. This in particular implies by basic degree theory (see [@Rickman-book]) that if $x_t \in B_{f_t}$ for all $t > t_0$ and $x_t \to x_0$ as $t \to t_0$, then $x_0 \in B_{f_{t_0}}$. Similarly we see that if $x_0 \in B_{f_{t_0}}$, then there must exist a continuous path $t \mapsto x_t \in B_{f_t}$ such that $x_t \to x_0$ as $t \to t_0$. So if $f$ is a path-type map at $x_0 \in B_{f}$, then $\dim_{\mathcal{T}}(B_f) \geq 1$, and a similar conclusion holds under the assumptions of Lemma \[lemma:ReverseMidlevelLemma\]. Moreover, we can deduce the following:
\[coro:DimensionEstimates\] Let $\Omega \subset {\mathbb R}^n$ be a domain and let $f \colon \Omega \to {\mathbb R}^n$ be a branched cover. Suppose that for some $k = 2, \ldots, n-2$ all the lower dimensional normal domains are manifolds with boundary. Then $\dim_{\mathcal{T}}(B_f) \geq (n-k)$.
Construction of a quasiregular mapping {#sec:EdenMap}
======================================
Our main results, Theorem \[thm:maintheorem\] and Corollary \[coro:GlobalMainTheorem\], can be used to produce examples of quasiregular mappings between manifolds. We give one such construction in this section.
We first note that the manifold $\mathbb{CP}^1$ is homeomorphic to $\widehat{\mathbb C}$ and $(\widehat{\mathbb C})^n$ is quasiregularly elliptic via e.g. the Alexander mapping, see [@Rickman-book]. Additionally, the composition of quasiregular mappings is still quasiregular. Thus in order to prove quasiregular ellpiticity of $\mathbb{CP}^n$, it suffices to construct a quasiregular mapping $(\mathbb{CP}^1)^n \to \mathbb{CP}^n$.
We first construct a branched covering $f \colon (\mathbb{CP}^1)^n \to \mathbb{CP}^n$. Consider the polynomial $$\begin{aligned}
p(u,v)=(z_1 u + w_1 v)\dots(z_n u + w_n v).
\end{aligned}$$ The coefficients of each term are homogeneous polynomials in $([z_i:w_i])_{i=1}^n$, so in particular the coefficients define a continuous map $f \colon (\mathbb{CP}^1)^n\to \mathbb{CP}^n$. By the definition of the mapping, $f$ is locally injective outside the set $$\begin{aligned}
B_f
= \{([z_1:w_1],\dots,[z_n:w_n]) : [z_i:w_i]=[z_j:w_j] \text{ for } i \ne j \}
\end{aligned}$$ and at each point $x \in B_f$, $f$ is $k$-to-$1$ for some $k = k(x) < \infty$. Thus $f$ is discrete. To see that $f$ is open, we note that away from $B_f$ the mapping is open by local injectivity and on the branch set $B_f$, $f$ is locally equivalent to a polynomial, and is thus an open map. Thus we conclude that $f$ is a branched cover.
Again by the definition of $f$, it is clear that $B_f$ has locally a simplicial structure. Since $f$ is locally a polynomial, we see that $f(B_f)$ is also locally topologically equivalent to an $(n-2)$ simplicial complex in $\mathbb{CP}^n$. Thus by Theorem \[thm:maintheorem\] $f$ is locally equivalent to a PL mapping and hence topologically equivalent to a quasiregular mapping. A similar argument as in Lemma \[lemma:localtoglobal\] implies that there exists PL structures on $(\mathbb{CP}^1)^n$ and $\mathbb{CP}^n$ so that $f$ is equivalent to a PL map. That is, there exists a map, $\tilde f \colon X \to Y$ such that $X$ and $Y$ are PL manifolds and the following diagram commutes: $$\begin{tikzcd}
(\mathbb{CP}^1)^n \arrow{r}{f} \arrow[swap]{d}{\phi} & \mathbb{CP}^n \arrow{d}{\psi} \\
X \arrow{r}{\tilde f} & Y
\end{tikzcd}$$ where the mappings $\phi$ and $\psi$ are homeomorphisms. The spaces $X$ and $Y$ have a PL structure and so they also have a quasiconformal structure. When the dimension is not $4$, that is, $n \ne 2$, by [@Sullivan] there is in fact a unique quasiconformal structure. Thus we can identify $X$ and $Y$ with $\times_{i=1}^n \mathbb{CP}^1$ and $\mathbb{CP}^n$, respectively. In the case $n=4$, a direct computation of the maps shows the same result. Thus we conclude that there exists a quasiregular mapping $$\begin{aligned}
\tilde f \colon (\mathbb{CP}^1)^n \to \mathbb{CP}^n
\end{aligned}$$ and we conclude this implies that $\mathbb{CP}^n$ is quasiregularly elliptic for all $n \geq 2$.
\[remark:HeinonenRickman\] In [@HeinonenSemmes Question 28] Heinonen and Semmes ask the following. (See also [@HeinonenRickman2].) *Let $f \colon {\mathbb S}^n \to {\mathbb S}^n$ be a branched cover. Does there exist homeomorphisms $h_1,h_2 \colon {\mathbb S}^3 \to {\mathbb S}^3$ such that $h_1 \circ f \circ h_2$ is a quasiregular mapping?* The methods in this section offer an advance in the understanding of the problem; indeed, the techniques here can be used to show that for $n \geq 4$ any branched cover $f \colon {\mathbb S}^n \to {\mathbb S}^n$ with $f B_f$ contained in a simplicial $(n-2)$-complex is, up to a conjugation by homeomorphisms, a quasiregular mapping.
**Acknowledgments.**
The project started at the 2017 Rolf Nevanlinna Colloquium, and the authors would like to thank the event and its organizers for an inspiring atmosphere.
A major part of the proofs were done while the first named author was visiting UCLA and he would like to extend his gratitude both to the university for their hospitality and to the second named author for accommodation.
Both of the authors extend their gratitude to Mario Bonk and Pekka Pankka for discussions on the topic and to Mike Miller who has patiently answered their questions about algebraic topology.
[[Sto]{}28]{}
Martina Aaltonen and Pekka Pankka. Local monodromy of branched covers and dimension of the branch set. , 42(1):487–496, 2017.
Bonk M. and Meyer D. . American Mathematical Soc., 2017.
Martin R. Bridson and Andr[é]{} Haefliger. , volume 319 of [ *Grundlehren der Mathematischen Wissenschaften \[Fundamental Principles of Mathematical Sciences\]*]{}. Springer-Verlag, Berlin, 1999.
James W. Cannon. The recognition problem: what is a topological manifold? , 84(5):832–866, 1978.
Alberto Candel and Lawrence Conlon. , volume 23 of [*Graduate Studies in Mathematics*]{}. American Mathematical Society, Providence, RI, 2000.
Philip T. Church and Erik Hemmingsen. Light open maps on [$n$]{}-manifolds. , 27:527–536, 1960.
Philip T. Church and J. G. Timourian. Differentiable maps with small critical set or critical set image. , 27(6):953–971, 1978.
Freedman M. The topology of four-dimensional manifolds. , 17(3):357–453, 1982.
Guillemin V. and Pollack A. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974.
Allen Hatcher. . Cambridge University Press, Cambridge, 2002.
Juha Heinonen. The branch set of a quasiregular mapping. In [*Proceedings of the [I]{}nternational [C]{}ongress of [M]{}athematicians, [V]{}ol. [II]{} ([B]{}eijing, 2002)*]{}, pages 691–700. Higher Ed. Press, Beijing, 2002.
Juha Heinonen and Seppo Rickman. Quasiregular maps [$\mathbf S^3\to \mathbf S^3$]{} with wild branch sets. , 37(1):1–24, 1998.
Juha Heinonen and Seppo Rickman. Geometric branched covers between generalized manifolds. , 113(3):465–529, 2002.
Juha Heinonen and Stephen Semmes. Thirty-three yes or no questions about mappings, measures, and metrics. , 1:1–12, 1997.
Vitali Kapovitch. Perelman’s stability theorem. In [*Surveys in differential geometry. [V]{}ol. [XI]{}*]{}, volume 11 of [*Surv. Differ. Geom.*]{}, pages 103–136. Int. Press, Somerville, MA, 2007.
Robion C. Kirby and Martin G. Scharlemann. Eight faces of the [P]{}oincaré homology [$3$]{}-sphere. In [*Geometric topology ([P]{}roc. [G]{}eorgia [T]{}opology [C]{}onf., [A]{}thens, [G]{}a., 1977)*]{}, pages 113–146. Academic Press, New York-London, 1979.
Kleiner B. and Lott J. Notes on Perelman’s papers , 12(5):2587–2855, Mathematical Sciences Publishers, 2008.
Rami Luisto. A characterization of [BLD]{}-mappings between metric spaces. , 27(3):2081–2097, 2017.
Rami Luisto and Pekka Pankka. Stoïlow’s theorem revisited. , 2017.
Olli Martio, Seppo Rickman, and Jussi Väisälä. Topological and metric properties of quasiregular mappings. , (488):31, 1971.
Olli Martio and Uri Srebro. On the local behavior of quasiregular maps and branched covering maps. , 36:198–212 (1980), 1979.
Grisha Perelman. Alexandrov spaces with curvatures bounded from below [II]{}. Preprint, 1991.
Yu. G. Reshetnyak. , volume 73 of [ *Translations of Mathematical Monographs*]{}. American Mathematical Society, Providence, RI, 1989. Translated from the Russian by H. H. McFaden.
Seppo Rickman. , volume 26 of [*Ergebnisse der Mathematik und ihrer Grenzgebiete (3) \[Results in Mathematics and Related Areas (3)\]*]{}. Springer-Verlag, Berlin, 1993.
Colin P. Rourke and Brian J. Sanderson. . Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69.
Smale S. . , World Scientific, 251–268, 2007.
Simion [Stoïlow]{}. , 45:347–382, 1928.
Dennis Sullivan. Hyperbolic geometry and homeomorphisms. In [*Geometric topology ([P]{}roc. [G]{}eorgia [T]{}opology [C]{}onf., [A]{}thens, [G]{}a., 1977)*]{}, pages 543–555. Academic Press, New York-London, 1979.
Jussi V[ä]{}is[ä]{}l[ä]{}. Discrete open mappings on manifolds. , 392:10, 1966.
[^1]: The first author was partially supported by a grant of the Finnish Academy of Science and Letters, the Academy of Finland (grant 288501 ‘*Geometry of subRiemannian groups*’) and by the European Research Council (ERC Starting Grant 713998 GeoMeG ‘*Geometry of Metric Groups*’).
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23.5cm -1cm
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
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[**April 23, 2000**]{}
3.cm
[**On the behavior of $F_2$\
and its logarithmic slopes** ]{}
1.cm
[.5em A.B. Kaidalov[^1] ]{}
[*ITEP, B. Cheremushkinskaya 25,\
117259 Moscow, Russia*]{}
[.5em C. Merino[^2] ]{}
[*Departamento de Física de Partículas\
Universidade de Santiago de Compostela\
15706 Santiago de Compostela, Spain*]{}
[.5em D. Pertermann[^3] ]{}
[*Physics Department, Univ-GH-Siegen\
D-57068 Siegen, Germany*]{}
.5cm [**Abstract** ]{}
> It is shown that the CKMT model for the nucleon structure function $F_2$, taken as the initial condition for the NLO evolution equations in perturbative QCD, provides a good description of the HERA data when presented in the form of the logarithmic slopes of $F_2$ vs $x$ and $Q^2$ (Caldwell-plot), in the whole available kinematic ranges. Also the results obtained for the behavior of the gluon component of a nucleon are presented.
The CKMT model
==============
The CKMT model [@ckmt] for the parametrization of the nucleon structure function $F_2$ is a theoretical model based on Regge theory which provides a consistent formulation of this function in the region of low $Q^2$, and describes the experimental data on $F_2$ in that region.
The CKMT model [@ckmt] proposes for the nucleon structure functions $$F_2(x,Q^2) = F_S(x,Q^2) + F_{NS}(x,Q^2),
\label{eq:eq1}$$ the following parametrization of its two terms in the region of small and moderate $Q^2$. For the singlet term, corresponding to the Pomeron contribution: $$F_S(x,Q^2) = A\cdot x^{-\Delta(Q^2)}\cdot(1-x)^{n(Q^2)+4}
\cdot\left({Q^2\over Q^2+a}\right)^{1+\Delta(Q^2)},
\label{eq:eq2}$$ where the $x$$\rightarrow$0 behavior is determined by an effective intercept of the Pomeron, $\Delta$, which takes into account Pomeron cuts and, therefore (and this is one of the main points of the model), it depends on $Q^2$. This dependence was parametrized [@ckmt] as : $$\Delta (Q^2) = \Delta_0\cdot\left(1+{\Delta_1\cdot Q^2
\over Q^2+\Delta_2}\right).
\label{eq:eq3}$$ Thus, for low values of $Q^2$ (large cuts), $\Delta$ is close to the effective value found from analysis of hadronic total cross-sections ($\Delta$$\sim$0.08), while for high values of $Q^2$ (small cuts), $\Delta$ takes the bare Pomeron value, $\Delta$$\sim$0.2-0.25. The parametrization for the non-singlet term, which corresponds to the secondary reggeon (f, $A_2$) contribution, is: $$F_{NS}(x,Q^2) = B\cdot x^{1-\alpha_R}\cdot(1-x)^{n(Q^2)}
\cdot\left({Q^2\over Q^2+b}\right)^{\alpha_R},
\label{eq:eq4}$$ where the $x$$\rightarrow$0 behavior is determined by the secondary reggeon intercept $\alpha_R$, which is in the range $\alpha_R$=0.4-0.5. The valence quark contribution can be separated into the contribution of the u ($B_u$) and d ($B_d$) valence quarks, the normalization condition for valence quarks fixes both contributions at one given value of $Q^2$ (we use $Q_v^2=2.GeV^2$ in our calculations). For both the singlet and the non-singlet terms, the behavior when $x$$\rightarrow$1 is controlled by $n(Q^2)$, with $n(Q^2)$ being $$n(Q^2) = {3\over2}\cdot\left(1+{ Q^2
\over Q^2+c}\right),
\label{eq:eq5}$$ so that, for $Q^2$=0, the valence quark distributions have the same power, given by Regge intercepts, as in the Quark Gluon String Model [@kaidalov] or in the Dual Parton Model [@dpm], $n$(0)=$\alpha_R$(0)$-$$\alpha_N$(0)$\sim$ 3/2, and the behavior of $n(Q^2)$ for large $Q^2$ is taken to coincide with dimensional counting rules.
The total cross-section for real ($Q^2$=0) photons can be obtained from the structure function $F_2$ using the following relation: $$\sigma^{tot}_{\gamma p}(\nu) = \left[{4\pi^2\alpha_{EM}\over Q^2}
\cdot F_2(x,Q^2)\right]_{Q^2=0}.
\label{eq:eq6}$$ The proper $F_2(x,Q^2)$$\sim$$Q^2$ behavior when $Q^2$$\rightarrow$0, is fulfilled in the model due to the last factors in equations \[eq:eq2\] and \[eq:eq4\]. Thus, the $\sigma^{tot}_{\gamma p}(\nu)$ has the following form in the CKMT model: $$\sigma^{tot}_{\gamma p}(\nu) = 4\pi^2\alpha_{EM}
\cdot\left(A\cdot a^{-1-\Delta_0}\cdot(2m\nu)^{\Delta_0}
+(B_u+B_d)\cdot b^{-\alpha_R}\cdot(2m\nu)^{\alpha_R-1}\right).
\label{eq:eq6a}$$
The parameters were determined [@ckmt] from a joint fit of the $\sigma^{tot}_{\gamma p}$ data and the NMC data [@nmc] on the proton structure function in the region $1 GeV^2 \leq Q^2 \leq 5 GeV^2$, and a very good description of the experimantal data available was obtained.
The next step in this approach is to introduce the QCD evolution in the partonic distributions of the CKMT model and thus to determine the structure functions at higher values of $Q^2$. For this, the evolution equation in two loops in the $\overline{\mbox{MS}}$ scheme with $\Lambda=200.MeV$ was used [@ckmt].
The results obtained by taking into account the QCD evolution in this way are [@ckmt] in a very good agreement with the experimental data on $F_2(x,Q^2)$ at high values of $Q^2$.
When the publication of the data [@h1; @zeus] on $F_2$ from HERA at low and moderate $Q^2$ provided the opportunity to include in the fit of the parameters of the model experimental points from the kinematical region where the CKMT parametrization should give a good description without need of any perturbative QCD evolution, one proceeded [@newpaper] to add these new data on $F_2$ from H1 and ZEUS at low and moderate $Q^2$, to those from NMC [@nmc] and E665 [@e665] collaborations, and to data [@cross] on cross-sections for real photoproduction, into a global fit which allowed the test of the model in wider regions of $x$ and $Q^2$. One took as initial condition for the values of the different parameters those obtained in the previous fit [@ckmt], and although the quality of the fit is not very sensitive to small changes in the values of the parameters, the best fit has been found for the values of the parameters given in Table\[tab:tab1\].
1.cm 1.cm
The quality of the description provided by the CKMT model of all the experimental data on $\sigma^{tot}_{\gamma p}$ and $F_2$, and, in particular, of the the new experimental data from HERA is very high, with a value of $\chi^2/d.o.f.$ for the global fit, $\chi^2/d.o.f.$=106.95/167, where the statistical and systematic errors have been treated in quadrature, and where the relative normalization among all the experimental data sets has been taken equal to 1.
Thus, by taking into account the general features of the CKMT model described above, we use the CKMT model to describe the experimental data in the region of low $Q^2$ ($0< Q^2< Q^2_0$), and then we take this parametrization as the initial condition at $Q^2_0$ to be used in the NLO QCD evolution equation to obtain $F_2$ at values of $Q^2$ higher than $Q^2_0$. In order to determine the distributions of gluons in a nucleon the CKMT model assumes [@ckmt] that the only difference between distributions of sea-quarks and gluons is in the $x\rightarrow 1$ behavior. Following [@fmartin] we write it in the form $$xg(x,Q^2)=Gx\bar{q}(x,Q^2)/(1-x),
\label{neweq1}$$ where $x\bar{q}(x,Q^2)$ is proportional to the expression in equation \[eq:eq2\]. The constant $G$ is determined from the energy-momentum conservation sum rule.
We have performed our calculations for two different values of $Q_0^2=2.GeV^2$ and $Q_0^2=4.GeV^2$. We also show our results in the shape of both the $dF_2/dlnQ^2$ and the $dlnF2/dln(1/x)$ slopes in order to compare with the experimental data when given in the so-called Caldwell-plot. This approach provides a smooth transition from the region of small $Q^2$, which is governed by the physics of Regge theory, to a region of large $Q^2$, where the effects of QCD-evolution are important.
The way we proceed to calculate $F_2$, and its logarithmic derivatives $dF_2/dlnQ^2$, and $dlnF2/dln(1/x)$, is the following (see Appendix A for all the technical details on how the NLO QCD evolution has been performed):
- In the region $0< Q^2\le Q^2_0$ we use the pure CKMT model for $F_2$.
- For $Q^2_0< Q^2\le charm$ $threshold$ [@grv], we make the QCD evolution of $F_2$ at NLO in the $\overline{\mbox{MS}}$ scheme for a number of flavours $n_f=3$, and we take as the starting parametrization that given by the CKMT model. For $Q^2_0$ we have used in this calculation two different values: $Q^2_0=2.GeV^2$, and $Q^2_0=4.GeV^2$.
- When $charm$ $threshold < Q^2\le \bar{Q}^2=50. GeV^2$, also the QCD evolution of $F_2$ is implemented at NLO in the $\overline{\mbox{MS}}$ scheme for a number of flavours $n_f=3$, using the parton distribution functions for the $u,d,s$ quarks, and by including the charm contribution via photon-gluon fusion.
- For values of $Q^2> \bar{Q}^2$, QCD evolution is computed at NLO in the $\overline{\mbox{MS}}$ scheme, but now with a number of flavours $n_f=4$, and by using the parton distribution functions for the $u,d,s$, and $c$ quarks.
One has to note that in the treatment of the charm contribution we have followed reference [@haakman].
Results
========
The results we have obtained are presented in figures 1 to 9.
Figure 1 shows $F_2(x,Q^2)$ as a function of $x$ for several values of $Q^2$, from $Q^2=0.6 GeV^2$ to $Q^2=17. GeV^2$. The dotted lines correspond to the pure CKMT model without any perturbative evolution, while the full lines run for the evoluted CKMT parametrization. When for a given value of $Q^2$ two full lines are depicted, the bold (solid) one has been obtained by taking the starting point for the QCD evolution as $Q_0^2=2. GeV^2$ ($Q_0^2=4. GeV^2$). Experimental points in this figure are from E665 [@e665], H1 [@H1caldwell], and ZEUS [@caldw] collaborations.
In Figures 2.a and 2.b, we present the comparison of the pure CKMT parametrization of $F_2$ with the low $Q^2$ data of E665, ZEUS-BPC95, and ZEUS-BPT97, as compiled in [@amelung] and [@zhokin]. One sees that the agreement between the CKMT model and the experimental data in this region of low $Q^2$ is good.
In Figure 3 (Caldwell-plot), the slope $dF_2/dlnQ^2$ is shown as a function of $x$, and compared with the $a+blnQ^2$ fit to the ZEUS $F_2$ data in bins of $x$. This plot was considered as an evidence for a transition from hard to soft regime of QCD in the region of $Q^2\sim 5.GeV^2$ (see for example [@amueller]). This question has been studied theoretically in references [@levin; @jenkovszky]. Figure 3 shows that the CKMT model is in a good agreement with experimental points in the whole region of $x$ and $Q^2$. One problem with the presentation of the data in Figure 3 is a strong correlation between $x$ and $Q^2$ values for the data points. It follows from the formulas of CKMT model for $dF_2(x,Q^2)/dlnQ^2$ given in Appendix B that for a fixed value of $Q^2$ this quantity monotonically increases as $x\rightarrow 0$. The existence of a maximun of $dF_2(x,Q^2)/dlnQ^2$ in Figure 3 is related to the correlation between $Q^2$ and $x$ in the region of small $x$ (or $Q^2$). The same conclusion was achieved in reference [@levin], and recently confirmed by experimental data [@zhokin].
Figures 4 and 5 show the slope $dlnF2/dln(1/x)$ as a function of $Q^2$ compared to the fits $F_2=Ax^{-\Delta_{eff}}$ of the the ZEUS [@caldw] and H1 [@H1caldwell] data, respectively. In Figure 4, as the $x$ range of the BPC95 data is restricted, also the E665 [@e665] data were included in [@caldw], and are now also taken into account. This slope is sometimes interpreted as the $\Delta_{eff}$ of the Pomeron exchange, $\Delta_{eff}=dlnF2/dln(1/x)$. Let us note that in our approach $\Delta_{eff}$ for $Q^2 > Q^2_0$ can not be interpreted as an effective Pomeron intercept, because the QCD evolution leads to a substantial increase of $\Delta_{eff}$ as $Q^2$ increases. On the other hand this effect should decrease as $x\rightarrow 0$.
In the experimental fits, each $Q^2$ bin corresponds to a average value of $x$, $<x>$, calculated from the mean value of $ln(1/x)$ weighted by the statistical errors of the corresponding $F_2$ values in that bin. Even though we can proceed as in the experimental fits, and we get a very good agreement with the data, since the estimation of $<x>$ is in some sense artificial and arbitrary, and it introduces unphysical wiggles when drawing one full line connecting the different bins, we made for all the $Q^2$ bins in this figures the choice of the smallest $x$ in the data, instead of considering a different $<x>$ for each $Q^2$. This choice is based on the fact that the ansatz $\Delta_{eff}=dlnF2/dln(1/x)$ is actually valid for small $x$, and it results in a smooth curve except for the jump in the region around $Q^2\sim 50.GeV^2$, where the evolution procedure changes (again, see Appendix A for more details).
Since the structure function $F_2$ in the region of low $x$ is determined at large extent by the gluon component, we present our prediction for the behavior of this gluon component. Thus, Figure 6 shows the gluon density distribution as a function of $Q^2$ calculated by performing the NLO QCD evolution of the CKMT model, and its comparison with the H1 Collaboration data in reference [@adloff], and Figure 7 represents the gluon densities at $\mu^2=25.GeV^2$ as a function of $x$ calculated by evoluting the CKMT model at NLO in the QCD evolution, and compared to those determined from H1 DIS and photoproduction data. Experimental data on $D^*$ meson cross-section measurements are from references [@zhokin; @adloff]. Figure 8 shows the behavior of $xg(x,\mu^2)$ at $\mu^2=200~GeV^2$ as a function of $Q^2$ to be compared with the H1-dijets results [@zhokin; @wobisch]. Finally, Figure 9 shows the prediction of the CKMT model for $xg(x,Q^2)$ as a function of $x$ at the values of $Q^2$ measured both by H1 and ZEUS collaborations.
A satisfactory agreement with the experiment is obtained in the whole ranges of $x$ and $Q^2$ where experimental data are available, showing that the experimental behavior of $F_2$, its logarithmic slopes, and its gluon component can be described by using as initial condition for the QCD evolution equation a model of $F_2$ where the shadowing effects which are important at low values of $Q^2$ are included, like the CKMT model.
Conclusions
===========
The CKMT model for the parametrization of the nucleon structure functions provides a very good description of all the available experimental data on $F_2(x,Q^2)$ at low and moderate $Q^2$, including the recent small-$x$ HERA points.
An important ingredient of the model is the dependence of an effective intercept of the Pomeron on $Q^2$. It has been shown recently [@elenagf] that such a behavior is naturally reproduced in a broad class of models based on reggeon calculus, which describes simultaneously the structure function $F_2$ and the diffractive production by virtual photons.
Use of the CKMT model as the initial condition for the QCD-evolution equations in the region of $Q^2=2.\div5.GeV^2$ leads to a good description of all available data in a broad region of $Q^2$, including the logarithmic slopes of the structure function $F_2(x,Q^2)$, $dF_2(x,Q^2)/dlnQ^2$ and $dlnF_2(x,Q^2)/dln(1/x)$. Thus an unified description of the data on $F_2$ for all values of $Q^2$ is achieved.
A.B.K. acknowledges support of a NATO grant OUTR.LG971390 and the RFBR grant 98-02-17463, and C.M. was partially supported by CICYT (AEN99-0589-C02-02).
[99]{} A. Capella, A.B. Kaidalov, C. Merino, and J. Tran Than Van, .
A.B. Kaidalov, and .\
A.B. Kaidalov and K.A. Ter-Martirosyan, .
A. Capella, U. Sukhatme, C.-I. Tan, and J. Tran Than Van, .
P. Amaudruz [*et al*]{} (New Muon Collaboration), .
C. Adloff [*et al*]{} (H1 Collaboration), .
J. Breitweg [*et al*]{} (ZEUS Collaboration), .
A.B. Kaidalov and C. Merino, [*hep-ph/9806367*]{} and [*Eur. Phys. J. C*]{} [**10**]{} 153 (1999).
M.R. Adams [*et al*]{} (E665 Collaboration), FERMILAB-Pub 1995/396, and .
D.O. Caldwell [*et al*]{}, .\
M. Derrick [*et al*]{} (ZEUS Collaboration), , and .\
S. Aid [*et al*]{} (H1 Collaboration), .
F. Martin, .
M. Glück, E. Reya, and A. Vogt, .
L.P.A. Haakman, A.B. Kaidalov, and J.H. Koch, [*hep-ph/9704203*]{}, and [*Eur. Phys. J. C*]{} [**1**]{}, 547 (1999).
S. Aid [*et al*]{} (H1 Collaboration), [*DESY-96-039*]{}, [*hep-ex/9603004*]{}, contribution to the Proceedings of the XXXI Rencontres de Moriond: QCD and High Energy Hadronic Interactions, March 1996, Les Arcs (France), edited by J. Tran Thanh Van, Editions Frontières, Gif-sur-Yvette (France), 1996 M93, pages 349-355, and .
A. Caldwell, DESY Theory Workshop, DESY, Hamburg (Germany), October 1997.\
J. Breitweg [*et al*]{} (ZEUS Collaboration), [*DESY-98-121*]{}, [*hep-ex/9809005*]{}, and [*Eur. Phys. J. C*]{} [**7**]{}, 609 (1999).
C. Amelung (ZEUS Collaboration), contribution to the Proceedings of the 7th International Workshop on Deep Inelastic Scattering and QCD (DIS99), DESY Zeuthen, Germany, April 19-23 1999, edited by by J. Blümlein and T. Riemann, .
A. Zhokin, on behalf of the H1 and ZEUS collaborations, contribution to the Proceedings of the XXIX International Symposium on Multiparticle Dynamics (ISMD99), Brown University, Providence, RI 02912, USA, August 9-13 1999, edited by I. Sarcevic and C.-I. Tang, to be published in World Scientific.
A.H. Mueller, contribution to the Proceedings of the 6th International Workshop on Deep Inelastic Scattering and QCD (DIS98), Brussels, Belgium, April 4-8 1998, edited by Gh. Coremans and R. Roosen, World Scientific, pages 3-19.
E. Gotsman, E. Levin, and U. Maor, , and .
P. Desgrolard, L.L. Jenkovszky, A. Lengyel, and F. Paccanoni, [*hep-ph/9903397*]{}, and .
C. Adloff [*et al*]{} (H1 Collaboration), .
M. Wobisch (H1 Collaboration), Proceedings of the 7th International Workshop on Deep Inelastic Scattering and QCD (DIS99), DESY Zeuthen, Germany, April 19-23 1999, edited by by J. Blümlein and T. Riemann, .
A. Capella, E.G. Ferreiro, A.B. Kaidalov, and C.A. Salgado, to be published.\
A.B. Kaidalov, contribution to the Proceedings of the XXIX International Symposium on Multiparticle Dynamics (ISMD99), Brown University, Providence, RI 02912, USA, August 9-13 1999, edited by I. Sarcevic and C.-I. Tang, to be published in World Scientific.
W. A. Bardeen, A. J. Buras, D. W. Duke and T. Muta, Phys. Rev. [**D18**]{} (1978) 3998.
M. Glück, E. Reya, and A. Vogt, .
For the reader convenience we present here some technical remarks concerning the NLO QCD calculation of $F_2(x,Q^2)$.
For sufficiently large $Q^2 >1~GeV^2$, the structure function $F_2(x,Q^2)$ can be expressed by perturbative parton distributions. In leading order (LO) perturbation theory, the expression is given as $$\label{apx01}
\frac{1}{x}F_2(x,Q^2) = x \sum_q e_q^2 \{ q(x,Q^2) + \bar{q}(x,Q^2) \},$$ where $q$ and $\bar{q}$ denote the quark and anti-quark distribution functions, $e_q^2$ the square of the quark electric charge, and the sum runs over all quark flavors included [@grv]. On the other hand, with $F_2(x,Q^2)$ given in eqs. (\[eq:eq1\]-\[eq:eq5\]), and making reasonable assumptions concerning the flavor structure of the QCD-sea, one can extract from $F_2(x,Q^2)$ the different parton distribution functions, including that of the gluon component [@ckmt]. Generally, the calculation of $F_2(x,Q^2)$ at $Q^2 \gg 1~GeV^2$ requires a $Q^2$-evolution à la DGLAP [@DGLAP00]. The procedure consists in the solution of the LO-DGLAP equations for the parton distribution functions using reasonable initial distributions at a starting value $Q^2=Q_0^2$ ($1~GeV^2 < Q_0^2 < 5~GeV^2$). Using eq.(\[apx01\]), the resulting quark distributions at $Q^2$ can be recombined to $F_2$ at this virtuality.
By the evolution of the CKMT-model we mean the application of this procedure to the model discussed in this paper. As mentioned above, the CKMT-model of $F_2(x,Q^2)$ is valid within $0 \le Q^2 < 5~GeV^2$. Due to the good aggreement with experimental data the parton distributions extracted from $F_2^{CKMT}$ at a $Q_0^2$ in the range given above seem to be reasonable initial distributions for an evolution to higher $Q^2$.
In next to leading order (NLO), the relation between $F_2(x,Q^2)$ and the parton distribution functions is more complicated and depends on the renormalization scheme. The calculations presented here are performed in the $\rm \overline{MS}$-scheme [@Bardeen:1978yd]. In this context, the structure function is given by [@grv] as $$\label{apx02}
\frac{1}{x}F_2(x,Q^2) = \sum_q e_q^2
\left\{ q(x,Q^2)+\bar{q}(x,Q^2) + \frac{\alpha_s(Q^2)}{2\pi}
\left[ C_{q,2}*(q+\bar{q}) + 2\cdot C_{g,2}*g \right]
\right\},$$ where $q,\bar{q}$ and $g$ are the NLO quark, anti-quark and gluon distribution functions, respectively. $\alpha_s$ denotes the strong coupling constant in NLO. The convolutions $C*q$ and $C*g$ are defined as $$\protect\label{apx03}
C*q = \int_x^1 \frac{dy}{y} C \left( \frac{x}{y} \right) q(y,Q^2).$$ The Wilson coefficients $C_{q,g,2}(z)$ are given by $$\begin{aligned}
\protect\label{apx04}
C_{q,2}(z) = \frac{4}{3}\left[ \frac{1+z^2}{1-z}\left( \ln\frac{1-z}{z}
- \frac{3}{4} \right) + \frac{1}{4}(9+5z) \right]_+, &&
\nonumber \\
C_{g,2}(z) = \frac{1}{2}\left[ (z^2 + (1-z)^2)\ln\frac{1-z}{z}
- 1 + 8z(1-z) \right].\end{aligned}$$ Here, the integral over a $[\cdot]_+$-distribution is defined as described in [@grv2]: $$\begin{aligned}
\protect\label{apx05}
C_+*q &=& \int_x^1 \frac{dy}{y} C \left( \frac{x}{y} \right)_+ q(y,Q^2)
\nonumber \\
&=& \int_x^1 \frac{dy}{y} C \left( \frac{x}{y} \right)
\left[ q(y,Q^2) - \frac{x}{y} q(x,Q^2)\right]
- q(x,Q^2) \int_0^x dy C(y).\end{aligned}$$
There are alternative renormalization schemes as, for instance, the $DIS$-scheme [@grv]. Here, the form of eq. (\[apx01\]) is kept for NLO also, i.e. $$\label{apx06}
\frac{1}{x}F_2(x,Q^2) = x \sum_q e_q^2 \{ q_{DIS}(x,Q^2) + \bar{q}_{DIS}(x,Q^2) \}.$$ The relation between the $\overline{MS}$- and the $DIS$-distributions is given by $$\begin{aligned}
\protect\label{apx07}
\stackrel{(-)}{q}_{DIS}(x,Q^2) &=& \stackrel{(-)}{q}(x,Q^2) +
\frac{\alpha_s(Q^2)}{2\pi}\left[ C_{q,2}*\stackrel{(-)}{q} +
C_{g,2}*g \right] +
O(\alpha_s^2),
\\
g_{DIS}(x,Q^2) &=& g(x,Q^2) - \frac{\alpha_s(Q^2)}{2\pi}
\left[\sum_q C_{q,2}*(q+\bar{q}) + 2f\cdot C_{g,2}*g \right] +
O(\alpha_s^2). \nonumber\end{aligned}$$ The parameter $f$ denotes the number of active flavors in the sea.
Our procedure to extract the parton-distributions from $F_2^{CKMT}$ is based on the LO-formula eq. (\[apx01\]). Therefore, in NLO, we extract the $DIS$-distributions. Now, the task is to calculate the $\overline{MS}$-distributions at $Q^2=Q_0^2$ . This can be done using a first order approximation in $\alpha_s(Q^2)/2\pi$: $$\begin{aligned}
\protect\label{apx08}
\stackrel{(-)}{q}(x,Q_0^2) &\approx& \stackrel{(-)}{q}_{DIS}(x,Q_0^2) -
\frac{\alpha_s(Q_0^2)}{2\pi}\left[ C_{q,2}*\stackrel{(-)}{q}_{DIS} +
C_{g,2}*g_{DIS} \right]
\\
g(x,Q_0^2) &\approx& g_{DIS}(x,Q_0^2) + \frac{\alpha_s(Q_0^2)}{2\pi}
\left[\sum_q C_{q,2}*(q_{DIS}+\bar{q}_{DIS}) +
2f\cdot C_{g,2}*g_{DIS} \right].
\nonumber\end{aligned}$$
In summary, the $Q^2$-evolution of $F_2^{CKMT}$ works as follows:
1. One chooses an appropriate value $Q^2=Q_0^2 > 1~GeV^2$ as a starting point for the evolution. These are $Q_0^2= 2~GeV^2$ or $Q_0^2= 4~GeV^2$ in our calculations.
2. At $Q^2=Q_0^2$, one extracts the NLO parton distribution functions from $F_2^{CKMT}$ . The relation between these parton distributions and the structure function is given by eq.(\[apx06\]) which is formally the same as eq.(\[apx01\]) in LO. So the resulting parton distributions are the $DIS$-functions, i.e. $q_{DIS}(x,Q_0^2),~\bar{q}_{DIS}(x,Q_0^2)$ and $g_{DIS}(x,Q_0^2)$.
3. Using eq.(\[apx08\]) one calculates the $\overline{MS}$-distributions $q(x,Q_0^2),~\bar{q}(x,Q_0^2)$ and $g(x,Q_0^2)$.
4. These $\overline{MS}$-functions serve as initial distributions in a numerical procedure to solve the NLO-DGLAP-equations in the $\overline{MS}$-scheme for a certain value $Q^2 > Q_0^2$. The result are the evoluted $\overline{MS}$-parton distributions $q(x,Q^2),~\bar{q}(x,Q^2)$ and $g(x,Q^2)$.
5. Finally, using eq.(\[apx02\]), the structure function $F_2^{CKMT}(x,Q^2)$ can be recalculated.
The charm production is of particular interest. Following refs. [@grv; @haakman], the assumption of a “massless” charm quark produced above the threshold $Q^2_c=4m_c^2$ ($m_c^2$ – charm quark mass) via the usual DGLAP-evolution is not realistic. This procedure is useful in the range of high $Q^2 \gg Q_c^2$ only. In the intermediate region $Q_c^2 < Q^2 < \bar{Q^2}=50~GeV^2$, the charm is treated via a photon-gluon fusion process. The corresponding contribution to the structure function is defined as $$\label{apx20}
\frac{1}{x} F_2^c(x,Q^2,m_c^2) ~=~
2e_c^2~\frac{\alpha_s(\mu^2)}{2\pi} \int_{ax}^1\frac{dy}{y}~\cdot
C_{g,2}^c \left( \frac{x}{y},\frac{m_c^2}{Q^2} \right) \cdot g(y,\mu^2),$$ where $\mu^2=4m_c^2$, $a=1~+~4m_c^2/Q^2$, and, the coefficient $C_{g,2}^c (Z,R)$ is given by $$\begin{aligned}
\label{apx21}
C_{g,2}^c(Z,R) & = &
\frac{1}{2}
\Big\{ [Z^2 + (1-Z)^2 + 4ZR(1-3Z) -8Z^2R^2]
\ln{\frac{1+V}{1-V}} \nonumber \\
&+&V [-1 + 8Z(1-Z) - 4ZR(1-Z)] \Big\},
\end{aligned}$$ with $V^2=1-4RZ/(1-Z)$. So, $F_2$ in total is given by eq.(\[apx02\]) where the sum runs over $q=u,d,s$ plus eq.(\[apx20\]). The contributions of the botom and top quarks are neglected here. Precisely, the charm threshold is defined as discussed in refs. [@grv; @haakman], $$\label{apx23}
W^2 \equiv Q^2 (1/x~-~1) \ge Q_c^2 = 4m_c^2.$$
The $Q^2$-dependence of $F_2$ can be summarized as follows:
i) $Q^2 < Q_0^2$:
: In the low $Q^2$ region, $F_2$ is calculated as given in the pure CKMT-model, eqs.(\[eq:eq1\]–\[eq:eq5\]).
ii) $Q_0^2 < Q^2 < Q_c^2$:
: Below the charm threshold $F_2$ is calculated using eq.(\[apx02\]) from NLO QCD evoluted parton distributions in the $\overline{MS}$-scheme. The number of flavors is $f=3$ ($u,d,s$).
iii) $Q_c^2 < Q^2 < \bar{Q}=50~GeV^2$:
: Above the charm threshold $F_2$ is determined from eqs.(\[apx02\]) and (\[apx20\]). Note that the number of flavors active in the evolution is again $f=3$ ($u,d,s$). However, $f=4$ after the charm is produced. This is important for the calculation of $\alpha_s$.
iv) $Q^2 > \bar{Q}^2=50~GeV^2$:
: In the high $Q^2$-region, $F_2$ is given by eq.(\[apx02\]). The charm is produced as “massless” quark in the evolution process. Generally, the number of flavors is $f=4$.
The threshold $\bar{Q}^2$ where the charm production in the evolution process is more important than the photon-gluon fusion is discussed in detail in [@haakman]. The value of $50~GeV^2$ is chosen to guarantee a transition as smooth as possible. This method works better for $x \rightarrow 0$ than for $x \rightarrow 1$. This explains the small wiggles in some of the figures at $Q^2=50~GeV^2$.
For low $Q^2$, the “pure” CKMT-model is used, i.e. the one defined in eqs.(\[eq:eq1\]-\[eq:eq5\]). Here, the calculation of the slopes as $dF_2(x,Q^2)/d\ln{Q^2}$ and $d\ln{F_2}(x,Q^2)/d\ln{(1/x)}=\Delta_{eff}$ is straightforward. Considering $x$ and $Q^2$ as independent variables one gets $$\begin{array}{rcl}
\frac{\textstyle dF_2(x,Q^2)}{\textstyle dlnQ^2} =& F_S(x,Q^2) &
[\frac{\Delta_2}{Q^2+\Delta_2}
\left(\Delta(Q^2)-\Delta_0\right)
ln\frac{Q^2}{x(Q^2+a)}\\
&&
+\frac{c}{Q^2+c}
\left(n(Q^2)-\frac{3}{2}\right)ln(1-x)
+\frac{a~\left(1+\Delta(Q^2)\right)}{Q^2+a}]\\
+ &F_{NS}(x,Q^2)&
[\frac{c}{Q^2+c}\left(n(Q^2)-\frac{3}{2}\right)ln(1-x)
+\frac{b~\alpha_R(0)}{Q^2+b}],\\ & &
\end{array}
\label{apx09}$$ which in the limit $Q^2\rightarrow 0$ takes the form $$\begin{array}{rcl}
\frac{\textstyle dF_2(x,Q^2)}{\textstyle dlnQ^2} & \sim &
\left(1+\Delta_0\right) F_S(x,Q^2) + \alpha_R(0) F_{NS}(x,Q^2).
\end{array}
\label{apx10}$$
Also, if one considers the case when W is fixed one can take $x\sim C\cdot Q^2$, and then, up to constant factors, one gets: $$\begin{array}{rcl}
\frac{\textstyle dF_2(x,Q^2)}{\textstyle dlnQ^2} =& F_S(x,Q^2) &
[-\frac{\Delta_2}{Q^2+\Delta_2}\left(\Delta(Q^2)-\Delta_0\right)
ln(Q^2+a)\\
&&
-\Delta(Q^2)+\frac{c}{Q^2+c}\left(n(Q^2)-\frac{3}{2}\right)ln(1-Q^2)\\
&&
-\frac{Q^2n(Q^2)}{1-Q^2}+\frac{a~(1+\Delta(Q^2))}{Q^2+a}]\\
+ & F_{NS}(x,Q^2) &
[\frac{c}{Q^2+c}(n(Q^2)-\frac{3}{2})ln(1-Q^2)\\
&&
+\frac{b~\alpha_R(0)}{Q^2+b}+(1-\alpha_R(0))-\frac{Q^2n(Q^2)}{1-Q^2}].\\
&&
\end{array}
\label{apx11}$$ Now, if one takes W fixed with $Q^2\sim x\rightarrow 0$, one can easily see that this equation simply reduces to: $$\frac{dF_2(x,Q^2)}{dlnQ^2} \sim F_2(x,Q^2).
\label{apx12}$$
The calculations presented in the paper are based on the assumption of independent $x$ and $Q^2$, i.e. eqs.(\[apx09\], \[apx10\]). In this context, the effective x-slope $\Delta_{eff}=d\ln{F_2(x,Q^2)}/d\ln{(1/x)}$ is given by $$\begin{array}{rl}
F_2(x,Q^2) \cdot
\frac{\textstyle d\ln{F_2(x,Q^2)}}{\textstyle d\ln{(1/x)}} =&
[\Delta(Q^2) + \frac{x}{1-x} (n(Q^2)+4)] \cdot F_S\\
+&[\alpha_R(0) - 1 + \frac{x}{1-x} n(Q^2) + \frac{x~B_d}{B_u + B_d(1-x)}]
\cdot F_{NS}.
\end{array}
\label{apx13}$$
For $Q^2 > Q_0^2$, the slopes have to be calculated from the evoluted structure function. Here, there are two fundamental procedures, the pure numerical and the mainly analytical calculations. The pure numerical procedure is very simple: $$\begin{array}{rl}
\frac{\textstyle dF_2(x,Q^2)}{\textstyle d\ln{Q^2}} \approx &
Q^2 \cdot
\frac{1}{2\delta Q^2} \cdot [F_2(x,Q^2 + \delta Q^2) - F_2(x,Q^2 - \delta Q^2)],
\end{array}
\label{apx14}$$ $$\begin{array}{rl}
\frac{\textstyle d\ln{F_2(x,Q^2)}}{\textstyle d\ln{(1/x)}} \approx &
(-1)\cdot \frac{\textstyle x}{\textstyle F_2(x,Q^2)} \cdot
\frac{1}{2\delta x} \cdot [F_2(x + \delta x,Q^2) - F_2(x - \delta x,Q^2)].
\end{array}
\label{apx15}$$ $F_2(x,Q^2)$ is the evoluted structure function whereas $\delta Q^2$ and $\delta x$ denote the corresponding increments which are fixed to be $10^{-3}\cdot Q^2$ or $10^{-3}\cdot x$ in the calculations presented. For low $Q^2$, we have checked this procedure comparing the values of eqs. (\[apx14\],\[apx15\]) with those calculated using eqs. (\[apx09\],\[apx13\]). The agreement is very good which, in some cases, is demonstrated by identical numbers. This numerical procedure is the method used to determine the effective x-slope $\Delta_{eff}=d\ln{F_2(x,Q^2)}/d\ln{(1/x)}$ of the evoluted structure function. In the case of $dF_2(x,Q^2)/d\ln{Q^2}$ there is, in addition, the way of mainly analytical calculations. If the parton distribution functions are known their derivatives concerning $Q^2$ can be calculated from the DGLAP-equations. Instead of $Q^2$ the parameter $$\begin{array}{rcl}
S = \ln \left\{ \frac{T}{T_o} \right\}, &
T = \ln(Q^2/\Lambda_{QCD}^2), &
T_o = \ln(Q_0^2/\Lambda_{QCD}^2)
\end{array}
\label{apx16}$$ is often used in perturbation theory. In terms of $S$ $$\frac{dF_2}{d\ln{Q^2}} =
\frac{1}{\ln(Q^2/\Lambda_{QCD}^2)} \frac{dF_2}{dS},
\label{apx17}$$ and in the $\overline{MS}$-scheme $$\begin{aligned}
\protect\label{apx18}
\frac{1}{x} \frac{dF_2(x,S)}{dS} &=& \sum_q e_q^2
\Bigg\{ \frac{dq(x,S)}{dS}+\frac{d\bar{q}(x,S)}{dS}
\nonumber \\
&& + \frac{\alpha_s(Q^2)}{2\pi}
\left[ C_{q,2}*(\frac{dq}{dS}+\frac{d\bar{q}}{dS}) +
2\cdot C_{g,2}*\frac{dg}{dS} \right]
\nonumber \\
&& + \frac{1}{2\pi}\frac{d\alpha_s(Q^2)}{dS}
\left[ C_{q,2}*(q+\bar{q}) + 2\cdot C_{g,2}*g \right]
\Bigg\}.\end{aligned}$$
The numerical integration procedure for solving the DGLAP-equations used in the work presented here gives the evoluted parton distributions and their derivatives on $S$ as the output. In NLO, $d\alpha_s(Q^2)/dS$ is simple to calculate, $$\begin{aligned}
\protect\label{apx19}
\frac{\alpha_s(T)}{2\pi} & = & \frac{2}{\beta_0T}\left(1 -
\frac{\beta_1}{\beta_0^2}\frac{\ln(T)}{T}
\right), \nonumber \\
\frac{1}{2\pi}\frac{d\alpha_s(T)}{dT} &=& -\frac{1}{T}\cdot
\frac{\alpha_s(T)}{2\pi} + \frac{2\beta_1}{\beta_0^2 T^3}(\ln(T)-1),\\
\frac{1}{2\pi}\frac{d\alpha_s}{dS} & = &
T\cdot \frac{1}{2\pi}\frac{d\alpha_s}{dT}. \nonumber \end{aligned}$$ Thus, with the derivatives $dq/dS$, $d\bar{q}/dS$ and $dg/dS$ one gets the $Q^2$-derivative of $F_2$. This method is called as “mainly analytical” (it includes a numerical integration procedure).
Eq.(\[apx18\]) is valid below the charm threshold [@grv; @haakman], i.e. $Q_0^2 < Q^2 < Q_c^2$, and in the high $Q^2$-region where the charm can be considered as a “massless” dynamical quark [@haakman]. As described above, the charm is treated via a photon-gluon fusion process in the range $Q_c^2 < Q^2 < \bar{Q^2}=50~GeV^2$ [@grv; @haakman]. From eq. (\[apx20\]) the charm slope contribution can be detrmined as $$\label{apx22}
\frac{1}{x} \frac{dF_2^c(x,Q^2,m_c^2)}{d\ln{Q^2}} ~=~
2e_c^2~\frac{\alpha_s(\mu^2)}{2\pi} \int_{ax}^1\frac{dy}{y}~\cdot
\frac{dC_{g,2}^c}{dlnQ^2} \left( \frac{x}{y},\frac{m_c^2}{Q^2} \right)
\cdot g(y,\mu^2).$$ The total slope is the sum of eqs. (\[apx18\]) and (\[apx22\]).
We have calculated the $Q^2$-slope of the evoluted $F_2$ in the perturbative region ($Q^2 \ge Q_0^2$) using both, the numerical and the analytical methods. The values are in agreement although the difference increases somewhat in the region near to $Q_0^2$. The values presented in the figures are from the numerical calculation.
[**Figure 1.**]{} $F_2$ as a function of $x$ computed in the CKMT model for twelve different values of $Q^2$, and compared with the following experimental data (see [@caldw] for the experimental references): ZEUS SVX95 (black circles), H1 SVX95 (white triangles), ZEUS BPC95 (white squares), E665 (white diamonds), and ZEUS 94 (white circles). The dotted line is the theoretical result obtained with the pure CKMT model, and the bold (solid) line is the result obtained with the NLO QCD-evoluted CKMT model when one takes $Q^2_0=2. GeV^2$ ($Q^2_0=4. GeV^2$).
[**Figure 2.**]{} $F_2$ as a function of $x$ computed in the CKMT model for six (a) and five (b) different low values of $Q^2$, and compared with the following experimental data (see [@caldw; @amelung; @zhokin] for the experimental references): ZEUS BPT97 (black circles), ZEUS BPC95 (white circles), and E665 (white squares). The theoretical result has been obtained with the pure CKMT model.
[**Figure 3.**]{} $dF_2/dlnQ^2$ as a function of $x$ computed by performing the NLO QCD perturbative evolution of the CKMT model (see appendices A and B for details on the calculation), and compared with the fit of the ZEUS $F_2$ data in bins of $x$ to the form $a+blnQ^2$ (see reference [@caldw] and references therein for more details on the data and the experimental fit). The dotted line is the theoretical result obtained with the pure CKMT model, and the bold (solid) line is the result obtained with the NLO QCD-evoluted CKMT model when one takes $Q^2_0=2. GeV^2$ ($Q^2_0=4. GeV^2$).
[**Figure 4.**]{} $dlnF_2/dln(1/x)$ as a function of $Q^2$ calculated by performing the NLO QCD evolution of the CKMT model, and compared to the fit $F_2=Ax^{-\Delta_{eff}}$ of the ZEUS [@caldw] and the E665 [@e665] data with $x<0.01$. For details on the CKMT calculation, see Appendices A and B. The dotted line is the theoretical result obtained with the pure CKMT model, and the bold (solid) line is the result obtained with the NLO QCD-evoluted CKMT model when one takes $Q^2_0=2. GeV^2$ ($Q^2_0=4. GeV^2$).
[**Figure 5.**]{} $dlnF_2/dln(1/x)$ as a function of $Q^2$ calculated by performing the NLO QCD evolution of the CKMT model, and compared to the fit $F_2=Ax^{-\Delta_{eff}}$ of the H1 data [@H1caldwell]. For details on the CKMT calculation, see Appendices A and B. The dotted line is the theoretical result obtained with the pure CKMT model, and the bold (solid) line is the result obtained with the NLO QCD-evoluted CKMT model when one takes $Q^2_0=2. GeV^2$ ($Q^2_0=4. GeV^2$).
[**Figure 6.**]{} Gluon density distribution as a function of $Q^2$ calculated by performing the NLO QCD evolution of the CKMT model, and compared with the H1 Collaboration data in reference [@adloff]. $g(x, Q^2)$ is plotted and not $xg(x, Q^2)$, in order to show more clearly the evolution with the scale. In the theoretical calculation, the bold (solid) line has been obtained by taking a value of $Q^2_0$ at the starting point of the QCD evolution, $Q^2_0=2. GeV^2$ ($Q^2_0=4. GeV^2$).
[**Figure 7.**]{} Gluon densities at $\mu^2=25. GeV^2$ as a function of $x$ calculated by performing the NLO QCD evolution of the CKMT model, and compared to those determined from H1 DIS data (black dots) and from H1 photoproduction data (stars). Experimental data on $D^*$ meson cross-section measurements are from references [@zhokin; @adloff]. In the theoretical calculation, the solid (dotted) line corresponds to a value of $Q^2_0$ at the starting point of the QCD evolution, $Q^2_0=2. GeV^2$ ($Q^2_0=4. GeV^2$).
[**Figure 8.**]{} Gluon density at $\mu^2=200. GeV^2$ as a function of $x$ calculated by performing the NLO QCD evolution of the CKMT model, to be compared with that obtained from the analysis of the H1 di-jet data [@zhokin; @wobisch]. In the theoretical calculation, the solid (dotted) line has been obtained by taking a value of $Q^2_0$ at the starting point of the QCD evolution, $Q^2_0=2. GeV^2$ ($Q^2_0=4. GeV^2$).
[**Figure 9.**]{} Prediction of the behavior of $xg(x,Q^2)$ as a function of $x$ for several values of $Q^2$ measured both by H1 and ZEUS collaborations. The experimental points are not shown since the analysis of the more recent data is not completed. The solid (dotted) lines have been obtained by taking a value of $Q^2_0$ at the starting point of the QCD evolution, $Q^2_0=2. GeV^2$ ($Q^2_0=4. GeV^2$).
[^1]: E-mail: kaidalov@vxitep.itep.ru
[^2]: E-mail: merino@fpaxp1.usc.es
[^3]: E-mail: pertermann@physik.uni-siegen.de
|
---
abstract: |
Recent results by Alagic and Russell have given some evidence that the Even-Mansour cipher may be secure against quantum adversaries with quantum queries, if considered over other groups than $(\mathbb{Z}/2)^n$. This prompts the question as to whether or not other classical schemes may be generalized to arbitrary groups and whether classical results still apply to those generalized schemes.
In this paper, we generalize the Even-Mansour cipher and the Feistel cipher. We show that Even and Mansour’s original notions of secrecy are obtained on a one-key, group variant of the Even-Mansour cipher. We generalize the result by Kilian and Rogaway, that the Even-Mansour cipher is pseudorandom, to super pseudorandomness, also in the one-key, group case. Using a Slide Attack we match the bound found above. After generalizing the Feistel cipher to arbitrary groups we resolve an open problem of Patel, Ramzan, and Sundaram by showing that the $3$-round Feistel cipher over an arbitrary group is not super pseudorandom.
Finally, we generalize a result by Gentry and Ramzan showing that the Even-Mansour cipher can be implemented using the Feistel cipher as the public permutation. In this last result, we also consider the one-key case over a group and generalize their bound.
author:
- 'Hector B. Hougaard[^1]'
bibliography:
- 'Articlebib.bib'
title: ' How to Generate Pseudorandom Permutations Over Other Groups: Even-Mansour and Feistel Revisited'
---
=1
Introduction
============
In [@EM], Even and Mansour introduced and proved security for the DES inspired block cipher scheme we now call the Even-Mansour (EM) scheme. Given a public permutation, $P$, over $n$-bit strings, with two different, random, secret, $n$-bit keys $k_1$ and $k_2$, a message $x\in \lbrace 0,1 \rbrace^n$ could be enciphered as $$\begin{aligned}
EM_{k_1,k_2}^P (x) = P(x \oplus k_1)\oplus k_2,\end{aligned}$$ with an obvious decryption using the inverse public permutation. The scheme was minimal, in the sense that they needed to XOR a key before and after the permutation, otherwise the remaining key could easily be found. As an improvement, Dunkelman, Keller, and Shamir [@DKS] showed that there was only a need for a single key and the scheme would still retain an indistinguishability from random, i.e. it was pseudorandom. As another consideration of block ciphers, the Feistel cipher construction of Luby and Rackoff [@LubyR] showed how to build pseudorandom permutations from pseudorandom functions.
Eventually, Kuwakado and Morii showed that both the EM scheme [@BrokenEM] and the Feistel scheme [@BrokenFeistel] could be broken by quantum adversaries with quantum queries. Rather than discard these beautiful constructions entirely, Alagic and Russell [@Gorjan] considered whether it would be possible to define the two-key EM scheme over Abelian groups in order to retain security against quantum adversaries with quantum queries. What they showed was a security reduction to the Hidden Shift Problem, over certain groups, such as $\mathbb{Z}/2^n$ and $S_n$. This result inspires us to ask whether the EM and Feistel schemes can be generalized over all groups, and if so, whether or not we can get pseudorandomness in some model.
Prior Work
----------
In extension of their simplification of the EM scheme, Dunkelman, Keller, and Shamir [@DKS] attacked the construction using variants of slide attacks in order to show that the security bound was optimal. They further considered other variants of the EM scheme, such as the Addition Even-Mansour with an Involution as the Permutation (two-keyed). Also Kilian and Rogaway [@Kilr] were inspired by DESX and EM to define their $FX$ construction, of which the EM scheme is a special case.
As referred to above, Kuwakado and Morii were able to break the EM scheme [@BrokenEM] and the $3$-round Feistel scheme [@BrokenFeistel] on $n$-bit strings, using Simon’s algorithm, if able to query their oracle with a superposition of states. Kaplan et al. [@Kaplan], using Kuwakado and Morii’s results, showed how to break many classical cipher schemes, which in turn incited Alagic and Russell [@Gorjan].
In their work with the Hidden Shift Problem, [@Gorjan] posit that a Feistel cipher construction over other groups than the bit strings might be secure against quantum adversaries with quantum queries. Many Feistel cipher variants exist, with different relaxations on the round functions, see for example [@NaorReingold] and [@PatelRamzanSundaram], the latter of which also considered Feistel ciphers over other groups. Vaudenay [@Vaudenay] also considered Feistel ciphers over other groups in order to protect such ciphers against differential analysis attacks by what he called decorrelation.
Removed from the schemes considered below and with a greater degree of abstraction, Black and Rogaway [@BlackRogaway] consider ciphers over arbitrary domains. In general, on the question of the existence of quantum pseudorandom permutations, see [@Zhandry].
Summary of Results
------------------
We work in the Random Oracle Model and consider groups $G$ in the family of finite groups, $\mathcal{G}$. We consider pseudorandom permutations, given informally as the following.
**\[Informal\]** A keyed permutation $P$ on a group $G$ is a **Pseudorandom Permutation (PRP)** on $G$ if it is indistinguishable from a random permutation for all probabilistic distinguishers having access to only polynomially many permutation-oracle queries.
A **Super Pseudorandom Permutation (SPRP)** is a permutation where the distinguisher is given access to the inverse permutation-oracle as well.
We define the **Group Even-Mansour (EM) scheme** on $G$ to be the encryption scheme having the encryption algorithm $$\begin{aligned}
E_k(m) = P(m\cdot k) \cdot k,\end{aligned}$$ where $m\in G$ is the plaintext and $k\in G$ is the uniformly random key.
We define two problems for the Group Even-Mansour scheme: **Existential Forgery** (EFP) and **Cracking** (CP). In EFP, the adversary must eventually output a plaintext-ciphertext pair which satisfies correctness. In CP, the adversary is given a ciphertext and asked to find the corresponding plaintext.
It holds that for our Group EM scheme, the probability that an adversary succeeds in the EFP is polynomially bounded:
\[IntuitiveEFP\] **\[Informal\]** If $P$ is a uniformly random permutation on $G$ and $k\in G$ is chosen uniformly at random. Then, for any probabilistic adversary $\mathcal{A}$, the success probability of solving the EFP is negligible, specifically, bounded by $$\begin{aligned}
O\left( \frac{st}{|G|} \right),\end{aligned}$$ where $s$ and $t$ are the amount of encryption/decryption- and permutation/inverse permutation-oracle queries, respectively.
By a basic reduction, and for the latter, by an inference result, we also get that
**\[Informal\]** If $P$ is a super pseudorandom permutation on $G$ and $k\in G$ is chosen uniformly at random. For any probabilistic adversary $\mathcal{A}$, the success probability of solving the EFP is negligible.
**\[Informal\]** If $P$ is a super pseudorandom permutation on $G$ and $k\in G$ is chosen uniformly at random. For any probabilistic adversary $\mathcal{A}$, the success probability of solving the CP is negligible.
With the same bound as in Theorem \[IntuitiveEFP\], we find that
**\[Informal\]** For any probabilistic adversary $\mathcal{A}$, limited to polynomially many encryption- and decryption-oracle queries and polynomially many permutation- and inverse permutation-oracle queries, the Group EM scheme over a group $G$ is a super pseudorandom permutation.
We then apply a Slide Attack, to find an attack which matches the bound given above.
Considering the **Group Feistel cipher**, whose encryption algorithm consists of multiple uses of the round function $$\begin{aligned}
\mathcal{F}_f(x,y) = (y,x\cdot f(y)),\end{aligned}$$ where $f$ is a pseudorandom function on $G$, we show that the **$3$-round Feistel cipher** is pseudorandom but is not super pseudorandom, regardless of the underlying group $G$. We then note that the $4$-round Feistel cipher is super pseudorandom as proven in [@PatelRamzanSundaram].
Finally, we consider the **Group Even-Mansour scheme instantiated using a $4$-round Feistel cipher** over $G^2 = G\times G$, which uses the encryption algorithm $$\begin{aligned}
\Psi_k^{f,g}(m) = \mathcal{F}_{g,f,f,g}(m \cdot k)\cdot k,\end{aligned}$$ where $f$ and $g$ are modelled as random functions, $m\in G^2$ the plaintext, and $k \in G^2$ is a uniformly random key. We then show one of our main results:
**\[Informal\]** For any probabilistic $4$-oracle adversary $\mathcal{A}$ with at most
- $q_c$ queries to the $\Psi$- and inverse $\Psi$-oracles (or random oracles),
- $q_f$ queries to the $f$-oracle, and
- $q_g$ queries to the $g$-oracle,
we have that the success probability of $\mathcal{A}$ distinguishing between $\Psi$ and a random oracle, is bounded by $$\begin{aligned}
(2q_c^2 +4q_fq_c + 4q_gq_c + 2q_c^2 - 2q_c)|G|^{-1} + 2\cdot \begin{pmatrix}
q_c \\ 2
\end{pmatrix}(2|G|^{-1} + |G|^{-2}).\end{aligned}$$
We may also rewrite our main theorem as the following:
**\[Informal\]** For any $4$-oracle adversary $\mathcal{A}$, with at most $q$ total queries, we have that the success probability of $\mathcal{A}$ distinguishing between $\Psi$ and a random oracle, is bounded by $$\begin{aligned}
2(3q^2-2q)|G|^{-1} + (q^2-q)|G|^{-2}.\end{aligned}$$
We note that this main result is due to [@GentryRamzan], however, we consider a one-key group version and add details to their proof sketches.
Outline of Paper
----------------
In Section \[GenDefns\], we state the assumptions for this paper. In Section \[GenDefns\], we give definitions that hold for the paper in general, leaving specialized definitions to the various sections. In Section \[Even-MansourSection\], we introduce the generalized EM scheme over arbitrary groups, stating and proving some results about it. In Section \[FeistelSection\], we define the generalized Feistel cipher over arbitrary groups and prove a few small results about it. In Section \[ImplementEM\], we consider an implementation of the generalized EM scheme using the generalized Feistel cipher as the public permutation. In Section \[ConclusionSection\], we give our concluding remarks.
General Definitions {#GenDefns}
===================
In the following, we work in the Random Oracle Model such that we may assume the existence of a random permutation oracle on group elements. We let $\mathcal{G}$ be the family of all finite groups, e.g. a group $G\in\mathcal{G}$ is a pair of the set $G$ and operation $\cdot$ satisfying the group axioms. We also assume that for any group $G\in \mathcal{G}$, $|G|\leq 2^{poly(n)}$ for some $n\in \mathbb{N}$ and some polynomial $poly(\cdot)$.
We will need the concept of pseudorandom, which is also called indistinguishable from random, in several forms. On notation, we write $x\in_R X$ for an element chosen uniformly at random from a set $X$. In the following, we consider the positive integer $\lambda$ to be the security parameter, specified in unary per convention. We assume that for each $\lambda$ there exists a uniquely specified group $G(\lambda) = G_\lambda \in \mathcal{G}$ with size $|G_\lambda| \geq 2^\lambda$.
Let $F_{m,n}:G_\lambda \times G_m \rightarrow G_n$, for $G_m,G_n \in \mathcal{G}$, be an efficient, keyed function. $F_{m,n}$ is a **pseudorandom function (PRF)** if for all probabilistic distinguishers $\mathcal{A}$, limited to only polynomially many queries to the function-oracle, there exists a negligible function $negl(\cdot)$, such that $$\begin{aligned}
\left| \underset{k \in_R G_\lambda}{Pr}\left[ \mathcal{A}^{F_{m,n}(k,\cdot)}(\lambda)=1 \right] - \underset{\pi \in_R \mathfrak{F}_{G_m\rightarrow G_n}}{Pr}\left[ \mathcal{A}^{\pi(\cdot)}(\lambda)=1 \right] \right| \leq negl(\lambda),\end{aligned}$$ where $\mathfrak{F}_{G_m\rightarrow G_n}$ is the set of functions from $G_m$ to $G_n$.
If $F:G \times G \rightarrow G$ is a pseudorandom function, we say that it is a **pseudorandom function on $G$**.
Let $P:G_\lambda \times G \rightarrow G$ be an efficient, keyed permutation. $P$ is a **pseudorandom permutation (PRP)** if for all probabilistic distinguishers $\mathcal{A}$, limited to only polynomially many queries to the permutation-oracle, there exists a negligible function $negl(\cdot)$, such that $$\begin{aligned}
\left| \underset{k \in_R G_\lambda}{Pr}\left[ \mathcal{A}^{P(k,\cdot)}(\lambda)=1 \right] - \underset{\pi \in_R \mathfrak{P}_{G\rightarrow G}}{Pr}\left[ \mathcal{A}^{\pi(\cdot)}(\lambda)=1 \right] \right| \leq negl(\lambda),\end{aligned}$$ where $\mathfrak{P}_{G\rightarrow G}$ is the set of permutations on $G$.
Let $P:G_\lambda \times G \rightarrow G$ be an efficient, keyed permutation. $P$ is said to be a **super pseudorandom permutation (SPRP)** if for all probabilistic distinguishers $\mathcal{A}$, limited to only polynomially many queries to the permutation- and inverse permutation-oracles, there exists a negligible function $negl(\cdot)$, such that $$\begin{aligned}
\left| \underset{k \in_R G_\lambda}{Pr}\left[ \mathcal{A}^{P(k,\cdot),P^{-1}(k,\cdot)}(\lambda)=1 \right] - \underset{\pi \in_R \mathfrak{P}_{G\rightarrow G}}{Pr}\left[ \mathcal{A}^{\pi(\cdot),\pi^{-1}(\cdot)}(\lambda)=1 \right] \right| \leq negl(\lambda),\end{aligned}$$ where $\mathfrak{P}_{G\rightarrow G}$ is the set of permutations on $G$.
A (super) pseudorandom permutation $P:G \times G \rightarrow G$ is said to be a **(super) pseudorandom permutation on $G$**.
Even-Mansour {#Even-MansourSection}
============
We first remark that the results in this section were initially proven in a project prior to the start of the thesis but were further worked on to complement this thesis. Thus we have chosen to include parts of it, while this inclusion accounts for the brevity in certain results. We begin by defining the one-key Even-Mansour scheme over arbitrary groups, which we will refer to as the Group EM scheme.
We define the **Group Even-Mansour scheme** to be the triple of a key generation algorithm, encryption algorithm, and decryption algorithm. The key generation algorithm takes as input the security parameter $1^\lambda$, fixes and outputs a group $G\in_R \mathcal{G}$ with $|G|\geq 2^\lambda$, and outputs a key $k\in_R G$. The encryption algorithm $E_k(m)$ takes as input the key $k$ and a plaintext $m\in G$ and outputs $$\begin{aligned}
E_k(m) = P(m\cdot k) \cdot k,\end{aligned}$$ where $P$ is the public permutation. The decryption algorithm $D_k(c)$ takes as input the key $k$ and a ciphertext $c\in G$ and outputs $$\begin{aligned}
D_k(c) = P^{-1}(c \cdot k^{-1}) \cdot k^{-1},\end{aligned}$$ where $P^{-1}$ is the inverse public permutation. This definition satisfies correctness.
Two Forms of Security for the Group EM Scheme
---------------------------------------------
In this subsection, we prove classical results about our new scheme. We do so by considering Even and Mansour’s two notions of security: the Existential Forgery Problem and the Cracking Problem, the Cracking Problem being the stronger of the two.
In the **Existential Forgery Problem** (EFP), we consider the following game:
1. A group $G\in \mathcal{G}$ and a key $k\in_R G$ are generated.
2. The adversary $\mathcal{A}$ gets the security parameter, in unary, and the group $G$.
3. $\mathcal{A}$ receives oracle access to the $E_k, D_k, P,$ and $P^{-1}$ oracles.
4. $\mathcal{A}$ eventually outputs a pair $(m,c)$.
If $E_k(m)=c$, and $(m,c)$ has not been queried before, we say that $\mathcal{A}$ succeeds.
In the **Cracking Problem** (CP), we consider the following game:
1. A group $G\in \mathcal{G}$ and a key $k\in_R G$ are generated.
2. The adversary $\mathcal{A}$ gets the security parameter, in unary, and the group $G$.
3. $\mathcal{A}$ is presented with $E_k(m_0)=c_0\in_R G$.
4. $\mathcal{A}$ receives oracle access to the $E_k, D_k, P,$ and $P^{-1}$ oracles, but the decryption oracle outputs $\perp$ if $\mathcal{A}$ queries $c=c_0$.
5. $\mathcal{A}$ outputs a plaintext $m$.
If $D_k(c_0)=m$, then we say that $\mathcal{A}$ succeeds. The **success probability** is the probability that on a uniformly random chosen encryption $c_0 = E_k(m_0)$, $\mathcal{A}$ outputs $m_0$.
Even and Mansour show that polynomial-time EFP security infers polynomial-time CP security. There are no limiting factors prohibiting the problems and inference result from being employed on groups. In fact, there is nothing disallowing the use of the same proof of the EFP security for the EFP security of the one-key EM scheme, as noted in [@DKS], which we therefore omit. Indeed, by redefining notions in the [@EM] proof to take into account that we are working over a not necessarily abelian group, we are able to prove that the Group EM scheme satisfies the EFP notion of security, specifically the following.
\[MainEFP\] Assume $P\in_R\mathfrak{P}_{G\rightarrow G}$ and let the key $k\in_R G$. For any probabilistic adversary $\mathcal{A}$, the success probability of solving the EFP is bounded by $$\begin{aligned}
Succ(\mathcal{A}) = Pr_{k,P}\left[ EFP(\mathcal{A})=1\right] = O\left( \frac{st}{|G|} \right),\end{aligned}$$ where $s$ is the number of $E/D$-queries and $t$ is the number of $P/P^{-1}$-queries, i.e. the success probability is negligible.
By the Even and Mansour inference result, we get the corollary below.
Assume $P\in_R\mathfrak{P}_{G\rightarrow G}$ and let the key $k\in_R G$. For any probabilistic polynomial-time (PPT) adversary $\mathcal{A}$, the success probability of solving the Cracking Problem is negligible.
As Even and Mansour also note, the above results may be extended to instances where the permutation is a pseudorandom permutation by a simple reduction. Hence, we get the following two results.
Assume $P$ is a pseudorandom permutation on $G\in \mathcal{G}$ and let the key $k\in_R G$. For any probabilistic adversary $\mathcal{A}$ with only polynomially many queries to its oracles, the success probability of solving the Existential Forgery Problem is negligible.
Assume $P$ is a pseudorandom permutation on $G\in \mathcal{G}$ and let the key $k\in_R G$. For any probabilistic polynomial-time (PPT) adversary $\mathcal{A}$, the success probability of solving the Cracking Problem is negligible.
Pseudorandomness Property of the Group EM Scheme
------------------------------------------------
Although the above notions of security are strong, we are more interested in any pseudorandomness property the Group EM scheme offers us. Kilian and Rogaway [@Kilr] show that the one-key EM scheme satisfies the pseudorandom permutation property, i.e. with only an encryption oracle and the permutation oracles, the EM scheme is indistinguishable from random to any adversary with only polynomially many queries to its oracles. We note that they only show the pseudorandomness property, but state in their discussion section that their proof may be adapted to include a decryption oracle, i.e. that the one-key EM scheme satisfies the super pseudorandom permutation property. Having done the analysis with the decryption oracle, over an arbitrary group, we concur. However, we were also able to generalize the [@Kilr] proof to a one-key construction. This not entirely remarkable as the key $k$ will usually be different from its group inverse, hence we were able to use the same proof, but with adjustments to the games and their analysis. The proof is given in the appendix for posterity. For completeness, we present the result as the following theorem.
Assume $P\in_R\mathfrak{P}_{G\rightarrow G}$ and let the key $k\in_R G$. For any probabilistic adversary $\mathcal{A}$, limited to polynomially many $E/D$- and $P/P^{-1}$-oracle queries, the adversarial advantage of $\mathcal{A}$ is bounded by $$\begin{aligned}
\text{Adv}(\mathcal{A}) {\stackrel{\mathclap{\normalfont\mbox{\tiny def}}}{=}}\left| Pr\left[ \mathcal{A}_{E_k,D_k}^{P,P^{-1}} = 1 \right] - Pr\left[\mathcal{A}_{\pi,\pi^{-1}}^{P,P^{-1}} = 1 \right]\right| = \mathcal{O}\left(\frac{st}{|G|}\right).\end{aligned}$$ where $s$ is the number of $E/D$-queries and $t$ is the number of $P/P^{-1}$-queries, i.e. the success probability is negligible.
Stated simply,
For any probabilistic adversary $\mathcal{A}$, limited to polynomially many $E/D$- and $P/P^{-1}$-oracle queries, the Group EM scheme over a group $G$ is a super pseudorandom permutation.
By removing the decryption oracle, we get the following corollary:
For any probabilistic adversary $\mathcal{A}$, limited to polynomially many $E$- and $P/P^{-1}$-oracle queries, the Group EM scheme over a group $G$ is a pseudorandom permutation.
We see that in the group $((\mathbb{Z}/2\mathbb{Z})^n,\oplus)$, our Group EM scheme reduces to the one-key EM scheme given in [@DKS]. The proof given in [@DKS] proves the security of the scheme, and the proof given in [@Kilr] proves the pseudorandomness, equivalently to our claims.
It can be proven that a multiple round Group EM scheme is an SPRP because the security only depends on the last round, which is also an SPRP.
Slide Attack
------------
We would like to show that the security bound that we have found above is optimal, so we slightly alter the simple optimal attack on the Single-Key Even-Mansour cipher as constructed in [@DKS]. The original version works for abelian groups with few adjustments and [@DKS] also present another slide attack against a modular addition DESX construction.
Consider the one-key Group Even-Mansour cipher $$\begin{aligned}
E(x) = P(x \cdot k) \cdot k,\end{aligned}$$ over a group $G$ with binary operation $\cdot$, where $P$ is a publicly available permutation oracle, $x\in G$, and $k\in_R G$. Define the following values: $$\begin{aligned}
x=x, \hspace*{5pt} y=x \cdot k, \hspace*{5pt} z= P(y), \hspace*{5pt} w=E(x)=P(x \cdot k) \cdot k.\end{aligned}$$ We hereby have that $w\cdot y^{-1} = z \cdot x^{-1} $. Consider the attack which follows.
1. For $d = \sqrt{|G|}$ arbitrary values $x_i\in G$, $i=1,\ldots, d$, and $d$ arbitrary values $y_i\in G$, $i=1,\ldots, d$, query the $E$-oracle on the $x_i$’s and the $P$-oracle on the $y_i$’s. Store the values in a hash table as $$\begin{aligned}
(E(x_i)\cdot y_i^{-1}, P(y_i)\cdot x_i^{-1}, i),\end{aligned}$$ sorted by the first coordinate.
2. If there exists a match in the above step, i.e. $E(x_i)\cdot y_i^{-1} = P(y_i)\cdot x_i^{-1}$ for some $i$, check the guess that $k = x_i^{-1}\cdot y_i$.
It can be seen by the Birthday Problem[^2], that with non-negligible probability, there must exist a slid pair $(x_i,y_i)$ satisfying the above property, i.e. there exists $1\leq i \leq d$ such that $k = x_i^{-1} \cdot y_i$. For a random pair $(x,y)\in G^2$ it holds that $E(x) = P(y) \cdot x^{-1} \cdot y$ with probability $|G|^{-1}$, so we expect few, if any, collisions in the hash table, including the collision by the slid pair where the correct key $k$ is found. The data complexity of the attack is $d$ $E$-oracle queries and $d$ $P$-oracle queries. Hence the attack bound $d^2 = |G|$, which matches the lower bound given in Theorem \[MainEFP\] and Theorem \[PseudoEMbounded\]. We have therefore found that our scheme is optimal.
Feistel {#FeistelSection}
=======
We now consider the Feistel cipher over arbitrary groups, which we will call the Group Feistel cipher. The following is a complement to [@PatelRamzanSundaram] who treat the Group Feistel cipher construction with great detail. Our main accomplishment in this section is the settling of an open problem posed by them.
Definitions
-----------
We define a Feistel cipher over a group $(G,\cdot)$ as a series of round functions on elements of $G\times G=G^2$.
Given an efficiently computable but not necessarily invertible function $f: G \rightarrow G$, called a **round function**, we define the **1-round Group Feistel cipher** $\mathcal{F}_{f}$ to be $$\begin{aligned}
\mathcal{F}_{f}: G \times G &\longrightarrow G \times G,\\
(x,y) &\longmapsto (y, x \cdot f(y)).\end{aligned}$$ In the case where we have multiple rounds, we index the round functions as $f_i$, and denote the **$r$-round Group Feistel cipher** by $\mathcal{F}_{f_1,\ldots,f_r}$. We concurrently denote the input to the $i$’th round by $(L_{i-1},R_{i-1})$ and having the output $(L_i,R_i) = (R_{i-1}, L_{i-1}\cdot f_i(R_{i-1}))$, where $L_i$ and $R_i$ respectively denote the left and right parts of the $i$’th output.
Note that if $(L_i,R_i)$ is the $i$’th round output, we may invert the $i$’th round by setting $R_{i-1}:=L_i$ and then computing $L_{i-1}:= R_i \cdot (f_i(R_{i-1}))^{-1}$ to get $(L_{i-1},R_{i-1})$. As this holds for all rounds, regardless of the invertibility of the round functions, we get that an $r$-round Feistel cipher is invertible for all $r$.
Let $F:G_\lambda \times G \rightarrow G$ be a pseudorandom function. We define the keyed permutation $F^{(r)}$ as $$\begin{aligned}
F^{(r)}_{k_1,\ldots,k_r}(x,y) {\stackrel{\mathclap{\normalfont\mbox{\tiny def}}}{=}}\mathcal{F}_{F_{k_1},\ldots, F_{k_r}}(x,y).\end{aligned}$$ We sometimes index the keys as $1,2,\ldots, r$, or omit the key index entirely.
Results
-------
For completeness, we show some of the preliminary results for Group Feistel ciphers, not considered in [@PatelRamzanSundaram].
We first note that $F^{(1)}$ is *not* a pseudorandom permutation as $$\begin{aligned}
F^{(1)}_{k_1}(L_0,R_0) = (L_1,R_1) = (R_0,L_0\cdot F_{k_1}(R_0)),\end{aligned}$$ such that any distinguisher $\mathcal{A}$ need only compare $R_0$ to $L_1$.
Also $F^{(2)}$ is *not* a pseudorandom permutation: Consider a pseudorandom function $F$ on $G$. Pick $k_1,k_2\in_R G_\lambda$. Distinguisher $\mathcal{A}$ sets $(L_0,R_0)=(1,g)$ for some $g\in G$, where $1$ is the identity element of $G$, then queries $(L_0,R_0)$ to its oracle and receives,
$L_2 = L_0 \cdot F_{k_1}(R_0) = F_{k_1}(g)$ and $R_2 = R_0 \cdot F_{k_2}(L_0\cdot F_{k_1}(R_0)) = g \cdot F_{k_2}(F_{k_1}(g))$.
On its second query, the distinguisher $\mathcal{A}$ lets $L_0 \in G \setminus \lbrace 1\rbrace$ but $R_0=g$, such that it receives
$L_2 = L_0 \cdot F_{k_1}(R_0) = L_0 \cdot F_{k_1}(g)$ and $R_2 = g \cdot F_{k_2}(L_0 \cdot F_{k_1}(g))$.
As $\mathcal{A}$ may find the inverse to elements in $G$, $\mathcal{A}$ acquires $(F_{k_1}(g))^{-1}$, and by so doing, may compute $L_2 \cdot (F_{k_1}(g))^{-1} = L_0$. If $F^{(2)}$ were random, this would only occur negligibly many times, while $\mathcal{A}$ may query its permutation-oracle polynomially many times such that if $L_0$ is retrieved non-negligibly many times out of the queries, $\mathcal{A}$ is able to distinguish between a random permutation and $F^{(2)}$ with non-negligible probability.
As one would expect, the $3$-round Group Feistel cipher (see Figure \[3roundFeistel\]) is indeed a pseudorandom permutation.
[0.49]{}
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(p0) \[above of = g1, minimum width=5cm,minimum height=0.5cm,node distance=1cm\] ; (l0) \[above of = prik1,node distance=1cm\] [$L_0$]{}; (r0) \[right of = l0, node distance = 4cm\] [$R_0$]{}; (l0 |- p0.south) – (prik1.north); ($(g1.east)+(1.5cm,0)$) – +(0,0.75cm);
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\[$3$-round Group Feistel cipher.\][$3$-round Group Feistel cipher.]{} \[3roundFeistel\]
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(p0) \[above of = f1, minimum width=5cm,minimum height=0.5cm,node distance=1cm\] ; (l0) \[above of = xor1,node distance=1cm\] [$x^L$]{}; (mid0) \[above of = xor1,node distance = .5cm\] ; (kl0) \[right of = mid0,node distance=.3cm\] [$\cdot k^L$]{}; (r0) \[right of = l0, node distance = 4cm\] [$x^R$]{}; (kr0) \[right of = kl0,node distance= 4.05cm\] [$\cdot k^R$]{}; (l0 |- p0.south) – (xor1.north); ($(f1.east)+(1.5cm,0)$) – +(0,0.75cm);
(p4) \[below of = f4, minimum width=5cm,minimum height=0.5cm,node distance=1.75cm\] ; (l4) \[below of = xor4,node distance=1.75cm\] [$y^L$]{}; (kl4) \[below of = kl0,node distance=6.15cm\] [$\cdot k^L$]{}; (r4) \[right of = l4, node distance = 4cm\] [$y^R$]{}; (kr4) \[right of = kl4,node distance= 4.05cm\] [$\cdot k^R$]{}; (f4.east) -| +(1.5cm,-0.5cm) – ($(xor4) - (0,1cm)$) – (xor4 |- p4.north); (xor4.south) – ($(xor4)+(0,-0.5cm)$) – ($(f4.east) + (1.5cm,-1cm)$) – +(0,-0.5cm);
\[Group EM scheme with Feistel.\][Group EM scheme with Feistel.]{} \[GendGenRamPic\]
If $F$ is a pseudorandom function on $G$, then $F^{(3)}$ is a pseudorandom permutation on $G$.
The proof of this proposition can be generalized from the proof given in Katz and Lindell [@KL] of the analogous result over bit-strings with XOR, with no difficulties. We therefore omit it here.
Among the considerations in [@PatelRamzanSundaram], they showed that the $3$-round Feistel cipher over abelian groups was not super pseudorandom, but left as an open problem a proof over non-abelian groups. We present such a proof now.
The $3$-round Group Feistel cipher is not super pseudorandom.
The proof is a counter-example using the following procedure:
1. Choose two oracle-query pairs in $G\times G$: $(L_0,R_0)$ and $(L'_0,R_0)$ where $L_0\neq L'_0$.
2. Query the encryption oracle to get $(L_3,R_3)$ and $(L'_3,R'_3)$.
3. Query $(L''_3,R''_3)= (L'_3,L_0\cdot (L'_0)^{-1} \cdot R'_3)$ to the decryption oracle.
4. If $R''_0=L'_3\cdot (L_3)^{-1} \cdot R_0$, guess that the oracle is $F^{(3)}$, else guess random.
For $F^{(3)}$, this algorithm succeeds with probability $1$. For a random permutation, this algorithm succeeds negligibly often.
For super pseudorandomness of the $4$-round Group Feistel cipher, we refer the reader to [@PatelRamzanSundaram]. In the paper, they show a strong result using certain hash functions as round functions, from which the following is a corollary.
Let $G$ be a group, with characteristic other than $2$, and let $f,g: G_\lambda \times G \rightarrow G$ be pseudorandom functions. Then, for any adversary $\mathcal{A}$ with polynomially many queries to its $E/D$-oracles, the family $\mathcal{P}$ of permutations on $G\times G$ consisting of permutations of the form $F^{(4)}=\mathcal{F}_{g,f,f,g}$ are indistinguishable from random, i.e. super pseudorandom permutations (SPRPs).
Implementing the Group Even-Mansour Scheme {#ImplementEM}
==========================================
Now that we have shown that both the Even-Mansour scheme and the Feistel cipher are generalizable to arbitrary groups, we might consider how to implement one given the other. Gentry and Ramzan [@GentryRamzan] considered exactly this for the two-key EM scheme over $(\mathbb{Z}/2\mathbb{Z})^n$. However, their paper only had sketches of proofs and refer to another edition of the paper for full details. As we are unable to find a copy in the place that they specify it to exist, and as we generalize their result non-trivially, we have decided to fill in the details while generalizing their proof.
In this section, we consider a generalized version of the Gentry and Ramzan [@GentryRamzan] construction, namely, the Group Even-Mansour scheme on $G^2$ instantiated with a $4$-round Group Feistel cipher as the public permutation: $$\begin{aligned}
\Psi_{k}^{f,g}(x)=\mathcal{F}_{g,f,f,g}(x\cdot k)\cdot k,\end{aligned}$$ where $k=(k^L,k^R)\in G^2$ is a key consisting of two subkeys, chosen independently and uniformly at random, and $f$ and $g$ are round functions on $G$, modelled as random function oracles, available to all parties, including the adversary. We consider the operation $x\cdot k$ for $x=(x^L,x^R)\in G^2$, to be the coordinate-wise group operation, but do not otherwise discern between it and the group operation $\cdot$ on elements of $G$. In the following, we shall follow the proof in [@GentryRamzan] closely. However, we make quite a few modifications, mostly due to the nature of our generalization. Note that we consider a one-key scheme, as opposed to the two-key version in [@GentryRamzan] (see Figure \[GendGenRamPic\].) Our main theorem for this section is the following.
\[GentryRamzanMain\] Let $f,g$ be modelled as random oracles and let the subkeys of $k=(k^L,k^R)\in G^2$ be chosen independently and uniformly at random. Let $\Psi_k^{f,g}(x)=\mathcal{F}_{g,f,f,g}(x\cdot k)\cdot k$, and let $R\in_R \mathfrak{P}_{G^2 \rightarrow G^2}$. Then, for any probabilistic $4$-oracle adversary $\mathcal{A}$ with at most
- $q_c$ queries to $\Psi$ and $\Psi^{-1}$ (or $R$ and $R^{-1}$),
- $q_f$ queries to $f$, and
- $q_g$ queries to $g$,
we have $$\begin{aligned}
&\left| Pr\left[ \mathcal{A}^{\Psi,\Psi^{-1},f,g} = 1\right] - Pr\left[ \mathcal{A}^{R,R^{-1},f,g} = 1\right] \right| \\
&\hspace*{20pt}\leq (2q_c^2 +4q_fq_c + 4q_gq_c + 2q_c^2 - 2q_c)|G|^{-1} + 2\cdot \begin{pmatrix}
q_c \\ 2
\end{pmatrix}(2|G|^{-1} + |G|^{-2}).\end{aligned}$$
Definitions
-----------
Before we can begin the proof, we will need several definitions all of which are identical to the [@GentryRamzan] definitions, up to rewording.
Let $P$ denote the permutation oracle (either $\Psi$ or $R$), $\mathcal{O}^f$ and $\mathcal{O}^g$ the $f$ and $g$ oracles, respectively. We get the transcripts: $T_P$, the set of all $P$ queries, $T_f$, the set of all $f$ queries, and $T_g$, the set of all $g$ queries, i.e. the sets $$\begin{aligned}
T_P &= \lbrace \langle x_1,y_1 \rangle, \langle x_2,y_2 \rangle, \cdots, \langle x_{q_c},y_{q_c} \rangle \rbrace_P, \\
T_f &= \lbrace \langle x'_1,y'_1 \rangle, \langle x'_2,y'_2 \rangle, \cdots, \langle x'_{q_f},y'_{q_f} \rangle \rbrace_f, \\
T_g &= \lbrace \langle x''_1,y''_1 \rangle, \langle x''_2,y''_2 \rangle, \cdots, \langle x''_{q_g},y''_{q_g} \rangle \rbrace_g.\end{aligned}$$ We discern between two types of oracle queries: Cipher queries $(+,x)=P(x)$ and $(-,y)=P^{-1}(y)$; Oracle queries $(\mathcal{O}^f,x')$ and $(\mathcal{O}^g,x'')$, respectively $f$- and $g$-oracle queries.
As we have no bounds on the computational complexity of the adversary $\mathcal{A}$, we may assume that $\mathcal{A}$ is deterministic, as we did in the proof of Theorem \[PseudoEMbounded\]. Hence, we may consider an algorithm $C_\mathcal{A}$ which, given a set of $\mathcal{A}$’s queries, can determine $\mathcal{A}$’s next query.
For $0\leq i \leq q_c$, $0\leq j \leq q_f$, and $0\leq k \leq q_g$, the $i+j+k+1$’st query by $\mathcal{A}$ is $$\begin{aligned}
C_\mathcal{A}\left[ \lbrace \langle x_1,y_1 \rangle, \ldots, \langle x_{i},y_{i} \rangle \rbrace_P, \lbrace \langle x'_1,y'_1 \rangle, \ldots, \langle x'_{j},y'_{j} \rangle \rbrace_f, \lbrace \langle x''_1,y''_1 \rangle, \ldots, \langle x''_{k},y''_{k} \rangle \rbrace_g \right]\end{aligned}$$ where the upper equality case on the indexes is defined to be $\mathcal{A}$’s final output.
Let $\sigma = (T_P,T_f,T_g)$ be a tuple of transcripts with length $q_c,q_f,q_g$, respectively. We say that $\sigma$ is a **possible $\mathcal{A}$-transcript** if for every $1\leq i \leq q_c, 1 \leq j \leq q_f$, and $1\leq k \leq q_g$, $$\begin{aligned}
C_\mathcal{A}\left[ \lbrace \langle x_1,y_1 \rangle, \ldots, \langle x_{i},y_{i} \rangle \rbrace_P, \lbrace \langle x'_1,y'_1 \rangle, \ldots, \langle x'_{j},y'_{j} \rangle \rbrace_f, \lbrace \langle x''_1,y''_1 \rangle, \ldots, \langle x''_{k},y''_{k} \rangle \rbrace_g \right] \\ \hspace*{70pt}\in \lbrace (+,x_{i+1}), (-,y_{i+1}), (\mathcal{O}^f, x'_{j+1}), (\mathcal{O}^g,x''_{k+1})\rbrace .\end{aligned}$$
Let us define two useful ways in which we may answer $\mathcal{A}$’s queries other than what we have already defined.
Let $\tilde{\Psi}$ be the process where the $\Psi$- and $\Psi^{-1}$ cipher query oracles use $f$ and $g$, and $\mathcal{O}^f$ uses $f$, but $\mathcal{O}^g$ is replaced by $\mathcal{O}^h$ for another, independent, random function $h$.
Let $\tilde{R}$ denote the process which answers all queries using $f$ and $g$, but answers the $i$’th query as follows.
1. If $\mathcal{A}$ queries $(+,x_i)$ and there exists $1 \leq j < i$, such that the $j$’th query-answer pair has $x_j=x_i$, return $y_i:=y_j$.
2. If $\mathcal{A}$ queries $(-,y_i)$ and there exists $1 \leq j < i$, such that the $j$’th query-answer pair has $y_j=y_i$, return $x_i:=x_j$.
3. Otherwise, return uniformly chosen element in $G^2$.
The latter definition may not be consistent with any function or permutation, so we formalize exactly this event.
\[inconsistent\] Let $T_P$ be a possible $\mathcal{A}$-cipher-transcript. $T_P$ is **inconsistent** if for some $1\leq i < j \leq q_c$ there exist cipher-pairs such that either
- $x_i = x_j$ but $y_i \neq y_j$, or
- $x_i \neq x_j$ but $y_i=y_j$.
Any $\sigma$ containing such a transcript $T_P$ is called **inconsistent**.
Assume from now on that $\mathcal{A}$ never repeats any part of a query if the answer can be determined from previous queries, i.e. every possible $\mathcal{A}$-transcript $\sigma$ is consistent such that if $i\neq j$, then $x_i \neq x_j$, $y_i\neq y_j$, $x'_i \neq x'_j$, and $x''_i \neq x''_j$.
Let $T_\Psi, T_{\tilde{\Psi}}, T_{\tilde{R}}, T_R$ denote the transcripts seen by $\mathcal{A}$ when its cipher queries are answered by $\Psi, \tilde{\Psi}, \tilde{R}, R$, respectively, and oracle queries by $\mathcal{O}^f$ and $\mathcal{O}^g$ (noting that in the case of $\tilde{\Psi}$, the function in the $\mathcal{O}^g$ has been replaced by another random function, $h$.) We also note that using this notation, we have that $\mathcal{A}^{\Psi,\Psi^{-1},f,g} = C_\mathcal{A}(T_\Psi)$ (and likewise for $\tilde{\Psi}, \tilde{R}$, and $R$.)
Lemmas
------
Now, let us begin finding results that will aid us in proving our main theorem. First, we will compare the distributions of $\tilde{R}$ and $R$, using a result by Naor-Reingold[^3]. Afterwards, we shall consider when the distributions of $\Psi$ and $\tilde{\Psi}$ are equal. Lastly, we shall consider when the distributions of $\tilde{\Psi}$ and $\tilde{R}$ are equal. Combining these results will allow us to prove our main theorem.
We remark that whenever we write $k= (k^L,k^R)\in_R G^2$, we mean that the subkeys are chosen independently and uniformly at random.
\[Lem37\] $\left| \underset{\tilde{R}}{Pr}\left[ C_\mathcal{A}(T_{\tilde{R}})=1 \right] - \underset{R}{Pr}\left[ C_\mathcal{A}(T_{R})=1 \right] \right| \leq \begin{pmatrix}
q_c \\ 2
\end{pmatrix}\cdot |G|^{-2}$.
Let $\sigma$ be a possible and consistent $\mathcal{A}$-transcript, then $$\begin{aligned}
\underset{R}{Pr}\left[ T_R = \sigma \right] = \begin{pmatrix}
|G|^{2} \\ q_c \end{pmatrix} = \underset{\tilde{R}}{Pr} \left[ T_{\tilde{R}} = \sigma \mid T_{\tilde{R}} \textit{ is consistent} \right],\end{aligned}$$ simply because the only difference between $T_R$ and $T_{\tilde{R}}$ is in the cipher queries, and when $T_{\tilde{R}}$ is consistent, we have no overlap on the query-answer pairs, hence we need only consider how to choose $q_c$ elements from $|G|^2$ many possible elements, without replacement. Let us now consider the probability of $T_{\tilde{R}}$ being inconsistent. If $T_{\tilde{R}}$ is inconsistent for some $1\leq i < j \leq q_c$ then either $x_i=x_j$ and $y_i \neq y_j$, or $x_i \neq x_j$ and $y_i = y_j$. For any given $i,j$, this happens with at most probability $|G|^{-2}$, because if $x_i=x_j$ is queried, then the $\tilde{R}$-oracle would return the corresponding $y_i=y_j$, but if $x_i \neq x_j$ is queried, then the $\tilde{R}$-oracle would return a uniformly random element (and likewise if $y_i=y_j$ or $y_i\neq y_j$ were queried to the inverse $\tilde{R}$-oracle.) Hence, $$\begin{aligned}
\underset{\tilde{R}}{Pr} \left[ T_{\tilde{R}} \textit{ is inconsistent} \right] \leq \begin{pmatrix}
q_c \\ 2 \end{pmatrix} \cdot |G|^{-2}.\end{aligned}$$ We thereby get that,
$$\begin{aligned}
&\left| \underset{\tilde{R}}{Pr}\left[ C_\mathcal{A}(T_{\tilde{R}})=1 \right] - \underset{R}{Pr}\left[ C_\mathcal{A}(T_{R})=1 \right] \right| \\
&\leq \left| \underset{\tilde{R}}{Pr}\left[ C_\mathcal{A}(T_{\tilde{R}})=1 | T_{\tilde{R}} \textit{ is consistent} \right] - \underset{R}{Pr}\left[ C_\mathcal{A}(T_{R})=1 \right]\right| \cdot \underset{\tilde{R}}{Pr}\left[ T_{\tilde{R}} \textit{ is consistent} \right] \\
&\hspace*{10pt} + \left| \underset{\tilde{R}}{Pr}\left[ C_\mathcal{A}(T_{\tilde{R}})=1 | T_{\tilde{R}} \textit{ is inconsistent} \right] - \underset{R}{Pr}\left[ C_\mathcal{A}(T_{R})=1 \right]\right| \cdot \underset{\tilde{R}}{Pr}\left[ T_{\tilde{R}} \textit{ is inconsistent} \right] \\
&\leq \underset{\tilde{R}}{Pr}\left[ T_{\tilde{R}} \textit{ is inconsistent} \right] \\
&\leq \begin{pmatrix} q_c \\ 2 \end{pmatrix} \cdot |G|^{-2},\end{aligned}$$
as the distribution over $R$ is independent of the (in)consistency of $T_{\tilde{R}}$.
Let us now focus on the distributions of $T_\Psi$ and $T_{\tilde{\Psi}}$, to show that they are identical unless the input to $g$ in the cipher query to $\Psi$ is equal to the oracle input to $h$ in $\mathcal{O}^h$. In order to do so, we first define the event $\textsf{BadG}(k)$.
For every specific key $k=(k^L,k^R)\in_R G^2$, we define $\textsf{BadG}(k)$ to be the set of all possible and consistent $\mathcal{A}$-transcripts $\sigma$, satisfying at least one of the following:
- $\exists i,j, 1\leq i \leq q_c, 1 \leq j \leq q_g$, such that $x_i^R \cdot k^R = x''_j$, or
- $\exists i,j, 1\leq i \leq q_c, 1 \leq j \leq q_g$, such that $y_i^L \cdot (k^L)^{-1} = x''_j$.
\[Lem39\] Let $k=(k^L,k^R)\in_R G^2$. For any possible and consistent $\mathcal{A}$-transcript $\sigma=(T_P,T_f,T_g)$, we have $$\begin{aligned}
\underset{k}{Pr}\left[ \sigma \in \textsf{BadG}(k) \right] \leq \frac{2q_gq_c}{|G|}.\end{aligned}$$
We know that $\sigma \in \textsf{BadG}(k)$ if one of **BG1** or **BG2** occur, hence, using the union bound, $$\begin{aligned}
\underset{k}{Pr}\left[ \sigma \in \textsf{BadG}(k) \right] &= \underset{k}{Pr}\left[ \textbf{BG1} \text{ occurs } \vee \textbf{BG2} \text{ occurs } | \sigma \right] \\
&\leq \underset{k}{Pr}\left[ \textbf{BG1} \text{ occurs } | \sigma \right] + \underset{k}{Pr}\left[ \textbf{BG2} \text{ occurs } | \sigma \right] \\
&\leq q_gq_c\cdot |G|^{-1} + q_gq_c\cdot |G|^{-1} \\
&= 2q_gq_c\cdot |G|^{-1}.\end{aligned}$$
\[NotBadPsitoBarPsi\] Let $\sigma$ be a possible and consistent $\mathcal{A}$-transcript, then $$\begin{aligned}
\underset{\Psi}{Pr}\left[ T_{\Psi} = \sigma | \sigma \not\in \textsf{BadG}(k) \right] = \underset{\tilde{\Psi}}{Pr}\left[ T_{\tilde{\Psi}} = \sigma \right].\end{aligned}$$
We want to show that the query answers in the subtranscripts of the games $\Psi$ and $\tilde{\Psi}$ are equally distributed, under the condition that neither of the events **BG1** nor **BG2** occur in game $\Psi$. Fix the key $k=(k^L,k^R)\in_R G^2$. Recall that the adversary does not query an oracle if it can determine the answer from previous queries.
In both games, for any $\mathcal{O}^f$-oracle query $x' \in G$, the query answer will be equally distributed in both games as the underlying random function $f$ is the same in both games.
In game $\Psi$, an $\mathcal{O}^g$-oracle query, $x''\in G$, will have a uniformly random answer as $g$ is a random function. Likewise, in game $\tilde{\Psi}$, an $\mathcal{O}^g$-oracle query, $x'' \in G$, will have a uniformly random answer as $h$ is a random function.
Consider now the permutation oracle $P= \mathcal{F}_{g,f,f,g}(x\cdot k)\cdot k$. We consider a query-answer pair $\langle x, y \rangle \in T_P$ for $x,y\in G^2$.
In both games, $x^R\cdot k^R$ will be the input to the first round function, which is $g$. In game $\tilde{\Psi}$ the output is always a uniformly random element, newly selected by $g$. In game $\Psi$, if $x^R\cdot k^R$ has already been queried to the $\mathcal{O}^g$-oracle, the output of the round function is the corresponding oracle answer, else it is a uniformly random element, newly selected by $g$. As the former event in game $\Psi$ never occurs because the event **BG1** never occurs, the distributions are equal.
As both games have access to the same random function $f$, the second and third round function outputs will have equal distributions.
In both games, $y^L\cdot (k^L)^{-1}$ will be the input to the fourth round function, which is again $g$. In game $\tilde{\Psi}$ the output is always a uniformly random element, newly selected by $g$, unless $y^L\cdot (k^L)^{-1} = x^R\cdot k^R$, in which case the output is equal to the output of the first round function. In game $\Psi$, if $x^R\cdot k^R$ has already been queried to the $\mathcal{O}^g$-oracle, but not as input to the first round function, the output of the round function is the corresponding oracle answer. If $y^L\cdot (k^L)^{-1} = x^R\cdot k^R$, then the output is equal to the output of the first round function, else it is a uniformly random element newly selected by $g$. As the former event in game $\Psi$ never occurs because the event **BG2** never occurs, the distributions are equal.
As $\mathcal{A}$ does not ask a query if it can determine the answer based on previous queries, we see that the inverse permutation oracle, using $P^{-1}$, yields analogous distributions. Thus, the distributions for the two games must be equal.
Let us show that the distributions of $T_{\tilde{\Psi}}$ and $T_{\tilde{R}}$ are identical, unless the same value is input to $f$ on two separate occasions. Here we also define when a key is “bad” as we did above, but altered such that it pertains to our current oracles.
For every specific key $k=(k^L,k^R)\in_R G^2$ and function $g\in_R \mathfrak{F}_{G\rightarrow G}$, define $\textsf{Bad}(k,g)$ to be the set of all possible and consistent $\mathcal{A}$-transcripts $\sigma$ satisfying at least one of the following events:
- $\exists 1\leq i < j \leq q_c$, such that $$\begin{aligned}
x_i^L\cdot k^L \cdot g(x_i^R\cdot k^R) = x_j^L \cdot k^L \cdot g(x_j^R \cdot k^R)\end{aligned}$$
- $\exists 1\leq i < j \leq q_c$, such that $$\begin{aligned}
y_i^R\cdot (k^R)^{-1} \cdot \left(g(y_i^L\cdot (k^L)^{-1})\right)^{-1} = y_j^R\cdot (k^R)^{-1} \cdot \left(g(y_j^L\cdot (k^L)^{-1})\right)^{-1}\end{aligned}$$
- $\exists 1\leq i , j \leq q_c$, such that $$\begin{aligned}
x_i^L\cdot k^L \cdot g(x_i^R\cdot k^R) = y_j^R\cdot (k^R)^{-1} \cdot \left(g(y_j^L\cdot (k^L)^{-1})\right)^{-1}\end{aligned}$$
- $\exists 1\leq i \leq q_c, 1\leq j \leq q_f$, such that $$\begin{aligned}
x_i^L\cdot k^L \cdot g(x_i^R\cdot k^R) = x'_j\end{aligned}$$
- $\exists 1\leq i \leq q_c, 1\leq j \leq q_f$, such that $$\begin{aligned}
y_i^R\cdot (k^R)^{-1} \cdot \left(g(y_i^L\cdot (k^L)^{-1})\right)^{-1} = x'_j\end{aligned}$$
\[Lem42\] Let $k=(k^L,k^R)\in_R G^2$. For any possible and consistent $\mathcal{A}$-transcript $\sigma$, we have that $$\begin{aligned}
\underset{k,g}{Pr}\left[ \sigma \in \textsf{Bad}(k,g)\right] \leq \left( q_c^2+2q_fq_c + 2\cdot \begin{pmatrix} q_c \\ 2 \end{pmatrix} \right)\cdot |G|^{-1}.\end{aligned}$$
We have that $\sigma \in \textsf{Bad}(k,g)$ if it satisfies a $\textit{\textbf{Bi}}$ for some $\textbf{i}=\lbrace1,\ldots,5\rbrace$. Using that $k^L,k^R$ are uniform and independently chosen, and $g\in_R \mathfrak{F}_{G\rightarrow G}$, we may achieve an upper bound on the individual event probabilities, and then use the union bound.
There are $\begin{pmatrix} q_c \\ 2 \end{pmatrix}$ many ways of picking $i,j$ such that $1\leq i< j \leq q_c$, also, $q_fq_c$ many ways of picking $i,j$ such that $1\leq i \leq q_c, 1 \leq j \leq q_f$, and $q_c^2$ many ways of picking $i,j$ such that $1 \leq i,j\leq q_c$. The probability that two elements chosen from $G$ are equal is $|G|^{-1}$, so we may bound each event accordingly and achieve, using the union bound, that
$$\begin{aligned}
\underset{k,g}{Pr}\left[ \sigma \in \textsf{Bad}(k,g)\right] &= \underset{k,g}{Pr}\left[ \bigvee_{i=1}^5 \textit{\textbf{Bi}} \text{ occurs } | \sigma \right] \\
&\leq \sum_{i=1}^5 \underset{k,g}{Pr}\left[ \textit{\textbf{Bi}} \text{ occurs } | \sigma \right] \\
&\leq \begin{pmatrix} q_c \\ 2 \end{pmatrix}\cdot |G|^{-1} + \begin{pmatrix} q_c \\ 2 \end{pmatrix}\cdot |G|^{-1} + q_c^2 \cdot |G|^{-1} + q_fq_c \cdot |G|^{-1} + q_fq_c \cdot |G|^{-1} \\
&= \left( q_c^2 + 2q_fq_c + 2\begin{pmatrix} q_c \\ 2 \end{pmatrix} \right) \cdot |G|^{-1}.\end{aligned}$$
\[Lem43\] Let $\sigma$ be a possible and consistent $\mathcal{A}$-transcript, then $$\begin{aligned}
\underset{\tilde{\Psi}}{Pr}\left[ T_{\tilde{\Psi}} = \sigma | \sigma \not\in \textsf{Bad}(k,g)\right] = \underset{\tilde{R}}{Pr}\left[ T_{\tilde{R}} = \sigma \right].\end{aligned}$$
The following proof is based on the proof in [@GentryRamzan] which refers to [@NaorReingold] for the first part of their argument. We need the generalization of this argument and so also include it.
Since $\sigma$ is a possible $\mathcal{A}$-transcript, we have for all $1\leq i \leq q_c, 1\leq j \leq q_f, 1\leq k \leq q_g$: $$\begin{aligned}
C_\mathcal{A}\left[ \lbrace \langle x_1,y_1 \rangle, \ldots, \langle x_{i},y_{i} \rangle \rbrace_P, \lbrace \langle x'_1,y'_1 \rangle, \ldots, \langle x'_{j},y'_{j} \rangle \rbrace_f, \lbrace \langle x''_1,y''_1 \rangle, \ldots, \langle x''_{k},y''_{k} \rangle \rbrace_g \right] \\ \hspace*{70pt}\in \lbrace (+,x_{i+1}), (-,y_{i+1}), (\mathcal{O}^f, x'_{j+1}), (\mathcal{O}^g,x''_{k+1})\rbrace .\end{aligned}$$ Therefore, $T_{\tilde{R}}=\sigma$ if and only if $\forall 1\leq i \leq q_c, \forall 1\leq j \leq q_f$, and $\forall 1\leq k \leq q_g$, the $i,j,k$’th respective answers $\tilde{R}$ gives are $y_i$ or $x_i$, and $x'_j$ and $x''_k$, respectively. As $\mathcal{A}$ never repeats any part of a query, we have, by the definition of $\tilde{R}$, that the $i$’th cipher-query answer is an independent and uniform element of $G^2$, and as $f$ and $g$ were modelled as random function oracles, so too will their oracle outputs be independent and uniform elements of $G$. Hence, $$\begin{aligned}
\underset{\tilde{R}}{Pr}\left[ T_{\tilde{R}} = \sigma \right] = |G|^{-(2q_c+q_f+q_g)}.\end{aligned}$$
For the second part of this proof, we fix $k,g$ such that $\sigma \not\in \textsf{Bad}(k,g)$ and seek to compute $\underset{f,h}{Pr}\left[ T_{\tilde{\Psi}} = \sigma \right]$. Since $\sigma$ is a possible $\mathcal{A}$-transcript, we have that $T_{\tilde{\Psi}}= \sigma$ if and only if
- $y_i = \mathcal{F}_{g,f,f,g}(x_i\cdot k)\cdot k$ for all $1\leq i \leq q_c$,
- $y'_j = f(x'_j)$ for all $1\leq j \leq q_f$, and
- $y''_k = g(x''_k)$ for all $1\leq k \leq q_g$ (note that $g=h$ here.)
If we define $$\begin{aligned}
X_i &:= x_i^L\cdot k^L \cdot g(x_i^R\cdot k^R) \\
Y_i &:= y_i^R \cdot (k^R)^{-1} \cdot \left(g(y_i^L\cdot (k^L)^{-1})\right)^{-1},\end{aligned}$$ then $(y_i^L, y_i^R) = \tilde{\Psi}(x_i^L,x_i^R)$ if and only if
$k^R\cdot f(X_i) = (x_i^R)^{-1} \cdot Y_i$ and $X_i\cdot f(Y_i) = y_i^L \cdot (k^L)^{-1}$,
where the second equality of the latter is equivalent to $(k^L)^{-1}\cdot (f(Y_i))^{-1} = (y_i^L)^{-1}\cdot X_i$. Observe that, for all $1 \leq i < j \leq q_c$, $X_i \neq X_j$ (by ***B1***) and $Y_i \neq Y_j$ (by ***B2***.) Similarly, $1 \leq i < j \leq q_c$, $X_i \neq Y_j$ (by ***B3***.) Also, for all $1 \leq i \leq q_c$ and for all $1 \leq j \leq q_f$, $x'_j \neq X_i$ (by ***B4***) and $x'_j \neq Y_i$ (by ***B5***.) Hence, $\sigma \not\in \textsf{Bad}(k,g)$ implies that all inputs to $f$ are distinct. This then implies that $Pr_{f,h}\left[ T_{\tilde{\Psi}} = \sigma \right] = |G|^{-(2q_c+q_f+q_g)}$ as $h$ was also modelled as a random function, independent from $g$. Thus, as we assumed that $k$ and $g$ were chosen such that $\sigma \not\in \textsf{Bad}(k,g)$, $$\begin{aligned}
\underset{\tilde{\Psi}}{Pr}\left[ T_{\tilde{\Psi}} = \sigma | \sigma \not\in \textsf{Bad}(k,g)\right] &= |G|^{-(2q_c+q_f+q_g)} = \underset{\tilde{R}}{Pr}\left[ T_{\tilde{R}} = \sigma \right].\end{aligned}$$
Proof of Theorem \[GentryRamzanMain\]
-------------------------------------
To complete the proof of Theorem \[GentryRamzanMain\], we combine the above lemmas into the following probability estimation.
Let $\Gamma$ be the set of all possible and consistent $\mathcal{A}$-transcripts $\sigma$ such that $\mathcal{A}(\sigma)=1$. In the following, we ease notation, for the sake of the reader. We let $\textsf{BadG}(k)$ be denoted by $BadG$ and $\textsf{Bad}(k,g)$ by $Bad$. Furthermore, we abbreviate inconsistency as $incon.$. Let us consider the cases between $\Psi,\tilde{\Psi}$ and $\tilde{R}$.
$$\begin{aligned}
&\left| Pr_{\Psi}\left[ C_\mathcal{A}(T_\Psi)=1\right] - Pr_{\tilde{\Psi}}\left[ C_\mathcal{A}(T_{\tilde{\Psi}})=1\right] \right| \\
&\leq \left| \sum_{\sigma\in\Gamma} \left( Pr_{\Psi}\left[ T_\Psi = \sigma \right] - Pr_{\tilde{\Psi}}\left[ T_{\tilde{\Psi}} = \sigma \right]\right) \right| + Pr_{\tilde{\Psi}}\left[ T_{\tilde{\Psi}} \hspace*{4pt} incon.\right] \\
&\leq \sum_{\sigma\in \Gamma} \left| Pr_\Psi \left[ T_{\Psi} = \sigma \mid \sigma \not\in BadG \right] - Pr_{\tilde{\Psi}}\left[ T_{\tilde{\Psi}} = \sigma \right] \right| \cdot Pr_k \left[ \sigma \not\in BadG \right] \\
&\hspace*{15pt}+ \left| \sum_{\sigma \in \Gamma} \left( Pr_{\Psi} \left[ T_\Psi = \sigma \mid \sigma \in BadG \right] - Pr_{\tilde{\Psi}} \left[ T_{\tilde{\Psi}} = \sigma \right]\right) \cdot Pr_k \left[ \sigma \in BadG \right]\right| \\
&\hspace*{30pt}+ Pr_{\tilde{\Psi}}\left[ T_{\tilde{\Psi}} \hspace*{4pt} incon. \right] \\
&\leq \left| \sum_{\sigma \in \Gamma} \left( Pr_{\Psi} \left[ T_\Psi = \sigma \mid \sigma \in BadG \right] - Pr_{\tilde{\Psi}} \left[ T_{\tilde{\Psi}} = \sigma \right]\right) \cdot Pr_k \left[ \sigma \in BadG \right]\right| + q_c(q_c-1)|G|^{-1},\end{aligned}$$
where we in the last estimate used Lemma \[NotBadPsitoBarPsi\] and a consideration of the maximal amount of possible inconsistent pairs.
At the same time,
$$\begin{aligned}
&\left| Pr_{\tilde{\Psi}}\left[ C_\mathcal{A}(T_{\tilde{\Psi}})=1\right] - Pr_{\tilde{R}}\left[ C_\mathcal{A}(T_{\tilde{R}})=1\right] \right| \\
&\leq \left| \sum_{\sigma\in\Gamma} \left( Pr_{\tilde{\Psi}}\left[ T_{\tilde{\Psi}} = \sigma \right] - Pr_{\tilde{R}}\left[ T_{\tilde{R}} = \sigma \right]\right) \right| + Pr_{\tilde{R}}\left[ T_{\tilde{R}} \hspace*{4pt} incon.\right] + Pr_{\tilde{\Psi}}\left[ T_{\tilde{\Psi}} incon. \right] \\
&\leq \sum_{\sigma\in \Gamma} \left| Pr_{\tilde{\Psi}} \left[ T_{\tilde{\Psi}} = \sigma \mid \sigma \not\in Bad \right] - Pr_{\tilde{R}}\left[ T_{\tilde{R}} = \sigma \right] \right| \cdot Pr_k \left[ \sigma \not\in Bad \right] \\
&\hspace*{15pt} + \left| \sum_{\sigma \in \Gamma} \left( Pr_{\tilde{\Psi}} \left[ T_{\tilde{\Psi}} = \sigma \mid \sigma \in Bad \right] - Pr_{\tilde{R}} \left[ T_{\tilde{R}} = \sigma \right]\right) \cdot Pr_k \left[ \sigma \in Bad \right]\right| \\
&\hspace*{30pt}+ Pr_{\tilde{R}}\left[ T_{\tilde{R}} \hspace*{4pt} incon. \right] + Pr_{\tilde{\Psi}}\left[ T_{\tilde{\Psi}} incon. \right] \\
&\leq \left| \sum_{\sigma \in \Gamma} \left( Pr_{\tilde{\Psi}} \left[ T_{\tilde{\Psi}} = \sigma \mid \sigma \in Bad \right] - Pr_{\tilde{R}} \left[ T_{\tilde{R}} = \sigma \right]\right) \cdot Pr_k \left[ \sigma \in Bad \right]\right| + \begin{pmatrix} q_c \\ 2 \end{pmatrix}|G|^{-2} + 2\begin{pmatrix} q_c \\ 2 \end{pmatrix}|G|^{-1},\end{aligned}$$
where we in the last estimate used Lemma \[Lem43\] and the proof of Lemma \[Lem37\].
Let us use the above in a temporary estimate,
$$\begin{aligned}
&\left| Pr_\Psi\left[ C_\mathcal{A}(T_\Psi)=1\right] - Pr_R\left[C_\mathcal{A}(T_R)=1\right] \right| \nonumber \\
&= \left| Pr_\Psi\left[ C_\mathcal{A}(T_\Psi)=1\right] - Pr_{\tilde{\Psi}}\left[C_\mathcal{A}(T_{\tilde{\Psi}})=1\right] \right| \nonumber \\
&\hspace*{40pt}+ \left| Pr_{\tilde{\Psi}}\left[ C_\mathcal{A}(T_{\tilde{\Psi}})=1\right] - Pr_{\tilde{R}}\left[C_\mathcal{A}(T_{\tilde{R}})=1\right] \right| \nonumber \\
&\hspace*{80pt}+ \left| Pr_{\tilde{R}}\left[ C_\mathcal{A}(T_{\tilde{R}})=1\right] - Pr_R\left[C_\mathcal{A}(T_R)=1\right] \right| \nonumber \\
&\leq \left| \sum_{\sigma \in \Gamma} \left( Pr_{\Psi} \left[ T_\Psi = \sigma \mid \sigma \in BadG \right] - Pr_{\tilde{\Psi}} \left[ T_{\tilde{\Psi}} = \sigma \right]\right) \cdot Pr_k \left[ \sigma \in BadG \right]\right| + q_c(q_c-1)|G|^{-1} \nonumber\\
&\hspace*{30pt}+ \left| \sum_{\sigma \in \Gamma} \left( Pr_{\tilde{\Psi}} \left[ T_{\tilde{\Psi}} = \sigma \mid \sigma \in Bad \right] - Pr_{\tilde{R}} \left[ T_{\tilde{R}} = \sigma \right]\right) \cdot Pr_k \left[ \sigma \in Bad \right]\right| + \begin{pmatrix} q_c \\ 2 \end{pmatrix}|G|^{-2} + 2\begin{pmatrix} q_c \\ 2 \end{pmatrix}|G|^{-1} \nonumber\\
&\hspace*{60pt}+ \begin{pmatrix} q_c \\ 2 \end{pmatrix}|G|^{-2} \label{GenRamMainEstimate},\end{aligned}$$
where we in the last estimate also used Lemma \[Lem37\].
We may assume WLOG that
$$\begin{aligned}
\sum_{\sigma \in \Gamma} Pr_\Psi\left[ T_\Psi = \sigma \mid \sigma \in BadG \right] \cdot Pr_k\left[ \sigma \in BadG \right] \leq \sum_{\sigma \in \Gamma} Pr_{\tilde{\Psi}}\left[ T_{\tilde{\Psi}} = \sigma\right] \cdot Pr_k\left[ \sigma \in BadG \right]\end{aligned}$$
and likewise,
$$\begin{aligned}
\sum_{\sigma \in \Gamma} Pr_{\tilde{\Psi}}\left[ T_{\tilde{\Psi}} = \sigma \mid \sigma \in BadG \right] \cdot Pr_k\left[ \sigma \in BadG \right] \leq \sum_{\sigma \in \Gamma} Pr_{\tilde{R}}\left[ T_{\tilde{R}} = \sigma\right] \cdot Pr_k\left[ \sigma \in BadG \right],\end{aligned}$$
such that by Lemma \[Lem39\], respectively Lemma \[Lem42\], we get the following continued estimate from (\[GenRamMainEstimate\]), using the triangle inequality and that $|\Gamma|\leq |G|^{2q_c+q_f+q_g}$ (every combination of query elements).
$$\begin{aligned}
&\left| Pr_\Psi\left[ C_\mathcal{A}(T_\Psi)=1\right] - Pr_R\left[C_\mathcal{A}(T_R)=1\right] \right| \\
&\leq 2 \sum_{\sigma \in \Gamma} Pr_{\tilde{\Psi}} \left[T_{\tilde{\Psi}} = \sigma \right]\cdot Pr_k \left[ \sigma \in BadG \right] + 2q_c(q_c-1)|G|^{-1} \\
&\hspace*{30pt}+ 2\sum_{\sigma \in \Gamma} Pr_{\tilde{R}}\left[ T_{\tilde{R}}= \sigma\right] \cdot Pr_k \left[\sigma \in Bad \right] \\
&\hspace*{60pt}+ 2\begin{pmatrix} q_c \\ 2 \end{pmatrix}|G|^{-2} \\
&\leq 2|\Gamma|\cdot |G|^{-(2q_c+q_f+q_g)}\cdot \max_{\sigma \in \Gamma} Pr_k \left[ \sigma \in BadG \right] + 2q_c(q_c-1)|G|^{-1} \\
&\hspace*{30pt}+ 2|\Gamma|\cdot |G|^{-(2q_c+q_f+q_g)}\cdot\max_{\sigma\in\Gamma} Pr_k \left[ \sigma \in Bad \right] \\
&\hspace*{60pt}+ 2\begin{pmatrix} q_c \\ 2 \end{pmatrix}|G|^{-2} \\
&\leq 4q_gq_c\cdot |G|^{-1} + 2q_c(q_c-1)|G|^{-1} + 2\left(q_c^2 + 2q_fq_c + 2\begin{pmatrix} q_c \\ 2 \end{pmatrix}\right)|G|^{-1} + 2\begin{pmatrix} q_c \\ 2 \end{pmatrix}|G|^{-2} \\
&= (2q_c^2+4q_gq_c + 4q_fq_c + 2q_c^2-2q_c)|G|^{-1} + 2\begin{pmatrix} q_c \\ 2 \end{pmatrix}\left(2|G|^{-1} + |G|^{-2}\right).\end{aligned}$$
If we denote the total amount of queries as $q=q_c+q_f+q_g$, then we may quickly estimate and reword the main theorem as:
Let $f,g$ be modelled as random oracles, let $k=(k^L,k^R)\in_R G^2$, let $\Psi_k^{f,g}(x)=\mathcal{F}_{g,f,f,g}(x\cdot k)\cdot k$, and let $R\in_R \mathfrak{P}_{G^2 \rightarrow G^2}$. Then, for any $4$-oracle adversary $\mathcal{A}$, with at most $q$ total queries, we have $$\begin{aligned}
\left| Pr\left[ \mathcal{A}^{\Psi,\Psi^{-1},f,g} = 1\right] - Pr\left[ \mathcal{A}^{R,R^{-1},f,g} = 1\right] \right| \leq 2(3q^2-2q)|G|^{-1} + (q^2-q)|G|^{-2}.\end{aligned}$$
Given Theorem \[GentryRamzanMain\], we get, by using that $q_f,q_g\geq 0$, $$\begin{aligned}
&q_c^2 +2q_fq_c + 2q_gq_c + q_c^2 - q_c \\
&= 2(q_c^2 + q_fq_c + q_gq_c) - q_c \\
&\leq 2(q_c^2 + q_fq_c + q_gq_c) + (2(q_f+q_g)^2 + 2(q_fq_c+q_gq_c) -q_f-q_g)-q_c \\
&= 2(q_c^2 + 2q_fq_c + q_f^2 + 2q_fq_g + 2q_gq_c+ q_g^2) - (q_c+q_f+q_g) \\
&= 2(q_c+q_f+q_g)^2 - (q_c+q_f+q_g) \\
&= 2q^2-q.\end{aligned}$$ As $2\cdot\begin{pmatrix} q_c \\ 2 \end{pmatrix} =q_c^2-q_c \leq q^2-q$, we get the final estimate by some reordering.
Conclusion {#ConclusionSection}
==========
We generalized the Even and Mansour scheme as well as the Feistel cipher to work over arbitrary groups and proved that classical results pertain to the group versions. Based on the work in [@Gorjan], we hope that this opens avenues to proving that classical schemes may be made quantum secure by generalizing them to certain groups. For further work, we suggest generalizing other classical schemes and using the underlying group structures to do Hidden Shift reductions.
The author would like to thank his thesis advisor Gorjan Alagic for the topic, enlightening questions and answers, as well as the encouragements along the way. The author would also like to thank the Department of Mathematical Sciences, at the University of Copenhagen, for lending their facilities during the writing process.
Super Pseudorandomness of the Group EM Scheme
=============================================
In the following, we assume that the adversary $\mathcal{A}$ is unbounded computationally, but may only make polynomially many queries to the $E/D$- and $P/P^{-1}$-oracles, where all oracles act as black boxes and $P$ is a truly random permutation. We intend to play the “pseudorandom or random permutation game”: $\mathcal{A}$ is given an encryption oracle $E$ (with related decryption oracle $D$) which is randomly chosen with equal probability from the following two options:
1. A random key $k\in_R G$ is chosen uniformly and used to encrypt as $E(m)=E_k(m)=P(m\cdot k )\cdot k$, or
2. A random permutation $\pi\in_R \mathfrak{P}_{G \rightarrow G}$ is chosen and used to encrypt as $E(m)=\pi(m)$.
The adversary wins the game if it can distinguish how $E$ was chosen, with probability significantly better than $1/2$. More explicitly, we wish to prove the following for the group Even-Mansour scheme.
\[PseudoEMbounded\] Assume $P\in_R\mathfrak{P}_{G\rightarrow G}$ and let the key $k\in_R G$. For any probabilistic adversary $\mathcal{A}$, limited to polynomially many $E/D$- and $P/P^{-1}$-oracle queries, the adversarial advantage of $\mathcal{A}$ is bounded by $$\begin{aligned}
\label{AdvA}
\text{Adv}(\mathcal{A}) {\stackrel{\mathclap{\normalfont\mbox{\tiny def}}}{=}}\left| Pr\left[ \mathcal{A}_{E_k,D_k}^{P,P^{-1}} = 1 \right] - Pr\left[\mathcal{A}_{\pi,\pi^{-1}}^{P,P^{-1}} = 1 \right]\right| = \mathcal{O}\left(\frac{st}{|G|}\right).\end{aligned}$$ where $s$ is the total number of $E/D$-queries and $t$ is the total number of $P/P^{-1}$-queries, i.e. the success probability is negligible.
We may assume that $\mathcal{A}$ is deterministic (in essence, being unbounded computationally affords $\mathcal{A}$ the possibility of derandomizing its strategy by searching all its possible random choices and picking the most effective choices after having computed the effectiveness of each choice. For an example, see [@DingDong].) We may also assume that $\mathcal{A}$ never queries a pair in $S_s$ or $T_t$ more than once, where $S_i$ and $T_i$ are the sets of $i$ $E/D$- and $P/P^{-1}$-queries, respectively. Let us define two main games, that $\mathcal{A}$ could play, through oracle interactions (see next page for the explicit game descriptions.)
Note that the steps in italics have no impact on the response to $\mathcal{A}$’s queries, we simply continue to answer the queries and only note if the key turns bad, i.e. we say that a key $k$ is **bad w.r.t. the sets $S_s$ and $T_t$** if there exist $i,j$ such that either $m_i \cdot k = x_j$ or $c_i \cdot k^{-1} = y_j$, and $k$ is **good** otherwise. There are at most $\frac{2st}{|G|}$ bad keys.
**Game R**: We consider the random game which corresponds to the latter probability in (\[AdvA\]), i.e. $$\begin{aligned}
P_R := Pr\left[ \mathcal{A}_{\pi,\pi^{-1}}^{P,P^{-1}}=1\right].\end{aligned}$$
From the definition of **Game R**, we see that, letting $Pr_R$ denote the probability when playing **Game R**, $$\begin{aligned}
\label{PR}
Pr_R \left[ \mathcal{A}_{E,D}^{P,P^{-1}}=1\right] = P_R,\end{aligned}$$ as we are simply giving uniformly random answers to each of $\mathcal{A}$’s queries.
\[GamesXandR\] **Notation:** We let $S^1_i = \lbrace m | (m,c)\in S_{i} \rbrace, \hspace*{5pt} S^2_i = \lbrace c | (m,c)\in S_{i} \rbrace, T^1_i = \lbrace x | (x,y)\in T_{i} \rbrace,$ and $\hspace*{5pt} T^2_i = \lbrace y | (x,y)\in T_{i} \rbrace.$
------------------------------------------------------------------------
**GAME R:** Initially, let $S_0$ and $T_0$ be empty and flag unset. Choose $k\in_R G$, then answer the $i+1$’st query as follows:
**$E$-oracle query with $m_{i+1}$:**\
**1.** Choose $c_{i+1}\in_R G\setminus S^2_i$.\
**2.** *If $P(m_{i+1}\cdot k)\in T^2_i$, or $P^{-1}(c_{i+1}\cdot k^{-1})\in T^1_i$, then set flag to **bad**.*\
**3.** Define $E(m_{i+1})=c_{i+1}$ (and thereby also $D(c_{i+1})=m_{i+1}$) and return $c_{i+1}$.\
**$D$-oracle query with $c_{i+1}$:**\
**1.** Choose $m_{i+1}\in_R G\setminus S^1_i$.\
**2.** *If $P^{-1}(c_{i+1}\cdot k^{-1})\in T^1_i$, or $P(m_{i+1}\cdot k)\in T^2_i$, then set flag to **bad**.*\
**3.** Define $D(c_{i+1}) = m_{i+1}$ (and thereby also $E(m_{i+1})=c_{i+1}$) and return $m_{i+1}$.\
**$P$-oracle query with $x_{i+1}$:**\
**1.** Choose $y_{i+1}\in_R G\setminus T^2_i$.\
**2.** *If $E(x_{i+1}\cdot k^{-1})\in S^2_i$, or $D(y_{i+1}\cdot k)\in S^1_i$, then set flag to **bad**.*\
**3.** Define $P(x_{i+1}) = y_{i+1}$ (and thereby also $P^{-1}(y_{i+1})=x_{i+1}$) and return $y_{i+1}$.\
**$P^{-1}$-oracle query with $y_{i+1}$:**\
**1.** Choose $x_{i+1}\in_R G\setminus T^1_i$.\
**2.** *If $D(y_{i+1}\cdot k)\in S^1_i$, or $E(x_{i+1}\cdot k^{-1})\in S^2_i$, then set flag to **bad**.*\
**3.** Define $P^{-1}(y_{i+1}) = x_{i+1}$ (and thereby also $P(x_{i+1})=y_{i+1}$) and return $x_{i+1}$.
**GAME X:** Initially, let $S_0$ and $T_0$ be empty and flag unset. Choose $k\in_R G$, then answer the $i+1$’st query as follows:
**$E$-oracle query with $m_{i+1}$:**\
**1.** Choose $c_{i+1}\in_R G\setminus S^2_i$.\
**2.** If $P(m_{i+1}\cdot k)\in T^2_i$ then redefine $c_{i+1} := P(m_{i+1}\cdot k)\cdot k$ *and set flag to **bad***. Else if $P^{-1}(c_{i+1}\cdot k^{-1})\in T^1_i$, *then set flag to **bad** and* goto Step 1.\
**3.** Define $E(m_{i+1})=c_{i+1}$ (and thereby also $D(c_{i+1})=m_{i+1}$) and return $c_{i+1}$.\
**$D$-oracle query with $c_{i+1}$:**\
**1.** Choose $m_{i+1}\in_R G\setminus S^1_i$.\
**2.** If $P^{-1}(c_{i+1}\cdot k^{-1})\in T^1_i$ then redefine $m_{i+1} := P^{-1}(c_{i+1}\cdot k^{-1})\cdot k^{-1}$ *and set flag to **bad***. Else if $P(m_{i+1}\cdot k)\in T^2_i$, *then set flag to **bad** and* goto Step 1.\
**3.** Define $D(c_{i+1}) = m_{i+1}$ (and thereby also $E(m_{i+1})=c_{i+1}$) and return $m_{i+1}$.\
**$P$-oracle query with $x_{i+1}$:**\
**1.** Choose $y_{i+1}\in_R G\setminus T^2_i$.\
**2.** If $E(x_{i+1}\cdot k^{-1})\in S^2_i$ then redefine $y_{i+1} := E(x_{i+1}\cdot k^{-1})\cdot k^{-1}$ *and set flag to **bad***. Else if $D(y_{i+1}\cdot k)\in S^1_i$, *then set flag to **bad** and* goto Step 1.\
**3.** Define $P(x_{i+1}) = y_{i+1}$ (and thereby also $P^{-1}(y_{i+1})=x_{i+1}$) and return $y_{i+1}$.\
**$P^{-1}$-oracle query with $y_{i+1}$:**\
**1.** Choose $x_{i+1}\in_R G\setminus T^1_i$.\
**2.** If $D(y_{i+1}\cdot k)\in S^1_i$ then redefine $x_{i+1} := D(y_{i+1}\cdot k)\cdot k$ *and set flag to **bad***. Else if $E(x_{i+1}\cdot k^{-1})\in S^2_i$, *then set flag to **bad** and* goto Step 1.\
**3.** Define $P^{-1}(y_{i+1}) = x_{i+1}$ (and thereby also $P(x_{i+1})=y_{i+1}$) and return $x_{i+1}$.
**Game X**: Consider the experiment which corresponds to the game played in the prior probability in (\[AdvA\]) and define this probability as $$\begin{aligned}
P_X := Pr\left[ \mathcal{A}_{E_k,D_k}^{P,P^{-1}}=1\right].\end{aligned}$$ We define **Game X**, as outlined above. Note that again the parts in italics have no impact on the response to $\mathcal{A}$’s queries, however, this time, when a key becomes *bad*, we choose a new random value repeatedly for the response until the key is no longer *bad*, and then reply with this value. Intuitively, **Game X** behaves like **Game R** except that **Game X** checks for consistency as it does not want $\mathcal{A}$ to win on some collision. It is non-trivial to see that, letting $Pr_X$ denote the probability when playing **Game X**, $$\begin{aligned}
\label{PX}
Pr_X \left[ \mathcal{A}_{E,D}^{P,P^{-1}}=1\right] = P_X.\end{aligned}$$ The proof is given in Appendix \[ExplainX\].
We have defined both games in such a way that their outcomes differ only in the event that a key turns *bad*. Thus, any circumstance which causes a difference in the instructions carried out by the games, will also cause both games to set the flag to *bad*. Let $BAD$ denote the event that the flag gets set to *bad* and the case that the flag is not set to *bad* by $\neg BAD$, then the two following lemmas follow from the previous statement.
\[BadisBad\] $Pr_R\left[ BAD \right] = Pr_X\left[ BAD \right]$ and $Pr_R\left[ \neg BAD \right] = Pr_X\left[ \neg BAD \right]$.
\[Notbadisnotbad\] $Pr_R\left[ \mathcal{A}_{E,D}^{P,P^{-1}}=1| \neg BAD \right] = Pr_X\left[ \mathcal{A}_{E,D}^{P,P^{-1}}=1| \neg BAD \right]$.
Using these two lemmas we are able to prove the lemma:
$\text{Adv}(\mathcal{A}) \leq Pr_R\left[ BAD \right]$.
This is because, using (\[PR\]), (\[PX\]), and lemmas \[BadisBad\] and \[Notbadisnotbad\], $$\begin{aligned}
\text{Adv}(\mathcal{A}) &= |P_X - P_R| \\
&= \left| Pr_X\left[ \mathcal{A}_{E,D}^{P,P^{-1}}=1 \right] - Pr_R\left[ \mathcal{A}_{E,D}^{P,P^{-1}}=1 \right] \right| \\
&= | Pr_X\left[ \mathcal{A}_{E,D}^{P,P^{-1}}=1 | \neg BAD \right]\cdot Pr_X\left[\neg BAD \right] \\
&\hspace*{25pt}+ Pr_X\left[ \mathcal{A}_{E,D}^{P,P^{-1}}=1 | BAD \right]\cdot Pr_X\left[ BAD \right] \\
&\hspace*{45pt} - Pr_R\left[ \mathcal{A}_{E,D}^{P,P^{-1}}=1 | \neg BAD \right]\cdot Pr_R\left[\neg BAD \right] \\
&\hspace*{65pt}- Pr_R\left[ \mathcal{A}_{E,D}^{P,P^{-1}}=1 | BAD \right]\cdot Pr_R\left[ BAD \right] | \\
&= \left| Pr_R\left[ BAD \right]\cdot \left( Pr_X\left[ \mathcal{A}_{E,D}^{P,P^{-1}}=1 | BAD \right] - Pr_R\left[ \mathcal{A}_{E,D}^{P,P^{-1}}=1 | BAD \right]\right) \right| \\
&\leq Pr_R\left[ BAD \right].\end{aligned}$$
Let us now define yet another game, **Game R’**.
------------------------------------------------------------------------
**GAME R’:** Initially, let $S_0$ and $T_0$ be empty and flag unset. Answer the $i+1$’st query as follows: **$E$-oracle query with $m_{i+1}$:**\
**1.** Choose $c_{i+1}\in_R G\setminus S^2_i$.\
**2.** Define $E(m_{i+1}):=c_{i+1}$ (and thereby also $D(c_{i+1}):=m_{i+1}$) and return $c_{i+1}$.\
**$D$-oracle query with $c_{i+1}$:**\
**1.** Choose $m_{i+1}\in_R G\setminus S^1_i$.\
**2.** Define $D(c_{i+1}) := m_{i+1}$ (and thereby also $E(m_{i+1}):=c_{i+1}$) and return $m_{i+1}$.\
**$P$-oracle query with $x_{i+1}$:**\
**1.** Choose $y_{i+1}\in_R G\setminus T^2_i$.\
**2.** Define $P(x_{i+1}) := y_{i+1}$ (and thereby also $P^{-1}(y_{i+1}):=x_{i+1}$) and return $y_{i+1}$.\
**$P^{-1}$-oracle query with $y_{i+1}$:**\
**1.** Choose $x_{i+1}\in_R G\setminus T^1_i$.\
**2.** Define $P^{-1}(y_{i+1}) := x_{i+1}$ (and thereby also $P(x_{i+1}):=y_{i+1}$) and return $x_{i+1}$.\
*After all queries have been answered, choose $k\in_R G$. If there exists $(m,c)\in S_{s}$ and $(x,y)\in T_{t}$ such that $k$ becomes bad then set flag to **bad**.*
------------------------------------------------------------------------
This game runs as **Game R** except that it does not choose a key until all of the queries have been answered and then checks for badness of the flag (by checking whether or not the key has become bad). It can be shown that the flag is set to ***bad*** in **Game R** if and only if the flag is set to ***bad*** in **Game R’** (by a consideration of cases (see Appendix \[App:ReqRR\].)) Hence, we get the following lemma.
$Pr_R\left[ BAD \right] = Pr_{R'} \left[ BAD \right]$.
Using the above lemma, we now only have to bound $Pr_{R'} \left[ BAD \right]$ in order to bound $\text{Adv}(\mathcal{A})$, but as the adversary queries at most $s$ elements to the $E/D$-oracles and at most $t$ elements to the $P/P^{-1}$-oracles, and the key $k$ is chosen uniformly at random from $G$, we have that the probability of choosing a bad key is at most $2st/|G|$, i.e. $$\begin{aligned}
\text{Adv}(\mathcal{A}) \leq Pr_{R'} \left[ BAD \right] = \mathcal{O}\left( \frac{st}{|G|} \right).\end{aligned}$$
Restating the theorem, we get:
For any probabilistic adversary $\mathcal{A}$, limited to polynomially many $E/D$- and $P/P^{-1}$-oracle queries, the generalized EM scheme over a group $G$ is a super pseudorandom permutation.
Proof of probability of Game X {#ExplainX}
==============================
Recall the definition of $S^1_i, S^2_i, T^1_i$ and $T^2_i$ (see p. .) We write $S_s$ and $T_t$ to denote the final transcripts. We drop the index $i$ if it is understood. We begin by defining **Game X’**.
------------------------------------------------------------------------
**GAME X’:** Initially, let $S_0$ and $T_0$ be empty. Choose $k\in_R G$, then answer the $i+1$’st query as follows:
**$E$-oracle query with $m_{i+1}$:**\
**1.** If $P(m_{i+1}\cdot k)\in T^2_i$ return $P(m_{i+1}\cdot k)\cdot k$\
**2.** Else choose $y_{i+1}\in_R G\setminus T^2_i$, define $P(m_{i+1}\cdot k) = y_{i+1}$, and return $y_{i+1}\cdot k$.\
**$D$-oracle query with $c_{i+1}$:**\
**1.** If $P^{-1}(c_{i+1}\cdot k^{-1}) \in T^1_i$, return $P^{-1}(c_{i+1}\cdot k^{-1})\cdot k^{-1}$.\
**2.** Else choose $x_{i+1}\in_R G\setminus T^1_i$, define $P^{-1}(c_{i+1}\cdot k^{-1})=x_{i+1}$, and return $x_{i+1}\cdot k^{-1}$.\
**$P$-oracle query with $x_{i+1}$:**\
**1.** If $P(x_{i+1})\in T^2_i$, return $P(x_{i+1})$.\
**2.** Else choose $y_{i+1}\in_R G\setminus T^2_i$, define $P(x_{i+1})=y_{i+1}$, and return $y_{i+1}$.\
**$P^{-1}$-oracle query with $y_{i+1}$:**\
**1.** If $P^{-1}(y_{i+1})\in T^1_i$, return $P^{-1}(y_{i+1})$.\
**2.** Else choose $x_{i+1}\in_R G\setminus T^1_i$, define $P^{-1}(y_{i+1})=x_{i+1}$, and return $x_{i+1}$.
------------------------------------------------------------------------
Notice that the only difference between **Game X’** and the game defining $P_X$ is that the latter has defined all values for the oracles beforehand while the former “defines as it goes.” Still, an adversary cannot tell the difference between playing the **Game X’** or the game defining $P_X$. Thus, $Pr_{X'}\left[ \mathcal{A}_{E,D}^{P,P^{-1}}=1 \right] = P_X$.
What we wish to show is that $$\begin{aligned}
Pr_X \left[ \mathcal{A}_{E,D}^{P,P^{-1}}=1\right] = Pr_{X'} \left[ \mathcal{A}_{E,D}^{P,P^{-1}}=1\right],\end{aligned}$$ i.e. that no adversary $\mathcal{A}$ may distinguish between playing **Game X** and playing **Game X’**, even negligibly. We will do this by showing that no adversary $\mathcal{A}$ may distinguish between the outputs given by the two games. As both games begin by choosing a uniformly random key $k$ and as we show that for this value the games are identical, we hereby assume such a key $k$ to be a fixed, but arbitrary, value for the remainder of this proof.
Considering the definitions of **Game X** and **Game X’**, we see that the two games define their $E/D$- and $P/P^{-1}$-oracles differently: the former defining both, while the latter defines only the $P/P^{-1}$-oracle and computes the $E/D$-oracle. We show that **Game X** also answers its $E/D$-oracle queries by referring to $P/P^{-1}$, although not directly.
Given the partial functions $E$ and $P$ in **Game X**, i.e. functions having been defined for all values up to and including the $i$’th query, define the partial function $\widehat{P}$ as the following. $$\begin{aligned}
\widehat{P}(x) {\stackrel{\mathclap{\normalfont\mbox{\tiny def}}}{=}}\begin{cases}
P(x) & \text{if } P(x) \text{ is defined,} \\
E(x\cdot k^{-1}) \cdot k^{-1} & \text{if } E(x\cdot k^{-1}) \text{ is defined, and} \\
\text{undefined} & \text{otherwise.}
\end{cases}\end{aligned}$$ Using the above definition, defining a value for $E$ or $P$ implicitly defines a value for $\widehat{P}$. The first question is, whether or not $\widehat{P}$ is well-defined, i.e. whether there are clashes of values (that is, differences between values differing by other than $\cdot k$ (or $\cdot k^{-1}$)) for some $x$ for which both $P(x)$ and $E(x\cdot k^{-1})$ are defined.
Let $E$ and $P$ be partial functions arising in **Game X**, then the partial function $\widehat{P}$ is well-defined.
Proof by induction on the number of “Define” steps in **Game X** (i.e. steps $E-3, D-3, P-3,$ and $P^{-1}-3$) as these are the steps where $\widehat{P}$ becomes defined. The initial case of the induction proof is trivial as $S_0$ and $T_0$ are empty such that no values may clash. Suppose now that in step $E-3$ we define $E(m)=c$. The only possibility that $\widehat{P}$ becomes ill-defined will occur if the new $E(m)$ value clashes with a prior defined $P(m\cdot k)$ value: If $P(m \cdot k)$ was not defined, then no clashes can arise. If $P(m\cdot k)$ was defined, then by step $E-2$, the value is $E(m)\cdot k^{-1}$, such that there is no clash.
For $D-3$, the argument is similar as $E(m)$ will become defined as well. Although, for the case where $P(m\cdot k)$ is defined, step $D-2$ forces a new uniformly random value of $m$ to be chosen until no clash occurs.
Analogously, for $P$ and $P^{-1}$, no clashes will arise, hence, $\widehat{P}$ must be well-defined.
We may also consider $\widehat{P}$ in **Game X’**, in the sense that when we define a value for $P$ in the game, we implicitly define a value for $\widehat{P}$ where $\widehat{P}(x)=P(x)$ as $E(x\cdot k^{-1})=P(x)$ in **Game X’**.
We wish now to show that the oracle query-answers of $E, D, P,$ and $P^{-1}$ in **Game X**, expressed in terms of $\widehat{P}$, correspond exactly to those in **Game X’**.
**Case 1: $E$-oracle query.** Beginning with **Game X**, we first note that **Game X** never defines $E(m)$ unless $m$ has been queried to the $E$-oracle, or alternately, the $D$-oracle has been queried with a $c$ such that $E(m)=c$. However, as $\mathcal{A}$ never repeats a query if it can guess the answer, i.e. never queries any part of an already defined $E/D$-oracle pair, we may assume that $E(m)$ is undefined when $m$ is queried. Therefore, we see that concurrently with $m$ being queried, we have that $\widehat{P}(m\cdot k)$ will be defined if and only if $P(m\cdot k)$ is defined, and $\widehat{P}(m\cdot k) = P(m\cdot k)$. Let us consider the two cases: when $\widehat{P}(m\cdot k)$ is defined and when it is undefined.
- When $\widehat{P}(m\cdot k)$ is defined, then **Game X** returns $c = \widehat{P}(m\cdot k)\cdot k$. Setting $E(m)=c$ leaves $\widehat{P}$ unchanged, i.e. the value $\widehat{P}(m\cdot k)$ remains the same, unlike the next case.
- When $\widehat{P}(m \cdot k)$ is undefined, then **Game X** repeatedly chooses ${c\in_R G \setminus S^2}$ uniformly until $P^{-1}(c\cdot k^{-1})$ is undefined, i.e. the set $U=\lbrace c\in G | P^{-1}(c\cdot k^{-1})\not\in T^1\rbrace$. It follows that $y=c\cdot k^{-1}$ is uniformly distributed over $G \setminus \widehat{T}^2$.[^4] This can be seen by showing that $S^2 \cup U^\complement = \widehat{T}^2\cdot k$, where the only non-triviality in the argument follows from the definition of $\widehat{P}$. In this case, setting $E(m)=c$ also sets $\widehat{P}(m\cdot k)=y$, in contrast to the prior case as it is now defined.
We now consider the same query on **Game X’**.
- When $\widehat{P}(m\cdot k)=P(m\cdot k)$ is defined, $c=P(m\cdot k)\cdot k$ is returned, and $\widehat{P}$ is unchanged.
- When $\widehat{P}(m\cdot k)=P(m\cdot k)$ is undefined, we choose $y\in_R G\setminus T^2 = G\setminus \widehat{T}^2$, $\widehat{P}(m\cdot k)$ is set to $y$, and $c=y\cdot k$ is returned.
Thus, the behaviour of **Game X** and **Game X’** are identical on the $E$-oracle queries.
We will be briefer in our arguments for the following $3$ cases as the arguments are similar.
**Case 2: $D$-oracle query.** Here we again assume that no element of an $E/D$-oracle pair $(m,c)$, such that $E(m)=c$, has been queried before. Like in the above case, we see that, as $\widehat{P}(m \cdot k) = P(m\cdot k)$, we also have $\widehat{P}^{-1}(c \cdot k^{-1}) = P^{-1}(c\cdot k^{-1})$.
- When $\widehat{P}^{-1}(c\cdot k^{-1})=P^{-1}(c\cdot k^{-1})$ is defined, then $m=P^{-1}(c\cdot k^{-1})\cdot k^{-1}$ is returned, leaving $\widehat{P}^{-1}(c \cdot k^{-1})$ unchanged in both games.
- If $\widehat{P}^{-1}(c\cdot k^{-1})=P^{-1}(c\cdot k^{-1})$ is undefined, then $x \in_R G \setminus \widehat{T}^1$ is chosen uniformly and $\widehat{P}^{-1}(c\cdot k^{-1}) = x$, in both cases.
Thus, the behaviour of **Game X** and **Game X’** are identical on the $D$-oracle queries.
**Case 3: $P$-oracle query.** Here we instead assume that no element of a $P/P^{-1}$-oracle pair $(x,y)$ such that $P(x)=y$, has been queried before.
- Using the definition of the $E$- and $P$-oracles in **Game X** and the definition of $\widehat{P}$ we see that $P(x)$ is defined if and only if $E(x\cdot k^{-1})$ is defined, but then this also holds if and only if $\widehat{P}(x)$ is defined (by the assumption in the beginning of case $3$). Hence, if $\widehat{P}(x)$ is defined, then $y=E(x\cdot k^{-1})\cdot k^{-1}=\widehat{P}(x)$. Indeed, both games secure this value.
- If $\widehat{P}(x)$ is undefined, then $y \in_R G \setminus \widehat{T}^2$ is chosen uniformly and $\widehat{P}(x)$ is defined to be $y$, in both cases.
Thus, the behaviour of **Game X** and **Game X’** are identical on the $P$-oracle queries.
**Case 4: $P^{-1}$-oracle query.** Again, we assume that no element of a $P/P^{-1}$-oracle pair $(x,y)$ such that $P(x)=y$, has been queried before.
- Using the definition of **Game X** and the definition of $\widehat{P}$, as well as our case $4$ assumption, we see that $\widehat{P}^{-1}(y)$ is defined if and only if $D(y\cdot k)$ is defined. Hence, if $\widehat{P}^{-1}(y)$ is defined, then $x=D(y\cdot k) \cdot k = \widehat{P}^{-1}(y)$. Indeed, both games secure this value.
- If $\widehat{P}^{-1}(y)$ is undefined, then $x \in_R G \setminus \widehat{T}^1$ is chosen uniformly and $\widehat{P}^{-1}(y)$ is defined to be $x$, in both cases.
Thus, the behaviour of **Game X** and **Game X’** are identical on the $P^{-1}$-oracle queries. Q.E.D.
Proof that the probability of Game R and Game R’ match {#App:ReqRR}
======================================================
Recall the definition of $S^1_i, S^2_i, T^1_i$ and $T^2_i$ (see p. ). We write $S_s$ and $T_t$ to denote the final transcripts. We also introduce the following definition.
\[overlapidentical\] We say that two $E/D$-pairs $( m_i,c_i )$ and $( m_j,c_j )$ **overlap** if $m_i=m_j$ or $c_i=c_j$. If $m_i=m_j$ and $c_i=c_j$, we say that the pairs are **identical**. Likewise for $P/P^{-1}$-pairs $( x_i,y_i )$ and $( x_j,y_j )$.
If two pairs overlap, then by the definition of the $E/D$- and $P/P^{-1}$-oracles, they must be identical. Therefore, WLOG, we may assume that all queries to the oracles are non-overlapping. Let us now prove the lemma.
$Pr_R\left[ BAD \right] = Pr_{R'} \left[ BAD \right]$.
We need to prove that **Game R** has its flag set to **bad** if and only if **Game R’** has its flag set to **bad**.
“$\Rightarrow$”: We want to show that there exists $(m,c)\in S_s$ and $(x,y)\in T_t$ such that either $m\cdot k = x$ or $c\cdot k^{-1} = y$ (i.e. such that $k$ becomes bad). We have to consider the $8$ cases where the flag is set to bad. All of the cases use an analogous argument to the following: If $P(m \cdot k)$ is defined then $P(m\cdot k) = y = P(x)$ for some $(x,y)\in T_{t}$ such that, as overlapping pairs are identical, $m\cdot k = x$.
“$\Leftarrow$”: We assume that there exists $(m,c)\in S_s$ and $(x,y)\in T_t$ such that $k$ becomes bad. i.e. such that either $m\cdot k = x$ or $c\cdot k^{-1} = y$. We need to check that in all four oracle queries, the flag in **Game R** is set to bad, which needs a consideration of 8 cases.
Assume that $m\cdot k = x$, then $$\begin{aligned}
E\text{-oracle on } m &: P(m\cdot k)=P(x) = y \in T_t^2, \\
D\text{-oracle on } c &: P(m\cdot k)=P(x) = y \in T_t^2, \\
P\text{-oracle on } x &: E(x\cdot k^{-1}) = E(m) = c \in S_s^2, \\
P^{-1}\text{-oracle on } y &: E(x\cdot k^{-1}) = E(m) = c \in S_s^2.\end{aligned}$$
Assume now that $c\cdot k^{-1} = y$, then $$\begin{aligned}
E\text{-oracle on } m &: P^{-1}(c\cdot k^{-1})=P^{-1}(y) = x \in T_t^1, \\
D\text{-oracle on } c &: P^{-1}(c\cdot k^{-1})=P^{-1}(y) = x \in T_t^1, \\
P\text{-oracle on } x &: D(y\cdot k) = D(c) = m \in S_s^1, \\
P^{-1}\text{-oracle on } y &: D(y\cdot k) = D(c) = m \in S_s^1.\end{aligned}$$
[^1]: This article is based on work done for my Master’s Thesis at the University of Copenhagen. For more details on the thesis, contact me at aehogo@gmail.com.
[^2]: Considering the approximation $p(n)\approx \tfrac{n^2}{2m}$, where $p(n)$ is the probability of there being a Birthday Problem collision from $n$ randomly chosen elements from the set of $m$ elements, then $p(\sqrt{|G|})\approx \tfrac{\sqrt{|G|}^2}{2|G|}=1/2$.
[^3]: The proof of the proposition follows the argument of Proposition 3.3 in [@NaorReingold].
[^4]: $\widehat{T}^1$ and $\widehat{T}^2$ are the corresponding sets on the query pairs of $\widehat{P}$.
|
---
abstract: 'We report on an X-ray observation of the Seyfert 1.5 galaxy Mrk 6 obtained with the EPIC instruments onboard . Archival PDS data from were also used to constrain the underlying hard power-law continuum. The results from our spectral analyses generally favor a double partial-covering model, although other spectral models such as absorption by a mixture of partially ionized and neutral gas cannot be firmly ruled out. Our best-fitting model consists of a power law with a photon index of $\Gamma=1.81^{+0.22}_{-0.20}$ and partial covering with large column densities up to $N_{\rm H}\sim10^{23}~{\rm cm}^{-2}$. We also detect a narrow emission line consistent with Fe K$\alpha$ fluorescence at $6.45^{+0.03}_{\rm -0.04}$ keV with an equivalent width of $93^{+26}_{\rm -20}$ eV. Joint analyses of , , and data further provide evidence for both spectral variability (a factor of $\sim2$ change in absorbing column) and absorption-corrected flux variations (by $\sim60$%) during the $\sim4$ year period probed by the observations.'
author:
- |
Stefan Immler, W. N. Brandt, Cristian Vignali, Franz E. Bauer,\
D. Michael Crenshaw, John J. Feldmeier, and Steven B. Kraemer
title: 'Probing the Complex and Variable X-ray Absorption of Markarian 6 with [*XMM-Newton*]{}'
---
Introduction
============
Markarian 6 (Mrk 6, IC 0450; $z=0.0188$; 77 Mpc distant for $H_0=75~{\rm km~s}^{-1}~{\rm Mpc}^{-1}$) is one of a handful of intermediate Seyferts of type 1.5, including NGC 4151 and Mrk 766, showing evidence for an ‘ionization cone’ rather than the ‘ionization halo’ expected for a type 1 viewing geometry (Meaburn, Whitehead, & Pedlar 1989; Kukula et al. 1996). A torus ‘atmosphere’ along the line of sight has been suggested to explain the presence of the ionization cone (e.g., Evans et al. 1993). The torus atmosphere is thought to be optically thick for ionizing radiation between the Lyman edge and soft X-rays, but it allows radiation outside of this spectral range (e.g., broad optical line emission from the Broad Line Region) to pass through with only moderate obscuration. Mrk 6 shows significant optical line profile variations on time-scales of months to years, suggesting that at least some of the gaseous material creating the lines can undergo coherent variations (e.g., Rosenblatt et al. 1992; Eracleous & Halpern 1993).
X-ray investigations of Seyfert galaxies are well suited to probing matter along the line of sight toward their central black hole regions. Prominent examples of intermediate Seyferts that have been well studied in X-rays include the ‘cousins’ of Mrk 6, NGC 4151 (e.g., Weaver et al. 1994; Ogle et al. 2000; Yang, Wilson, & Ferruit 2001; Schurch & Warwick 2002) and Mrk 766 (e.g., Leighly et al. 1996; Page et al. 1999, 2001; Matt et al. 2000; Branduardi-Raymont et al. 2001). In contrast, the only published X-ray study of Mrk 6 is that of Feldmeier et al. (1999; hereafter F99) using 0.6–9.5 keV data. These data revealed heavy and complex X-ray absorption that was best fit by a double partial-covering model with large column densities \[$\sim(3$–$20)\times10^{22}~{\rm cm}^{-2}$\], likely due to the torus atmosphere. However, detailed X-ray spectral modeling was limited by modest photon statistics and the fact that the absorption in Mrk 6 dominates the X-ray spectrum throughout most of the bandpass.
We capitalized on the superior throughput (about an order of magnitude higher than ) and good spectral resolution of the instruments onboard to investigate further the X-ray emission and absorption properties of Mrk 6. We also used the data to perform sensitive searches for temporal and spectral variability during the observation and in comparison with earlier and observations.
X-ray Observations and Data Reduction {#obs}
=====================================
Mrk 6 was observed with the European Photon Imaging Camera (EPIC) onboard (Jansen et al. 2001) with on-source exposure times of 31.8 ks (EPIC p-n) and 30.8 ks (EPIC MOS 1 and MOS 2). It was also observed with the Reflection Grating Spectrometer (RGS) for 46.4 ks (RGS 1) and 37.9 ks (RGS 2). The EPIC data allow sensitive imaging spectroscopy with moderate spectral ($E / \Delta E \sim20$–$50$) and angular ($6''$ FWHM) resolution. The parameters of the individual observations are listed in Tab. \[obs\_tab\].
Both the p-n and MOS data were acquired in ‘full-frame mode’ using the medium filter. The data were reduced using the Science Analysis System (SAS) v5.3.3 with the latest calibration products. Due to background flares at the end of the observation period, 7% of the p-n and 31% of the MOS 1 and MOS 2 data were rejected. After inspection of a source-free background spectrum extracted from CCD 1 on the p-n and MOS detectors, we selected data only in the 0.3–12 keV energy range due to enhanced background at low photon energies (up to a factor of $\sim5$). We furthermore excluded bad and hot pixels from the data. Since the cleaned RGS data did not yield enough photon statistics for spectral analysis, they were not used further.
Source counts were extracted for the p-n and MOS detectors within $30''$-radius circular regions (the 85% encircled-energy radius at 7.5 keV). The background was extracted locally from source-free regions. For the p-n we used a rectangular region of size $2\farcm4\times3\farcm8$ close to Mrk 6 on the same p-n chip, and for the MOS we used an annulus with inner and outer radii of $2\farcm5$ and $4\farcm3$. Given the observed count rates of $\sim 1.2$ and $\sim 0.4~{\rm cts~s}^{-1}$ for the p-n and MOS detectors, the pile-up fractions and dead times are estimated to be $\ls 2$%. The source counts were binned with a minimum of $50$ counts per bin to allow $\chi^2$ spectral fitting. We also constructed background-corrected light curves of Mrk 6 in the soft (0.3–2 keV), hard (2–12 keV), and broad (0.3–12 keV) bands by binning the events into 250 s intervals and excluding telemetry dropouts.
Archival MECS 2+3 (Medium Energy Concentrator Spectrometer) and LECS (Low Energy Concentrator Spectrometer) data are also used in the analyses below (see Tab. \[obs\_tab\]); these data cover the 1.3–10 keV (MECS) and 0.5–4.5 keV (LECS) energy bands. To extend the spectral range of our analyses to higher photon energies, which is useful to constrain the underlying power-law continuum, we furthermore used the PDS (Phoswich Detection System) data in the energy range 18–120 keV.
We checked for contamination of the PDS data due to other X-ray sources within its field of view ($1\fdg3$ FWHM). No strong X-ray source other than Mrk 6 is found within the $\sim30'$-diameter EPIC field of view in high-energy ($>5$ keV) images. The All-Sky Survey broad-band source catalog shows only two other X-ray sources (NGC 2256 and NGC 2258) located within the PDS field of view at large off-axis angles ($23\farcm2$ and $18\farcm1$). Both of these sources have 2–10 keV MECS fluxes much lower (only $\sim2$–$3\%$) than that of Mrk 6. Since the PDS effective area decreases sharply with increasing off-axis angle, their contamination to the Mrk 6 PDS spectrum is negligible.
Results
=======
Spectral Analysis {#spec_analysis}
-----------------
Spectral analysis was performed using XSPEC v11.2 (Arnaud 1996). All errors are quoted at the $90\%$ level of confidence for one parameter of interest ($\Delta \chi^2 = 2.71$; Avni 1976) and Galactic absorption ($N_{\rm H}=6.4\times10^{20}~{\rm cm}^{-2}$; Stark et al. 1992) is always implicitly included. The previous observation has shown that Mrk 6 is an intrinsically absorbed system ($\gs10^{22}$–$10^{23}~{\rm cm}^{-2}$; F99). We therefore used the 18–120 keV PDS data, which should be much less affected by absorption, to determine the shape of the underlying X-ray continuum. Fitting a power-law model to the PDS data (model 1 in Tab. \[spec\_tab\]) gives a best-fit photon index of $\Gamma=1.81^{+0.22}_{-0.20}$ and an unabsorbed 20–100 keV flux of $5.0\times10^{-11}~{\rm ergs~cm}^{-2}~{\rm s}^{-1}$.
We then fitted the and PDS spectra simultaneously. Due to uncertainties in the cross-calibration of the instruments (p-n and MOS: $\ls7\%$; Snowden 2002) as well as possible variability between the and observations, we left all normalizations for the spectral components free. All other spectral parameters were tied together. We first tried fitting a power-law model with simple (fully covering) intrinsic absorption. Such absorption could arise in the host galaxy, perhaps in gas associated with the irregularly distributed dust seen in the image of Malkan, Gorjian, & Tam (1998). It could also arise on smaller scales in the nuclear region. This model is statistically rejected ($\chi^2_{\nu}=3.24$; ${\rm d.o.f.=867}$). It also gives a photon index ($\Gamma\sim0.9$) well below that derived from our PDS analysis and expected intrinsically for a Seyfert galaxy ($\Gamma\sim1.6$–$2.2$; e.g., Nandra et al. 1997; Risaliti 2002).
We next tried fitting a power-law model with a single partial-covering absorption component. Partially covering absorption could arise from electron scattering of X-rays in the nucleus of Mrk 6 (see F99), and such absorption has been used successfully to explain the X-ray spectra of other intermediate Seyferts. This model also includes the simple intrinsic absorption used in the previous model, since again there could be absorption in the host galaxy after the X-rays have escaped the nucleus (this simple intrinsic absorption will be implicitly included in all of the following spectral fits). Finally, we added a Gaussian emission line to model Fe K$\alpha$ emission visible in the spectrum. This model is marginally acceptable ($\chi^2_{\nu}=1.08$; ${\rm d.o.f.=859}$) although it leaves clear systematic residuals below $\sim2$ keV and above $\sim40$ keV. Furthermore, like the previous model, it requires an implausibly low photon index of $\Gamma\sim1.3$. If we constrain the photon index to lie in the range 1.6–2.2, the fit is statistically unacceptable.
Given the failures of the two previous spectral models, we tried a model consisting of a power law and two partially covering absorbers (model 2 in Tab. \[spec\_tab\]). This model was successfully used by F99 to describe the data, and double partial covering could arise if electron scattering of X-rays provides several lines of sight into the nucleus. The photon index was fixed to the best-fit value derived from the PDS data ($\Gamma = 1.81$; our results are not changed materially if the photon index is left as a free parameter), and we also included a Gaussian Fe K$\alpha$ emission line. Despite some small systematic residuals below $\sim 2$ keV, a good overall fit to the and data is found ($\chi^2_{\nu}=0.96$; ${\rm d.o.f.=858}$; see Fig. \[spectrum\]). The absorption includes a high column density ($10.96^{+0.58}_{-0.42}\times10^{22}~{\rm cm}^{-2}$) component covering $(57\pm1)\%$ of the emitting region, as well as a lower column density ($2.46^{+0.07}_{-0.05}\times10^{22}~{\rm cm}^{-2}$) component with a higher covering percentage of ($93\pm1)\%$. The best-fit rest-frame energy of the Fe K$\alpha$ line is $6.45^{+0.03}_{\rm -0.04}$ keV with an equivalent width of $93^{+26}_{\rm -20}$ eV. There is no clear evidence for a broad Fe K$\alpha$ line, and the fitted line properties are consistent with those from . Given its properties, the observed line could originate via reprocessing in the outer part of the accretion disk (e.g., George & Fabian 1991) or perhaps in Broad Line Region clouds (see equation 5 in Eracleous, Halpern, & Livio 1996).
The Fe K$\alpha$ line is fairly weak for a Seyfert galaxy, suggesting that Compton reflection is unlikely to make a dominant contribution to the X-ray spectrum. To examine any effects of Compton reflection further, we replaced the simple power law in the previous fit with a power law plus neutral reflection (the ‘PEXRAV’ model in XSPEC assuming solar abundances). All other aspects of the model were unchanged. We obtain a relative reflection fraction (defined as $r=\Omega/2\pi$, where $\Omega$ is the angle subtended by the reflector) of $r=1.21^{+0.14}_{-0.16}$, an inclination angle of cos $i=0.94^{+0.01}_{-0.25}$, and a cut-off energy of $\sim 238$ keV (model 3 in Tab. \[spec\_tab\]). The fit quality is slightly improved compared to the previous fit ($\chi^2_{\nu}=0.95$; ${\rm d.o.f.=855}$), but there is no material change in the nature of the partially covering absorption. Due to both statistical and systematic uncertainties associated with this model, we do not consider the derived inclination angle to have clear physical significance.
We also tested a model which assumes that the nuclear X-ray emission is attenuated by both partially ionized and neutral material, dispersed along the line of sight. This model was proposed by Schurch & Warwick (2002) to characterize the complex X-ray absorption of NGC 4151, a Seyfert with many similarities to Mrk 6 (see §1). Although the model gives a relatively good fit to the 2–120 keV data, large residuals are visible below $\sim2$ keV ($\chi^2_{\nu}=1.14$; ${\rm d.o.f.=866}$; see Fig. \[spectrum\]). We obtain an ionization parameter of ${\rm log}~\xi = {\rm log}~(L_{\rm ion}/nr^2) = 2.42\pm0.01$ (with $L_{\rm ion}$ the source ionizing luminosity in the 0.0136–13.6 keV band, $n$ the number of hydrogen atoms/ions per $\rm cm^{-3}$, and $r$ the distance from the central source to the photoionized region), an ionized column density of $1.33^{+0.01}_{-0.02}\times10^{23}~{\rm cm}^{-2}$, and a neutral column density of $3.35^{+0.17}_{-0.11}\times10^{21}~{\rm cm}^{-2}$.
A slightly better fit ($\chi^2_{\nu}=1.09$; ${\rm d.o.f.=863}$) is obtained if the simple power law in the previous fit is replaced with a power law plus neutral reflection (model 4 in Tab. \[spec\_tab\]). As for the double partial-covering model, the reflection is not a dominant spectral component and the best-fit absorption parameters do not change materially.
We performed several additional fits to check the general robustness of our results. Models with no intrinsic absorption and Compton reflection cannot provide acceptable fits. Models with fully covering intrinsic absorption and Compton reflection can at best provide marginally acceptable fits. These models also suffer from physical consistency problems since the energy region near the Fe K$\alpha$ line is dominated by the reflection component. An Fe K$\alpha$ line with a large equivalent width of $\sim1$–$2$ keV is then expected (e.g., Matt, Brandt, & Fabian 1996) but not observed.
Using the double partial-covering model (model 2 in Tab. \[spec\_tab\]), we derive observed 0.3–2 keV and 2–10 keV fluxes of $5.3\times10^{-13}~{\rm ergs~cm}^{-2}~{\rm s}^{-1}$ and $1.2\times10^{-11}~{\rm ergs~cm}^{-2}~{\rm s}^{-1}$, respectively. These values are consistent with those in §2.3.5 of F99, given statistical and systematic uncertainties. The absorption-corrected 2–10 keV flux and luminosity are $2.0\times10^{-11}~{\rm ergs~cm}^{-2}~{\rm s}^{-1}$ and $1.4\times10^{43}~{\rm ergs~s}^{-1}$, respectively. The absorption-corrected 0.3–2 keV flux and luminosity are less certain due to the large required absorption correction; likely values are $\sim 1.5\times10^{-11}~{\rm ergs~cm}^{-2}~{\rm s}^{-1}$ and $\sim 1.1\times10^{43}~{\rm ergs~s}^{-1}$, respectively. Using the partially ionized and neutral absorption model (model 4 in Tab. \[spec\_tab\]), we derive an absorption-corrected 2–10 keV flux and luminosity $\sim36\%$ lower than from the double partial-covering model.
Temporal and Spectral Variability {#timing_analysis}
---------------------------------
After fitting constant models to light curves and inspecting the residuals, we do not find any rapid, large-amplitude X-ray variability during the or observations. While constant-model fits are statistically inconsistent with the p-n, MECS, and LECS light curves, there are no strong, systematic variations about these fits (the poor fit quality appears to be due to stochastic small-amplitude variability only slightly larger than the statistical noise). We can constrain the amplitude of any systematic variability to be $\ls15\%$ for the p-n, $\ls17\%$ for the MECS, and $\ls29\%$ for the LECS. Within the individual observations, no evidence for spectral variability is found upon analysis of hardness ratios (e.g., 2–12 keV to 0.3–2 keV) computed as a function of time. No variability was seen in the data (F99), consistent with our results here.
We have checked for inter-observation spectral variations via joint spectral fitting. A joint spectral analysis of the and data gives a fit with $\chi^2_{\nu}=1.04$ (${\rm d.o.f.=1595}$).[^1] This joint fit is statistically acceptable, indicating that we cannot prove spectral variability between the and observations. If we use the and data, however, we obtain $\chi^2_{\nu}=1.20$ (${\rm d.o.f.=1410}$). This joint fit can be rejected with $>99.9$% confidence, indicating spectral variability. Simultaneous and observations of the Seyfert 1 galaxies NGC 5548 (Pounds et al. 2003) and IC 4329A (Gondoin et al. 2001) have shown that excellent fits can be found in joint spectral analyses, ruling out major cross-calibration errors. The spectral differences between the and data of Mrk 6 are largest below $\sim 3$ keV where absorption dominates the spectral shape (see Fig. \[spectrum\_sax\]), so absorption variability seems a likely explanation. We therefore analyzed the data (MECS, LECS and PDS) individually, which gives best-fit column densities a factor of $\sim 2$ smaller compared to ($N_{\rm H}^1 =4.76^{+0.27}_{-0.25} \times 10^{22}~{\rm cm}^{-2}$; $N_{\rm H}^2 =1.37^{+0.08}_{-0.07} \times 10^{22}~{\rm cm}^{-2}$) and covering fractions which are consistent with our results \[$f_{\rm c}^1 =(54\pm1)\%$; $f_{\rm c}^2 =(95\pm1)\%$\]. The observed and fluxes are $1.1\times10^{-12}~{\rm ergs~cm}^{-2}~{\rm s}^{-1}$ and $2.4\times10^{-11}~{\rm ergs~cm}^{-2}~{\rm s}^{-1}$, respectively. These values are significantly higher than during the observation. The absorption-corrected 2–10 keV flux and luminosity are $3.2\times10^{-11}~{\rm ergs~cm}^{-2}~{\rm s}^{-1}$ and $2.1\times10^{43}~{\rm ergs~s}^{-1}$, respectively. The 2–10 keV luminosity of Mrk 6 during the observation was $\sim 60$% higher than during the observation.
Discussion and Summary {#discussion}
======================
The analyses above have substantially tightened the constraints upon the X-ray absorption and emission properties of Mrk 6. The spectrum has $\sim 7$ times as many counts and a wider bandpass than the earlier spectrum, and the 18–120 keV PDS data have provided the first reliable determination of the underlying X-ray continuum shape (a critical quantity for modeling of the X-ray absorption). The absorption measured in the new data can be fit acceptably with the same basic double partial-covering model used to fit the data (F99), providing substantially improved support for the applicability of this model to Mrk 6. The measured column densities are large with $N_{\rm H}$ up to $\sim 10^{23}$ cm$^{-2}$, so absorption controls the shape of the X-ray spectrum up to $\sim 6$ keV. As discussed in §4.2.1 of F99, the absorption seen in X-rays is substantially larger than expected from the optical reddening; the X-ray absorbing material may be dust poor. The small 0.3–1 keV residuals left by the double partial-covering model may be due to $\sim 5\times 10^{40}$ ergs s$^{-1}$ emission from ionized gas in the nucleus or host galaxy. These residuals are consistent with X-ray emission from the Narrow Line Region, similar to that observed in NGC 4151 (Ogle et al. 2000) and the Seyfert 2 galaxies NGC 1068 (Ogle et al. 2003) and Mrk 3 (Sako et al. 2000). The residuals around 0.8–0.9 keV could be Fe L emission, but this spectral complexity is difficult to model due to the limited low-energy photon statistics of the EPIC instruments and the insufficient signal-to-noise ratio of the RGS spectra.
Motivated by recent studies of NGC 4151 (Schurch & Warwick 2002), we also tried fitting the measured X-ray absorption with another physically plausible model consisting of both ionized and neutral columns of gas. This model provides an acceptable fit to the X-ray spectrum above 2 keV, but it leaves substantially larger residuals at lower energies than does the double partial-covering model (visible in the lower panel of Fig. \[spectrum\]). Again these residuals may be plausibly explained by emission from ionized gas in the nucleus or host galaxy.
Three pieces of evidence suggest that, at least at high energies, our X-ray observations have penetrated all the way to the black hole region of Mrk 6: (1) the 18–120 keV continuum shape measured by the PDS is consistent with those of Seyfert 1 galaxies (compare with §3.2 of Malaguti et al. 1999), (2) the relatively small equivalent width of the Fe K$\alpha$ line ($EW=93^{+26}_{-20}$ eV) indicates significant dilution of the reflection continuum at 6.4 keV by direct power-law emission from the black hole region, and (3) the relative 2–10 keV, \[O [iii]{}\], and far-infrared luminosities are consistent with those of Seyfert 1 galaxies (see §4.1 of F99). Given these results, we can have confidence that the 2–10 keV ($L_{2-10}=1.4\times10^{43}~{\rm ergs~s}^{-1}$ from ) and 20–100 keV ($L_{20-100}=3.5\times10^{43}~{\rm ergs~s}^{-1}$ from ) luminosities we have found for Mrk 6 represent those of the intrinsic X-ray continuum (rather than just scattered X-ray emission, for example). The X-ray luminosity of Mrk 6 is $\sim 5$ times the average X-ray luminosity of NGC 4151 ($L_{2-10}\sim 3\times10^{42}~{\rm ergs~s}^{-1}$; $L_{20-100}\sim 1\times10^{43}~{\rm ergs~s}^{-1}$; Yang et al. 2001; Schurch & Warwick 2002; Schurch 2002). Its lower X-ray flux arises only because it is $\sim 5.8$ times more distant.
Our analyses have also revealed significant X-ray absorption variability during the 1.5 yr between the and observations. In the double partial-covering model, the column densities of both partial covering components drop by a factor of $\sim2$ while the covering fractions do not change significantly. If the fitted partial covering in fact indicates that we have both direct, absorbed as well as electron-scattered, unabsorbed views of the nucleus, the observed changes seem physically plausible. The column densities along the direct line of sight could change due to gas motions in the torus atmosphere, while the scattered fraction would not change if scattering occurs on large spatial scales. The mechanism of absorption variability must allow the column density to undergo large fractional changes; such changes could perhaps arise due to bulk rotation of a torus or Poisson fluctuations in a relatively small number of obscuring ‘clouds’ along the line of sight. Our discovery of X-ray absorption variability from Mrk 6 (as well as the observed variability amplitude and timescale) is generally consistent with the absorption variability seen from other absorbed Seyferts (e.g., Risaliti, Elvis, & Nicastro 2002; Schurch & Warwick 2002).
Based on our results, further monitoring of the X-ray absorption variability of Mrk 6 is merited to refine understanding of its amplitude and timescale. Such monitoring could be significantly complemented by coordinated optical spectroscopy, given the known optical line profile variability (see §1). High-quality ultraviolet spectroscopy of Mrk 6 would allow connections to be made between the X-ray absorption and any ultraviolet absorption. The archival [*IUE*]{} data on Mrk 6 suggest intrinsic Mg [ii]{} absorption, but higher quality data are required for a proper study.
We thank L.A. Antonelli, M.C. Eracleous, and N.J. Schurch for helpful discussions. The project was supported by NASA grants NAG5-9940 and LTSA NAG5-8107 and NAG5-13035.
Arnaud, K.A. 1996, in ASP Conf. Ser. 101, Astronomical Data Analysis Software and Systems V, ed. G.H. Jacoby, & J.Barnes (San Francisco: ASP), 17 Avni, Y. 1976, , 210, 642 Branduardi-Raymont, G., Sako, M., Kahn, S.M., Brinkman, A.C., Kaastra, J.S., & Page, M.J. 2001, , 365, L140 Eracleous, M., & Halpern, J.P. 1993, , 409, 584 Eracleous, M., Halpern, J.P., & Livio, M. 1996, , 459, 89 Evans, I.N., Tsvetanov, Z., Kriss, G.A., Ford, H.C., Caganoff, S., & Koratkar, A. 1993, , 417, 82 Jansen, F., Lumb, D., Altieri, B., et al. 2001, A&A, 365, L1 Feldmeier, J.J., Brandt, W.N., Elvis, M., Fabian, A.C., Iwasawa, K., & Mathur, S. 1999, , 510, 167 (F99) George, I.M., & Fabian, A.C. 1991, , 249, 352 Gondoin, P., Barr, P., Lumb, D., Oosterbroek, T., Orr, A., & Parmar, A.N. 2001, , 378, 806 Kukula, M.J., Holloway, A.J., Pedlar, A., et al. 1996, , 280, 1283 Leighly, K.M., Mushotzky, R.F., Yaqoob, T., Kunieda, H., & Edelson, R. 1996, , 469, 147L Malaguti, G., et al. 1999, , 342, L41 Malkan, M.A., Gorjian, V., & Tam, R. 1998, , 117, 25 Matt, G., Brandt, W.N., & Fabian, A.C. 1996, , 280, 832 Matt, G., Perola, G.C., Fiore, F., Guainazzi, M., Nicastro, F., & Piro, L. 2000, , 363, 863 Meaburn, J., Whitehead, M.J., & Pedlar, J. 1989, , 241, 1P Nandra, K., George, I.M., Mushotzky, R.F., Turner, T.J., & Yaqoob, T. 1997, , 477, 602 Ogle, P.M., Marshall, H.L., Lee, J.C., & Canizares, C.R. 2000, , 545, L81 Ogle, P.M., Brookings, T., Canizares, C.R., Lee, C.J., & Marshall, H.L. 2003, , in press, astro-ph/0211406 Page, M.J., Carrera, F.J., Mittaz, J.P.D., & Mason, K.O. 1999, , 305, 775 Page, M.J., Mason, K.O., Carrera, F.J., et al. 2001, , 365, L152 Pounds, K.A., Reeves, J.N., Page, K.L., Edelson, R., Matt, G., & Perola, G.C. 2003, , submitted, astro-ph/0210288 Risaliti, G. 2002, , 386, 379 Risaliti, G., Elvis, M., & Nicastro, F. 2002, , 571, 234 Rosenblatt, E.I., Malkan, M.A., Sargent, W.L.W., & Readhead, A.C.S. 1992, , 81, 59 Sako, M., Kahn, S.M., Paerels, F., & Liedahl, D.A. 2000, , 334, 811 Schurch, N.J., & Warwick, R.S. 2002, , 334, 811 Schurch, N.J. 2002, private communication Snowden, S..L. 2002, in: New Visions of the X-ray Universe in the XMM-Newton and Chandra Era, ESTEC, Noordwijk, The Netherlands, astro-ph/0203311 Stark, A.A., Gammie, C.F., Wilson, R.W., Bally, J., Linke, R.A., Heiles, C., & Hurwitz, M. 1992, , 79, 77 Weaver, K.A. Yaqoob, T., Holt, S.S., Mushotzky, R.F., Matsuoka, M., & Yamauchi, M. 1994, , 436, L27 Yang, Y., Wilson, A.S., & Ferruit, P. 2001, , 563, 124
[lcccccc]{} & GIS & 75041000 & 1997-04-07 & 42.1\
& SIS & 75041000 & 1997-04-07 & 36.4\
& LECS & 51067001 & 1999-09-14 & 49.3\
& MECS & 51067001 & 1999-09-14 & 109.4\
& PDS & 51067001 & 1999-09-14 & 52.0\
& p-n & 0061540101 & 2001-03-27 & 31.8\
& MOS 1 & 0061540101 & 2001-03-27 & 30.8\
& MOS 2 & 0061540101 & 2001-03-27 & 30.8\
& RGS 1 & 0061540101 & 2001-03-27 & 46.4\
& RGS 2 & 0061540101 & 2001-03-27 & 37.9\
[lcccccc]{} Absorption, $N_{\rm H}$ ($10^{22}~{\rm cm}^{-2}$) & $\cdots$ & $0.11^{+0.01}_{-0.01}$ & $0.11^{+0.01}_{-0.01}$ & $0.34^{+0.02}_{-0.01}$\
Photon index, $\Gamma$ & $1.81^{+0.22}_{-0.20}$ & $1.81$ (fixed) & $\cdots$ & $\cdots$\
$A1$ (${\rm cts~keV}^{-1}~{\rm cm}^{-2}~{\rm s}^{-1}$) & $9.48^{+0.12}_{-0.10} \times 10^{-2}$ & $5.39^{+0.05}_{-0.06} \times 10^{-3}$ & $\cdots$ & $\cdots$\
Partial covering, $N_{\rm H}^1$ ($10^{22}~{\rm cm}^{-2}$) & $\cdots$ & $10.96^{+0.58}_{-0.42}$ & $8.05^{+0.69}_{-0.48}$ & $\cdots$\
Partial covering, $f_{\rm c}^1$ ($\%$) & $\cdots$ & $57^{+1}_{-1}$ & $56^{+2}_{-1}$ & $\cdots$\
Partial covering, $N_{\rm H}^2$ ($10^{22}~{\rm cm}^{-2}$) & $\cdots$ & $2.46^{+0.07}_{-0.05}$ & $2.27^{+0.03}_{-0.08}$ & $\cdots$\
Partial covering, $f_{\rm c}^2$ ($\%$) & $\cdots$ & $93^{+1}_{-1}$ & $91^{+1}_{-1}$ & $\cdots$\
Photon index, $\Gamma$ & $\cdots$ & $\cdots$ & 1.81 (fixed) & 1.81 (fixed)\
Cutoff energy (keV) & $\cdots$ & $\cdots$ & $238^{+499}_{-54}$ & $2735^{+383}_{-2368}$\
Inclination angle, cos $i$ & $\cdots$ & $\cdots$ & $0.94^{+0.01}_{-0.25}$ & $0.95^{+0.01}_{-0.19}$\
Reflection fraction, $r$ & $\cdots$ & $\cdots$ & $1.21^{+0.14}_{-0.16}$ & $1.33^{+0.21}_{-0.01}$\
$A2$ (${\rm cts~keV}^{-1}~{\rm cm}^{-2}~{\rm s}^{-1}$) & $\cdots$ & $\cdots$ & $4.68^{+0.02}_{-0.02} \times 10^{-3}$ & $4.59^{+0.03}_{-0.06} \times 10^{-3}$\
Warm absorption, $N_{\rm H}$ ($10^{22}~{\rm cm}^{-2}$) & $\cdots$ & $\cdots$ & $\cdots$ & $13.32^{+0.12}_{-0.13}$\
Ionization parameter, ${\rm log}~\xi$ & $\cdots$ & $\cdots$ & $\cdots$ & $2.42^{+0.01}_{-0.01}$\
Line energy, $E$ (keV) & $\cdots$ & $6.45^{+0.03}_{-0.04}$ & $6.47^{+0.03}_{-0.03}$ & $6.45^{+0.04}_{-0.03}$\
Line $EW$ (eV) & $\cdots$ & $93^{+26}_{-20}$ & $87^{+20}_{-26}$ & $84^{+19}_{-26}$\
$A3$ (${\rm cts~cm}^{-2}~{\rm s}^{-1}$) & $\cdots$ & $1.85^{+0.60}_{-0.37} \times 10^{-5}$ & $1.68^{+0.39}_{-0.53} \times 10^{-5}$ & $1.67^{+0.41}_{-0.33} \times 10^{-5}$\
$\chi^2/{\rm d.o.f.}~(\chi^2_{\nu})$ & 3.61/9 (0.40) & 824.2/858 (0.96) & 812.8/855 (0.95) & 937.3/863 (1.09)\
[ ]{}
[ ]{}
[^1]: The spectra used in this analysis were the same as those presented in F99 (see Tab. \[obs\_tab\]).
|
---
abstract: 'This paper describes the security weakness of a recently proposed secure communication method based on discrete-time chaos synchronization. We show that the security is compromised even without precise knowledge of the chaotic system used. We also make many suggestions to improve its security in future versions.'
address: 'Instituto de Física Aplicada, Consejo Superior de Investigaciones Científicas, Serrano 144—28006 Madrid, Spain'
author:
- 'G. Álvarez'
- 'F. Montoya'
- 'M. Romera'
- 'G. Pastor'
title: 'Cryptanalyzing a discrete-time chaos synchronization secure communication system'
---
,
,
,
Introduction
============
In recent years, a growing number of cryptosystems based on chaos have been proposed [@asocscs; @cc], many of them fundamentally flawed by a lack of robustness and security [@stusc; @cocborcr; @emmbc; @uamccs; @pwtcisea; @bcsugse; @bcscuas; @ccscurm; @coaces; @coacscs; @coaecc; @otsoacespwccifcp; @coadccuek]. In [@sdcudtcs], a secure communication system based on chaotic modulation using discrete-time chaos synchronization is proposed. Two different schemes of message encoding are presented. In the first scheme, the binary message ($m(i)=\pm 1$) is multiplied by the chaotic output signal of the transmitter and then sent to drive the receiver system. In the second scheme, the binary message is modulated by multiplication with the chaotic output signal and then is fed back to the transmitter system and simultaneously sent to the receiver system.
Discrete-time chaotic systems are generally described by a set of nonlinear difference equations. The first communication system based on modulation by multiplication can be described by:
$$\label{eq:modmultr}
\rm transmitter\left\{ \begin{array}{l}
x_1(i + 1) = 1-\alpha x_1^2(i) + x_2(i) \\
x_2(i + 1) = \beta x_1(i) \\
s(i) = x_1(i) \cdot m(i) \\
\end{array} \right.$$
$$\label{eq:modmulrc}
\rm receiver\left\{ \begin{array}{l}
\hat{x}_1(i + 1) = 1-\alpha s^2(i) + \hat{x}_2(i) \\
\hat{x}_2(i + 1) = \beta \hat{x}_1(i) \\
\hat{m}(i)=s(i)/\hat{x}_1(i)
\end{array} \right.$$
The communication scheme using modulation by multiplication and feedback, with a modification to avoid divergence due to feedback, is described by:
$$\label{eq:modmulfbtr}
\rm transmitter\left\{ \begin{array}{l}
x_1(i + 1) = 1-\alpha (s(i)- \left\lfloor{\frac{s(i)+P}{2P}}\right\rfloor 2P)^2 + x_2(i) \\
x_2(i + 1) = \beta x_1(i) + 0.05x_1(i)(m(i)-1) \\
s(i) = x_1(i) \cdot m(i) \\
\end{array} \right.$$
$$\label{eq:modmulfbrc}
\rm receiver\left\{ \begin{array}{l}
\hat{x}_1(i + 1) = 1-\alpha (s(i)- \left\lfloor{\frac{s(i)+P}{2P}}\right\rfloor 2P)^2 + \hat{x}_2(i) \\
\hat{x}_2(i + 1) =\beta \hat{x}_1(i)+0.05(s(i)-\hat{x}_1(i)) \\
\hat{m}(i)=s(i)/\hat{x}_1(i)
\end{array} \right.$$
with $P=(1+\sqrt{6.6})/2.8$.
Although the authors seemed to base the security of their cryptosystems on the chaotic behavior of the output of the Henon non-linear dynamical system, no analysis of security was included. It was not considered whether there should be a key in the proposed system, what it should consist of, what the available key space would be, what precision to use, and how the key would be managed.
In the next section we discuss the weaknesses of this secure communication system using the Henon attractor and make some suggestions to improve its security.
Attacks on the proposed system {#sec:attack}
==============================
The key space
-------------
Although it is not explicitly stated in [@sdcudtcs], it is assumed that the key is formed by the two parameters of the map, $\alpha$ and $\beta$. Thus, in [@sdcudtcs], the key is fixed to $k=\{\alpha,\beta\}=\{1.4,0.3\}$. However, in [@sdcudtcs] there is no information given about what the key space is. The key space is defined by all the possible valid keys. The size of the key space $r$ is the number of encryption/decryption key pairs that are available in the cipher system.
In this chaotic scheme the key space is nonlinear because all the keys are not equally strong. We say that a key is *weak* or *degenerated* if it is easier to break a ciphertext encrypted with this key than breaking a ciphertext encrypted with another key from the key space.
The study of the chaotic regions of the parameter space from which valid keys, i.e., parameter values leading to chaotic behavior, can be chosen is missing in [@sdcudtcs]. A possible way to describe the key space might be in terms of positive Lyapunov exponents. According to [@caitds p. 196], let $\mathbf{f}$ be a map of ${\mathbb{R}}^m$, $m\geq 1$, and $\{{\mathbf{x}}_0,{\mathbf{x}}_1, {\mathbf{x}}_2,\dots\}$ be a bounded orbit of $\mathbf{f}$. The orbit is chaotic if
1. it is not asymptotically periodic,
2. no Lyapunov exponent is exactly zero, and
3. the largest Lyapunov exponent is positive.
The largest Lyapunov exponent can be computed for different combinations of the parameters. If it is positive, then the combination can be used as a valid key. In Fig. \[fig:lyap\], the chaotic region for the Henon attractor used in [@sdcudtcs] has been plotted. This region corresponds to the keyspace. In general, parameters chosen from the lower white region give rise to periodic orbits, undesirable because the ciphertext is easily predictable. Parameters chosen from the upper white region give rise to unbounded orbits diverging to infinity, and hence the system can not work. Therefore, both regions should be avoided to get suitable keys. Only keys within the black region are good. And even within this region, there exist periodic windows, unsuitable for robust keys.
This type of irregular and often fractal chaotic region shared by most secure communication systems proposed in the literature is inadequate for cryptographic purposes because there is no easy way to define its boundary. And if the boundary is not mathematically and easily defined, then it is hard to find suitable keys within the key space. This difficulty in defining the key space discourages the use of the Henon map. Instead, complete chaoticity for any parameter value should be preferred. Piecewise linear (PWL) maps are a good choice because they behave chaotically for any parameter value in the useful interval [@spodplcmatricaprc].
Insensitivity to parameter mismatch
-----------------------------------
Both communication systems, the one based on modulation by multiplication and the one using modulation by multiplication and feedback, can only have valid keys carefully chosen from the chaotic region plotted in Fig. \[fig:lyap\] to avoid periodic windows and divergence. Due to low sensitivity to parameter mismatch, if the system key is fixed to $k=\{\alpha,\beta\}=\{1.4,0.3\}$ as in [@sdcudtcs], then any key $k'$ chosen from the same key space will decrypt the ciphertext into a message $m'$ with an error rate which is well below 50%. Fig. \[fig:error\] plots the bit error rate (BER) when the ciphertext encrypted with $k=\{\alpha,\beta\}=\{1.4,0.3\}$ is decrypted using keys $k'$ from the valid key space at a distance $d$ from $k$. For this experiment the Euclidean distance was chosen:
$$\label{eq:d}
d=\sqrt{(\alpha-\alpha\,')^2+(\beta-\beta\,')^2}$$
This insensitivity to parameter mismatch due to the coupling between transmitter and receiver renders the system totally insecure when the Henon map is used. A different map more sensitive to small differences in the parameter values should be used to grant security.
Brute force attacks
-------------------
A brute force attack is the method of breaking a cipher by trying every possible key. The quicker the brute force attack, the weaker the cipher. Feasibility of brute force attacks depends on the key space size $r$ of the cipher and on the amount of computational power available to the attacker. Given today’s computer speed, it is generally agreed that a key space of size $r<2^{100}$ is insecure.
However, this requirement might be very difficult to meet by this cipher because the key space does not allow for such a big number of different strong keys. For instance, Fig. \[fig:lyap\] was created using a resolution of $10^{-3}$, i.e., there are $1400\times 3000$ different points. To get a number of keys $r>2^{100}\simeq 10^{30}$, the resolution should be $10^{-15}$. However, with that resolution, thousands of keys would be equivalent, unless there is a strong sensitivity to parameter mismatch, which is usually lost by synchronization, even when using a different chaotic map.
Statistical analysis
--------------------
Fig. \[fig:error\]a shows that the error is upper bounded: BER$\leq0.33$. This is a consequence of the fact that the orbit followed by any initial point in the Henon attractor is not uniformly distributed, because in average it spends two thirds of the time above $x=0$. As a consequence, mixing the cleartext with the output of a function whose probability density is not uniform will result in a weak cryptosystem. In Fig. \[fig:map\] the Henon attractor is plotted. It can be observed that the distribution is far from flat because the orbit visits more often the region $x>0$. In average, two thirds of the iterates lie to the right of $x=0$ (depicted as a dashed line). This fact allows the attacker to guess in average two thirds of the encrypted bits, even with no knowledge about the transmitter/receiver structure.
To get a balanced distribution, the threshold should be moved to the right [@thaaakg]. Let $x_m$ denote the real value such that
$$\label{eq:media}
P(x_i\leq x_m)=P(x_i>x_m)=0.5.$$
A good estimation presented in [@thaaakg] is $\hat{x}_m=0.39912$, depicted as a dotted line in Fig. \[fig:map\]. However, this result is difficult to apply provided the way in which the Henon attractor is used by the cryptosystem. Therefore, it is seen again that the Henon map is a bad choice as a chaotic map for this communication scheme. A different map with a balanced distribution, i.e., whose orbit visits with equal frequency the regions above and below a certain level $x=0$, should be chosen to prevent statistical attacks.
Plaintext attacks
-----------------
In the previous sections we showed that the use of the Henon map is not advisable because of its inability to define a good key space, of its low sensitivity to parameter mismatch, and of its non uniformly distributed orbits. We are to show next that if a different map is used, the security of the communication system will not improve if the same key is used repeatedly for successive encryptions.
According to [@ctap p. 25], it is possible to differentiate between different levels of attacks on cryptosystems. In a known plaintext attack, the opponent possesses a string of plaintext, $p$, and the corresponding ciphertext, $c$. In a chosen plain text, the opponent has obtained temporary access to the encryption machinery, and hence he can choose a plain text string, $p$, and construct the corresponding cipher text string, $c$.
The cipher under study behaves as a modified version of the one-time pad [@ctap p. 50]. The one-time pad uses a randomly generated key of the same length as the message. To encrypt a message $m$, it is combined with the random key $k$ using the exclusive-OR operation bitwise. Mathematically,
$$\label{eq:pad}
c(i)=m(i)+k(i)\mod2,$$
where $c$ represents the encrypted message or ciphertext. This method of encryption is perfectly secure because the encrypted message, formed by XORing the message and the random secret key, is itself totally random. It is crucial to the security of the one-time pad that the key be as long as the message and never reused, thus preventing two different messages encrypted with the same portion of the key being intercepted or generated by an attacker.
Eq. (\[eq:modmultr\]) and Eq. (\[eq:modmulfbtr\]) are used to generate a keystream $\{x_1(1)=k(1),x_1(2)=k(2),x_1(3)=k(3),\ldots\}$. This keystream is used to encrypt the plain text string according to the rule
$$\label{eq:rule}
c(i)=k(i)\cdot m(i)$$
Therefore, if the attacker possesses the plaintext $m(i)$ and its corresponding ciphertext $c(i)$, he will be able to obtain $k(i)$. If the same key, i.e. the same parameter values, is used to encrypt any subsequent message in the future, it will generate an identical chaotic orbit, which is already known. As a consequence, when $c(i)$ and $k(i)$ are known in Eq. (\[eq:rule\]), $m(i)$ is readily obtained by the attacker.
Obviously, when using this cryptosystem, regardless of the choice of the chaotic map, the key can never be reused. A slight improvement to partially enhance security even when the key is reused consists of randomly setting the initial point of the chaotic orbit at the transmitter end. Synchronization will guarantee that the message is correctly decrypted by the authorized receiver. However, an eavesdropper would have more difficulty in using past chaotic orbits because they will diverge due to sensitivity to initial conditions.
Conclusions {#sec:conclusion}
===========
The proposed cryptosystem using the Henon map is rather weak, since it can be broken without knowing its parameter values and even without knowing the transmitter precise structure. However, the overall security might be highly improved if a different chaotic map with higher number of parameters is used. The inclusion of feedback makes it possible to use many different systems with non symmetric nonlinearity as far as the whole space is folded into a bounded domain to avoid divergence. However, to rigorously present future improvements, it would be desirable to explicitly mention what the key is, how the key space is characterized, what precision to use, how to generate valid keys, and also to perform a basic security analysis. For the present work [@sdcudtcs], the total lack of security discourages the use of this algorithm as is for secure applications.
[10]{}
T. Yang. A survey of chaotic secure communication systems. , 2:81–130, 2004.
G. Álvarez, F. Montoya, M. Romera, and G. Pastor. Chaotic cryptosystems. In Larry D. Sanson, editor, [*33rd Annual 1999 International Carnahan Conference on Security Technology*]{}, pages 332–338. IEEE, 1999.
K. M. Short. Steps toward unmasking secure communications. , 4:959–977, 1994.
T. Beth, D. E. Lazic, and A. Mathias. Cryptanalysis of cryptosystems based on remote chaos replication. In Yvo G. Desmedt, editor, [*Advances in Cryptology - CRYPTO ’94*]{}, volume 839 of [*Lecture Notes in Computer Science*]{}, pages 318–331. Springer-Verlag, 1994.
G. Pérez and H. A. Cerdeira. Extracting messages masked by chaos. , 74:1970–1973, 1995.
K. M. Short. Unmasking a modulated chaotic communications scheme. , 6:367–375, 1996.
H. Zhou and X. Ling. Problems with the chaotic inverse system encryption approach. , 44:268–271, 1997.
T. Yang, L. B. Yang, and C. M. Yang. Breaking chaotic switching using generalized synchronization: Examples. , 45:1062–1067, 1998.
T. Yang, L. B. Yang, and C. M. Yang. Breaking chaotic secure communications using a spectogram. , 247:105–111, 1998.
T. Yang, L. B. Yang, and C. M. Yang. Cryptanalyzing chaotic secure communications using return maps. , 245:495–510, 1998.
G. Álvarez, F. Montoya, M. Romera, and G. Pastor. Cryptanalysis of a chaotic encryption system. , 276:191–196, 2000.
G. Álvarez, F. Montoya, M. Romera, and G. Pastor. Cryptanalysis of a chaotic secure communication system. , 306:200–205, 2003.
G. Álvarez, F. Montoya, M. Romera, and G. Pastor. Cryptanalysis of an ergodic chaotic cipher. , 311:172–179, 2003.
S. Li, X. Mou, Y. Cai, Z. Ji, and J. Zhang. On the security of a chaotic encryption scheme: problems with computerized chaos in finite computing precision. , 153:52–58, 2003.
G. Álvarez, F. Montoya, M. Romera, and G. Pastor. Cryptanalysis of a discrete chaotic cryptosystem using external key. , 319:334–339, 2003.
Moez Feki, Bruno Robert, Guillaume Gelle, and Maxime Colas. Secure digital communication using discrete-time chaos synchronization. , 18:881–890, 2003.
K. Alligood, T. Sauer, and J. Yorke. . Springer, 1997.
S. Li, Q. Li, W. Li, X. Mou, and Y. Cai. Statistical properties of digital piecewise linear chaotic maps and their roles in cryptography and pseudo-random coding. In [*Cryptography and Coding - 8th IMA International Conference Proceedings*]{}, volume 2260 of [*Lecture Notes in Computer Science*]{}, pages 205–221. Springer-Verlag, 2001.
R. Forré. The henon attractor as a keystream generator. In [*Advances in Cryptology – EuroCrypt’91*]{}, volume 0547 of [ *Lecture Notes in Computer Science*]{}, pages 76–81. Springer-Verlag, 1991.
D. R. Stinson. . CRC Press, 1995.
Figures {#figures .unnumbered}
=======
![\[fig:lyap\]Chaotic region for the Henon attractor.](figure1)
![\[fig:error\]BER when decrypting the ciphertext with a key at a distance $d$ from the real encryption key, $k=\{\alpha,\beta\}=\{1.4,0.3\}$: (a) modulation by multiplication; (b) modulation by multiplication and feedback. Note the difference in scale.](figure2)
![\[fig:map\]100,000 successive points obtained by iteration of the Henon map for $\{\alpha,\beta\}=\{1.4,0.3\}$.](figure3)
|
---
abstract: 'To optimize fruit production, a portion of the flowers and fruitlets of apple trees must be removed early in the growing season. The proportion to be removed is determined by the bloom intensity, i.e., the number of flowers present in the orchard. Several automated computer vision systems have been proposed to estimate bloom intensity, but their overall performance is still far from satisfactory even in relatively controlled environments. With the goal of devising a technique for flower identification which is robust to clutter and to changes in illumination, this paper presents a method in which a pre-trained convolutional neural network is fine-tuned to become specially sensitive to flowers. Experimental results on a challenging dataset demonstrate that our method significantly outperforms three approaches that represent the state of the art in flower detection, with recall and precision rates higher than $90\%$. Moreover, a performance assessment on three additional datasets previously unseen by the network, which consist of different flower species and were acquired under different conditions, reveals that the proposed method highly surpasses baseline approaches in terms of generalization capability.[^1], [^2]'
author:
- |
Philipe A. Dias\
Marquette University\
Electrical and Computer Engineering\
Milwaukee, Wisconsin, USA\
[philipe.ambroziodias@marquette.edu]{}
- |
Amy Tabb\
USDA-ARS-AFRS\
Kearneysville, West Virginia, USA\
[amy.tabb@ars.usda.gov]{}
- |
Henry Medeiros\
Marquette University\
Electrical and Computer Engineering\
Milwaukee, Wisconsin, USA\
[henry.medeiros@marquette.edu]{}
bibliography:
- 'techrefs\_v2.bib'
title: Apple Flower Detection using Deep Convolutional Networks
---
Introduction
============
Various studies have established the relationships between bloom intensity, fruit load and fruit quality [@Forshey1986; @Link2000]. Together with factors such as climate, bloom intensity is especially important to guide thinning, which consists of removing some flowers and fruitlets in the early growing season. Proper thinning directly impacts fruit market value, since it affects fruit size, coloration, taste and firmness.
Despite its importance, there has been relatively limited progress so far in automating bloom intensity estimation. Currently, this activity is typically carried out manually with the assistance of rudimentary tools. More specifically, it is generally done by inspecting a random sample of trees within the orchard and then extrapolating the estimates obtained from individual trees to the remainder of the orchard [@Gongal2016]. As the example in Figure \[fig:bigexample\] illustrates, obstacles that hamper this process are: 1) manual tree inspection is time-consuming and labor-intensive, which contributes to making labor responsible for more than $50\%$ of apple production costs [@Singh2010]; 2) estimation by visual inspection is characterized by large uncertainties and is prone to errors; 3) extrapolation of the results from the level of the inspected trees to the row or parcel level relies heavily on the grower’s experience; and 4) inspection of a small number of trees does not provide information about the spatial variability which exists in the orchard, making it difficult to develop and adopt site-specific crop load management strategies that could lead to optimal fruit quality and yield.
![**Best viewed in color.** Example of image from a flower detection dataset used in this paper. []{data-label="fig:bigexample"}](Figures/Figure1.jpeg)
With the goal of introducing more accurate and less labor intensive techniques for the estimation of bloom intensity, machine vision systems using different types of sensors and image processing techniques have been proposed [@Kapach2012]. Most existing methods, which are mainly based on simple color thresholding, have their applicability hindered especially by variable lighting conditions and occlusion by leaves, stems or other flowers [@Gongal2015].
Inspired by successful works using convolutional neural networks (CNNs) in multiple computer vision tasks, we propose a novel method for apple flower detection based on features extracted using a CNN. In our approach, an existing CNN trained for saliency detection is fine-tuned to become particularly sensitive to flowers. This network is then used to extract features from portraits generated by means of superpixel segmentation. After dimensionality reduction, these features are fed into a pre-trained classifier that ultimately determines whether each image region contains flowers or not. The proposed method significantly outperformed state-of-the-art approaches on four datasets composed of images acquired under different conditions.
Our main contributions are:
1. a novel CNN-based flower detection algorithm;
2. an extensive evaluation on a challenging dataset acquired under realistic and uncontrolled conditions;
3. an analysis of the generalization capability of the proposed approach on additional datasets previously unseen by the evaluated models.
The remainder of paper is organized as follows. Section \[sec:relatedwork\] discusses the most relevant existing approaches for automated flower and fruit detection. Our proposed approach is described in Section \[sec:methods\], which also includes a description of three baseline comparison methods as well as implementation details. Experiments performed to evaluate the impact of specific design choices are described in Section \[sec:results\], followed by an extensive comparison of our optimal model against the baseline methods on four different datasets. Our concluding remarks are presented in Section \[sec:conclusion\].
Related Work {#sec:relatedwork}
============
While existing techniques employed for flower detection are based only on color information, methods available for fruit quantification exploit more modern computer vision techniques. For this reason, in this section we first review the most relevant works on automated flower detection, followed by a discussion of the relevant literature on fruit quantification. Moreover, to make this article self-contained and therefore accessible to a wider audience, we also provide a brief introduction to the fundamentals of CNNs.
Flower and fruits quantification
--------------------------------
Aggelopoulou and colleagues presented in [@Aggelopoulou2011] one of the first works using computer vision techniques to detect flowers. That method is based on color thresholding and requires image acquisition at specific daylight times, with the presence of a black cloth screen behind the trees. Thus, although its reported error in predicted yield is relatively low ($18\%$), such approach is applicable only for that controlled scenario.
Similar to the work of Thorp and Dierig [@Thorp2011] for identification of *Lesquerella* flowers, the technique described by Hočevar et al. in [@Hocevar2014] does not require a background screen, but it is still not robust to changes in the environment. The image analysis procedure is based on hard thresholding according to color (in the HSL color space) and size features, such that parameters have to be adjusted whenever changes in illumination (daylight/night), in flowering density (high/low concentration) or in camera position (far/near trees) occur.
Horton and his team described in [@Horton2017] a system for peach bloom intensity estimation that uses a different imaging approach. Based on the premise that the photosynthetic activity of this species increases during bloom period, the system relies on multispectral aerial images of the orchard, yielding an average detection rate of $84.3\%$ for $20$ test images. Similarly to the aforementioned methods, the applicability of this method also has the intrinsic limitation of considering only color/spectral information (thresholding near-infrared and blue bands), such that its performance is sensitive to changes in illumination conditions.
More advanced computer vision techniques have been employed for fruit quantification [@Kapach2012]. A multi-class image segmentation for agrovision is proposed by Hung et al. in [@Hung2013], classifying image pixels into leaves, almonds, trunk, ground and sky. Their method combines sparse autoencoders for feature extraction, logistic regression for label associations and conditional random fields to model correlations between pixels. Some other methods are based on support vector machine (SVM) classifiers that use information obtained from different shape descriptors and color spaces as input [@Das2015; @Ji2012]. Compared to existing methods for flower detection, these methods are more robust since morphological characteristics are taken into account. As many other shape-based and spectral-based approaches [@Linker2012; @Wachs2010; @Wang2012; @Dorj2017], these techniques are, however, still limited by background clutter and variable lighting conditions in orchards [@Gongal2016].
Recent works on fruit quantification include the use of metadata information. Bargoti and colleagues in [@Bargoti2016] built on [@Hung2013] to propose an approach that considers pixel positions, orchard row numbers and the position of the sun relative to the camera. Similarly, Cheng et al. [@Cheng2017] proposed the use of information such as fruit number, fruit area, area of apple clusters and foliage area to improve accuracy of early yield prediction, especially in scenarios with significant occlusion. However, the inclusion of metadata is highly prone to overfitting, particularly when limited training data is available and the variability of the training set is hence low [@Bargoti2016].
Deep learning
-------------
Following the success of Krizhevsky’s model [@Krizhevsky2012] in the ImageNet 2012 Challenge, deep learning methods based on CNNs became the dominant approach in many computer vision tasks. The architecture of traditional CNNs consists of a fixed-size input, multiple convolutional layers, pooling (downsampling) layers and fully connected layers [@Guo2016]. Winner of the ImageNet 2013 Classification task, the Clarifai model is one such network [@Zeiler2014]. Illustrated on the right side of Figure \[fig:flow\], it takes input image portraits of size $227 \times 227$ pixels, which traverse a composition of $5$ convolutional layers (C1-C5) and $3$ fully connected layers (FC6-FC7 and the softmax FC8). Each type of layer plays a different role within the CNN architecture: while convolutional layers allow feature extraction, the latter fully connected layers act on this information to perform classification.
In computer vision and image processing, a *feature* corresponds to information that is meaningful for describing an image and its regions of interest for further processing. Feature extraction is therefore crucial in image analysis, since it represents the transition from pictorial (qualitative) to nonpictorial (quantitative) data representation [@Marques2011]. Rather than relying on hand-engineered features (e.g. HOG [@Dalal2005Histograms]), deep CNNs combine multiple convolutional layers and downsampling techniques to learn hierarchical features, which are a key factor for the success of these models [@LeCun2015]. As described in [@Zeiler2014], the convolutional layers C1-C2 learn to identify low-level features such as corners and other edge/color combinations. The following layers C3-C5 combine this low-level information into more complex structures, such as motifs, object parts and finally entire objects.
Traditional deep CNNs are composed of millions of learned parameters (over $60$ million in AlexNet [@Krizhevsky2012]), such that large amounts of labeled data are required for network training. Deep learning models became feasible relatively recently, after the introduction of large publicly available datasets, of graphics processing units (GPUs), and of training algorithms that exploit GPUs to efficiently handle large amounts of data [@LeCun2015]. Nevertheless, gathering domain specific training data is an expensive task. One alternative to reduce the required amount of labeled data is data augmentation, a technique proven to benefit the training of multiple machine learning models [@Wong2016]. It is typically performed by applying transformations such as translation, rotation and color space shifts to pre-labeled data.
In addition, transfer learning approaches such as fine-tuning have been investigated. Earlier layers of a deep network tend to contain more generic information (low-level features), which are then combined by the latter layers into task specific objects of interest. Thus, a network that can recognize different objects present in a large dataset must contain a set of low-level descriptors robust enough to characterize a wide range of patterns. Under this premise, fine-tuning procedures typically aim at adjusting the higher-level part of a network pre-trained on a large generic dataset, rather than training the full network from scratch. This greatly reduces the need for task-specific data, since only a smaller set of parameters has to be refined for the particular application [@Yosinski2014].
At the classification side, most CNN architectures employ fully connected layers for final categorization. They determine which features are mostly correlated to each specific class employing a logistic regression classifier. For scenarios in which the output is binary, consistent albeit small improvements on popular datasets have been demonstrated by replacing the final CNN layer by a SVM classifier [@Tang2013]. SVM models tend to generalize better than logistic regression, since they target a solution that not only minimizes the training error, but also maximizes the margin distance between classes.
Following the success of CNNs on image classification tasks, the work of Girschick et al. [@Girshick2014] introduced the concept of region-based CNNs (R-CNN), outperforming by a large margin previous hand-engineered methods for object detection. In that work, a CNN is first pre-trained on a large auxiliary dataset (ImageNet) and then fine-tuned using a smaller but more specific dataset (PASCAL dataset for object detection). The Faster R-CNN proposed in [@Ren2015] improved this model by replacing selective search [@Uijlings2013] with the concept of Region Proposal Network (RPN), which shares convolutional layers with the classification network. Both modules compose a single, unified network for object detection.
Recent works adapt the Faster R-CNN for fruit detection. Bargoti and Underwood in [@Bargoti2016b] present a Faster R-CNN trained for detection of mangoes, almonds and apples fruits on trees. Stein et al. in [@Stein2016] extended this model for tracking and localization of mangoes, combining it with a monocular multi-view tracking module that relies on a GPS system. Sa et al. in [@Sa2016_DeepFruits] applied the Faster R-CNN to RGB and near-infrared multi-modal images. Each modality was fine-tuned independently, with optimal results obtained using a late fusion approach. Still in the context of agricultural applications, CNNs have been also successfully used for plant identification from leaf vein patterns [@Grinblat2016].
In summary, existing methods for flower identification are based on hand-engineered image processing techniques that work only under specific conditions. Color and size thresholding parameters composing these algorithms have to be readjusted in case of variations of lightning conditions, camera position with respect to the orchard (distance and angle), or expected bloom intensity. Recent techniques employed for fruit quantification exploit additional features and machine learning strategies, providing insights to further develop strategies for flower detection. Aiming at a technique for flower identification that is robust to clutter, changes in illumination and applicable for different flower species, we therefore propose a novel method in which an existing CNN trained for saliency detection is fine-tuned to become particularly sensitive to flowers.
Proposed Approach {#sec:methods}
=================
In this section, we first describe the prediction steps performed by our method, i.e., the sequence of operations applied to an image in order to detect the presence of flowers. Subsequently, we describe the fine-tuning procedure carried out to obtain the core component of our model: a CNN highly sensitive to flowers. We conclude with a discussion of alternative flower detection approaches against which we evaluate our proposed method and a brief mention of relevant details regarding the implementation of our methods.
In the discussion that follows, we will refer to our proposed approach for flower detection as the *CNN+SVM* method. As illustrated in Figure \[fig:flow\], our CNN+SVM method consists of three main steps: i) computation of region proposals; ii) feature extraction using our fine-tuned CNN, which follows the Clarifai architecture [@Zeiler2014]; and iii) final classification of each region according to the presence of flowers. The operations that comprise these steps are described in detail below. In our description, we make reference to Algorithm \[alg:pseudocode\], which lists the operations performed by our method on each input image. The sensitivity of the method to specific design choices is detailed in Section \[sub:design\].
*1) Step 1 - Region proposals:* The first step in the proposed method consists of generating region proposals by grouping similar nearby pixels into superpixels, which are perceptually meaningful clusters of variable size and shape (Line \[ln:seg\] of Algorithm \[alg:pseudocode\]). To this end, we use the *simple linear iterative clustering* (SLIC) superpixel algorithm. Currently one of the most widely-used algorithms for superpixel segmentation, it adapts *k*-means clustering to group pixels according to a weighted distance measure that considers both color and spatial proximity [@Achanta2012]. For additional information on superpixel approaches, we refer the reader to the review provided in [@Stutz2016]. The second leftmost image in Figure \[fig:flow\] illustrates the superpixels $s_i \in \mathbb{S}$ generated by the SLIC algorithm when applied to a typical image obtained in an orchard.
{width="\textwidth"}
Although other approaches such as the Faster R-CNN [@Ren2015] provide a unified architecture in which both region proposal and classification modules can be fine-tuned for a specific task, they have more parameters that need to be learned in a supervised manner. Since in most cases flowers are salient with respect to its surrounding background, an unsupervised, local-context based approach such as superpixel segmentation should be sufficient to obtain region proposals suitable for flower detection.
\[alg:pseudocode\]
Segment $I$ into set of superpixels $\mathbb{S}$ using SLIC. \[ln:seg\] Crop smallest squared portrait $p_i$ enclosing $s_i$. \[ln:crop\] \[ln:pad\] \[ln:feat\] Obtain $\hat{f}_i$ by performing PCA analysis on ${f}_i$. \[ln:pca\] Classify $s_i$ by applying pre-trained SVM on $\hat{{f}}_i$. \[ln:svm\]
Once the image is segmented into superpixels, as Algorithm \[alg:pseudocode\] indicates, we iterate over each superpixel in the image. Since the input size required by the Clarifai CNN model is $227 \times 227$, we first extract the smallest square portrait enclosing the superpixel under analysis (Line \[ln:crop\]), which we denote $p_i$. The output of this step is illustrated in the third leftmost image of Figure \[fig:flow\] for one superpixel. The background surrounding the superpixel of interest within a portrait is then padded with the training set mean, i.e. the average RGB color of all images composing the dataset (greenish color). Finally, the portrait is resized to $227 \times 227$ (Line \[ln:pad\]). The resulting region proposal, $\hat{p}_i$, is illustrated in the fourth image of Figure \[fig:flow\].
*2) Step 2 - Feature extraction:* In the feature extraction step (Line \[ln:feat\]), each of the portraits generated above is mean-centered and then evaluated individually by our CNN. The mean-centering step consists of subtracting from the portrait the same average training set RGB mean used for padding its background. This procedure is commonly employed to facilitate training convergence of deep learning models, since it ensures similarly ranged features within the network. For each input portrait, we collect as features the output of the rectified linear unit (ReLU) associated with the first fully connected layer (FC6). With a dimensionality of $N=4,096$, the feature vector $f_i \in \mathbb{R}^N$ collected at this stage of the network encapsulates the hierarchical features extracted by layers $C1-C5$, which contain the key information required for accurate classification.
*3) Step 3 - Classification:* To classify each proposed region as containing a flower or not, we first perform principal component analysis (PCA) to reduce the feature dimensionality to a value $k<N$ such that the new feature vector $\hat{f}_i \in \mathbb{R}^k$ (Line \[ln:pca\]). As demonstrated in our experimental evaluation in Section \[ss:dimensionality\] a value of $k=69$, which corresponds to approximately $94\%$ of the original variance of the data, provides performance levels virtually identical to those of the original features. Finally, based on these features a pre-trained SVM model binary classifies superpixels according to the presence of flowers (Line \[ln:svm\]). Details on SVM training are provided in the next section.
{width="\textwidth"}
Network fine-tuning and SVM training {#sub:tuning}
------------------------------------
Based on the techniques introduced by Girshick et al. in [@Girshick2014] and Zhao et al. in [@Zhao2015] for object and saliency detection, in our model an existing CNN architecture is made particularly sensitive to flowers by means of fine-tuning. In the work of Zhao et al. [@Zhao2015], the Clarifai model [@Zeiler2014] was adopted as the starting point and fine-tuned for saliency detection. We further tuned Zhao et al.’s model for flower identification using labeled portraits from our training set, which we describe below.
The generation of training samples for network tuning takes place similarly to prediction. For each labeled image composing the training set, we computed region proposals according to Step 1 described above. Using these training examples, $10,000$ backpropagation training iterations were performed in order to minimize the network classification error. After fine-tuning, we computed the CNN features of the training examples, reduced their dimensionality to $k=69$, and used them to train the SVM classifier.
#### Image dataset
Images of apple trees were collected using a camera model Canon EOS 60D under natural daylight illumination (i.e. uncontrolled environment). This dataset, which we refer to as *AppleA*, is composed of a total of $147$ images with resolution of $5184 \times 3456$ pixels acquired under multiple angles and distances of capture. Figure \[fig:imgsamples\] shows some images that comprise this dataset. For performance evaluation and learning purposes, the entire dataset was labeled using a MATLAB GUI in which the user selected only superpixels that contain parts of flowers in approximately half of its total area. As summarized in Table \[tab:labels\], the labeled images were randomly split into training and validation sets composed of $100$ and $47$ images, respectively. This corresponds to a total of $91,488$ training portraits (i.e. superpixels) and $42,430$ validation ones. The training examples were used to fine-tune the network and train the SVM as described above. The validation examples were used in the performance evaluation discussed in Sections \[sub:design\] and \[sub:comparison\].
-- ------- ----------------- -------------------- -----------
[Positives]{} [Negatives]{} [Total]{}
$100$ $3,691$ ($4\%$) $87,797$ ($96\%$) $91,488$
$47$ $1,719$ ($4\%$) $40,711$ ($96\%$) $42,430$
$147$ $5,410$ ($4\%$) $128,508$ ($96\%$) $133,918$
-- ------- ----------------- -------------------- -----------
: Statistics of the training and validation dataset (*AppleA*).[]{data-label="tab:labels"}
#### Data augmentation
According to our labeling, only $4\%$ of the samples contain flowers (positives). Imbalanced datasets represent a problem for supervised machine learning approaches, since overall accuracy measures become biased towards recognizing mostly the majority class [@Visa2005]. In our case, that means the learner would present a bias towards classifying the portraits as negatives. To overcome this situation and increase the amount of training data, we quadrupled the number of positive samples using data augmentation. As illustrated in Figure \[fig:augmentation\], this was accomplished by mirroring each positive sample with respect to: (i) the vertical axis, (ii) the horizontal axis, and (iii) both axes.
![**Best viewed in color.** Example of data augmentation. a) Original portrait. b) Portrait mirrored with respect to the vertical axis, c) the horizontal axis, d) and both axes.[]{data-label="fig:augmentation"}](Figures/Figure4.pdf)
#### Parameters’ optimization
Support vector machines are supervised learning models that search for a hyperplane that maximizes the margin distance to each class. This characteristic allows SVM models to generalize better than classifiers such as the ones based on logistic regression. For non-linearly separable data, kernel functions such as the popular radial basis function (RBF, or *Gaussian*) are used. We refer to [@BenHur2010; @Hsu2008] for further details on the formulation of SVMs.
Two main parameters control the performance of SVMs with a Gaussian kernel function, the regularization cost $C$ and the width of the Gaussian kernel $\gamma$. By regulating the penalty applied to misclassifications, the parameter $C$ controls the trade-off between maximizing the margin with which two classes are separated and the complexity of the separating hyperplane. The parameter $\gamma$ regulates the flexibility of the classifier’s hyperplane. For both parameters, excessively large values can lead to overfitting.
The optimization of $C$ and $\gamma$ is a problem without straightforward numerical solution. Therefore, it is typically solved using grid search strategies [@BenHur2010; @Hsu2008] in which multiple parameter combinations are evaluated according to a performance metric. We adopt this strategy in this work.
Comparison Approaches {#sub:comparison_description}
---------------------
As has been noted in Section \[sec:relatedwork\], current algorithms for automated identification of flowers are mostly based on binarization by thresholding information from different color-spaces (typically RGB or HSV) [@Aggelopoulou2011; @Thorp2011], occasionally combined with size filtering [@Hocevar2014]. We implement three alternative baseline approaches which reflect the state of the art in fruit/flower detection. The first implementation, which we call the *HSV* method, replicates the algorithm described by Hocevar and his team in [@Hocevar2014]. Images are binarized at pixel-level based on HSV color information, followed by filtering according to minimum and maximum cluster sizes.
We refer to the second baseline implementation as *HSV+Bh*. Similar to our proposed approach, the starting point for this method is the generation of superpixels using the SLIC algorithm. We then compute a $100$-bin histogram of each superpixel in the HSV color space, which has the advantage of dissociating brightness from chromaticity and saturation. Studies on human vision and color-based image retrieval have demonstrated that most of the color information is contained in the hue channel, with the saturation playing a significant role in applications where identifying white (or black) objects is important [@Gonzalez2006; @Stricker95hist]. In our experiments, we construct a single 1-D histogram consisting of $100$ bins, which corresponds to the concatenation of a 50-bin hue channel histogram, a 40-bin saturation histogram and a 10-bin value histogram. Afterwards, we use the Bhattacharyya distance [@Bhattacharyya1943] to compare each superpixel histogram against the histograms of all positive samples composing the training set. We compute the Bhattacharyya distance using a Gaussian kernel function, as formulated in [@Hoak2017; @Hoseinnezhad2012]. The average Bhattacharyya distance is taken as the likelihood that the superpixel includes a flower, and superpixels with distance lower than an optimal threshold are classified as flowers.
Since the technique described above is based on the average Bhattacharyya distance in the HSV color space, it makes no distinction between poorly and highly informative training sample features. Its ability to make accurate classification decisions is therefore limited in such complex feature spaces. Inspired by works on fruit quantification [@Das2015; @Ji2012], we extend this method by combining the same HSV histograms with an SVM classifier for apple flower detection. That is, rather than determining whether a superpixel contains a flower based on the Bhattacharyya distance, we train an SVM classifier that uses the HSV histograms as inputs. We call this approach the *HSV+SVM* method.
Implementation Details {#sec:paropt}
----------------------
Most image analysis tasks were performed using MATLAB R2016b. Additionally, we used the open source Caffe framework [@Jia2014a] for fine-tuning and extracting features from the CNN. To reduce computation times by exploiting GPUs, we used the cuSVM software package for SVM training and prediction [@Carpenter2009].
Experiments and Results {#sec:results}
=======================
Experiments were performed with three main goals. Our optimal CNN+SVM model extracts features from the CNN’s first fully connected layer ($FC6$), reduces feature dimensionality to $69$, and performs final classification using SVM. Thus, we first evaluated the impact of these specific design choices on the final performance of our method. We then compared it against the three baseline methods (HSV, HSV+Bh and HSV+SVM). Finally, we evaluated the performance of the proposed approach on previously unseen datasets to determine its generalization capability.
As described in Section \[sub:tuning\], our datasets are severely imbalanced. In such scenarios, evaluations of performance using only accuracy measurements and ROC curves may be misleading, since they are insensitive to changes in the rate of class distribution. We therefore perform our analysis in terms of precision-recall curves (PR) and the corresponding $F_1$ score [@Fawcett2006]. Precision is normalized by the number of positives rather than the number of true negatives, so that false positive detections have the same relative weight as true positives. While the maximum $F_1$ score indicates the optimal performance of a classifier, the area under the respective PR curve (AUC-PR) corresponds to its expected performance across a range of decision thresholds, such that a model with higher AUC-PR is more likely to generalize better.
Analysis of design choices {#sub:design}
--------------------------
In order to validate our design choices, we performed experiments to evaluate how the final performance of the classifier is affected by: (i) the dimensionality of the feature space; (ii) the point where features are collected from the CNN; and (iii) the type of input portrait.
### Dimensionality analysis {#ss:dimensionality}
PCA is one of the most widespread techniques for dimensionality reduction. It consists of projecting $N$-dimensional input data onto a $k$-dimensional subspace in such a way that this projection minimizes the reconstruction error (i.e. $L_2$ norm between original and projected data) [@Mohri2012]. PCA can be performed by computing the eigenvectors and eigenvalues of the covariance matrix and ranking principal components according to the obtained eigenvalues [@Smith2002].
In this application, the original dimensionality corresponds to the number of elements in the CNN vectors extracted from a fully connected layer, i.e., $N=4,096$ as represented for the two last layers in Figure \[fig:flow\]. The first two columns of Table \[tab:pcaf\] show the reduced dimensionality $k$ of the feature vector and the corresponding ratio of the total variance of the $N$-dimensional dataset that is retained at that dimensionality for layer $FC6$. As the table indicates, the first most significant dimension alone already covers almost half of the total variance, and $23$ dimensions are sufficient to cover nearly $90\%$ of it.
-------- ---------- ------------------- ---------- ---------- -------------------
$1 $ $48.3\%$ $90.4\%$ $92.2\%$ $88.6\%$ $96.5\%$
$2 $ $63.7\%$ $91.4\%$ $92.7\%$ $90.2\%$ $94.0\%$
$5 $ $79.9\%$ $91.9\%$ $92.3\%$ $91.5\%$ $96.9\%$
$15 $ $87.4\%$ $91.5\%$ $92.6\%$ $90.4\%$ $94.3\%$
$23 $ $89.6\%$ $\mathbf{92.1\%}$ $92.9\%$ $91.2\%$ $95.2\%$
$69 $ $93.8\%$ $91.9\%$ $92.6\%$ $91.2\%$ $\mathbf{97.2}\%$
$150 $ $95.8\%$ $91.3\%$ $92.7\%$ $90.0\%$ $97.1\%$
$300 $ $97.2\%$ $91.6\%$ $91.6\%$ $91.7\%$ $97.2\%$
$500 $ $98.0\%$ $91.8\%$ $91.8\%$ $91.8\%$ $95.0\%$
$1080$ $99.0\%$ $91.7\%$ $91.5\%$ $91.8\%$ $94.9\%$
-------- ---------- ------------------- ---------- ---------- -------------------
: Classification performance according to the number of principal components (dimensions) selected after applying PCA to the extracted features. Best results in terms of $F_1$ and AUC-PR are shown in boldface. []{data-label="tab:pcaf"}
We investigated then how samples are mapped into the lower dimensional feature space. Figure \[fig:proj13Ex\] shows the projections in dimensions $1$ and $2$ as well as dimensions $1$ and $3$. These plots illustrate how the convolutional network maps the samples into a space where it is possible to differentiate between multiple clusters. With *dim.* as an abbreviation for *dimension*, let $\downarrow$ denote low dimensionality values and $\uparrow$ high values, respectively. The following clusters can be identified: flowers ($\downarrow$ dim.1, $\uparrow$ dim.2); grass/floor ($\uparrow$ dim.1, $\uparrow$ dim.2); branches/leaves ($\downarrow$ dim.2); sky ($\uparrow$ dim.3). This indicates that positive and negative samples are almost linearly separable even for $2D$ projections of the original feature space.
{width="\textwidth"}
To quantitatively assess how the classification performance is affected by the dimensionality of the feature space, we trained SVM classifiers for different numbers of dimensions. For each dimensionality, Table \[tab:pcaf\] presents the optimal performance metrics and corresponding AUC-PR. As expected, these results demonstrate that the impact of dimensionality on the optimal performance of our method is rather low. A very good performance is already obtained using a 2D feature space, with both $F_1$ score and AUC-PR only around $0.7\%$ and $3.2\%$ lower than the highest obtained values, respectively. In terms of optimal recall and precision, this is equivalent to missing extra $4$ positive samples out of $1,719$, while including more $19$ false-positives out of $40,711$. Moreover, the table shows that a dimensionality of $69$ is nearly optimal: the performance in terms of optimal $F_1$ score is only $0.2\%$ lower than the highest obtained value ($23$ dimensions) and it is optimal in terms of AUC-PR.
Although in the discussion above we present results obtained using SVMs, such a high separability even for low dimensionalities indicates that the final prediction accuracy of our model is almost independent of the type of classifier employed. This conjecture is validated in the next subsection, where we demonstrate that the performance of our system does not change significantly by either including an additional fully connected layer to our CNN or by carrying out classification using the using network’s softmax layer directly.
### Feature analysis
As explained in Section \[sec:methods\], after fine-tuning the model, we use it to extract features that allow the classification of superpixels according to the presence of flowers within them. Three combinations of features and classification mechanisms were investigated: (A) predict using solely the neural network, by means of its softmax output layer; (B) train a SVM classifier on features collected after the last fully connected layer (FC7); (C) train a SVM classifier on features collected after the first fully connected layer (FC6). Figure \[fig:score\] shows the points where features are collected and how classification scores are computed using these features. Following the notation used in Figure \[fig:flow\], C1-C5 correspond to the convolutional layers of the fine-tuned Clarifai network, FC6-FC7 are the fully connected layers, and FC8 is the softmax layer.
![Diagram illustrating how classification scores are computed using the extracted features.[]{data-label="fig:score"}](Figures/Figure6.pdf)
For approaches B and C, features are collected from the output of the rectified linear units (ReLUs) located right after the respective fully connected layers. The same sequence of operations is performed for both methods B and C, i.e., the framework is the same regardless of whether the features are collected from the last (FC7) or first fully connected layer (FC6). Based on the results obtained in the previous section, for both cases $69$ dimensions are kept after PCA analysis.
Results obtained for classification on the validation set are summarized in Table \[tab:NnFCx\] and Figure \[fig:prNnFCx\]. As Figure \[fig:prNnFCx\] indicates, all three approaches show very similar performance. A closer inspection of Table \[tab:NnFCx\] reveals that the SVM-based approaches slightly outperform the direct use of the neural network softmax layer both in terms of optimal $F_1$ score and AUC-PR. The performances obtained with methods B and C are very similar for both metrics. We therefore opted for method C, which uses features extracted from the earlier layer FC6 and provides slight increases in both optimal $F_1$ score and AUC-PR.
--------- ---------- ---------- ---------- ----------
A (NN) $96.9\%$ $90.6\%$ $91.7\%$ $89.6\%$
B (FC7) $97.2\%$ $91.6\%$ $91.8\%$ $91.4\%$
C (FC6) $97.3\%$ $91.9\%$ $92.6\%$ $91.2\%$
--------- ---------- ---------- ---------- ----------
: Classification performance according to the CNN layer at which features are collected - Methods A, B, C.[]{data-label="tab:NnFCx"}
![**Best viewed in color.** PR curves illustrating the performance on the validation set according to the CNN layer at which features are collected. *NN* stands for prediction using solely the network softmax output layer, while *FC6* and *FC7* correspond to SVM classifiers trained on features collected at the first and second fully connected layers, respectively.[]{data-label="fig:prNnFCx"}](Figures/Figure7_Rev1-eps-converted-to.pdf)
### Different types of portraits
Using superpixels for region proposal computation and subsequent generation of portraits implies that our goal is to evaluate whether the superpixel itself is composed of flowers or not. In order to assess the influence of the local context surrounding the superpixel on the classification results, in addition to the approach based on replacing the region around the superpixel with the mean RGB value, two alternative approaches for portrait generation were considered. The first consists of retaining the unmodified image area surrounding the superpixel, whereas the second corresponds to blurring the background surrounding the superpixel with a low-pass filter. For all three cases, the portrait is mean-centered before being fed into the neural network. The three types of evaluated portraits are illustrated in Figure \[fig:portraits\].
![**Best viewed in color.** Example of the three types of portrait evaluated. a) *Original*; b) Blurred background (*Blur*); c) Mean padded background.[]{data-label="fig:portraits"}](Figures/Figure8.pdf)
![**Best viewed in color.** Classification performance according to the portrait adjustment strategy. *Original* stands for portraits evaluated without any further adjustment, *blur* corresponds to portraits where the background is blurred, and *mean padding* denotes the strategy of padding the background with the training set mean.[]{data-label="fig:prportraits"}](Figures/Figure9_Rev1-eps-converted-to.pdf)
Figure \[fig:prportraits\] shows the PR curves obtained for each portrait type. The best performance is obtained with mean-padded portraits, a behavior explained by the existence of cases such as the ones illustrated in Figure \[fig:mistakes\]. The superpixels highlighted in the images on the top row do not contain flowers in more than $50\%$ of their area and should therefore not be classified as flowers. However, these superpixels are surrounded by flowers, as depicted in the corresponding figures in the bottom row, and hence the approach of simply cropping a square region around the superpixel leads to cases in which the portrait contains a well-defined flower. As a consequence, features extracted from the CNN for the entire portrait will indicate the presence of flowers and therefore lead to high confidence false positives, which explain the non-maximal precision ratios in the upper-left part of the respective PR curve. This problem is eliminated by mean-padding the background.
![**Best viewed in color.** Examples of superpixels incorrectly classified for *Original* and *Blur* portraits. The superpixels are shown in the top row and the bottom row shows the entire portraits enclosing the superpixels.[]{data-label="fig:mistakes"}](Figures/Figure10.pdf)
Comparison against baseline methods {#sub:comparison}
-----------------------------------
The analysis in Section \[sub:design\] above validates the design choices of our optimal CNN+SVM model described in Section \[sec:methods\]. That is, our optimal model uses mean-padded portraits and $69$-dimensional features obtained from the FC6 layer of the CNN. In this section, we compare this optimal CNN+SVM model against the three baseline methods described in Section \[sub:comparison\_description\].
The parameters of all four methods were optimized using a grid search, as described in Section \[sec:paropt\]. Optimization of the SVM hyperparameters based on $F_1$ score resulted in the following values for regularization factor ($C$) and RBF kernel bandwidth ($\gamma$): HSV+SVM ($C = 180; \gamma = 10$); CNN+SVM ($C = 30$; $\gamma = 10^{-4}$). For the HSV+Bh method, we performed an analogous grid search to optimize the standard deviation associated with the Gaussian kernel function, obtaining an optimal parameter of $\sigma = 5$. For the HSV method, we performed an extensive grid search on our training dataset to selected an optimal set of threshold values. This procedure indicated that pixels composing flowers are distributed over the entire H range of $[0,255]$, with optimal ranges of S within $[0,32]$, V within $[139,255]$, minimum size of $1,200$ pixels and maximum size of $45,000$ pixels.
Once the optimal parameters for all the classification models were determined, we evaluated the overall performance of each method using 10-fold cross-validation. All the $133,918$ samples composing the full *AppleA* dataset were combined and divided into 10 folds containing $13,391$ samples each. A total of 10 iterations was performed, in which each subsample was used exactly once as validation data.
The final PR curves associated with each method are shown in Figure \[fig:PR\]. Table \[tab:results\] provides the AUC-PR for each method along with the metrics obtained for the optimal models as determined by the $F_1$ score.
--------- ------------------- ------------------- ---------- ----------
HSV $54.9\%$ $54.1\%$ $58.3\%$ $50.4\%$
HSV+Bh $61.6\%$ $64.6\%$ $56.9\%$ $60.5\%$
HSV+SVM $92.9\%$ $87.1\%$ $88.4\%$ $87.8\%$
CNN+SVM $\mathbf{97.7\%}$ $\mathbf{93.4\%}$ $92.0\%$ $92.7\%$
--------- ------------------- ------------------- ---------- ----------
: Summary of results obtained for our approach (CNN+SVM) and the three baseline methods (HSV, HSV+Bh and HSV+SVM). Best results in terms of $F_1$ and AUC-PR are shown in boldface.[]{data-label="tab:results"}
![**Best viewed in color.** Precision-recall (PR) curve illustrating the performance of our proposed approach (CNN+SVM) in comparison with the three baseline methods (HSV, HSV+Bh, and HSV+SVM).[]{data-label="fig:PR"}](Figures/Figure11_Rev1-eps-converted-to.pdf)
The HSV method, which closely replicates existing approaches for flower detection, performs poorly in terms of both recall and precision. Such low performance is expected for methods that rely solely on color information. Since these techniques do not consider morphology or higher-level context to characterize flowers, they are very sensitive to changes in illumination and to clutter. Small performance improvements are obtained using the HSV+Bh method, which replaces pixel-wise hard thresholding by HSV histogram analysis at superpixel level, thereby incorporating a limited amount of context information into its classification decisions.
As illustrated by the results obtained with the HSV+SVM method, the use of an SVM classifier on the same HSV color features leads to dramatic improvements in both $F_1$ and AUC-PR ratios (around $20\%$ and $30\%$, respectively). Rather than giving the same importance to all histogram regions, the SVM classifier is capable of distinguishing between poorly and highly informative features. However, as depicted in Figure \[fig:hsvmistake\], the precision of this method is still compromised by gross errors such as classifying parts of tree branches as flowers, since it does not take into account any morphological information.
![**Best viewed in color.** Example of classification results obtained using (left) the baseline HSV+SVM method and (right) our proposed CNN+SVM method. Some examples of false positives generated by the HSV+SVM method that our approach correctly classifies can be seen on the branches near the left border of the image.[]{data-label="fig:hsvmistake"}](Figures/Figure12.pdf)
The proposed approach (CNN+SVM) outperforms both baseline methods by extracting features using a convolutional neural network. Differently from the previous methods, the hierarchical features evaluated within the CNN take into account not only color but also morphological/spatial characteristics from each superpixel. Our results demonstrate the effectiveness of this approach, with significant improvements in both recall and precision ratios that culminate in an optimal $F_1$ score higher than $92\%$ and AUC-PR above $97\%$ for the evaluated dataset. Figure \[fig:imgsamples\] shows examples of the final classification yielded by this method.
Performance on additional datasets
----------------------------------
To evaluate the generalization capability of our method, we assessed its performance on three additional datasets, composed of $20$ images each and illustrated in Figure \[fig:extrasets\]. We compare the results of our method with the performance of the best performing baseline approach (HSV+SVM). No dataset-specific adjustment of parameters is performed for our method nor for the baseline (HSV+SVM), i.e. both methods are assessed with the same optimal configuration obtained for the *AppleA* dataset.
{width="\textwidth"}
Two of the additional datasets also correspond to apple trees, but with a blue background panel positioned behind the trees to visually separate them from other rows of the orchard, a common practice in agricultural vision systems. We denote the first dataset *AppleB*, which is composed of images with resolution $2704 \times 1520$ acquired using a camera model GoPro HERO5. In this dataset there is a substantial number of occlusions between branches, leaves and flowers.
The second dataset, which we call *AppleC*, is composed of images with resolution $2456 \times 2058$ acquired with a camera model JAI BB-500GE. In this dataset occlusions are less frequent but the saturation color component of the images is concentrated in a much narrower range of the spectrum than in the original *AppleA* dataset. The contrast between objects such as flowers and leaves is therefore significantly lower.
The third additional dataset contains images of peach flowers (we therefore call it *Peach*) with resolution $2704 \times 1520$ acquired using a camera model GoPro HERO5. Peach blossoms show a noticeable pink hue in comparison to the mostly white apple flowers composing the training dataset. Additionally, images were acquired during an overcast day, such that in comparison to the training set (*AppleA*) the illumination is lower and the sky composing the background is gray instead of blue. Although the main scope of this work is on apple flower detection, we ultimately aim at a highly generalizable system that can be applied by fruit growers of different crops without the need for species-specific adjustments. In fruit orchards, each species of tree is typically constrained to specific areas. Hence, rather than differentiating between flower species, it is preferable to have a system that can distinguish between flowers and non-flower elements (e.g. leaves, branches, sky) regardless of species. Thus, this dataset represents a good evaluation of detection robustness.
#### Transfer learning steps
For all three additional datasets, both feature extraction and final classification were performed using the same parameters obtained by training with the *AppleA* dataset, without any dataset specific fine-tuning. Our transfer learning strategy relies solely on generic pre-processing operations that approximate the characteristics of the previously unseen images to those of the training samples.
Our first pre-processing step consists of removing the different backgrounds of the additional datasets. Whether the background is composed of a blue panel (*AppleB* and *AppleC*) or a gray sky (*Peach*), background identification for subsequent subtraction can be performed by means of texture analysis. For each image we compute the corresponding local entropy, which is then binarized using Otsu’s threshold [@Otsu1979] to identify low texture clusters. We then apply morphological size filtering to the binarized image and model the background as a multimodal distribution.
To model the background, we compute the RGB-mean of the $n$ largest (in terms of number of pixels) low texture clusters to build a $n$-modal reference set. The likelihood that remaining low texture clusters belong to the background is estimated as the Euclidean distance between their means and the nearest reference in the RGB space. This metric allows differentiating between low texture components composing the background from the ones composing flowers, without any dataset specific color thresholding. For the *AppleB* and *Peach* datasets we adopted a bimodal distribution, where the modes correspond to the blue panel/gray sky and trunk/branches. Since the blue panel in the background of images composing the *AppleC* dataset is reflective, shadows are visible and therefore we included a third mode to automatically filter these undesired elements out. Automatically determining the number of background components is part of our future work.
Afterwards, histogram equalization and histogram matching are performed on the saturation channel of each image. While equalization aims at spreading the histogram components, histogram matching consists in approximating its distribution to the characteristic form of the training set channel distribution [@Marques2011]. Finally, to mitigate the effects of illumination discrepancies, we subtract the difference between the mean of the value channel components in the input image and in the training set.
Figure \[fig:PRsExtraSets\] shows the PR curves summarizing the performance on these datasets of our method (CNN+SVM) in comparison with the best performing baseline approach (HSV+SVM). The proposed method provides AUC-PR above $85\%$ for all datasets, significantly outperforming the baseline method. Since the HSV+SVM method relies solely on color information, its results are acceptable only for the *AppleB* dataset, the one that most closely resembles the training dataset. Its performance is notably poor for the *Peach* set, as this species differs to a great extent from apple flowers in terms of color. A large performance difference is also evident for the *AppleC* dataset, in which flowers and leaves share more similar color components than in the training set. Table \[tab:generalization\] shows that the proposed approach also outperforms the baseline by a large margin in terms of optimal $F_1$ score and the corresponding precision and recall values.
{width="\textwidth"}
** $\mathbf{F_1}$ **Recall** **Precision**
---- --------- ------------------- ------------ ---------------
HSV+SVM $70.7\%$ $69.8\%$ $71.6\%$
CNN+SVM $\mathbf{80.2\%}$ $81.9\%$ $78.5\%$
HSV+SVM $48.6\%$ $37.9\%$ $68.0\%$
CNN+SVM $\mathbf{82.2\%}$ $81.2\%$ $83.3\%$
HSV+SVM $49.0\%$ $61.3\%$ $40.8\%$
CNN+SVM $\mathbf{79.9\%}$ $81.5\%$ $78.3\%$
: Summary of results obtained for our approach (CNN+SVM) and the best baseline method (HSV+SVM) for the three additional datasets. Best results in terms of $F_1$ are shown in boldface.[]{data-label="tab:generalization"}
Additionally, it is noteworthy that a large number of superpixels classified as false positives by our proposed approach (CNN+SVM) correspond to regions where flowers are indeed present, but compose less than $50\%$ of the corresponding superpixel total area. This is illustrated in Figure \[fig:badsppx\], which contains examples for the three additional datasets. In other words, the sensitivity of the feature extractor to the presence of flowers is very high and the final performance would be improved if the region proposals were more accurate.
![**Best viewed in color.** Example of false positives caused by poor superpixel segmentation.[]{data-label="fig:badsppx"}](Figures/Figure15.pdf)
Conclusion {#sec:conclusion}
==========
In this work, we introduced a novel approach for apple flower detection, which is based on deep learning techniques that represent the state of the art for computer vision applications. In comparison with existing methods, which are mainly based solely on color analysis and have limited applicability in scenarios involving changes in illumination or occlusion levels, the hierarchical features extracted by our CNN effectively combine both color and morphological information, leading to significantly better performances for all the cases under consideration. Experiments performed on four different datasets demonstrated that the proposed CNN-based model allows accurate flower identification even in scenarios of different flower species and illumination conditions, with optimal recall and precision rates near $80\%$ even for datasets significantly dissimilar from the training sequences.
As part of our future work, we intend to explore existing datasets and state-of-the-art models for semantic image segmentation. Particularly successful strategies consist of end-to-end architectures that, without external computation of region proposals, generate pixel dense prediction maps for inputs with arbitrary size [@Shelhamer2016_FCN; @Chen2017DeepLab; @Lin2017RefineNet].
Moreover, similar to the approach proposed in [@Stein2016] for fruits, we intend to extend our module for flower tracking and localization based on probabilistic approaches that use the estimated motion between frames (e.g. particle filtering [@Mozhdehi2017]) to predict the location of flowers. To extend the applicability of our model to the detection of fruitlets as well as other flower species, we will consider additional transfer learning approaches such as data augmentation by affine transformations and the use of external datasets.
[^1]: The citation information for this article is: P. A. Dias, A. Tabb, and H. Medeiros, “Apple flower detection using deep convolutional networks,” Computers in Industry, vol. 99, pp. 17 $-–$ 28, Aug. 2018. DOI: 10.1016/j.compind.2018.03.010
[^2]: Mention of trade names or commercial products in this publication is solely for the purpose of providing specific information and does not imply recommendation or endorsement by the U.S. Department of Agriculture. USDA is an equal opportunity provider and employer.
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---
abstract: 'Rare earth ($R$) half-Heusler compounds, $R$BiPt, exhibit a wide spectrum of novel ground states. Recently, GdBiPt has been proposed as a potential antiferromagnetic topological insulator (AFTI). We have employed x-ray resonant magnetic scattering to elucidate the microscopic details of the magnetic structure in GdBiPt below $T_N$ = 8.5K. Experiments at the Gd $L_2$ absorption edge show that the Gd moments order in an antiferromagnetic stacking along the cubic diagonal \[111\] direction satisfying the requirement for an AFTI, where both time-reversal symmetry and lattice translational symmetry are broken, but their product is conserved.'
author:
- 'A. Kreyssig$^1$'
- 'M.G. Kim$^1$'
- 'J.W. Kim$^2$'
- 'S.M. Sauerbrei$^1$'
- 'S.D. March$^1$'
- 'G.R. Tesdall$^1$'
- 'S.L. Bud’ko$^1$'
- 'P.C. Canfield$^1$'
- 'R.J. McQueeney$^1$'
- 'A.I. Goldman$^1$'
bibliography:
- 'gdbipt.bib'
title: 'Magnetic order in GdBiPt studied by x-ray resonant magnetic scattering'
---
The discovery of three-dimensional topological-insulating states in binary alloys (Bi$_{1-x}$Sb$_x$)[@Fu_2007; @Hsieh_2008] and compounds (Bi$_2$Se$_3$, Bi$_2$Te$_3$, Sb$_2$Te$_3$)[@Xia_2009; @Zhang_2009; @Chen_2009], which feature an insulating gap in the bulk but with topologically protected conducting states on the surfaces or edges, has opened a new frontier for fundamental condensed matter physics research.[@Hasan_and_Kane] As pointed out in several papers, the novel properties of this class of materials offer potential for technological breakthroughs in quantum computing and magneto-electronic applications.[@Hasan_and_Kane; @Moore_2010; @Li_2010] Over the past year, attention has turned towards investigations of new phenomena that arise when topological insulators (TI) also manifest, or are in close proximity to, other phenomena including magnetic order and superconductivity.[@Li_2010; @Hosur_2010; @Mong_2010; @Hasan_and_Kane] Recently, the Heusler and half-Heusler compounds have been subject to intense scrutiny because of their potential as TI with tunable electronic properties.[@Chadov_2010; @Lin_2010; @Xiao_2010; @Li_2011] Specifically, Mong *et al.*[@Mong_2010] have proposed that GdBiPt may provide the first realization of an antiferromagnetic topological insulator (AFTI), where both time-reversal symmetry and lattice translational symmetry are broken, but their product is conserved. Predictions for this class of TI include gapped states on some surfaces, gapless states on others, and novel one-dimensional metallic states along step edges on the gapped surfaces.[@Mong_2010]
More generally, Heusler and half-Heusler compounds exhibit a wide spectrum of novel ground states.[@Graf_2011] The rare earth ($R$) half-Heusler compounds, $R$BiPt, feature magnetic ordering (GdBiPt),[@Canfield_1991] superconductivity (LaBiPt, YBiPt)[@Goll_2008; @Butch_2011] and heavy-fermion behavior (YbBiPt)[@Fisk_1991] . Although the low-temperature ground states of the $R$BiPt system (for $R$ = Ce, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, and Yb) have been characterized as antiferromagnetic through thermodynamic and transport measurements, there have been few magnetic structure determinations for this series.[@Wosnitza_2006] GdBiPt has the highest $T_N$ of the series at approximately 8.5K[@Canfield_1991] and, since the orbital angular moment $L$ = 0 for the $S$-state Gd ion, the magnetic structure in the absence of crystalline electric field effects may be directly investigated. However, the high neutron-absorption cross section for naturally occurring Gd is problematic for conventional magnetic diffraction experiments.
Here we describe the magnetic order of GdBiPt below $T_N$ = 8.5K determined by x-ray resonant magnetic scattering (XRMS) at the Gd $L_2$ absorption edge. GdBiPt crystallizes in the MgAgAs-type structure (cubic space group $F\,\overline{4}\,3\,m$, $a$ = 6.68${\AA}$ with Gd, Bi and Pt at the 4$c$, 4$d$ and 4$a$ sites, respectively; see Fig. \[fig1\]).[@Dwight_1974; @Robinson_1994] The structure may be viewed as three sets of elementally pure, interpenetrating face-centered cubic lattices. We find that the commensurate magnetic order doubles the cubic unit cell along the diagonal \[111\] direction, characterized by a propagation vector $\textbf{q}_m$ = ($\frac{1}{2}\,\frac{1}{2}\,\frac{1}{2}$), so that alternating ferromagnetic (111) planes of Gd are antiferromagnetically coupled along the \[111\] direction. This structure is quite similar to the model B magnetic structure for an AFTI via spin-orbit coupling as described by Mong *et al.*,[@Mong_2010] but we find that the moment direction in GdBiPt is not parallel to the magnetic propagation vector as is found, for example, in MnSbCu[@Forster_1968] or CeBiPt.[@Wosnitza_2006]
![(Color online) Crystal structure of GdBiPt.[]{data-label="fig1"}](fig1 "fig:"){width="0.75\linewidth"}\
Single crystals of GdBiPt were solution-grown using a Bi flux and emerged with sizeable facets perpendicular to the \[001\] direction and smaller facets perpendicular to \[111\]. High-purity Gd (obtained from Ames Laboratory), Pt, and Bi were placed in an alumina crucible in the ratio Gd:Pt:Bi=3:3:94, sealed in a silica ampule, and slowly cooled from 1170$^{\circ}$C to 600$^{\circ}$C over 200 hours. At 600$^{\circ}$C, the excess Bi solution was decanted from the GdBiPt crystals.[@Canfield_1992] The dimensions of the single crystal studied in the XRMS measurements were approximately 3$\times$3$\times$2mm$^3$ with a large as-grown facet perpendicular to \[001\]. The measured mosaicity of the crystal was less than 0.01degrees full-width-at-half-maximum (FWHM), attesting to the high quality of the sample. The XRMS experiment was performed on the 6ID-B beamline at the Advanced Photon Source at the Gd $L_2$-edge ($E$ = 7.934keV). The incident radiation was linearly polarized perpendicular to the vertical scattering plane ($\sigma$-polarized) with a beam size of 0.5mm (horizontal) $\times$ 0.2mm (vertical). In this configuration, dipole resonant magnetic scattering rotates the plane of linear polarization into the scattering plane ($\pi$-polarization). For some of the measurements, pyrolytic graphite PG (006) was used as a polarization analyzer to suppress the charge and fluorescence background relative to the magnetic scattering signal. For measurements of the magnetic reflections, the sample was mounted at the end of the cold finger of a closed-cycle cryogenic refrigerator with the ($H H L$) plane coincident with the scattering plane.
![(Color online) Resonant magnetic scattering from the GdBiPt single crystal. (a) Rocking scans ($\theta$) through the ($\frac{1}{2}$ -$\frac{1}{2}$ $\frac{13}{2}$) magnetic peak position above (open circles) and below (filled circles) $T_N$ taken in $\sigma$-$\pi$ scattering geometry. (b) Energy scan through the Gd $L_2$ absorption edge at the ($\frac{1}{2}$ -$\frac{1}{2}$ $\frac{13}{2}$) magnetic peak position at $T$ = 4.7K (blue filled circles) along with the measured x-ray fluorescence from the sample (black filled circles).[]{data-label="fig2"}](fig2 "fig:"){width="0.88\linewidth"}\
Measurements of the diffraction from the sample performed in the $\sigma$-$\pi$ scattering geometry using the PG (006) polarization analyzer are shown in Fig. \[fig2\]. For temperatures above $T_N$ = 8.5K, only Bragg peaks consistent with the chemical structure[@Dwight_1974; @Robinson_1994] of GdBiPt were observed. However, upon cooling below $T_N$, additional Bragg scattering at half-integer values of ($H K L$) was found as shown in Fig. \[fig2\](a). The magnetic origin of these peaks was confirmed by energy scans through the Gd $L_2$ absorption edge and from the temperature dependence of the diffraction peak intensity as described below.
![(Color online) (a) $M/H$ and its temperature derivative for GdBiPt. (b) The magnetic intensity measured while scanning temperature at the maximum of the ($\frac{1}{2}\,\frac{1}{2}\,\frac{9}{2}$) diffraction peak without a polarization analyzer (open small circles) and the integrated intensity of the ($\frac{1}{2}$ -$\frac{1}{2}$ $\frac{13}{2}$) diffraction peak measured at selected temperatures using the polarization analyzer (open large squares). The solid line is a power law fit to the integrated intensity data as described in the text.[]{data-label="fig3"}](fig3 "fig:"){width="0.88\linewidth"}\
The energy scan Fig. \[fig2\](b) was performed with the diffractometer set at the magnetic peak position and is typical of resonant magnetic scattering at the $L$ edges of rare-earth compounds.[@JWK_2005] At the $L_2$ edge of rare-earth elements, the resonance primarily involves electric dipole ($E$1) transitions from the 2$p_\frac{1}{2}$ core level to the empty 5$d$ states, seen as the strong line just at, or slightly below the maximum in the measured fluorescence intensity. The weaker feature below the $E$1 resonance in Fig. \[fig2\](b) is likely due to the electric quadrupole ($E$2) transition from the 2$p_\frac{1}{2}$ core level to the 4$f$ states that are pulled below the Fermi energy because of the presence of the core hole in the resonance process.
The temperature dependence of the magnetic scattering, along with the corresponding magnetization measurements performed on a sample from the same batch using a Quantum Design Magnetic Properties Measurement System, are shown in Fig. \[fig3\]. The magnetic order parameter was measured at the ($\frac{1}{2}\,\frac{1}{2}\,\frac{9}{2}$) peak position as the sample temperature was increased during a temperature scan in the absence of the polarization analyzer. These data were supplemented by measurements of the integrated intensity of the ($\frac{1}{2}$ -$\frac{1}{2}$ $\frac{13}{2}$) magnetic Bragg peak at selected temperatures and with polarization analysis. The line in Fig. \[fig3\](b) describes a fit to the integrated intensity data using a power law of the form $I\,\sim\,(1-\frac{T}{T_N})^{2\beta}$ yielding $T_N$ = 8.52$\pm$0.05K and $\beta$ = 0.33$\pm$0.02. The close proximity of $T_N$ determined from our scattering measurements and the peak in d\[($M/H$)T\]/d$T$ (Ref.) at $T$ = 8.6K, again confirms the magnetic origin of the Bragg scattering with a propagation vector of $\textbf{q}_m$ = ($\frac{1}{2}\,\frac{1}{2}\,\frac{1}{2}$). Systematic $M$ versus $H$ measurements (not shown) demonstrate in addition that no spontaneous ferromagnetic moment is present.
![(Color online) Integrated intensity in azimuth scans through the (-$\frac{1}{2}$ -$\frac{1}{2}$ $\frac{13}{2}$) magnetic Bragg peak. Measured data are depicted by full black circles. Full and dashed lines represent calculations for selected magnetic moment directions in a single magnetic domain, and for intensity from equally populated domains averaged over the three possible symmetry-equivalent magnetic moment orientations, respectively.[]{data-label="fig4"}](fig4 "fig:"){width="0.88\linewidth"}\
Having established the nature of the magnetic ordering in GdBiPt, we now describe our attempt to determine the direction of the ordered magnetic moment. The angular dependence of the resonant magnetic intensity $I(\psi)$ for the incident $\sigma$-polarized beam depends upon the component of the magnetic moment along the scattered beam direction and can be written as $I(\psi)_{(\textbf{Q},\,\alpha,\,\beta)}~=~C~[{\widehat{\textbf{m}}\cdot\widehat{\textbf{k}^\prime}(\psi)_{(\textbf{Q})}}]^2~A(\psi)_{(\textbf{Q},\,\alpha,\,\beta)}$ where C is an overall scale factor that accounts for the resonant scattering matrix element and incident beam intensity, $\widehat{\textbf{m}}$ and $\widehat{\textbf{k}^\prime}$ represent the magnetic moment and scattered beam directions, respectively, and $A$ accounts for the absorption correction.[@Detlefs_1997] The sample geometry required off-specular scattering measurements of the magnetic peaks. That is, the angle $\alpha$ of the incident beam $\textbf{k}$ with respect to the sample surface is different from the angle $\beta$ of the outgoing beam $\textbf{k}^\prime$ with respect to the sample surface.[@You_1999] For the azimuth angle $\psi$ scans shown in Fig. \[fig4\], the diffractometer was set at the position of the magnetic Bragg peak and the crystal was rotated around the scattering vector $\textbf{Q}\,=\,\textbf{k}^\prime-\textbf{k}$ thereby rotating $\widehat{\textbf{k}^\prime}$ with respect to $\widehat{\textbf{m}}$ while leaving $\textbf{Q}$ fixed. This yields an azimuth dependence of the intensity which is specific to a given magnetic moment direction. Note, that the absorption correction $A$ also depends on the azimuth angle $\psi$.
For a cubic lattice, the determination of the ordered moment direction is significantly complicated by the presence of domains that arise from symmetry-equivalent magnetic propagation vectors and moment directions. For the observed magnetic Bragg peaks at ($H\,K\,L$) with $H$, $K$, and $L$ half integers, four symmetry-equivalent $\{\frac{1}{2}\,\frac{1}{2}\,\frac{1}{2}\}$ propagation vectors exist: ($\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{2}$), (-$\frac{1}{2}$ -$\frac{1}{2}$ $\frac{1}{2}$), ($\frac{1}{2}$ -$\frac{1}{2}$ -$\frac{1}{2}$), and (-$\frac{1}{2}$ $\frac{1}{2}$ -$\frac{1}{2}$). Fortunately, for GdBiPt in the cubic space group $F\,\overline{4}\,3\,m$, only one propagation vector contributes to a particular magnetic reflection \[e.g. the magnetic Bragg peak (-$\frac{1}{2}$-$\frac{1}{2}$$\frac{13}{2}$) is generated by the propagation vector $\textbf{q}_m$ = (-$\frac{1}{2}$-$\frac{1}{2}$$\frac{1}{2}$) from the (006) zone center\]. The measured data for the (-$\frac{1}{2}$-$\frac{1}{2}$$\frac{13}{2}$) magnetic Bragg peak show two important features in the azimuth scan presented in Fig. \[fig4\]: a distinct minimum with almost no intensity close to $\psi$ = 0, and an increase in intensity by more than an order of magnitude as $\psi$ is varied by $\pm30^\circ$. Both features are in strong contrast to the expected $\psi$ dependence of the intensity for magnetic moments parallel to the propagation vector $\textbf{q}_m$ = (-$\frac{1}{2}$-$\frac{1}{2}$$\frac{1}{2}$) as illustrated in Fig. \[fig4\] by the bold black line with a maximum close to $\psi$ = 0. Therefore, we can exclude that the moments are parallel to the propagation vector in GdBiPt.
In Fig. \[fig4\], calculated curves are also shown for other moment directions. Unfortunately, for each of the depicted moment directions, three different symmetry-equivalent orientations can occur yielding three magnetic domains. The dashed lines in Fig. \[fig4\] represent the calculated $\psi$ dependence of the intensity if we include all such domains for a given moment direction with equal population. We again find poor agreement between the domain-averaged calculations for moments along the set of {1 1 1}, {1 1 0} and {0 0 1} directions. However, calculations assuming the presence of only a single domain within the probed volume, with one specified moment direction (either \[-1 1 1\] or \[-1 1 0\] for the (-$\frac{1}{2}$-$\frac{1}{2}$$\frac{13}{2}$) Bragg peak in Fig. \[fig4\]) come much closer to describing the measured data. This behavior clearly indicates that the magnetic domains are large; smaller than the footprint of the incident beam on the sample (approximately 0.5$\times$0.5mm$^2$), but of the same order of magnitude. Similar large magnetic domains have been noted in previous XRMS work on GdNi$_2$Ge$_2$ as well.[@JWK_2005b] Nevertheless, a unique determination of the moment direction is not possible based on the available data. A more precise determination of the moment direction may be possible from measurements with much smaller incident beam dimensions and/or control of domain populations.[@JWK_2005b]
Summarising the experimental results, below $T_N$ = 8.5K the magnetic Gd moments order in a commensurate antiferromagnetic structure in GdBiPt that can be described as doubling the cubic unit cell along the diagonal \[111\] direction, so that alternating ferromagnetic (111) planes of Gd are antiferromagnetically coupled along the \[111\] direction. The moments are not aligned parallel to this diagonal \[111\] direction.
In contrast to GdBiPt, CeBiPt is an antiferromagnet characterized by a propagation vector $\textbf{q}_m$ = (100) and the ordered moments are collinear with the propagation vector along \[100\],[@Wosnitza_2006] but with a reduced moment that may, in part, be attributed to crystalline electric field (CEF) effects.[@Goll_2007] Unfortunately, XRMS measurements do not allow a direct extraction of the ordered moment in GdBiPt, but earlier specific heat measurements[@Canfield_1991] estimated an entropy of $\sim$0.8$\mathcal{R}$ln8 associated with the magnetic transition close to the value expected for full moment ordering without CEF effects. The entropy associated with the corresponding magnetic transitions for the Nd, Tb and Dy compounds were considerably less than $\mathcal{R}$ln(2$J$+1) expected for the full Hund’s rule $J$ multiplet, indicating the importance of CEF effects in these compounds. The magnetic structures for $R$ = Nd, Sm, Tb, Dy, Ho, Er, Tm, and Yb, have not yet been identified by neutron or XRMS measurements and such measurments are planned.
Finally, we comment on our results in light of the proposal that GdBiPt may be an AFTI candidate.[@Mong_2010] The AFTI state may be derived from either magnetic ordering in a pre-existing strong TI (model A in Ref.) or, alternatively, for specific antiferromagnetic ordering schemes that induce spin-orbit coupling in the system (model B in Ref.). Given previous ARPES measurements[@Liu_2011] above $T_N$, which do not find direct evidence for band inversion in GdBiPt, it seems unlikely that GdBiPt is itself a strong TI. However, the magnetic structure determined here is consistent with the alternative model B presented by Mong *et al.*[@Mong_2010] The doubling along the cubic diagonal direction represents the broken lattice translational symmetry (by order of two) and the ordering of each magnetic moment breaks the time-reversal symmetry, however, the product of both symmetry operations is conserved for the determined magnetic order.
We acknowledge valuable discussions with A. Kaminski, J.E. Moore, R.S.K. Mong, P.J. Ryan, and J.C. Lang. This work was supported by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, U.S. Department of Energy. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. DE-AC02-07CH11358. Use of the Advanced Photon Source was supported by the U.S. DOE under Contract No. DE-AC02-06CH11357.
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---
abstract: 'The standard approach for optimization of XPath queries by rewriting using views techniques consists in navigating inside a view’s output, thus allowing the usage of only one view in the rewritten query. Algorithms for richer classes of XPath rewritings, using intersection or joins on node identifiers, have been proposed, but they either lack completeness guarantees, or require additional information about the data. We identify the tightest restrictions under which an XPath can be rewritten in polynomial time using an intersection of views and propose an algorithm that works for any documents or type of identifiers. As a side-effect, we analyze the complexity of the related problem of deciding if an XPath with intersection can be equivalently rewritten as one without intersection or union. We extend our formal study of the view-based rewriting problem for XPath by describing also (i) algorithms for more complex rewrite plans, with no limitations on the number of intersection and navigation steps inside view outputs they employ, and (ii) adaptations of our techniques to deal with XML documents without persistent node Ids, in the presence of XML keys. Complementing our computational complexity study, we describe a proof-of-concept implementation of our techniques and possible choices that may speed up execution in practice, regarding how rewrite plans are built, tested and executed. We also give a thorough experimental evaluation of these techniques, focusing on scalability and the running time improvements achieved by the execution of view-based plans.'
author:
- BOGDAN CAUTIS ALIN DEUTSCH IOANA ILEANA NICOLA ONOSE
bibliography:
- 'biblio-tcs.bib'
title: 'Rewriting XPath Queries using View Intersections: Tractability versus Completeness'
---
Nested Intersection {#sec:nestedinter}
===================
APPENDIX {#appendix .unnumbered}
========
Proof of Theorem \[th:completenessUF-es\] (fragment for PTIME) {#sec:proof1}
==============================================================
The proof is organized as follows. We first show that $\proc{Apply-Rules}$ is complete over DAG patterns in which the root and the output node are connected by a path having only /-edges (Lemma \[l:onetokenq\_nviews\]).
We then consider the complementary case when all the branches in parallel (the compensated views) have at least one //-edge in the main branch. For clarity, we prove completeness progressively, starting with the case of intersecting two views under certain restrictions: their root tokens have the same main branch, their result tokens have the same main branch as well (Lemma \[lem:2-comp-skel\]). We then extend to the case of arbitrary many views, with these restrictions (Lemma \[lem:n-comp-skel\]). Then we consider the general case, which will rely on results proven for the limited cases.
We now consider the case when all the branches in parallel (the compensated views) have at least one //-edge in the main branch.
We start by proving the following lemma.
\[lem:2-comp-skel\] For two patterns $v_1, v_2$ s.t. their root and result tokens have the same main branch, the DAG pattern $d=dag(v_1
\cap v_2)$ is union-free iff rewrites $d$ into a tree.
Let us first consider what rule steps may apply in order to refine $d$. First, since we are dealing with patterns with root and result tokens having the same main branch, R1 steps will first apply, coalescing the root and result tokens of the two branches. At this point, the only rules that remain applicable are R6 and R7. This is because we do not have nodes with incoming (or outgoing) /-edges and //-edges simultaneously and R5 will only apply to predicates starting by a //-edge.
We argue now that $d$ is union-free iff rule R7 applies on it.
Note that since we only have 2 parallel branches an application of rule R7 would immediately yield a tree. So the *if* direction is straightforward. For the *only if* direction, if R7 does not apply this translates into *$(\dagger)$ there is no mapping (not necessarily root-mapping) between the intermediary part of $v_1$ and the intermediary part of $v_2$.*
Assuming that $(\dagger)$ holds, rule R6 remains the only option. So, possibly after some applications of R6, followed by applications of R1 collapsing entire tokens, we obtain a refined DAG $d$ as illustrated in Figure \[fig:completeness1\] (only the main branches of $d$ are illustrated). $p_r$ has the common main branch following the root (may have several tokens if R6 applied) and $t_o$ denotes the result token. $t_1$ and $t_2$ denote the two sibling /-patterns for which R6 no longer applies. As $t_1$, $t_2$ are *dissimilar* we have that $t_1 \not \equiv t_2$.
We show that $d$ is not union-free by the following approach: we build two interleavings, $p'$ and $p''$, that do not contain one another, and then show that by assuming the existence of a third interleaving $p$ that contains both we obtain the contradiction $t_1 \equiv t_2$.
We continue with the following observation which follows easily from the restriction on usage of //-edges in predicates: given two /-patterns $t_1$ and $t_2$ from , if $t_1$ does not map in $t_2$ then, for any tree pattern $q$ of the form $\dots//t_2'//\dots$ with $t_2'$ being an isomorphic copy of $t_2$, we have that $t_1$ does not map into $t_2'$.
The following steps will implicitly use this observation.
Let $p_1=t_1//p_3$ denote the left branch and let $p_2=t_2//p_4$ denote the right branch in $d$. Because of $(\dagger)$, there is no mapping (not necessarily root-mapping) between $p_1$ and $p_2$.
Let $sf_1$ be the maximal token-suffix of $p_1$ that can map into $p_2$, and let $pr_1$ be the remaining part (i.e., a token-prefix). Note that $pr_1$ cannot be empty, so we can write it as $p_1=pr_1//sf_1$.
Similarly, let $sf_2$ denote the maximal token-suffix of $p_2$ that can map into $p_1$, and let $pr_2$ denote the remaining part, non-empty as well. So we can write $p_2$ as $p_2=pr_2//sf_2$.
We build $p'$ and $p''$ as follows: $$\begin{aligned}
p' & = & p_r'//pr_2'//p_1'//t_o'=p_r'//pr_2'//pr_1'//sf_1'//t_o'\\
p''&=& p_r''//pr_1''//p_2''//t_o''=p_r''//pr_1''//pr_2''//sf_2''//t_o''\end{aligned}$$ where the $\#'$, $\#''$ parts are isomorphic copies of the $\#$ parts of $d$.
Note that $pr_1'$ (resp. $pr_2''$) starts by token $t_1' \equiv t_1$ (resp. $t_2'' \equiv t_2$).
These two queries are obviously in $interleave(d)$. Moreover, there can be no containment mapping between $p'$ and $p''$ since, by the way $sf_1$ and $sf_2$ were defined, $pr_1'$ (resp. $pr_2''$) could only map in $pr_1''$ (resp. $pr_2'$).
So neither $p'$ nor $p''$ can be the interleaving that reduces all the others. We show in the following that no other interleaving $p$ of $d$ can reduce both $p'$ and $p''$ unless $t_1 \equiv t_2$.
Let us assume that such a $p$ exists. Without loss of generality, let $p$ be of the form $$p=p_r//m//t_o.$$ (interleavings that are not of this kind will not remain in the normal form of $d$).
We assume a containment mapping $\phi'$ from $p$ to $p'$ and another one $\phi''$ from $p$ to $p''$. Obviously, $v_1,v_2$ must have containment mappings into $p$, since $p \equiv v_1 \cap
v_2$. In particular, their sub-sequences $p_1$ and $p_2$ have images in the $m$ part of $p$. Let $\psi'$ and $\psi''$ be these containment mappings.
With a slight abuse of notation, let $\psi'(pr_1)$ denote the minimal token-prefix of $m$ within which the image under $\psi'$ of the $pr_1$ part of $v_1$ occurs. $\psi'(pr_1)$ is well defined because $v_1$ and $v_2$, and hence $p_1$ and $p_2$, are in , hence the image of a token of $p_1$ and of its predicates is included into a token of $m$. In other words, $\psi'(pr_1)$ starts with the root token of $m$ and ends with the token into which the output token of $pr_1$ maps. Similarly, let $\psi''(pr_2)$ denote the minimal token-prefix of $m$ within which the image under $\psi''$ of the $pr_2$ part of $v_2$ occurs.
We can thus write $p$ in two forms, as $$\begin{aligned}
p& =& p_r//\psi'(pr_1) \dots \psi'(sf_1)//t_o \\
p & = &p_r//\psi''(pr_2)\dots\psi'(sf_2)//t_o\end{aligned}$$ Next, we argue that in the containment mapping $\phi''$ of $p$ in $p''$, we must have $\phi''(\psi'(pr_1))=pr_1''$. Similarly, we must have that $\phi'(\psi''(pr_2))=pr_2'$. This follows easily from the way $sf_1$ and $sf_2$ were defined. (For instance, no node of $\psi'(pr_1)$ can map below $pr_2''$ in $p''$, otherwise $\textit{sf}_1$ would not be maximal. And $\mb{p_r}=\mb{p_r''}$ and $|\psi'(pr_1) \geq pr_1''|$, hence no node of $\psi'(pr_1)$ can map higher than $pr_1''$ either, otherwise $p_r$ would not map into $p''$.) And it then implies that $\psi'(pr_1) \equiv pr_1$ and $\psi''(pr_2) \equiv pr_2$.
But since $m$ starts by both the token-prefix $\psi'(pr_1)$ and by $\psi''(pr_2)$, hence by token-prefixes $pr_1$ and $pr_2$, $pr_1$ and $pr_2$ should at least start by the same token. *Hence $t_1 \equiv t_2$, which is a contradiction.*
In other words, we showed that $d$ is union-free iff, after a sequence of R1 steps, R7 applies, transforming the pattern into a tree. As we know from Lemma \[l:termination\] that also terminates, it follows that $d$ is union-free iff rewrites $d$ into a tree.
For $d=v_1 \cap v_2$, where $v_1$ and $v_2$ are two skeleton patterns such that their root and result tokens have the same main branch, with the previous notations, we can also easily prove the following.
\[cl:shape\] All the interleavings of are of the form $p_r//\dots//t_o$.
We know so far that the intersection $d$ of two skeleton queries such that their root and result tokens have the same main branch is union-free iff rewrites $d$ into a tree. Moreover, this happens iff there is a mapping between the intermediary part of one into the intermediary part of the other. If $d$ is not union-free, the result is a union of queries having the same root and result tokens, as described in Lemma \[cl:shape\].
We now consider intersections of more than two patterns.
\[lem:n-comp-skel\] Given patterns $v_1, \dots, v_n$ s.t. their root and result tokens have the same main branch, the DAG pattern $d=dag(v_1 \cap \dots\cap v_n)$ is union-free iff rewrites $d$ into a tree. If the skeletons are of the form $v_i = p_r//p_i//t_o$, $1\leq i \leq n$, then $d$ is union-free iff there is a query among them, $v_j$, having an intermediary part $p_j$ such that all other $p_i$ map into $p_j$.
We prove this by induction on the number of patterns (Lemma \[lem:2-comp-skel\] covers $n=2$).
As in the case of Lemma \[lem:2-comp-skel\], we first rewrite $d$ by rule R1, coalescing the root and result tokens of the parallel branches. At this point, the only rules that remain applicable are R6 and R7.
Let us now assume that some run of terminates on $d$ without outputting a tree. Then, it is easy to check that will also stop in the particular run, in which we start by applying only R7 until it does not apply anymore.
We show in the following that $d$ resulting from this run is not union-free.
We continue with $d$ obtained, as said previously, possibly after some applications of R7 that removed some of the branches in parallel, yielding a DAG pattern as the one illustrated in Figure \[fig:completeness3\]. $2\leq k\leq n$ denotes the number of remaining branches in parallel and $i_1,
\dots i_k$ denote these branches. Without loss of generality, let these be the intermediary parts of $v_1, \dots, v_k$ respectively.
Note that we are now in a setting in which $d \equiv v_1 \cap \dots \cap v_k$ and the following holds: *$(\dagger)$ there is no mapping between the intermediary parts of any of $v_1, \dots, v_k$.*
Next, starting from the DAG pattern $d$ in Figure \[fig:completeness3\], by $(\dagger)$, only rule R6 is applicable. For convenience, we assume that R6 steps are applied by a slightly different strategy: we take an R6 step only if it applies to *all* the parallel branches simultaneously. At some point, this process will stop as well and we obtain a refined $d$ as illustrated in Figure \[fig:completeness2\] (only the main branches are given). Let $p_i=t_i//r_i$ denote the branches in parallel. Note that the $t_i$ tokens cannot all be equivalent (recall that in similar patterns must be equivalent).
Let us assume towards a contradiction that *$d$ is union-free* and let $q$ be the interleaving such that $q\equiv d$. Without loss of generality, let $q$ be of the form $q=p_r//t//m//t_o$, where $t$ is the token immediately following $p_r$ (the $m$ part might be empty).
Without loss of generality, let us also assume that $t_1 \not \equiv t$ (we know that there must be at least one such token among $t_1, \dots, t_k$.) We show that by assuming $q\equiv d$ we obtain the contradiction $t \equiv
t_1$.
For $p_1$ chosen in this way, let $d'$ denote the DAG pattern obtained from $d$ by removing its $p_1$ branch. Introducing for each $i$ the pattern $v_i'=p_r//p_i//t_o,$ by $(\dagger)$ all incomparable, note that $d'$ can be seen as $d'=\dagp{v_2' \cap \dots \cap v_k'}$ and note also that $d \equiv d' \cap v_1' = d' \cap (p_r//t_1//r_1//t_o)$. By the inductive hypothesis, $d'$ is not union-free, i.e., there is some $x \geq 2$ and some patterns $q_1, \dots, q_x$, which are some incomparable interleavings of $d'$ (such their root and result tokens have the same main branch), all of the form $p_r//\dots//t_o$ (by induction, from Lemma \[cl:shape\]), such that $d' \equiv q_1 \cup \dots \cup q_x$.
So can conclude that $d \equiv v_1' \cap (q_1 \cup q_2 \cup \dots \cup q_m) = (v_1' \cap q_1) \cup (v_1' \cap q_2) \cup \dots \cup (v_1' \cap q_m)$.
Note now that we cannot have $v_1' \sqsubseteq q_i$, for any $q_i$, since this would mean that $v_1' \sqsubseteq v_2', \dots, v_k'$, in contradiction with ($\dagger$).
We proceed by an exhaustive case analysis:
***Case 1:*** *for all $q_i$, we have $q_i \not \sqsubseteq v_1'$.*
In this case, each intersection of two given above will not be union-free. This follows easily from Lemmas \[lem:2-comp-skel\] and \[cl:shape\], since $v_1'$ and $q_i$ have the same root tokens and result tokens (there is no containment mapping between them, so there can be no mapping between their intermediary parts).
Hence any interleaving resulting from some DAG pattern $d_i=dag(q_1
\cap v'_i)$ cannot even reduce all the other interleavings of $d_i$, so $d$ cannot be union-free in this case, since $d=\cup_i d_i$. This case can be thus discarded.
***Case 2:*** *at least two interleavings of $d'$, say $q_1$ and $q_2$, are such that $q_1 \sqsubseteq v_1$ and $q_2 \sqsubseteq v_1$.*
We can thus reformulate $d$ as $d \equiv q_1 \cup q_2 \cup (v_1' \cap q_3) \cup \dots\cup (v_1'\cap q_m)$. Now, each DAG pattern $v_1' \cap q_j$ is not union-free and, moreover, their interleavings cannot contain $q_1$ or $q_2$ (since $q_1, q_2 \not \sqsubseteq q_j$ in the first place). Also, obviously, $q_2 \not \sqsubseteq q_1$ and $q_1 \not \sqsubseteq q_2$. So again $d$ can not be union-free and this case can be discarded as well.
***Case 3:*** *exactly one of the interleavings of $d'$ , call it $q_1$, is contained in $v_1'$ ($q_1 \sqsubseteq v_1'$).*
In this case, $d$ can be reformulated as $ d \equiv q_1 \cup(v_1' \cap q_2) \cup \dots\cup (v_1'\cap q_m)$ and cannot be union-free unless it is in fact equivalent to $q_1$. This means that for all other $q_i$’s we must have $v_1' \cap
q_i \sqsubseteq q_1$. Of course, $q_1$ should be equivalent (isomorphic modulo minimization, by Lemma \[lem:equiv-iso\]) to $q$, the interleaving of $d$ for which we supposed $d \equiv q$, i.e. $q_1\equiv p_r//t//m//t_o$.
We continue by assuming for instance that $v_1' \cap q_2 \sqsubseteq q_1$.
Recall that $v_1'$ is of the form $v_1'=p_r//p_1//t_o$ and let $q_2$ be of the form $q_2=p_r//m_2//t_o$. Since $q_2 \not \sqsubseteq v_1'$ and they have the same root and result tokens, there is no mapping from $p_1$ into $m_2$. Consequently, let $sf_1$ denote the maximal token-suffix of $p_1$ that can map into $m_2$, and let $pr_1$ denote the remaining part (i.e., a token-prefix). Since $pr_1$ cannot be empty, we can write $v_1'$ as $v_1'=p_r//pr_1'//sf_1//t_o$ where $pr_1'$ is an isomorphic copy of $pr_1$.
Let us now consider the interleaving $u$ of $v_1' \cap q_2$, of the form $u=p_r//pr_1''//m_2//t_o$ where $pr_1''$ is an isomorphic copy of $pr_1$ as well.
As we assumed that $u \sqsubseteq v_1'\cap q_2 \sqsubseteq q_1$, there must exist a containment mapping $\psi$ from $q_1$ to $u$.
Since $q_1 \sqsubseteq v_1'$, let $\phi$ be a containment mapping from $v_1'$ into $q_1$. So we have $v_1' \stackrel{\phi}{\longrightarrow} q_1 \stackrel{\psi}{\longrightarrow} u$.
In particular, $\phi$ must map the $pr_1'//sf_1$ part of $v_1'$ in the $t//m$ part of $q_1$. With a slight abuse of notation, let $\phi(pr_1')$ denote the minimal token-prefix of $t//m$ within which the image under $\phi$ of $pr_1'$ occurs. In other words $\phi(pr_1')$ starts with the root token of $t$ and ends with the token into which the output token of $pr_1'$ is mapped. (Again, $\phi(pr_1')$ is well defined because all patterns are skeletons and tokens can only map strictly inside tokens.)
We can thus write $q_1$ as $q_1=p_r//\phi(pr_1') \dots
\phi(sf_1)//t_o$.
Next, we argue that in the containment mapping $\psi$ of $q_1$ into $u$, we must have $\psi(\phi(pr_1'))=pr_1''$. (This follows easily from the definition of $sf_1$.) And this implies that $\phi(pr_1')
\equiv pr_1'' \equiv pr_1$. Hence $q_1$ and $v_1'$ start by some common non-empty token-prefix. Since one of them starts by $t$ and the other by $t_1$ this means in the end that *$t \equiv t_1$, which is a contradiction.*
**Remark.** We can also generalize Lemma \[cl:shape\] as follows: the interleavings of are of the form $p_r//\dots//t_o$ (see Figure \[fig:completeness2\]).
So we know for now that is complete for the case of DAG patterns that are defined as the intersection of skeleton queries when their root and result tokens have the same main branch. Such an intersection is union-free iff there is a query $v_i$ among them having an intermediary part into which all the other intermediary parts map. If this is not the case, the DAG is equivalent to a union of interleavings having the same root tokens and result tokens.
We are now ready to give sum up the results so far and conclude the completeness proof for .
We will show that, given $n$ (extended) skeletons $v_1, \dots, v_n$, all having several tokens, is complete for deciding union-freedom for the DAG pattern $$d=\dagp{v_1 \cap \dots\cap v_n}.$$
We first rewrite $d$ by R1 steps. We obtain after this phase a DAG pattern $d$ in which the root token of $d$ may have several main branch nodes with outgoing //-edges. Similarly, the result token may have several nodes with incoming //-edges. If this is not the case, neither for the root token nor for the result token, then we know that the algorithm is in this case complete by Lemma \[lem:n-comp-skel\].
Let us assume that some run of ends without a tree. We can easily prove that in this case the following run would also stop without yielding a tree:
- first refine by rules R2, R3 and R4 the root token and the result token w.r.t. their outgoing/incoming //- edges,
- then rewrite out some of the branches in parallel by applying R7.
We continue assuming that we do not obtain a tree by the above run. At this point, $d$ is a DAG pattern as the one illustrated in Figure \[fig:completeness4\], where $t_r$ denotes the root token (ending with node $n_r$) and $t_o$ denotes the result token (starting with $n_o$). Rules R2, R3 and R4 no longer apply, hence each //-edge outgoing from a node of $t_r$ that is ancestor of $n_r$ cannot be refined into connecting it to a lower node in $t_r$. Similar for //-edges incoming for nodes of $t_o$ that are descendants of $n_o$.
The intermediary branches $i_1, \dots, i_k$ denote those that start from $n_r$ and end at $n_o$ (we use this notation, even if there may be no such $i_1, \dots, i_k$ and $k=0$). The other branches in parallel, $i_{k+1},\dots, i_{k+l}$, denote those that do not obey both conditions. If $l=0$, i.e. there are no such branches, we fall again in the case handled by Lemma \[lem:n-comp-skel\], for which the algorithm is complete. We continue with the assumptions that $k \geq
0$ and $l \geq 1$ as well.
We next prove that $d$ is not union-free.
We introduce some additional notation. For each $i_{j}$, $k+1 \leq j
\leq k+l$, such that $i_j$ starts above $n_r$, let $n_{j}^r$ denote the node in $t_r$ that is sibling of the first node in $i_j$ (i.e., $n_{j}^r$ and the first node in $i_j$ have the same parent node, a node in $t_r$). Note that $n_{j}^r$ is ancestor-or-self of $n_r$. Let $n_{j}^o$ denote the node of $t_o$ that is “parent-sibling” of $i_j$ (they have the same child node). $n_{j}^o$ is defined if $i_j$ ends below $n_o$ and it is descendant-or-self of $n_o$.
For each $i_j$, by $pr_j$ we denote its maximal token-prefix that can map in $\tp{d}{n_{j}^r/\dots/n_r}$. Similarly, for each $i_j$ by $sf_j$ we denote the maximal token-suffix that can map in $\tp{d}{n_o/\dots/n_{j}^o}$.
Note that $pr_j$ and $sf_j$ cannot overlap since in this case $i_j$ would have been rewritten away by R7.
We can thus write each $i_j$ as $i_j=pr_j//m_j//sf_j,~ \textrm{for}~ k+1\leq j \leq l+1$.
Now, we consider a second DAG pattern $d'$ obtained from $d$ by replacing each $i_j$ branch by $m_j$, connected now by //-edges to $n_r$ and $n_o$ (Figure \[fig:completeness5\]), instead of the parent of $n_{j}^r$ and the child of $n_{j}^o$.
We argue now that the set of interleavings of $d'$ is included in the set of interleavings of $d$ (set inclusion). Moreover, $d$ is union-free only if $d'$ is union-free, such that if $d'\equiv p$, for an interleaving $p$, then $p$ is the only candidate for $d\equiv p$. First, it is straightforward that all the interleavings of $d'$ are interleavings of $d$ as well. The particularity of $d'$ is that its interleavings do not modify the tokens $t_r$ and $t_o$. More precisely each interleaving will be of the form $t_r//\dots//t_o$. Moreover, by the way $d'$ was defined and given that no R2, R3 or R4 steps applied on $d$, we argue that all other interleavings of $d$ will either (a) be redundant, i.e. contained in those of $d'$, (b) add some predicate on $t_r$ or $t_o$ or (c) have a longer root token (resp. result token) than $t_r$ (resp. $t_o$). But this means that an interleaving $p \in \nf{d} - \nf{d'}$ cannot have a containment mapping into an interleaving of the form $t_r//\dots//t_o$. Hence it cannot be equivalent to $d$. So the only interleaving $p$ candidates for $p\equiv d$ are those of $\nf{d'}$. From this it follows that $d$ can be union-free only if $d'$ is union-free.
Note now that by Lemma \[lem:n-comp-skel\] $d'$ is union-free iff there exists some $m_j$ into which all $i_1, \dots, i_k$ and all other $m_i$’s map. This is because of the assumption that among $i_1,\dots,i_k$ there is no branch $i_j$ into which all other $i_i$’s map.
We continue towards showing that $d$ is not union-free with this assumption and let $m$ denote the branch into which all others map. Note that among $m_{k+1},\dots, m_{k+l}$ there can be more than one “copy” of $m$ (i.e., equivalent to $m$). By $m_c$ we denote all these copies. Among $i_1,\dots, i_k$ there is no copy of $m$ (otherwise R7 would have triggered).
Let $d' \equiv p=t_r//m//t_o$, for $m=t//m'$. We build next an interleaving $w$ of $d$ s.t. $w\not \sqsubseteq p$. W.l.g. let us assume that all the $m_c$ copies of $m$ are connected in $d$ to a node that is strict ancestor of $n_r$[^1]. Since R2 or R4 did not apply on these copies of $m$, it means that a strict prefix of the main branch of $m$’s root token $t$ maps in a suffix of the main branch of $t_r$, when the possibly non-empty preceding token-prefix $pr_j$ is collapsed somewhere “higher”.
Let $\psi$ denote the partial mapping from $t$ into $t_r$ that uses the maximal possible prefix of $t$ across all the copies $m_c$. Let $t$ be $t=t'/t''$, where $t'$ is this maximal prefix (not empty).
We are now ready to build $w$.
We build first the root token $t_r'$ of the $w$ interleaving as follows: let $t_r'$ denote an interleaving of $t_r$ and $t$ defined by the code $i=\mb{t_r}/\mb{t''}$, and $f_i$ defined as “identity” on $t_r$ and $t''$, and $f_i(n)=\psi(n)$ for the main branch nodes of $t'$.
We build the intermediary part $p$ of the $w$ interleaving as follows: starting from $i_x \in \{i_1, \dots, i_k, m'\}$ (or simply from $i_x \in \{i_1 \dots, i_k\}$ in the case $m'$ is empty), let us interpret them as the intermediary parts of the following skeleton patterns $ s_x = start//i_x//end$.
Let also $s$ denote the pattern $s=start//m//end$.
Let us now consider now the DAG pattern $d'=dag(\cap_x s_x)$. Since none of the $s_x$ patterns is equivalent to $s$, from Lemma \[lem:n-comp-skel\] we have that $d' \not \equiv s$. Moreover, since $s \sqsubseteq d'$ (because $s_x \sqsubseteq s$), we must have that $d' \not \sqsubseteq s$. In other words, there must exist an interleaving $w'$ of $d'$, of the form $start//p//end$ such that $w' \not \sqsubseteq s$. Finally, this means $p$ is such that while all the $i_1, \dots i_k, m'$ map into it, we have that $m$ does not map into it.
Finally, we define $w$ as $ w=t_r'//p//t_o$. It is easy to check that $w$ is an interleaving of $d$ ($d$ has a containment mapping into $w$) but $w \not \sqsubseteq
p=t_r//t//m'//t_o$. Hence $d$ is not union-free.
**Remark.** We can draw the following conclusions from the proof of Theorem \[th:completenessUF-es\]: When is applied to DAG patterns built from multi-token views from , after R1 steps, followed eventually by R2, R3 and R4 steps, we obtain the branches in parallel $i_1, \dots, i_k$ starting from the last node of the root token ($t_r$) and ending with the first node of the result token ($t_o$). Other branches in parallel may exist in $d$, but connected to other nodes of $t_r$ and $t_o$. Then, by eventually some R7 steps, the DAG pattern must become a tree, otherwise it is not union-free. Under the extended skeletons restrictions, R5 and R6 are not necessary for completeness. The resulting tree is $t_r//i_1//t_o$, where $i_1$ is one of the branches in parallel, into which all other, $i_2, \dots, i_k$ map.
Proof of Theorem \[th:completenessUF-desc-akin\] (Rewrite-plans for PTIME) {#sec:completenessUF-desc-akin}
==========================================================================
We give in this section the completeness proof for rewrite plans formed by akin patterns.
We show that, given $n$ akin tree patterns $v_1, \dots, v_n$, decides union-freedom for $d=dag(v_1 \cap \dots \cap v_n)$. Let each $v_j$ be defined as $v_j=t_r^j//i_j//t_o^j$.\
**Special case.** We start by considering the special case when the patterns have the same main branch for their result tokens as well.
By Lemmas \[lem:skel-necessary\] and \[lem:n-comp-skel\], we know that $d$ is union-free only if the intermediary parts $i_j$ are such that their skeletons map in the skeleton of one of them. Without loss of generality, let us assume that all $s(i_j)$ map in $s(i_1)$. We continue with this assumption.
First, the initial R1 steps coalesce the root and result tokens of the $n$ branches, yielding a DAG pattern similar to the one illustrated in Figure \[fig:completeness3\]. Then, the only rules that may be applicable are R6 and R7. Let us assume that stops outputting a pattern that is not a tree. We show that $d$ is not union-free.
If the algorithm stops without outputting a tree in some run, then it will also stop without outputting a tree in the following particular rewriting strategy
- we first apply R6 on the “biggest”branches in parallel $i_j,i_k$ such that $s(i_j)\equiv s(i_k)\equiv s(i_1)$, if any. It is straightforward that R6 must apply for these branches, coalescing entirely the two branches into one branch. After this phase, there will be no other parallel branch with skeleton $s(i_1)$, besides $i_1$ itself.
- Then, R6 is applied only if applicable on all the branches in parallel at once. This phase will terminate with a refined $d$ similar to the one illustrated in Figure \[fig:completeness2\], where $2 \leq k \leq n$, $p_r$ denotes the common part following the root (may have several tokens if R6 was applied) and $t_1, \dots t_k$ denote the sibling tokens on which R6 no longer applies (i.e., they are not all *similar* hence they do not all have the same skeleton).
Also, rule R7 is applied freely, and it can rewrite out some of the branches in parallel.
After this phase, while there exists a mapping from each $s(p_i)$ into $s(p_1)$, there is no mapping from $p_i$ into $p_1$ . Note also that, by the first phase of the rewriting strategy, we cannot have the opposite mapping from $s(p_1)$ into $s(p_i)$.
- Finally, rule R6 is applied only between $p_1$ on the one hand, and other branches $p_i$ on the other hand, while R7 is still applied freely.
We obtain a DAG pattern similar to the one in Figure \[fig:completeness6\]. Let us assume that besides $p_1$ there are $l$ remaining branches in parallel, connected by a //-edge either to $p_r$ or to various tokens of $p_1$.
Let $p_1=t_1//\dots//t_m$. For $i=1,l$, let $p_i'=t_i'//r_i'$ denote now these branches in parallel with (part of) $p_1$. For each $i=1,l$, let $t_i^1$ denote the token in $p_1$ that is sibling of the token $t_i'$. Note that $t_i'$ and $t_i^1$ must be dissimilar, hence will have different skeletons. Let $sf_i^1$ denote the token-suffix of $p_1$ that is in parallel with $p_i'$ and let $pr_i^1$ denote the rest of $p_1$ (a token-prefix). For each $i=1,l$, we can thus reformulate $p_1$ as $p_1=pr_i^1//sf_i^1$, where the root token of $sf_i^1$ is $t_i^1$. Note that for each $i$ we have that $s(p_i')$ maps into $s(sf_i^1)$, while the opposite is not true.
It is immediate that $d$ can be union-free, for some $c$ such that $d \equiv c$, only if $c$ is of the form $c=p_r//m//t_o$ where $s(m)=s(p_1)$, since this is the minimal skeleton for an interleaving.
All the candidate interleavings $c$ will be defined by the code $i=\mb{p_r//m//t_o}$ and some function $f_i:\mbn{d} \rightarrow
i$. What distinguishes the various $c$’s is the definition of $f_i$ on the nodes of the branches $p_1', \dots, p_l'$ (since the other main branch nodes in $d$ have only one possible image). We show next that for any such $f_i$ and associated interleaving $c$ we can build the $w$ witness with $w \not
\sqsubseteq c$.
Let $f_i$ be fixed and let $c$ denote the corresponding interleaving for code $i$ and function $f_i$. Note that we can interpret $f_i$ as a series of rewrite steps over $d$ that collapse the pairs of nodes $(n,
f_i(n))$, for all the nodes $n$ in the $p_1', \dots, p_l'$ branches, outputting as end result the tree pattern $c$. These steps do not modify the skeleton of $p_1$, hence can only bring some new predicates starting by //-edge.
Next, we describe how the interleaving $w \not \sqsubseteq c$ is built, from the current pattern $d$ of Figure \[fig:completeness6\].
Let $n_c \in \mb{c}$ denote the lowest main branch node in $c$’s $m$ part which has a subtree predicate $st$ that is not present (in other words, cannot be mapped) at the associated node $n_1$ in the $p_1$ part of $d$. $st$ must start with a //-edge and must come from a node of some (maybe several) branches $p_i'$. (We know that such a node $n_c$ must exist, otherwise the $p_i'$ branches would fully map in the corresponding branch in parallel $sf_i^1$ and rule R7 would have applied).
Without loss of generality, let $n_{i_1}',\dots, n_{i_s}'$, for $\{i_1,\dots, i_{s}\} \subseteq \{1,\dots, l\}$, denote the nodes from the branches $p_{i_1}', \dots, p_{i_s}'$ that are the “source” of $st$[^2]. So we have $f_i(n_1)=f_i(n_{i_1}')= \dots =f_i(n_{i_s}')=n_c$ and we can say that $n_c$ is the result of coalescing $n_1$ with $n_{i_1}', \dots n_{i_s}'$.
Now, we can see the left branch $p_1$ as being divided into two parts, the one down to the token of $n_1$ (that token included), denoted $p_{11}$, and the rest, denoted $p_{12}$. So we can write $p_1$ as $p_1=p_{11}//p_{12}$.
Similarly, for each $x \in \{i_1,\dots, i_{s}\}$ we can see each main branch $pr_{x}^{1}//p_{x}'$ as being divided into two parts, the one down to the token of $n_{x}'$ (that token included), denoted $p_{x1}'$, and the rest, denoted $p_{x2}'$. So we can write each main branch $pr_{x}^{1}//p_{x}'$ of $d$ as $pr_{x}^{1}//p_{x}'=p_{x1}'//p_{x2}'$.
Note that by the way $n_c$ was chosen (as the lowest node) we can conclude that by $f_i$ (on the main branch nodes) we can fully map $\tp{d}{p_{x2}'}$ into $\sub{d}{n_1}$, for all $x$ (i.e., there are no other added predicates below $n_1$’s level). It is also easy to see that while $s(p_{x1}')$ maps in $s(p_{11})$ (by $f_i$), the opposite is not true, otherwise R6 steps would have applied up to this point.
We are now ready to construct $w$. First, we obtain a part $p$ of the $w$ interleaving as follows: starting from the set of skeleton queries $s(p_{x1}')$, for all $x \in \{i_1,\dots, i_{s}\}$, let us interpret them as the intermediary parts of the following skeleton patterns $ s_x = start//s(p_{x1}')//end$.
Let also $s$ denote the skeleton pattern $s=start//s(p_{11})//end$.
Let us now consider the DAG pattern $d'=dag(\cap_x s_x)$. Since none of the skeleton patterns $s_x$ is equivalent to $s$, from Lemma \[lem:n-comp-skel\] we have that $d' \not \equiv s$. Moreover, since $s \sqsubseteq d'$ (because $s \sqsubseteq s_x$, by the way $c$ was defined), we must have that $d' \not \sqsubseteq s$. In other words, there must exist an interleaving $w'$ of $d'$, of the form $start//p//end$ such that $w' \not \sqsubseteq s$. Finally, this means $p$ is such that while all the $s(p_{x1}')$ map into it, we have that $s(p_{11})$ does not map into it. We will use this property. For each $p_{x1}'$, let $f_{x1}$ denote a mapping from $s(p_{x1}')$ into $p$.
Next, we obtain a second part of $w$ as follows. Let $pr_{11}$ denote the maximal token-suffix of $p_{11}$ such that $s(p_{11})$ can map in $p$, and let $sf_{11}$ denote the remaining part. $sf_{11}$ cannot be empty, so it is formed by at least the output token of $p_{11}$, the one with node $n_1$. So we can see $p_{11}$ as $p_{11}=pr_{11}//sf_{11}$.
Let $f_p$ denote a partial mapping from $s(p_{11})$ into $p$ that exhibits $sf_{11}$.
We will define $w$ by a code $i'$ and function $f_i'$ as follows:
- $i' = \lambda(p_r//p//sf_{11}//p_{12}//t_o)$,
- $f_i'$ maps nodes of into $i'$ positions as follows:
- $f_i'$ is “identity” for the main branch nodes of $p_r$, $t_o$, for the $sf_{11}$ part of the $p_{11}$ prefix of $p_1$ and for the $p_{12}$ suffix of $p_1$,
- for the remaining main branch nodes $n$ in $p_{11}$ (i.e., those of $pr_{11}$), $f_i'(n) = f_p(n)$,
- for the main branch nodes $n$ of the $p_{x1}'$ prefix of the $pr_x^1//p_x'$ branch in $d$, for $x \in \{i_1,\dots, i_{s}\}$, $f_i'(n)=f_{x1}(n)$
- for the remaining nodes $n$ in the $pr_x^1//p_x'$ branches (i.e. those in $p_{x2}'$), $f_i'(n)=f_i'(f_i(n))$.
- finally, for all the main branch nodes of the remaining branches $p_{y}'$, for $y \not \in \{i_1,\dots, i_{s}\}$, $f_i'(n)=f_i'(f_i(n))$.
(they go where their images under $f_i$ go.)
We now argue that $w$ is an interleaving of $d$ and $w \not \sqsubseteq c$. First, it is easy to check that $w$ is an interleaving for $d$. Recall that $c$ is s.t. $s(c) = s(p_r//p_1//t_o)=s(p_r//pr_{11}//sf_{11}//p_{12}//t_o)$. Second, it is also easy to check that $c$ can have a containment mapping in $w$ iff its $sf_{11}$ part maps in the $sf_{11}$ of $w$. But this is not possible because the $st$ subtree predicate is not present on the $f_i'(n_1)$ node of $w$ (which is found somewhere in the output token of the $sf_{11}$ part).\
**General case.** We now consider the general case, when the result tokens do not necessarily have the same main branch. After the possible rewrite R1(i) steps on the root tokens, and after the possible rewrite steps of R1(ii), R2(ii), R3(ii) and R4(ii) on the result tokens, we may now obtain a DAG pattern in which the branches in parallel may not be “connected” to $t_o$ at its highest node ($n_o$), but at some other node that is strict descendant of $n_o$. If this is not the case, then we are back to the special case discussed previously.
Otherwise, let us now consider the DAG pattern $d'$ obtained from $d$ by connecting the endpoints of the branches in parallel at $n_o$. We can easily see that the interleavings of $d'$ are all among those of $d$ and moreover, $d$ is union-free only if $d'$ is union-free, with $d' \equiv d\equiv p$, for some $p \in interleave(d')$. This is because the interleavings of $d$ that are not interleavings of $d'$ as well are those that add some predicates on $t_o$ that are not present in all the interleavings.
By Lemmas \[lem:skel-necessary\] and \[lem:n-comp-skel\], we know that $d$ is union-free only if the intermediary parts $i_j$ are such that their skeletons map in the skeleton of one of them, which in addition, in the current $d$ pattern, must start at $n_r$ and end at $n_o$. Without loss of generality, let us assume that this is $i_1$ (note that the $i_1$ branch will not be affected by the transformation from $d$ to $d'$).
From the special case, we know under what conditions $d'$ is union-free, and it is immediate that, when they hold, the interleaving $p$ is obtained from $i_1$ possibly by adding some predicates of the form $[.//\dots]$ to some of its main branch nodes. Importantly, each such predicate is added on the highest possible main branch node of $i_1$.
Finally, it is now easy to check that an interleaving $p$ of $d'$ obtained in this way will always have a containment mapping in any interleaving $p'$ of $d$: everything except the added predicates will map (by identity), while the added predicates (of the form $[.//\dots]$) will map in their respective occurrence in $p'$ (by necessity, found at a lower main branch node then in the one in $p$).
This ends the completeness proof of over unfoldings of rewrite plans that intersect only akin views from .
[^1]: The remaining cases when
- all the $m_c$ copies of $m$ are connected in $d$ to a node that is strict descendant of $n_o$, or
- all the $m_c$ copies of $m$ but one (we cannot have more than one, otherwise R7 would have triggered leaving only one) are connected in $d$ to a node that is strict ancestor of $n_r$ and all the $m_c$ copies of $m$ but one are connected in $d$ to a node that is strict descendant of $n_o$,
can be handled similarly.
[^2]: They have a predicate $st'$ into which $st$ maps.
|
---
abstract: |
We formulate the Schiffer’s conjecture in spectral geometry in the context of scattering theory. The problem is equivalent to finding a non-trivial solution in an interior transmission problem. We compare the back-scattering data of the perturbation along all incident angles. The uniqueness of the inverse scattering problem along each incident direction proves the Schiffer’s conjecture.\
MSC: 35P25/35R30/34B24.\
Keywords: inverse scattering/Helmholtz equation/Rellich’s lemma/interior transmission eigenvalue\
/Cartwright-Levinson theory.
author:
- 'Lung-Hui Chen$^1$'
title: |
A Proof of Schiffer’s Conjecture in Starlike Domain\
by Far-Field Patterns
---
Introduction
============
In this paper, we study the following inverse spectral problem: $$\begin{aligned}
\label{1.1}
\left\{\begin{array}{ll}
\Delta u+k^2u=0, & \hbox{ in }D ,\,k^2\in\mathbb{R}^+;\vspace{3pt}\\\vspace{3pt}
\frac{\partial u}{\partial \nu}=0,& \hbox{ on }\partial D;\\
u=1,&\hbox{ on }\partial D,
\end{array}\right.\end{aligned}$$ where $\nu$ is the unit outer normal; $D$ is a fixed starlike domain in $\mathbb{R}^3$ containing the origin with Lipschitz boundary $\partial D$. We interpret the model as the plane waves perturbed by the boundary condition which is specified by $D$, and satisfies the Helmholtz equation outside $D$. Let $u$ be a non-trivial eigenfunction with some $k^2\in\mathbb{R}^+$. We want to show that $D$ are actually balls centered at origin.
Here we prove the result as a special case of interior transmission problem [@Aktosun; @Cakoni2; @Chen; @Chen2; @Chen3; @Chen5; @Colton4; @Colton; @Colton3; @Colton2; @Colton5; @Kirsch86; @Kirsch; @L; @La; @Liu; @Mc; @Rynne]. In interior transmission problems, we look for a frequency so that a perturbed stationary wave behaves like or somewhere like a spherical Bessel function outside the perturbation. In Schiffer’s conjecture, we ask if there is a frequency so that a perturbed wave can stay in its initial shape traveling to infinity in constant speed. We refer to [@A; @Liu] and the reference there for the connections of interior transmission problem to other questions in mathematical science.
To give a point of view from scattering theory to (\[1.1\]), we take the incident wave field to be the time harmonic acoustic plane wave of the form $$u^i(x):=e^{ikx\cdot d},$$ $k\in\mathbb{R}^+$, $x\in\mathbb{R}^3$, and $d\in\mathbb{S}^2$ is the incident direction. The inhomogeneity is defined by the index of refraction $n\in\mathcal{C}^2(\mathbb{R}^3)$ of (\[1.1\]), and the wave propagation is governed by the following equation. $$\begin{aligned}
\label{122}
\left\{\begin{array}{ll}
\Delta u(x)+k^2n(x)u(x)=0,\,x\in\mathbb{R}^3;\vspace{4pt}\\\vspace{3pt}
u(x)=u^i(x)+u^s(x),\,x\in\mathbb{R}^3\setminus D; \\
\lim_{|x|\rightarrow\infty}|x|\{\frac{\partial u^s(x)}{\partial |x|}-iku^s(x)\}=0,
\end{array}\right.\end{aligned}$$ in which the third equation is the Sommerfeld’s radiation condition. Particularly, we have the following asymptotic expansion on the scattered wave field [@Colton2; @Isakov]. $$\label{U}
u^s(x)=\frac{e^{ik|x|}}{|x|}u_\infty(\hat{x};d,k)+O(\frac{1}{|x|^{\frac{3}{2}}}),\,|x|\rightarrow\infty,$$ which holds uniformly for all $\hat{x}:=\frac{x}{|x|}$, $x\in\mathbb{R}^3$, and $u_\infty(\hat{x};d,k)$ is known as the scattering amplitude or far-field pattern in the literature [@Colton2; @Kirsch86]. It has an expansion in spherical harmonics [@Colton2 p.35,Theorem 2.15] $$u_\infty(\hat{x};d,k)=\frac{1}{k}\sum_{n=0}^\infty\frac{1}{i^{n+1}}\sum_{m=-n}^na_n^mY_n^m(\hat{x}),$$ where we follow the notation in the reference.
Let us start with the Rellich’s representation in scattering theory. We expand the possible solution $u$ of (\[1.1\]) in a series of spherical harmonics near infinity by Rellich’s lemma [@Colton2 p.32, p.227]: $$\begin{aligned}
\label{1.2}
u(x;k)=\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}a_{l,m}(r)Y_l^m(\hat{x}),\end{aligned}$$ where $r:=|x|$, $r\geq R_0$ with a sufficiently large $R_0$; $\hat{x}=(\theta,\varphi)\in\mathbb{S}^2$. The summations converge uniformly and absolutely on suitable compact subsets away from $D$. The spherical harmonics $$\label{S}
Y_l^m(\theta,\varphi):=\sqrt{\frac{2l+1}{4\pi}\frac{(l-|m|)!}{(l+|m|)!}}
P_l^{|m|}(\cos\theta)e^{im\varphi},
\,m=-l,\ldots,l;\,l=0,1,2,\ldots,$$ form a complete orthonormal system in $\mathcal{L}^2(\mathbb{S}^2)$, in which $$P_n^m(t):=(1-t^2)^{m/2}\frac{d^mP_n(t)}{dt^m},\,m=0,1,\ldots,n,$$ where the Legendre polynomials $P_n$, $n=0,1,\ldots,$ form a complete orthogonal system in $L^2[-1,1]$. We refer this to [@Colton2 p.25]. By the orthogonality of the spherical harmonics, the family of functions $$\begin{aligned}
\label{1.5}
\{u_{l,m}(x;k)\}_{l,m}:=\{a_{l,m}(r)Y_l^m(\hat{x})\}_{l,m}\end{aligned}$$ satisfy the first equation in (\[1.1\]) independently for each $(l,m)$ in $r\geq R_0$ for sufficiently large $R_0$.
Now we consider the boundary condition given by the second and third equations in (\[1.1\]), and then extend the solutions $u_{l,m}(x;k)$ into $r\leq R_0$ as follows. Let $\hat{x}_0\in\mathbb{S}^2$ be any given incident direction that intersects $\partial D$ at $(\hat{R},\hat{x}_0)\in \mathbb{R}^+\times\mathbb{S}^2$. For the given $\hat{x}_0$, we impose the differential operator$$\nonumber
\Delta=\frac{1}{r^2}\frac{\partial}{\partial r}
r^2\frac{\partial}{\partial r}+\frac{1}{r^2\sin{\varphi}}\frac{\partial}{\partial \varphi}\sin\varphi\frac{\partial}{\partial \varphi}
+\frac{1}{r^2\sin^2{\varphi}}\frac{\partial^2}{\partial \theta^2}$$ on $u_{l,m}(x;k)$ and, accordingly, we have the following ODE: $$\begin{aligned}
\frac{d^2 a_{l,m}(r)}{dr^2}+\frac{2}{r}\frac{d a_{l,m}(r)}{dr}+(k^2-\frac{l(l+1)}{r^2})a_{l,m}(r)=0,\end{aligned}$$ which is solved by spherical Bessel functions and spherical Neumann functions. Let $$\label{118}
y_{l,m}(r):=ra_{l,m}(r),$$ so we obtain $$\begin{aligned}
\label{1.10}
\left\{\begin{array}{ll}
y_{l,m}''(r)+(k^2-\frac{l(l+1)}{r^2})y_{l,m}(r)=0;\vspace{5pt}\\
y_{l,m}(0)=0.
\end{array}\right.\end{aligned}$$
To give an initial condition, we apply the boundary conditions in (\[1.1\]) to $u_{l,m}(x;k)$ near the intersection points $\hat{R}$ along $\hat{x}_0$. We replace the boundary condition $\frac{\partial u}{\partial \nu}=0$ to be $\nabla u=0$. Hence, $$\begin{aligned}
\label{1.11}
&&[\frac{y_{l,m}(r;k)}{r}]'|_{r=\hat{R}}=0;\\
&&[\frac{y_{l,m}(r;k)}{r}]|_{r=\hat{R}}=1,\label{1.12}\end{aligned}$$ in which we assume there is no tangent point. The solutions $y_{l,m}(r;k)$ are independent of $m$, so we write $y_{l,m}(r;k)$ as $y_{l}(r;k)$. Hence, now we have following boundary conditions. $$\begin{aligned}
\label{1.13}
&&\frac{y_l'(\hat{R};k)}{\hat{R}}-\frac{y_l(\hat{R};k)}{\hat{R}^2}=0;\\
&&y_l(\hat{R};k)- \hat{R}=0.\label{1.14}\end{aligned}$$ If $k$ satisfies (\[1.13\]) and (\[1.14\]), then $y_l'(\hat{R};k)=1$. Thus, (\[1.11\]) and (\[1.12\]) equivalently satisfy $$\begin{aligned}
\label{113}
&&F_l(k;\hat{R}):=y_l(\hat{R};k)- \hat{R}=0;\\
&&G_l(k;\hat{R}):=y_l'(\hat{R};k)-1=0.\label{114}\end{aligned}$$ In the initial state, $y_l(\hat{R};k)$ is exactly the boundary defining function of $D$. Combining (\[1.10\]), (\[113\]), and (\[114\]), we consider the following eigenvalue problem at $\hat{R}$ for each fixed $\hat{x}\in\mathbb{S}^2$ and all $l\geq0$: $$\begin{aligned}
\label{15}
\left\{\begin{array}{ll}
y_l''(r;k)+(k^2-\frac{l(l+1)}{r^2})y_l(r;k)=0,\,0<r<\infty;\vspace{4pt}\\
\vspace{3pt}
y_{l}(0;k)=0;\\ \vspace{3pt}
F_l(k;\hat{R})=0;\\ \vspace{3pt}
G_l(k;\hat{R})=0.
\end{array}\right.\end{aligned}$$ This is a two-way initial value problem starting at $r=\hat{R}$ inward and outward. The eigenvalue $k$ passes through to the infinity by the uniqueness of the ODE and defines the far-field patterns near infinity. There is an one-to-one correspondence between the far-field patterns and the radiating solution of the Helmholtz equation. The $y_l(r;k)$ depends on the incident angle $\hat{x}$. Most important of all, we will examine the zero set of $y_{l}(0;k)=0$ which constitutes the eigenvalues of (\[15\]). The solutions $\{y_l(r;k)\}_{l\geq0}$ is a family of entire functions of exponential type [@Carlson; @Carlson2; @Carlson3; @Po]. For each $l\geq0$, it behaves like sine functions in complex plane with zero set asymptotically approaching the zero set of sine functions for each incident direction. The Weyl’s law of the eigenvalues of (\[15\]) in many settings are found in [@Chen; @Chen3; @Chen5] as a direct consequence of the Cartwright-Levinson theory in value distribution theory [@Boas; @Cartwright; @Cartwright2; @Koosis; @Levin; @Levin2]. In particular, we can find that the density of the zero set for each incident direction is related to the radius $\hat{R}$ as a spectral invariant. Rellich’s lemma indicates that all perturbations behave like spherical waves near the infinity, by which we prove **a special case** of Schiffer’s conjecture.
\[11\] Let $D$ be a starlike domain as assumed in (\[1.1\]) under radiation condition (\[122\]). If there is an eigenvalue $k_0^2\in\mathbb{R}^+$, $k_0^2\geq1$ of (\[1.1\]), then $D$ is an open ball centered at the origin.
Singular Sturm-Liouville Theory
===============================
Here we collect the asymptotic behaviors for $y_l(r;k)$ and $y_l'(r;k)$. For $l\geq0$, we apply the results from [@Carlson; @Carlson2; @Carlson3; @Po]. Let $z_l(\xi;k)$ be the solution of $$\begin{aligned}
\label{21}
\left\{
\begin{array}{ll}
-z_l''(\xi)+\frac{l(l+1)z_l(\xi)}{\xi^2}+p(\xi)z_l(\xi)=k^2z_l(\xi);\vspace{9pt}\\
z_l(1;k)=-b;\,z_l'(1;k)=a,\,a,\,b\in\mathbb{R},
\end{array}
\right.\end{aligned}$$ where $p(\xi)$ is square integrable; the real number $l\geq-1/2$. In general, $$\begin{aligned}
\label{123}
|z_l(\xi;k)+b\cos{k(1-\xi)}+a\frac{\sin{k(1-\xi)}}{k}|\leq \frac{K(\xi)}{|k|}\exp\{|\Im k|[1-\xi]\},\,|k|\geq1,\end{aligned}$$ where $$\label{124}
K(\xi)\leq\exp\{\int_\xi^1\frac{|l(l+1)|}{t^2}+|p(t)|dt\},\,0\leq\xi\leq 1.$$ This explains the behaviors of solutions $z_l(\xi;k)$ and $z_l'(\xi;k)$ for all $l$ in unit interval. For its application in (\[113\]) and (\[114\]), we take $$b=-\hat{R};\,a=1$$ for each incident direction, and the problem (\[21\]) in interval $[0,\hat{R}]$.
Outside the domain $D$, we consider (\[15\]) as an initial problem starting at $\hat{R}$ to the infinity. If $p(\xi)\equiv0$, then we consider the following special case: $$\begin{aligned}
\label{2.4}
v_l''(\xi)+[k^2-\frac{l(l+1)}{\xi^2}]v_l(\xi)=0.\end{aligned}$$ The solutions of (\[2.4\]) are essentially Bessel’s functions with a basis of two elements. The variation of parameters formula leads to the following asymptotic expansions: For $\xi>0$ and $\Re k\geq0$, there is a constant $C$ so that $$\begin{aligned}
&&|v_l(\xi,k)-\frac{\sin\{k\xi-l\frac{\pi}{2}\}}{k^{l+1}}|\leq C|k|^{-(l+1)}\frac{\exp\{|\Im k|\xi\}}{|k\xi|};\label{2.5}\\
&&|v_l'(\xi,k)-\frac{\cos\{k\xi-l\frac{\pi}{2}\}}{k^{l}}|\leq C|k|^{-l}\frac{\exp\{|\Im k|\xi\}}{|k\xi|}.\label{2.6}\end{aligned}$$ We refer these estimates to [@Carlson2 Lemma3.2,Lemma3.3], and the we find that a solution of the initial value problem of (\[2.4\]) is a linear combination of (\[2.5\]) and (\[2.6\]).
Cartwright-Levinson Theory
==========================
We take the following vocabularies from entire function theory [@Boas; @Cartwright; @Cartwright2; @Koosis; @Levin; @Levin2] to describe the asymptotic behavior of the eigenvalues of (\[15\]).
Let $f(z)$ be an integral function of order $\rho$, $N(f,\alpha,\beta,r)$ be the number of the zeros of $f(z)$ inside the angle $[\alpha,\beta]$, and $|z|\leq r$. We define the density function as $$\label{Den}
\Delta_f(\alpha,\beta):=\lim_{r\rightarrow\infty}\frac{N(f,\alpha,\beta,r)}{r^{\rho}},$$ and $$\Delta_f(\beta):=\Delta_f(\alpha_0,\beta),$$ with some fixed $\alpha_0\notin E$, in which $E$ is at most a countable set [@Levin; @Levin2].
Let us define $$\hat{\Delta}(\xi):=\Delta_{z_l(\xi;k)}(-\epsilon,\epsilon),\,b=-\hat{R},$$ as the density of the zero set along $\hat{x}$.
The entire functions $y_l(\xi;k)$ and $y_l'(\xi;k)$ are of order one and of type $\hat{R}-\xi$.
From (\[123\]), we have $$\label{3.4}
y_l(\xi;k)=-\hat{R}\cos{k(\hat{R}-\xi)}-\frac{\sin{k(\hat{R}-\xi)}}{k}+O( \frac{K(\xi)}{|k|}\exp\{|\Im k|[\hat{R}-\xi]\}),\,|k|\geq1.$$ To find the type of an entire function, we compute the following definition of Lindelöf’s indicator function [@Levin; @Levin2]
Let $f(z)$ be an integral function of finite order $\rho$ in the angle $[\theta_1,\theta_2]$. We call the following quantity as the indicator of the function $f(z)$. $$h_f(\theta):=\lim_{r\rightarrow\infty}\frac{\ln|f(re^{i\theta})|}{r^{\rho}},
\,\theta_1\leq\theta\leq\theta_2.$$
We find that if $k=|k|e^{i\theta}$, then $$\label{3.6}
h_{y_l(\xi;k)}(\theta)=|(\hat{R}-\xi)\sin\theta|,\,\theta\in[0,2\pi],
\,0<\xi<\hat{R}.$$ When referring more details to [@Chen; @Chen3; @Chen5; @Cartwright2; @Levin; @Levin2], we find more examples in [@Cartwright2 p.70]. The maximal value of $h_{y_l(\xi;k)}(\theta)$ gives the type of an entire function [@Levin p.72], which is $(\hat{R}-\xi)$. A similar proof holds for $y_l'(\xi;k)$.
More importantly, the indicator function (\[3.6\]) leads to the following Cartwright’s theory [@Levin p.251].
\[34\] We have the following asymptotic behavior of the zero set of $y_l(\xi;k)$. $$\hat{\Delta}(\xi)=\frac{\hat{R}-\xi}{\pi}.$$
We observe in (\[3.4\]) that $|y_l(\xi;k)|$ is bounded on the real axis. Hence, it is in Cartwright’s class. All of the properties in [@Levin p.251] hold.
Letting $\xi=0$, we obtain the eigenvalue density of (\[15\]) in $\mathbb{C}$. Moreover, they are all real.
The eigenvalues $k$ of (\[15\]) are all real.
For $l=0$, the result is classic [@Carlson2; @Po]. In our case, $y_l(\xi;k)$ is real for $k\in0i+\mathbb{R}$. Furthermore, the asymptotic behavior of (\[3.4\]) proves the lemma, which is a special case of Bernstein’s theorem in entire function theory [@Duffin Theorem 1]. A step-by-step proof is provided in [@Chen5 Lemma 2.6].
Proof of Theorem \[11\]
=======================
Let $k_0^2$ be an eigenvalue of (\[1.1\]), as assumed in Theorem \[11\]. Particularly, from (\[1.2\]) we have $$\begin{aligned}
\label{4.1}
&&u(x;k_0)=\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}
a_{l,m}(r;k_0)Y_l^m(\hat{x});\\
&&u_{l,m}(x;k_0)=a_{l,m}(r;k_0)Y_l^m(\hat{x}),
\,\hat{x}\in\mathbb{S}^2,\label{4.2}\end{aligned}$$ in which the coefficient $a_{l,m}(r;k_0)$ does not depend on the incident direction $\hat{x}\in\mathbb{S}^2$ for sufficiently large $|x|:=r$: The functions in (\[4.1\]) solve the Helmholtz equation in $r\geq R_0$. As a result of the uniqueness of the ODE (\[15\]), the solutions $y_l(r;k_0)$ extend both outward to the infinity and inward to the origin for all $l\geq0$. For the given eigenvalue $k_0^2$, the equation (\[15\]) holds for all incident directions $\hat{x}\in\mathbb{S}^2$ and for all $l\geq0$.
The representation in (\[4.1\]) is unique in $\mathbb{R}^3$: If there is another eigenvalue $k'$ of (\[15\]) from incident angle $x'\neq \hat{x}\in\mathbb{S}^2$ with the solution $$u_{l,m}'(x;k'):=a_{l,m}'(r;k')Y_l^m(\hat{x}),$$ then the analytic continuation of Helmholtz equation [@Colton2 p.18] implies that $$\label{43}
a_{l,m}'(r;k')=a_{l,m}(r;k_0).$$ With the uniqueness of the ODE (\[1.10\]), $k_0$ or $k'$ satisfies (\[15\]) individually along its own incident direction inward to the origin. Therefore, Lemma \[34\] provides an eigenvalue density $$\hat{\Delta}(0)=\frac{\hat{R}}{\pi},\,\hat{x}\in\mathbb{S}^2.$$
The ODE (\[15\]) holds for all $\xi\geq\hat{R}$ and $l\geq 0$. In particular, we apply the estimates (\[2.5\]) and (\[2.6\]): $$\begin{aligned}
&&|v_l(\xi,k)-\frac{\sin\{k(\xi-\hat{R})-l\frac{\pi}{2}\}}{k^{l+1}}|\leq C|k|^{-(l+1)}\frac{\exp\{|\Im k|\xi\}}{|k\xi|};\\
&&|v_l'(\xi,k)-\frac{\cos\{k(\xi-\hat{R})-l\frac{\pi}{2}\}}{k^{l}}|\leq C|k|^{-l}\frac{\exp\{|\Im k|\xi\}}{|k\xi|}.\end{aligned}$$ Therefore, the initial value problem (\[21\]), with $p\equiv 0$, $b=\hat{R}$, and $a=1$, provides the asymptotic behavior for the solution: $$\nonumber
y_l(\xi;k)=\hat{R}\frac{\cos\{k(\xi-\hat{R})-l\frac{\pi}{2}\}}{k^l}
+O\{\frac{\exp\{|\Im k|\xi\}}{k^{l}(k\xi)}\}.$$ That is, $$\label{4.6}
k^ly_l(\xi;k)=\hat{R}\cos\{k(\xi-\hat{R})-l\frac{\pi}{2}\}
[1+O\{\frac{1}{k\xi}\}],\,\hat{R}<\xi<\infty,$$ outside the zeros of $\cos\{k(\xi-\hat{R})-l\frac{\pi}{2}\}$. This is classic in Sturm-Liouville theory [@Carlson; @Carlson2; @Po].
The given eigenvalue $k_0$ satisfies (\[15\]), for all $l\geq0$ and all $\hat{x}\in\mathbb{S}^2$, and (\[4.6\]). Therefore, $$k_0^ly_l(\xi;k_0)=\hat{R}\cos\{k_0(\xi-\hat{R})-l\frac{\pi}{2}\}
[1+O\{\frac{1}{k_0\xi}\}],\,0<\xi<\infty.$$ We choose $l\uparrow\infty$ and so $\xi\uparrow\infty$ such that $\xi=\hat{R}+\frac{l\pi}{2k_0}>R_0$ for any large $R_0$. Thus, for large $l$, $$k_0^ly_l(\xi;k_0)=\hat{R}+O(\frac{1}{\xi}),\,
\xi=\hat{R}+\frac{l\pi}{2k_0},\,|k_0|\geq1.$$ Using (\[118\]), (\[1.2\]) and the uniqueness of the Helmholtz equation, as shown in (\[43\]), the far-field patterns [@Colton2 (2.49)] are asymptotically the same periodic functions for each $\hat{x}\in\mathbb{S}^2$. In particular, the boundary defining function $\hat{R}$ is constant to $\hat{x}\in\mathbb{S}^2$, and Theorem \[11\] is thus proven.
The author declares there is no conflicts of interest regarding the publication of this paper. The research does not involve any human participant and/or animals, and no further informed consent is required.
[widest-label]{} M. Agranovsky, Integral geometry and spectral analysis, In “Topics in mathematical analysis”, World Scientific Publishing Co, (281–320) 2008. T. Aktosun, D. Gintides and V. G. Papanicolaou, The uniqueness in the inverse problem for transmission eigenvalues for the spherically symmetric variable-speed wave equation, Inverse Problems, V. 27, 115004 (2011). R.P. Boas, Entire functions, Academic Press, New York, 1954.
F. Cakoni, D. Colton and D. Gintides, The interior transmission eigenvalue problem, SIAM J. Math. Anal. 42, 2912–2921 (2010).
R. Carlson, Inverse spectral theory for some singular Sturm-Liouville problems, Journal of Differential Equations, V.106, 121–140 (1993). R. Carlson, A Borg-Levinson theorem for Bessel operators, Pacific Journal of Mathematics, Vol. 177, No. 1, 1–26 (1997). R. Carlson, Inverse Sturm-Liouville problems with a singularity at zero, Inverse problems, 10, 851–864 (1994). M. L. Cartwright, On the directions of Borel of functions which are regular and of finite order in an angle, Proc. London Math. Soc. Ser. 2 Vol. 38, 503–541 (1933). M. L. Cartwright, Integral functions, Cambridge University Press, Cambridge, 1956. L. -H. Chen, An uniqueness result with some density theorems with interior transmission eigenvalues, Forthcoming in “Applicable Analysis”. L. -H. Chen, An Inverse Uniqueness on the Index of Refraction with a Transition Region, Forthcoming in “Applicable Analysis”. L. -H. Chen, A uniqueness theorem on the eigenvalues of spherically symmetric interior transmission problem in absorbing medium, Complex Var. Elliptic Equ. 60, no. 2, 145–167 (2015).
L. -H. Chen, On the inverse spectral theory in a non-homogeneous interior transmission problem, Forthcoming in “Complex Variables and Elliptic Equations”. D. Colton and P. Monk, The inverse scattering problem for time harmonic acoustic waves in a penetrable medium, Q. Jl. Mech. appl. Math. Vol. 40, 189–212 (1987). D. Colton and P. Monk, The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium, Q. Jl. Mech. appl. Math. Vol. 41, 97–125 (1988). D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem, Inverse problems and imaging, V. 1, no. 1, 13–28 (2007). D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, 2rd ed. Applied mathematical science, V. 93, Springer–Verlag, Berlin, 2013. D. Colton, A. Kirsch, and L. Päivärinta, Farfield patterns for acoustic waves in an inhomogeneous medium, SIAM J. Math. Anal. 20, 1472–1483 (1989). R. Duffin and A. C. Schaeffer, Some properties of functions of exponential type, Bull. Amer. Math. Soc, Providence, 44, 236–240 (1938).
V. Isakov, Inverse problems for partial differential equations, Applied Mathematical Sciences, V. 127, Springer-Verlag, New York, 1998. A. Kirsch, The denseness of the far field patterns for the transmission problem, IMA J. Appl. Math, 37, no. 3, 213–225 (1986). A. Kirsch, On the existence of transmission eigenvalues, Inverse problems and imaging, 3, 155–172 (2009). P. Koosis, The logarithmic integral I, Cambridge University Press, New York, 1997.
E. Lakshtanov and B. Vainberg, Bounds on positive interior transmission eigenvalues, Inverse Problems, V. 28, No. 10, 0266–5611 (2012). E. Lakshtanov and B. Vainberg, Weyl type bound on positive interior transmission eigenvalues, Comm. Partial Differential Equations, 39, no. 9, 1729–1740 (2014). B. Ja. Levin, Distribution of zeros of entire functions, revised edition, Translations of mathematical mongraphs, American mathematical society, Providence, 1972.
B. Ja. Levin, Lectures on entire functions, Translation of mathematical monographs, V. 150, AMS, Providence, 1996. H. Y. Liu, Schiffer’s conjecture, interior transmission eigenvalues and invisibility cloaking: singular problem vs. nonsingular problem, Geometric analysis and integral geometry, , Contemp. Math, 598, Amer. Math. Soc., Providence, RI, 147–154(2013). J. R. McLaughlin and P. L. Polyakov, On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues, Jour. Differential Equations, 107, 351–382 (1994). J. Pöschel and E. Trubowitz, Inverse spectral theory, Academic Press, Orlando, 1987.
B. P. Rynne and B. D. Sleeman, The interior transmission problem and inverse scattering from inhomogeneous media, SIAM J. Math. Anal. 22, no.6, 1755–1762 (1991).
|
---
abstract: 'Commute Time Distance (CTD) is a random walk based metric on graphs. CTD has found widespread applications in many domains including personalized search, collaborative filtering and making search engines robust against manipulation. Our interest is inspired by the use of CTD as a metric for anomaly detection. It has been shown that CTD can be used to simultaneously identify both global and local anomalies. Here we propose an accurate and efficient approximation for computing the CTD in an incremental fashion in order to facilitate real-time applications. An online anomaly detection algorithm is designed where the CTD of each new arriving data point to any point in the current graph can be estimated in constant time ensuring a real-time response. Moreover, the proposed approach can also be applied in many other applications that utilize commute time distance.'
author:
- 'sanjay.chawla@sydney.edu.au'
bibliography:
- 'CommuteDistance\_arxiv.bib'
title: |
Online Anomaly Detection Systems\
Using Incremental Commute Time
---
commute time distance; incremental commute time; random walk; anomaly detection;
Introduction
============
Commute Time Distance (CTD) is a random walk based metric on graphs. The $CTD(i,j)$ between two nodes $i$ and $j$ is the [*expected*]{} number of steps a random walk starting at $i$ will take to reach $j$ for the first time and then return back to $i$. The fact that CTD is averaged over all paths (and not just the shortest path) makes it more robust to data perturbations.
CTD has found widespread applications in personalized search [@sarkar2008], collaborative filtering [@brand2005; @fouss2007] and making search engines robust against manipulation [@hopcroft2007]. Our interest is inspired by the use of CTD as a metric for anomaly detection. It has been shown that CTD can be used to simultaneously identify global, local and even collective anomalies in data [@khoa2010].
More advanced measures generally require more expensive computation. Estimating CTD involves the eigen decomposition of the graph Laplacian matrix and consequently has $O(n^3)$ time complexity which is impractical for large graphs. Saerens, Pirotte and Fouss [@saerens2004b] used subspace approximation, and Khoa and Chawla [@khoa2010] used graph sampling to reduce the complexity. Sarkar and Moore [@sarkar2007] introduced a notion of truncated commute time and a pruning algorithm to find nearest neighbors in commute time. They empirically demonstrated achieving a near-linear running time as a function of graph size. Spielman and Srivastava [@spielman2008] have proposed a near-linear time algorithm for approximating pairwise CTD in $O(\log n)$ time based on random projections.
However, there are many applications in practice which require the computation of CTD in an online fashion. When a new data point arrives, the application needs to respond quickly without recomputing everything from scratch. The algorithms noted above all work in a batch fashion and have a high computation cost for online applications.
We are interested in the following scenario: a dataset $D$ is given from an underlying domain of interest. For example, data from a network traffic log or environment or climate change monitoring. A new data point arrives and we want to determine if it is an anomaly with respect to $D$ in CTD. Intuitively a new data point is an anomaly if it is [*far away from its neighbors in CTD*]{}.
Consider the two graphs shown in Figure \[fig:CTD1\]. While the shortest path distance between node 1 and 2 is the same in both graphs, $CTD(1,2)$ increases after node 5 is added. This property of CTD can be used to great effect to detect both global and local outliers. However, the same property makes it challenging to calculate the CTD in an incremental manner.
Here we propose and compare two methods for computing CTD in an incremental fashion. The first method is based on an incremental update of the eigen decomposition of a Laplacian matrix. The second method uses the recursive definition of CTD based on hitting time. To the best of our knowledge both these methods are novel and their comparison provides revealing insights about CTD which are independent of the application domain.
The contributions of this paper are as follows:
- We make use of the characteristics of random walk to estimate CTD incrementally in constant time. The same approach could be used for estimating the hitting time incrementally.
- We propose a provably fast method to incrementally update the eigenvalues and eigenvectors of the graph Laplacian matrix. This method can be integrated with any technique requiring a graph spectral computation, such as spectral clustering.
- We design an online algorithm for anomaly detection using incremental CTD. The technique is verified by experiments in synthetic and real datasets. The experiments show the effectiveness of the proposed methods in terms of accuracy and performance.
The remainder of the paper is organized as follows. Section \[chapter:CTD\] reviews notation and concepts related to random walk and CTD, and a simple example to tie up all the definitions and ideas. In Sections \[chapter:incrementalLaplacian\] and \[chapter:incrementalCD\], we present two methods to incrementally approximate the CTD. In Section \[iCTDalgorithm\], we introduce an online anomaly detection algorithm which uses incremental CTD. In Section \[chapter:expres\], we evaluate our approach using experiments on synthetic and real datasets. Sections \[chapter:related\] covers related work. We conclude in Section \[chapter:conclusion\] with a summary and a direction for future research.
Commute Time Distance {#chapter:CTD}
=====================
We provide a self-contained introduction to random walks with an emphasis on CTD. Assume we are given a connected undirected and weighted graph $G=(V,E,W)$.
Let $i$ be a node in $G$ and $N(i)$ be its neighbors. The [*degree*]{} $d_{i}$ of a node $i$ is $\sum_{j \in N(i)}w_{ij}$. The [*volume*]{} $V_{G}$ of the graph is defined as $\sum_{i \in V}d_{i}$.
The transition matrix $M =(p_{ij})_{i,j \in V}$ of a random walk on $G$ is given by $$p_{ij} = \left\{\begin{array}{ll}\frac{w_{ij}}{d_{i}}, & \mbox{ if $(i,j) \in E$} \\0, & \mbox{ otherwise }
\end{array} \right.$$
Let $P_{0}$ be an initial distribution on $G$. Define $P_{t} = (M^{T})^{t}P_{0}$ for all $t \geq 0$. A distribution $P_{0}$ is [*stationary*]{} if $P_{1}=P_{0}$ [@lovasz1993].
The distribution $P_{0}$ defined by $\pi(v) = \frac{d_{v}}{V_{G}}$ for all $v \in V$ is a [*stationary distribution*]{} [@lovasz1993].
A random walk is [*time-reversible*]{} if for every pair of nodes $i,j \in V$, $\pi(i)p_{ij} = \pi(j)p_{ji}$ [@lovasz1993].
The Hitting Time $h_{ij}$ is the expected number of steps that a random walk starting at $i$ will take before reaching $j$ for the first time.
The Hitting Time can be defined in terms of the recursion $$h_{ij} = \left\{\begin{array}{ll}
1 + \sum_{l \in N(i)}p_{il}h_{lj} & \mbox{ if $ i \neq j$} \\
0 & \mbox{otherwise}
\end{array}\right.$$
The Commute Time Distance $c_{ij}$ between two nodes $i$ and $j$ is given by $c_{ij} = h_{ij} + h_{ji}$.
CTD is a metric: (i) $c_{ii} = 0$, (ii) $c_{ij} = c_{ji}$ and (iii) $c_{ij} \leq c_{ik} + c_{kj}$ [@klein1993].
Remarkably, CTD can be expressed in terms of the Laplacian of $G$.
Let $D$ be the diagonal degree matrix and $A$ be the adjacency matrix of $G$. The Laplacian of $G$ is the matrix $L = D - A$.
1. Let $e_{i}$ be the $V$ dimensional column vector with a 1 at location $i$ and zero elsewhere.
2. Let $(\lambda_{i}, v_{i})$ be the eigenpair of $L$ for all $i \in V$, i.e., $Lv_{i} = \lambda_{i}v_{i}$.
3. It is well known that $\lambda_{1}=0, v_{1} =(1,1,\ldots,1)^{\text{T}}$ and all $\lambda_{i} \geq 0$.
4. Assume $0 = \lambda_{1} \leq \lambda_{2} \ldots \leq \lambda_{|V|}$.
5. Then the pseudo-inverse of $L$ denoted by $L^{+}$ is $$L^{+} = \sum_{i=2}^{|V|}\frac{1}{\lambda_{i}}v_{i}v_{i}^{\text{T}}$$
$$\label{equa3}
c_{ij} = V_{G}(l_{ii}^{+}+l_{jj}^{+}-2l_{ij}^{+}) = V_{G}(e_{i}-e_{j})^{\text{T}}L^{+}(e_{i}-e_{j})$$
where $l_{ij}^{+}$ is the $(i,j)$ element of $L^{+}$ [@fouss2007].
Again, consider the graph $G$ shown in Figure \[fig:exp1\] where all the edge weights equal to 1. The sum of the degree of nodes, $V_{G}=8$. We will calculate the commute time $c_{12}$ in two different ways:
1. Using random walk: note that the expected number of steps for a random walk starting at node 1 and returning back to it is $\frac{V_{G}}{d_{1}} = \frac{8}{1} = 8$ [@lovasz1993]. But the walk from node 1 can only go to node 2 and then return from node 2 to 1. Thus $c_{12} = 8$.
2. Using algebraic approach: the Laplacian matrix is $$L= \left(
\begin{array}{rrrr}
1 & -1 & 0 & 0 \\
-1 & 3 & -1 & -1 \\
0 & -1 & 2 & -1 \\
0 & -1 & -1 & 2
\end{array} \right)$$ and the pseudo-inverse is $$L^{+} = \left(
\begin{array}{rrrr}
0.69 & -0.06 & -0.31 & -0.31 \\
-0.06 & 0.19 & -0.06 & -0.06 \\
-0.31 & -0.06 & 0.35 & 0.02 \\
-0.31 & -0.06 & 0.02 & 0.35 \\
\end{array} \right)$$ Now $c_{12} = V_{G}(e_{1}-e_{2})^{\text{T}}L^{+}(e_{1}-e_{2})$ and\
$$\left(\begin{array}{cccc} 1 & -1 & 0 & 0 \end{array}\right)
\left(\begin{array}{rrrr}
0.69 & -0.06 & -0.31 & -0.31 \\-0.06 & 0.19 & -0.06 & -0.06 \\
-0.31 & -0.06 & 0.35 & 0.02 \\-0.31 & -0.06 & 0.02 & 0.35 \end{array} \right)
\left(\begin{array}{r} 1 \\ -1 \\ 0 \\ 0 \\ \end{array} \right)
= 1$$
Thus $c_{12} = V_{G}\times 1 =8$.
Suppose we add a new node (labeled 5) to node 4 with a unit weight as in Figure \[fig:exp2\]. Then $c_{12}^{new}=V_{G}^{new}/d_{1}=10/1=10.$
The example in Figure \[fig:exp2\] shows that by adding an edge, i.e. making the “cluster” which contains node 2 denser, $c_{12}$ increases. This shows that CTD between two nodes captures not only the distance between them (as measured by the edge weights) but also their neighborhood densities. For the proof of this claim, see [@khoa2010]. This property of CTD has been used to simultaneously discover global and local anomalies in data - an important problem in the anomaly detection literature.
In the above example, we exploited the specific topology (degree one node) of the graph to calculate CTD efficiently. This can only work for very specific instances. The general, more widely used but slower approach for computing CTD is to use the Laplacian formula. A key contribution of this paper is that for incremental computation of CTD we can use insights from this example to accurately and efficiently compute the CTD in much more general situations.
Incremental Eigen Decomposition of Graph Laplacian {#chapter:incrementalLaplacian}
==================================================
In this section, we propose a method to incrementally update the eigensystem (eigenvalues and eigenvectors) of the Laplacian when a new node along with edges to its neighbors is added to the underlying graph. The unique feature of our approach as opposed to that of Ning et. al. [@ning2007] are (i) our emphasis is on handling the addition of a new node and the corresponding edges to its nearest neighbors as opposed to just weight updates on existing edges and (ii) simultaneous updating of all weight edges as opposed to one edge at a time.
Iterative incremental update of the Laplacian eigensystem
---------------------------------------------------------
We propose an algorithm based on the following proposition to incrementally update the eigensystem $(\lambda, v)$ of the Laplacian $L$ when a new node $i$ is added to the graph. Suppose there are $k$ edges $e = (i,j) \in E_{n}$ with weight $w_{e}$ added to the graph from $i$. Denote $\Delta L$ and ($\Delta \lambda, \Delta v$) be changes of $L$ and $(\lambda, v)$ resulting from the addition of $i$. Note that the size of matrix $L$ and its eigenvector $v$ change as mentioned in the Appendix.
\[prop2\] The solution of the eigensystem $$(L + \Delta L)(v + \Delta v) = (\lambda + \Delta \lambda)(v + \Delta v)$$ can be derived from the solution of the following set of simultaneous equations. $$\label{equa7}
\Delta\lambda = % \frac{v^{\text{T}}\Delta L(v + \Delta v)}{1+v^{\text{T}}\Delta v}
\frac{\sum_{e \in E_{n}}w_{e}[v(i)-v(j)][v(i)-v(j)+\Delta v(i)-\Delta v(j)]}{1 + v^{\text{T}}\Delta v}$$ $$\label{equa10}
\Delta v = K^{-1}h$$ where $$\label{equa8}
K = L + \Delta L - (\lambda + \Delta \lambda)I,$$ and $$\label{equa9}
h = (\Delta \lambda I - \Delta L)v.$$
For the proof, see Appendix.
Note that in general it is not practical to solve the system $\Delta v = K^{-1}h$ at the arrival of each new data point $i$. In practice, as noted by Ning. et. al. [@ning2007], we can set $\Delta v(k) =0$ for all components which are not $i$, its first or second order neighbors.
Denote $N_{i}=\{j|d(i,j) \leq 2\}$, where $d(i,j)$ is the shortest path between $i$ and $j$. Let $K_N$ be the matrix derived from $K$ after removing columns which do not correspond to nodes in $N_{i}$, $v_N$ and $\Delta v_N$ be the vectors derived from $v$ and $\Delta v$ after removing elements which do not correspond to nodes in $N_{i}$. Since $K_N$ is not a square matrix, we obtain: $$\label{equa11}
\Delta v = (K_N^{\text{T}}K_N)^{-1}K_N^{\text{T}}h,$$ $$\label{equa4}
\Delta\lambda = \frac{\sum_{e \in E_{n}}w_{e}[v(i)-v(j)][v(i)-v(j)+\Delta v(i)-\Delta v(j)]}{1 + v_N^{\text{T}}\Delta v_N}$$
Since $\Delta \lambda$ in Equation \[equa4\] depends on the value of $\Delta v$ in Equation \[equa11\] and vice versa, we can update the values of $\Delta \lambda$ and $\Delta v$ as follows. We initialize the values $\Delta v = 0$ to update the value of $\Delta \lambda$ and then using that to update $\Delta v$. The procedure is repeated until convergence. Algorithm \[iterativealgorithm\] gives the details.
**Input:** Laplacian matrix *L*, its eigenvalues *S* and eigenvectors *V*, weights *$w_{e}$* of all the new edges\
**Output:** New eigenvalues *$S_n$* and eigenvectors *$V_n$*\
Set $\Delta v=0$\
Update $\Delta \lambda$ using Equation \[equa4\]\
Update $\Delta v$ using Equation \[equa11\]\
Repeat steps 3 and 4 until there is no significant change in $\Delta \lambda$ or until the loop reaches a maximum iterations\
$v_n = v + \Delta v$, $\lambda_n = \lambda + \Delta \lambda$
Incremental Estimation of Commute Time Distance {#chapter:incrementalCD}
===============================================
In this section, we derive a new method for computing the CTD in an incremental fashion. This method uses the definition of CTD based on the hitting time. The basic intuition is to expand the hitting time recursion until the random walk has moved a few steps away from the new node and then use the [*old*]{} values. In Section \[chapter:expres\] we will show that this method results in remarkable agreement between the batch and online mode.
We deal with two cases shown in Figure \[fig:rank\].
1. Rank one perturbation corresponds to the situation when a new node connects with one other node in the existing graph.
2. Rank $k$ perturbation deals with the situation when the new node has $k$ neighbors in the existing graph.
The term [*rank*]{} is used as it corresponds to the rank of the perturbation matrix $\Delta L$.
Rank one perturbation
---------------------
\[prop3\] Let $i$ be a new node connected by one edge to an existing node $l$ in the graph $G$. Let $w_{il}$ be the weight of the new edge. Let $j$ be an arbitrary node in the graph $G$. Then $$\label{equa15}
c_{ij}\approx c_{lj}^{old} + \frac{V_G}{w_{il}}$$ where ‘old’ represents the CTD in graph $G$ before adding $i$.
(Sketch) Since the random walk needs to pass $l$ before reaching $j$, the commute distance from $i$ to $j$ is: $$\label{equa12}
c_{ij}=c_{il} + c_{lj}.$$ It is known that: $$\label{equa13}
c_{il}=\frac{(V_G + 2w_{il})}{w_{il}}$$ where $V_G$ is volume of graph $G$ [@khoa2010]. We also know $c_{lj}=h_{jl} + h_{lj}$ and $h_{jl}=h_{jl}^{old}$. The only unknown factor is $h_{lj}$. By definition: $$h_{lj} = 1 + \sum_{q \in N(l)}p_{lq}h_{qj}
= 1 + \sum_{q \in N(l), q\neq i}p_{lq}h_{qj} + p_{li}h_{ij}. \\$$
Since $h_{qj} \approx h_{qj}^{old}$, $p_{lq}=(1-p_{li})p_{lq}^{old}$, and $h_{ij}=1+h_{lj}$, $$\begin{split}
h_{lj} &\approx 1 + \sum_{q \in N(l), q\neq i}(1-p_{li})p_{lq}^{old}h_{qj}^{old} + p_{li}(1+h_{lj}) \\
&= 1 + (1-p_{li})\sum_{q \in N(l), q\neq i}p_{lq}^{old}h_{qj}^{old} + p_{li}(1+h_{lj}) \\
&= 1 + (1-p_{li})(h_{lj}^{old}-1) + p_{li}(1+h_{lj}).
\end{split}$$
After simplification, $h_{lj} = h_{lj}^{old} + \frac{2p_{li}}{1-p_{li}}.$
Then $c_{lj} \approx h_{jl}^{old} + h_{lj}^{old} + \frac{2p_{li}}{1-p_{li}}.$
Since there is only one edge connecting from $i$ to $G$, $i$ is likely an isolated point and thus $p_{li}\ll 1$. Then $$\label{equa14}
c_{lj}\approx h_{jl}^{old} + h_{lj}^{old} = c_{lj}^{old}.$$ As a result from equations \[equa12\], \[equa13\], and \[equa14\]: $$c_{ij}\approx \frac{(V_G + 2w_{il})}{w_{il}} + c_{lj}^{old} \approx c_{lj}^{old} + \frac{V_G}{w_{il}}$$
Rank $k$ perturbation
---------------------
The rank $k$ perturbation analysis is more involved but the final formulation is an extension of the rank one perturbation.
\[prop4\] Denote $l\in G$ be one of $k$ neighbors of $i$ and $j$ be a node in $G$. The approximate commute distance between nodes $i$ and $j$ is: $$\label{equa19}
c_{ij} \approx \sum_{l \in N(i)}p_{il}c_{lj}^{old} + \frac{V_G}{d_{i}}$$
For the proof, see Appendix.
Online Anomaly Detection Algorithm {#iCTDalgorithm}
==================================
We return to our original motivation for computing incremental CTD. We are given a dataset $D$ which is [*representative*]{} of the underlying domain of interest. We want to check if a new data point is an anomaly with respect to $D$. We will use the CTD as a distance metric.
This section describes an online anomaly detection system using the incremental update of the eigensystem of the Laplacian in Section \[chapter:incrementalLaplacian\] and the incremental estimation of commute time in Section \[chapter:incrementalCD\].
Generally CTD is robust against small changes or perturbation in data. Therefore, only the anomaly score of the new data point needs to be estimated and be compared with the anomaly threshold in the training data. This claim will be verified by experiment in Section \[chapter:expres\].
CTD-based anomaly detection
---------------------------
This section reviews the batch method based on CTD to detect anomalies [@khoa2010]. The method is described in Algorithm \[CTDalgorithm\]. First, a mutual $k_1$-nearest neighbor graph is constructed from the dataset. Then the graph Laplacian matrix $L$, its eigenvectors $V$ and eigenvalues $S$ are computed. Finally, a CTD distance-based anomaly detection with a pruning rule proposed by Bay and Schwabacher [@bay2003] is used to find the top $N$ anomalies. The anomaly score used is the average CTD of an observation to its $k_2$ nearest neighbors.
**Input:** Data matrix *X*, the numbers of nearest neighbors *$k_1$* (for building the $k$-nearest neighbor graph) and *$k_2$* (for estimating the anomaly score), the number of anomalies to return *N*\
**Output:** Top *N* anomalies
Construct the mutual $k$-nearest neighbor graph from the dataset (using $k_1$)\
Compute the graph Laplacian matrix $L$, its eigenvectors $V$ and eigenvalues $S$\
Find top $N$ anomalies using the CTD based technique with pruning rule (using $k_2$). Each CTD query uses Equation \[equa3\]\
Return top $N$ anomalies
[**Pruning Rule [@bay2003]:**]{} A data point is not an anomaly if its score (e.g. the average distance to its $k$ nearest neighbors) is less than an anomaly threshold. The threshold can be fixed or be adjusted as the score of the weakest anomaly found so far. Using the pruning rule, many non-anomalies can be pruned without carrying out a full nearest neighbors search.
Online Algorithms
-----------------
Algorithm \[iLEDalgorithm\] (denote as iLED) is a method to detect anomalies online using the eigenvalues and eigenvectors of the new Laplacian matrix which is updated incrementally. Algorithm \[iECTalgorithm\] (denote as iECT) on the other hand is a method to detect anomalies online using incremental estimation of commute time based on hitting time.
**Input:** Graph *G*, Laplacian matrix *L*, its eigenvalues *S* and eigenvectors *V*, the anomaly threshold *$\tau$* of the training set, and a test data point *p*\
**Output:** Determine if $p$ is an anomaly or not\
Add $p$ to $G$ using the mutual nearest neighbor graph, we have a new graph $G_n$\
Incrementally compute the new eigenvalues and eigenvectors of the new Laplacian $L_n$ using Algorithm \[iterativealgorithm\]\
Use Gram-Schmidt process [@golub1996] to orthogonalize the new eigenvectors\
Determine if $p$ is an anomaly or not by estimating its anomaly score using CTDs derived from the new eigenpairs. Use pruning rule with threshold $\tau$ to reduce the computation\
Return whether $p$ is an anomaly or not
**Input:** Graph *G*, Laplacian matrix *L*, its eigenvalues *S* and eigenvectors *V*, the anomaly threshold *$\tau$* of the training set, and a test data point *p*\
**Output:** Determine if $p$ is an anomaly or not\
Add $p$ to $G$ using the mutual nearest neighbor graph, we have a new graph $G_n$\
Determine if $p$ is an anomaly or not by estimating its anomaly score using incremental CTDs mentioned in Section \[chapter:incrementalCD\]. Use pruning rule with threshold $\tau$ to reduce the computation\
Return whether $p$ is an anomaly or not
When a new data point $p$ arrives, it is connected to graph $G$ created in the training phase. The CTDs are incrementally updated to estimate the anomaly score of $p$ using the approach in sections \[chapter:incrementalLaplacian\] and \[chapter:incrementalCD\]. For iLED algorithm, after updating the eigenvalues and eigenvectors, we use Gram-Schmidt process [@golub1996] to normalize and orthogonalize the eigenvectors.
Analysis
--------
First, we analyse the incremental eigen decomposition of the Laplacian in Section \[chapter:incrementalLaplacian\]. Here $n$ is the size of the original graph (Laplacian) and $N$ is the neighborhood size used in $K_N$ (i.e., cardinality of $N_i$). Note $N \ll n$.
It takes constant time to update $\Delta \lambda$ and $O(N^2n)$ to compute $X=K_N^{\text{T}}K_N$, $O(N^3)$ for $X^{-1}$, $O(Nn)$ for $y=K_N^{\text{T}}h$ and $O(N^2)$ for $\Delta v=X^{-1}y$. Since $N \ll n$, we obtain $O(n)$ time for the incremental update of eigenvalues and eigenvectors of the Laplacian.
On the other hand, incremental estimation of commute time update in Section \[chapter:incrementalCD\] requires $O(m)$ for each query of $c_{lj}^{old}$ where $m$ is the number of eigenvectors used. So if there are $k$ edges added to the graph, it takes $O(km)$ for each query of CTD.
Since we only need to compute the anomaly score of a test data point using the pruning rule with the threshold of anomaly score in the training set, it takes only $O(k_2)$ nearest neighbor search to determine if the test point is an anomaly or not where $k_2$ is the number of nearest neighbors for estimating the anomaly score. For each CTD query, it takes $O(m)$ for Algorithm \[iLEDalgorithm\] (iLED), and $O(km)$ for Algorithm \[iECTalgorithm\] (iECT). Therefore, iLED takes $O(n+k_2m)=O(n)$, and iECT takes $O(k_2km)=O(1)$ to determine if a new arriving point is an anomaly or not. Note that since $L$ and $K$ are sparse, we can get better than $O(n)$ for iLED.
Experiments and Results {#chapter:expres}
=======================
We report on the experiments carried out to determine and compare the effectiveness of the iECT and iLED methods. To recall, iECT uses the recursive definition of hitting time to calculate CTD while iLED uses the Laplacian definition.
### Approach {#approach .unnumbered}
We split a data set into two parts: training and test. We use Algorithm \[CTDalgorithm\] to compute the top $N$ anomalies in the training set and use the average distance of a data point to its $k_{2}$ nearest neighbor (in CTD) as its anomaly score. The weakest anomaly in the top $N$ set is one which has the smallest average distance to its nearest neighbors and is used as the threshold value $\tau$. Then the anomaly score of each instance $p$ in the test set is calculated based on its $k_{2}$ neighbors in the training set. If this score is greater than $\tau$ then the test instance is reported as an anomaly. During the time searching for the nearest neighbors of $p$, if its average distance to the nearest neighbors found so far is smaller than $\tau$, we can stop the search as $p$ is not anomaly (pruning rule).
### Data and Parameters {#data-and-parameters .unnumbered}
The experiments were carried out on synthetic as well as real datasets. We chose the number of nearest neighbors $k_1=10$ to build the mutual nearest neighbor graph, $k_2=20$ to estimate the anomaly score, the number of Laplacian eigenvectors $m=50$. In Algorithm \[iterativealgorithm\], the threshold to estimate the change of $\Delta \lambda$ was $10^{-6}$ and the maximum iterations was 5. The choice of parameters was determined from the experiments. In all experiments, the batch method was used as the benchmark. The anomaly threshold $\tau$ was set based on the training data. It was the score of the weakest anomaly in the top $N=50$ anomalies found by Algorithm \[CTDalgorithm\] in the training set.
Synthetic datasets
------------------
We created six synthetic datasets, each of which contained several clusters generated from Normal distributions and a number of random points generated from uniform distribution which might be anomalies. The number of clusters, the sizes, and the locations of the clusters were also chosen randomly. Each dataset was divided into a training set and a test set. There were 100 data points in every test set and half of them were random anomalies mentioned above.
### Experiments on Robustness {#experiments-on-robustness .unnumbered}
We first tested the robustness of CTD between nodes in an existing set when a new data instance is introduced. As $CTD(i,j)$ between nodes $i$ and $j$ is a measure of expected path distance, the hypothesis is that the addition of a new point will have minimal influence on $CTD(i,j)$ and thus the anomaly scores of data points in the existing set are relatively unchanged.
Table \[tab:table2\] shows the average, standard deviation, minimum, and maximum of anomaly scores of points in graph $G$ before and after a new data point was added to $G$. Graph $G$ was created from the 1,000 point dataset in Figure \[fig:dataset\]. The result with test point was averaged over 100 test points in the test set. The result shows that the anomaly scores of data instances in $G$ do not change much. This shows CTD is a robust measure, a small change or perturbation in the data will not result in large changes in CTD. Therefore, only the anomaly score of the new point needs to be estimated.
Average Std Min Max
-------------------- ----------- ----------- ------ --------------
Without test point 10,560.61 65,023.18 6.13 1,104,648.09
With test point 10,517.94 64,840.39 6.13 1,101,169.36
: Robustness of CTD. The anomaly scores of data instances in existing graph $G$ are relatively unchanged when a new point is added to $G$.
\[tab:table2\]
![1,000 points dataset with training and test sets[]{data-label="fig:dataset"}](1k_dataset.eps){width="40.00000%"}
### Experiments on Effectiveness {#experiments-on-effectiveness .unnumbered}
We applied iECT and iLED to all the datasets. The effectiveness of the iECT algorithm over iLED is shown in Figure \[fig:iLED\_iECT\_accuracy\]. There were 36 anomalies detected by batch method using CTD which are shown in Figure \[fig:dataset\]. iECT captured all of them and had 8 false positives whose scores were close to the threshold. iLED approach, on the other hand, had better precision with no false positives but worse recall with only 15 anomalies found. The reason was anomalies have more effect on the eigenvectors and eigenvalues of the graph Laplacian and thus iLED was unable to capture many of them.
![Accuracy of iLED and iECT in 1,000 points dataset. iECT detects anomalies better than iLED.[]{data-label="fig:iLED_iECT_accuracy"}](iL_iC_1k_accuracy.eps){width="40.00000%"}
To get a better understanding, Figure \[fig:iL\] shows the eigenvalues and eigenvectors of the new graph computed using batch method and iLED in 1,000 points dataset where the test point was an anomaly and a non-anomaly. The eigenvalue graph shows the top 50 smallest eigenvalues and the eigenvector graph shows the dot product of the top 50 smallest eigenvectors of the new graph Laplacian for both batch and iLED methods. When a new data point was a non-anomaly, the approximate eigenvalues and eigenvectors were significantly more accurate than those when a new point was an anomaly.
\
Table \[tab:table1\] shows the results in accuracy and performance of iLED and iECT in the synthetic datasets. Average score was the average anomaly score over 100 test points. The precision and recall were for the anomalous class. The time was the average time to process each of 100 test points. iECT generally had better approximation than iLED, did not miss any anomaly and had acceptable false alarms. iLED, on the other hand, did not have any false alarms but did miss many anomalies. Both of them were more efficient than the batch method. Note that the scores shown here were the anomaly scores with pruning rule and the scores for anomalies are always much higher than scores for normal points. Therefore the scores were dominated by the scores of anomalies and that is the reason why iLED had much lower scores. In fact, iLED is more accurate than iECT in estimating the scores of normal points.
Dataset
--------- -------------------- --------------- ------------ ---------- -------------------- --------------- ------------ ---------- -------------------- ----------
Size Avg Score Precision (%) Recall (%) Time (s) Avg Score Precision (%) Recall (%) Time (s) Avg Score Time (s)
1,000 $3.84 \times 10^4$ 100 41.7 0.13 $1.69 \times 10^5$ 81.8 100 0.20 $1.61 \times 10^5$ 0.17
10,000 $7.09 \times 10^5$ 100 53.2 1.28 $4.87 \times 10^6$ 95.9 100 1.42 $4.99 \times 10^6$ 2.04
20,000 $5.36 \times 10^6$ 100 83.3 2.64 $1.75 \times 10^7$ 80.0 100 2.96 $1.70 \times 10^7$ 4.53
30,000 $4.81 \times 10^6$ 100 39.6 3.68 $1.39 \times 10^8$ 96.0 100 4.33 $1.40 \times 10^8$ 7.13
40,000 $3.27 \times 10^6$ 100 15.6 4.44 $5.17 \times 10^7$ 71.1 100 5.19 $4.88 \times 10^7$ 9.05
50,000 $8.88 \times 10^6$ 100 32.5 5.70 $6.15 \times 10^7$ 87.0 100 6.61 $5.96 \times 10^7$ 11.60
\[tab:table1\]
There is an interesting dynamic at play between the anomaly, pruning rule, iECT, iLED, and the number of anomalies in the data. iECT was slightly slower than iLED in the experiment. The reason is we have many anomalies in the test set. We know that the pruning rule only works for non-anomalies. Moreover, iLED is faster per CTD query compared to iECT. Therefore, for anomalies, iLED is generally faster. Furthermore, because iLED tends to underestimate the scores of anomalies (and that was the reason it missed some anomalies), anomalies are treated as non-anomalies and the pruning kicks-in making it faster. In practise, since most of the test points are not anomalies, iECT will be more efficient than iLED. It is shown in Figure \[fig:iLED\_iECT\_time\] where except a few false alarms, iECT was generally faster than iLED in 50,000 points dataset when test instances were not anomalies. The same tendency also happened in other datasets used in the experiments.
![Performance of iLED and iECT in 50,000 points dataset. Except a few false alarms, iECT is generally faster than ilED when a test point is not an anomaly.[]{data-label="fig:iLED_iECT_time"}](iL_iC_50k_performance.eps){width="40.00000%"}
Real Datasets
-------------
### DBLP dataset
In this section, we evaluated the iECT method on the DBLP co-authorship network. Nodes are authors and edge weights are the number of collaborated papers between the authors. Since the graph is not fully connected, we extracted its biggest component. It has 344,800 nodes and 1,158,824 edges.
We randomly chose a test set of 50 nodes and removed them from the graph. We ensured that the graph remained connected. After training, each node was added back into the graph along with their associated edges.
Since the size of the graph is very large, normal training using the batch mode in Algorithm \[CTDalgorithm\] is not feasible. Instead we implemented the approximate method proposed by Spielman and Srivastava (SS) [@spielman2008] and used the underlying linear time CMG solver proposed by Koutis [@koutis2009]. The SS methods combines random projections with a linear time solver for diagonally dominant matrices to approximate the CTD. The SS method creates a matrix $Z$ from which CTD between two nodes can be computed in $k=O(\log n)$ time with provable accuracy. In practice we can use $k$ to be much smaller than $O(\log n)$ and still attain highly accurate results.
We trained the graph using the SS approach, stored the matrix $Z$ and used $Z$ to query the $c_{lj}^{old}$ in iECT algorithm. The batch method here is the CTD approximation using the matrix $Z_{new}$ created from the new graph after adding each test data point. The parameter for random projection was $k=200$.
The result shows that it took 0.0066 seconds on average over 50 test data points to detect whether each test point was an anomaly or not. The batch method, which is the fastest approximation of CTD to date, required 944 seconds on average to process each test data point. This dramatically highlights the constant time complexity of iECT algorithm and suggests that iECT is highly suitable for the computation of CTD in an incremental fashion. Since there was no anomaly in the random test set, we cannot report the detection accuracy here. The average anomaly score over all the test points of iECT was 1.1 times higher than the batch method. This shows the relatively high accuracy of iECT approximation even in a very large graph.
### KDD Cup 1999 datasets
We used the 10% dataset from the KDD cup 1999 competition provided by UCI Machine Learning Repository [@FrankAsuncion2010]. It was used to build detection tools of network attacks or intrusions. Since the dataset is huge and there are more anomalies than normal instances, we sampled 2,200 data points from it where there were 2,000 normal points and 200 anomalies (network intrusions). Categorical features were ignored and 38 numerical features were used. The dataset was divided into a training set and a test set with 100 data points.
iLED and iECT were applied on this dataset and min-max scaling was used as data normalization. iLED had a precision of 100% and a recall of 66.7% while iECT had a precision of 75% and a recall of 100%. The average anomaly scores of iLED and iECT were 2% and 1% lower than that of the batch method, respectively.
### NICTA datasets
The dataset is from a wireless mesh network which has seven nodes deployed by NICTA at the School of IT, University of Sydney [@zaidi2009]. It used a traffic generator to simulate traffic on the network. Packets were aggregated into one-minute time bins and the data was collected in 24 hours. There were 391 origin-destination flows and 1,270 time bins. Some anomalies were introduced to the network including DOS attacks and ping floods. The dataset was divided into a training set and a test set with 100 data points.
iLED and iECT were applied on this dataset and min-max scaling was used as data normalization. iLED had a precision of 100% and a recall of 27.3% while iECT had a precision of 84.6% and a recall of 100%. The average anomaly scores of iLED and iECT were 20% lower and 2% higher than that of the batch method, respectively. The tendency of the detection here of iLED and iECT are also similar to those of the synthetic datasets.
Summary and Discussion
----------------------
The experimental results on both synthetic and real datasets show that iECT generally has better detection ability than iLED. iECT has very high recall and acceptable precision. iLED, on the other hand, has very high precision but low recall. Both of them are faster than the batch method. The results on real datasets collected from different sources also have similar tendency showing the reliability and effectiveness of the proposed methods.
The experiments also reveal that iLED tends to underestimate the CTDs for anomalies while iECT tends to overestimate the CTDs for non-anomalies. It leads to a high precision for iLED and a high recall for iECT. If we can come up with a strategy to combine the strengths of the two methods, we can have a more accurate estimation. iECT is faster than iLED but it can only be used in case where a new test point is added and cannot be used when there are weigh updates in the graph. iLED on the other hand can be used in both cases by just changing the perturbation matrix.
Related work {#chapter:related}
============
Incremental learning using an update on eigen decomposition has been studied for a long time. Early work studied the rank one modification of the symmetric eigen decomposition [@golub1973; @bunch1978; @gu1994]. The authors reduced the original problem to the eigen decomposition of a diagonal matrix. Though they can have a good approximation of the new eigenpair, they are not suitable for online applications nowsaday since they have at least $O(n^2)$ computation for the update.
More recent approach was based on the matrix perturbation theory [@champagne1994; @agrawal2008]. They used the first order perturbation analysis of the rank-one update for a data covariance matrix to compute the new eigenpair. The algorithms have a linear time computation. The advantage of using the covariance matrix is if the perturbation involving an insertion of a new point, the size of the covariance matrix is unchanged. This approach cannot be applied directly to increasing matrix size due to an insertion of a new point. For example, in spectral clustering or CTD-based anomaly detection, the size of the graph Laplacian matrix increases when a new point is added to the graph.
Ning et. al [@ning2007] proposed an incremental approach for spectral clustering with application to monitor evolving blog communities. It incrementally updates the eigenvalues and eigenvectors of the graph Laplacian matrix based on a change of an edge weight on the graph using the first order error of the generalized eigen system.
Conclusion {#chapter:conclusion}
==========
The paper shows two novel approaches to compute CTD incrementally. The first one incrementally updates the eigenvectors and eigenvalues of the graph Laplacian matrix. It is linearly scaled and can be applied to the estimation of CTD incrementally or any application involving graph spectral computation. The second approach incrementally estimates CTD in constant time using the property of random walk and hitting time. We design novel anomaly detection algorithms using two approaches to detect anomalies online. The experimental results show the effectiveness of the proposed approaches in terms of performance and accuracy. It took less than 7 milliseconds on average to process a new arriving point in a graph of more than 300,000 nodes and one million edges. Moreover, the idea of this work can be extended in many applications which use the CTD and it is the direction for our future work.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors of this paper acknowledge the financial support of the Capital Markets CRC.
We provide relation between perturbation and incidence matrix, and the proof details of Propositions \[prop2\] and \[prop4\].
Incidence matrix and perturbation
---------------------------------
It is well known that the Laplacian of a graph can be expressed in terms of an incidence matrix.
Given a weighted graph $G=(V,E,W)$ and an arbitrary but fixed orientation of the edges, the incidence matrix $R$ is a $|V| \times |E|$ matrix where the columns of the matrix are defined as $$r_{e}(w) =
\begin{cases}
\sqrt{w} & \text{at location $v$ if $v$ is the head of $e$} \\
-\sqrt{w} & \text{at location $v$ if $v$ is the tail of $e$} \\
0 & \text{otherwise} \\
\end{cases}$$ Note that $r_{e}$ is a column vector of size $|V|$. We can also express each $r_{e} = \sqrt{w}u_{e}$ where $u_{e}$ is a column vector with entries 1, -1 at locations corresponding to head and tail of $e$ and 0 at other locations.
If $L$ is a graph Laplacian then $L=RR^{\text{T}}$ [@chung1997].
If an edge $e$ undergoes a similarity change $\Delta w_{e}$, the new graph Laplacian $L_n$ is $L_n=R_nR_n^{\text{T}}$ where $R_n$=\[$R$ $r_{e}(\Delta w_{e})$\] [@ning2007]. Therefore, a change in an edge weight can be represented by appending an incidence vector to $R$.
\[prop1\] Denote $R$ as an incidence matrix of a given graph $G=(V,E,W)$. Suppose a new node is added to the graph which results in $k$ new edges. Let $R_{n}$ represent the new $|V+1| \times |E+k|$ matrix and let $\Delta R$ represent the matrix of new incidence vectors of size $|V+1| \times k$. Then the new Laplacian $L_{n}$ can be expressed in terms of the old Lapalcian $L$ as $$L_{n} = \begin{bmatrix}L & 0 \\ 0 & 0\end{bmatrix} + \Delta L$$
Also note that if $(\lambda,v)$ is an eigenpair of $L$ then $\left(\lambda, \left(\begin{array}{c}v \\ 0 \end{array}\right)\right)$ is an eigenpair of $\begin{bmatrix}L & 0 \\ 0 & 0\end{bmatrix}$.
Let $E_{n}$ be the set of new edges resulting from the addition of a new node. Then $$\begin{split}
R_nR_n^{\text{T}} &= \left[\begin{pmatrix}R \\ 0\end{pmatrix}\Delta R\right]\left[\begin{pmatrix}R \\ 0\end{pmatrix}\Delta R\right]^{\text{T}} \\
&= \begin{bmatrix}L & 0 \\ 0 & 0\end{bmatrix} + \sum_{e \in E_{n}}w_{e}u_{e}u_{e}^{\text{T}}
= \begin{bmatrix}L & 0 \\ 0 & 0\end{bmatrix} + \Delta L = L_n
\end{split}$$
Incremental Update of Eigenvalues and Eigenvectors
--------------------------------------------------
**Proof for Proposition \[prop2\]:**
Given the eigen decomposition of the new Laplacian matrix: $$(L + \Delta L)(v + \Delta v) = (\lambda + \Delta \lambda)(v + \Delta v)$$
The perturbation is: $$\label{equa5}
\Delta L = \sum_{e \in E_{n}}w_{e}u_{e}u_{e}^{\text{T}}$$
Since $Lv = \lambda v$, $$\label{equa6}
L\Delta v + \Delta Lv + \Delta L\Delta v = \Delta \lambda v + \lambda\Delta v + \Delta \lambda\Delta v$$
Left multiply both sides by $v^\text{T}$: $$v^\text{T}L\Delta v + v^\text{T}\Delta Lv + v^\text{T}\Delta L\Delta v = v^\text{T}\Delta \lambda v + v^\text{T}\lambda\Delta v + v^\text{T}\Delta \lambda\Delta v$$
Since $v^{\text{T}}L = \lambda v^{\text{T}}$ ($L$ is symmetric): $$v^{\text{T}}\Delta \lambda(v + \Delta v) = v^{\text{T}}\Delta L (v + \Delta v)$$
Then we have the update of the eigenvalue $\lambda$: $$%\label{equa7}
\begin{split}
\Delta \lambda &= \frac{v^{\text{T}}\Delta L (v + \Delta v)}{v^{\text{T}}(v + \Delta v)}
= \frac{v^{\text{T}}\Delta L (v + \Delta v)}{1 + v^{\text{T}}\Delta v} \\
&= \frac{v^{\text{T}}\sum_{e \in E_{n}}w_{e}u_{e}u_{e}^{\text{T}} (v + \Delta v)}{1 + v^{\text{T}}\Delta v} \\
&= \frac{\sum_{e \in E_{n}}w_{e}[v(i)-v(j)][v(i)-v(j)+\Delta v(i)-\Delta v(j)]}{1 + v^{\text{T}}\Delta v} \\
\end{split}$$
From equation \[equa6\] we have: $$[L + \Delta L - (\lambda + \Delta \lambda)I]\Delta v = (\Delta \lambda I - \Delta L)v$$
Denotes $K= L + \Delta L - (\lambda + \Delta \lambda)I$ and $h=(\Delta \lambda I - \Delta L)v$, we have $\Delta v = K^{-1}h.$
Rank $k$ Perturbation
---------------------
\[lem1\] Denote $l\in G$ is one of $k$ neighbors of $i$ and $j$ is a node in $G$. We have: $$\sum_{l \in N(i)}p_{il}h_{li} = \frac{V_G}{d_{i}}+1.$$
Using the reversibility property of the random walk, it is easy to prove that the expected number of steps that a random walk which has just visited node $i$ will take before returning back to $i$ is $graph\text{-}volume/d_{i}$ [@lovasz1993].
In case of $i$, this distance equals to the distance from $i$ to one of its neighbors $l$ (one step) plus the hitting time $h_{li}$. Since the random walk goes from $i$ to $l$ with the probability $p_{il}$, we have $1 + \sum_{l \in N(i)}p_{il}h_{li} = \frac{V_G+2d_{i}}{d_{i}}$. Therefore, $\sum_{l \in N(i)}p_{il}h_{li} = \frac{V_G}{d_{i}}+1$.
**Proof for Proposition \[prop4\]:**
(Sketch) By definition, $$h_{ij} = 1 + \sum_{l \in N(i)}p_{il}h_{lj} = 1 + \sum_{l \in N(i)}p_{il}(1 + \sum_{q \in N(l)}p_{lq}h_{qj})$$
Using the same approach as the rank one case, $$\begin{split}
h_{ij} &= 1 + \sum_{l \in N(i)}p_{il}[1 + (1-p_{li})\sum_{q \in N(l), q\neq i}p_{lq}^{old}h_{qj}^{old} + p_{li}h_{ij}] \\
&= 1 + \sum_{l \in N(i)}p_{il}[1 + (1-p_{li})(h_{lj}^{old}-1) + p_{li}h_{ij}]
\end{split}$$
After a few manipulations, we have $$h_{ij} = \frac{1 + \sum_{l \in N(i)}p_{il}h_{lj}^{old}-\sum_{l \in N(i)}p_{il}p_{li}h_{lj}^{old}+\sum_{l \in N(i)}p_{il}p_{li}}{1-\sum_{l \in N(i)}p_{il}p_{li}}.$$
Because $\sum_{l \in N(i)}p_{il}p_{li} \ll 1$, $$\label{equa16}
h_{ij} \approx 1 + \sum_{l \in N(i)}p_{il}h_{lj}^{old}.$$
Since the commute distance between two nodes is the average of all possible path-length between them, $h_{ji} \approx \frac{1}{k}\sum_{l \in N(i)}(h_{jl} + h_{li})$. Instead of using the normal average, we take into account the probability $p_{il}$: $$\label{equa17}
h_{ji} \approx \sum_{l \in N(i)}p_{il}(h_{jl} + h_{li}) = \sum_{l \in N(i)}p_{il}h_{jl} + \sum_{l \in N(i)}p_{il}h_{li}$$
We have $h_{jl}\approx h_{jl}^{old}$. Moreover, from Lemma \[lem1\] we have $\sum_{l \in N(i)}p_{il}h_{li} = \frac{V_G}{d_{i}}+1$. Then from \[equa17\], $$\label{equa18}
h_{ji} \approx \sum_{l \in N(i)}p_{il}h_{jl}^{old} + \frac{V_G}{d_{i}}+1$$
As a result of equations \[equa16\] and \[equa18\], $$\begin{split}
c_{ij} &\approx 1+ \sum_{l \in N(i)}p_{il}c_{lj}^{old} + \frac{V_G}{d_{i}} +1
\approx \sum_{l \in N(i)}p_{il}c_{lj}^{old} + \frac{V_G}{d_{i}}
\end{split}$$
When $k=1$ (rank one case), the formula becomes Equation \[equa15\].
|
---
abstract: 'Any NLO calculation of a QCD final-state observable involves Monte Carlo integration over a large number of events. For DIS and hadron colliders this must usually be repeated for each new PDF set, making it impractical to consider many ‘error’ PDF sets, or carry out PDF fits. Here we discuss “a posteriori” inclusion of PDFs, whereby the Monte Carlo run calculates a grid (in $x$ and $Q$) of cross section weights that can subsequently be combined with an arbitrary PDF. The procedure is numerically equivalent to using an interpolated form of the PDF. The main novelty relative to prior work is the use of higher-order interpolation, which substantially improves the tradeoff between accuracy and memory use. An accuracy of about $0.01$% has been reached for the single inclusive cross-section in the central rapidity region $|y|<0.5$ for jet transverse momenta from $100$ to $5000 \mathrm{GeV}$. This method should facilitate the consistent inclusion of final-state data from HERA, Tevatron and LHC data in PDF fits, thus helping to increase the sensitivity of LHC to deviations from standard Model predictions.'
author:
- 'Tancredi Carli$^{1}$, Gavin P. Salam$^{2}$, Frank Siegert$^{1}$'
title: 'A posteriori inclusion of PDFs in NLO QCD final-state calculations[^1] '
---
Introduction
============
The Large Hadron Collider (LHC), currently under construction at CERN, will collide protons on protons with an energy of $7$ [TeV]{}. Together with its high collision rate the high available centre-of-mass energy will make it possible to test new interactions at very short distances that might be revealed in the production cross-sections of Standard Model (SM) particles at very high transverse momentum ($P_T$) as deviation from the SM theory.
The sensitivity to new physics crucially depends on experimental uncertainties in the measurements and on theoretical uncertainties in the SM predictions. It is therefore important to work out a strategy to minimize both the experimental and theoretical uncertainties from LHC data. For instance, one could use single inclusive jet or Drell-Yan cross-sections at low $P_T$ to constrain the PDF uncertainties at high $P_T$. Typical residual renormalisation and factorisation scale uncertainties in next-to-leading order (NLO) calculations for single inclusive jet-cross-section are about $5-10\%$ and should hopefully be reduced as NNLO calculations become available. The impact of PDF uncertainties on the other hand can be substantially larger in some regions, especially at large $P_T$, and for example at $P_T = 2000$ [GeV]{} dominate the overall uncertainty of $20\%$. If a suitable combination of data measured at the Tevatron and LHC can be included in global NLO QCD analyses, the PDF uncertainties can be constrained.
The aim of this contribution is to propose a method for consistently including final-state observables in global QCD analyses.
For inclusive data like the proton structure function $F_2$ in deep-inelastic scattering (DIS) the perturbative coefficients are known analytically. During the fit the cross-section can therefore be quickly calculated from the strong coupling (${\alpha_s}$) and the PDFs and can be compared to the measurements. However, final state observables, where detector acceptances or jet algorithms are involved in the definition of the perturbative coefficients (called “weights” in the following), have to be calculated using NLO Monte Carlo programs. Typically such programs need about one day of CPU time to calculate accurately the cross-section. It is therefore necessary to find a way to calculate the perturbative coefficients with high precision in a long run and to include ${\alpha_s}$ and the PDFs “a posteriori”.
To solve this problem many methods have been proposed in the past [@Graudenz:1995sk; @Kosower:1997vj; @Stratmann:2001pb; @wobisch; @wobisch; @zeus2005]. In principle the highest efficiencies can be obtained by taking moments with respect to Bjorken-$x$ [@Graudenz:1995sk; @Kosower:1997vj], because this converts convolutions into multiplications. This can have notable advantages with respect to memory consumption, especially in cases with two incoming hadrons. On the other hand, there are complications such as the need for PDFs in moment space and the associated inverse Mellin transforms.
Methods in $x$-space have traditionally been somewhat less efficient, both in terms of speed (in the ‘a posteriori’ steps — not a major issue here) and in terms of memory consumption. They are, however, somewhat more transparent since they provide direct information on the $x$ values of relevance. Furthermore they can be used with any PDF. The use of $x$-space methods can be further improved by using methods developed originally for PDF evolution [@Ratcliffe:2000kp; @Dasgupta:2001eq].
PDF-independent representation of cross-sections
================================================
Representing the PDF on a grid
------------------------------
We make the assumption that PDFs can be accurately represented by storing their values on a two-dimensional grid of points and using $n^{\mathrm{th}}$-order interpolations between those points. Instead of using the parton momentum fraction $x$ and the factorisation scale $Q^2$, we use a variable transformation that provides good coverage of the full $x$ and $Q^2$ range with uniformly spaced grid points:[^2] $$\label{eq:ytau}
y(x) = \ln \frac{1}{x} \; \; \; {\rm and} \; \; \;
\tau(Q^2) = \ln \ln \frac{Q^2}{\Lambda^2}.$$ The parameter $\Lambda$ is to be chosen of the order of $\Lambda_{\mathrm{QCD}}$, but not necessarily identical. The PDF $q(x,Q^2)$ is then represented by its values $q_{i_y,i_\tau}$ at the 2-dimensional grid point $(i_y \, \delta y, i_\tau \, \delta \tau)$, where $\delta
y$ and $\delta \tau$ denote the grid spacings, and obtained elsewhere by interpolation: $$\label{eq:interp}
q(x,Q^2) = \sum_{i=0}^n \sum_{\iota=0}^{n'} q_{k+i,\kappa+\iota} \,\,
I_i^{(n)}\left( \frac{y(x)}{\delta y} - k \right)\,
I_\iota^{(n')}\left( \frac{\tau(Q^2)}{\delta\tau}-\kappa \right),$$ where $n$, $n'$ are the interpolation orders. The interpolation function $I_i^{(n)}(u)$ is 1 for $u=i$ and otherwise is given by: $$\label{eq:Ii}
I_i^{(n)}(u) = \frac{(-1)^{n-i}}{i!(n-i)!} \frac{u (u-1) \ldots (u-n)}{u-i}\,.$$ Defining $\mathrm{int} (u)$ to be the largest integer such that $\mathrm{int}(u) \le u$, $k$ and $\kappa$ are defined as: $$\begin{aligned}
\label{eq:kchoice}
k(x) =& \mathrm{int} \left( \frac{y(x)}{\delta y} - \frac{n-1}{2} \right), &
\kappa(x) = \mathrm{int} \left( \frac{\tau(Q^2)}{\delta \tau} - \frac{n'-1}{2} \right).\end{aligned}$$ Given finite grids whose vertex indices range from $0\ldots N_y-1$ for the $y$ grid and $0\ldots N_\tau-1$ for the $\tau$ grid, one should additionally require that eq. (\[eq:interp\]) only uses available grid points. This can be achieved by remapping $k \to
\max(0,\min(N_y-1-n,k))$ and $\kappa \to
\max(0,\min(N_\tau-1-n',\kappa))$.
Representing the final state cross-section weights on a grid (DIS case)
-----------------------------------------------------------------------
Suppose that we have an NLO Monte Carlo program that produces events $m=1\dots N$. Each event $m$ has an $x$ value, $x_m$, a $Q^2$ value, $Q^2_m$, as well as a weight, $w_m$, and a corresponding order in ${\alpha_s}$, $p_m$. Normally one would obtain the final result $W$ of the Monte Carlo integration from:[^3] $$\label{eq:normalint}
W = \sum_{m=1}^N \,w_m \, \left( \frac{\alpha_s(Q_m^2)} {2\pi}\right)^{p_m} \, q(x_m,Q^2_m).$$
Instead one introduces a weight grid $W_{i_y,i_\tau}^{(p)}$ and then for each event updates a portion of the grid with:\
$i = 0\dots n,\; \iota = 0\dots n':$ $$\begin{aligned}
\label{eq:weight2evolve}
W_{k+i,\kappa + \iota}^{(p_m)} \to W_{k+i,\kappa + \iota}^{(p_m)} + w_m\,
I_i^{(n)} \left(\frac{y(x_m)}{\delta y} - k\right)
I_{\iota}^{(n')}\left(\frac{\tau(Q^2_m)}{\delta \tau} - \kappa \right), \\
\;\;\; {\rm where} \;\;\;
k \equiv k(x_m),\; \kappa \equiv \kappa(Q^2_m). \nonumber
\end{aligned}$$ The final result for $W$, for an arbitrary PDF, can then be obtained *subsequent* to the Monte Carlo run: $$\label{eq:WfinalxQ}
W = \sum_p \sum_{i_y} \sum_{i_\tau}
W_{i_y,i_\tau}^{(p)} \, \left( \frac{\alpha_s\left({Q^2}^{(i_\tau)}\right)}{2\pi}\right)^{p} q \!\left(x^{(i_y)}, {Q^2}^{(i_\tau)} \right)\,,$$ where the sums index with $i_y$ and $i_\tau$ run over the number of grid points and we have have explicitly introduced $x^{(i_y)}$ and ${Q^2}^{(i_\tau)}$ such that: $$\label{eq:xQdefs}
y(x^{(i_y)}) = i_y \, \delta y \quad {\rm and} \quad
\tau\left({Q^2}^{(i_\tau)}\right) = i_\tau \, \delta \tau.$$
Including renormalisation and factorisation scale dependence
-------------------------------------------------------------
If one has the weight matrix $W_{i_y,i_\tau}^{(p)}$ determined separately order by order in ${\alpha_s}$, it is straightforward to vary the renormalisation $\mu_R$ and factorisation $\mu_F$ scales a posteriori (we assume that they were kept equal in the original calculation).
It is helpful to introduce some notation relating to the DGLAP evolution equation: $$\label{eq:DGLAP}
\frac{d q(x,Q^2)}{d \ln Q^2} = \frac{\alpha_s(Q^2)}{2\pi} (P_0 \otimes q)(x,Q^2)
+ \left(\frac{\alpha_s(Q^2)}{2\pi}\right)^2 (P_1
\otimes q)(x,Q^2) + \ldots,$$ where the $P_0$ and $P_1$ are the LO and NLO matrices of DGLAP splitting functions that operate on vectors (in flavour space) $q$ of PDFs. Let us now restrict our attention to the NLO case where we have just two values of $p$, $p_{\mathrm{LO}}$ and $p_{\mathrm{NLO}}$. Introducing $\xi_R$ and $\xi_F$ corresponding to the factors by which one varies $\mu_R$ and $\mu_F$ respectively, for arbitrary $\xi_R$ and $\xi_F$ we may then write: $$\begin{aligned}
\label{eq:Wfinalxi}
W(\xi_R, \xi_F) = \sum_{i_y} \sum_{i_\tau}
\left(\frac{\alpha_s\left(\xi_R^2 {Q^2}^{(i_\tau)}\right)\,
}{2\pi}
\right)^{p_{\mathrm{LO}}}
W_{i_y,i_\tau}^{(p_{\mathrm{LO}})}
q \!\left(x^{(i_y)}, \xi_F^2 {Q^2}^{(i_\tau)} \right) +
\nonumber \\
\left(\frac{\alpha_s\left(\xi_R^2 {Q^2}^{(i_\tau)} \right)\,
}{2\pi}
\right)^{p_{\mathrm{NLO}}}
\left[
\left( W_{i_y,i_\tau}^{(p_{\mathrm{NLO}})} + 2\pi \beta_0 p_{\mathrm{LO}} \ln \xi_R^2
\,W_{i_y,i_\tau}^{(p_{\mathrm{LO}})}
\right) q \!\left(x^{(i_y)}, \xi_F^2 {Q^2}^{(i_\tau)} \right)
\right. \\\left.
- \ln \xi_F^2 \,W_{i_y,i_\tau}^{(p_{\mathrm{LO}})}
(P_0\otimes q) \!\left(x^{(i_y)}, \xi_F^2 {Q^2}^{(i_\tau)} \right)
\right] \,, \nonumber\end{aligned}$$ where $\beta_0 = (11 N_c - 2n_f)/(12\pi)$ and $N_c$ ($n_f$) is the number of colours (flavours). Though this formula is given for $x$-space based approach, a similar formula applies for moment-space approaches. Furthermore it is straightforward to extend it to higher perturbative orders.
Representing the weights in the case of two incoming hadrons
------------------------------------------------------------
In hadron-hadron scattering one can use analogous procedures with one more dimension. Besides $Q^2$, the weight grid depends on the momentum fraction of the first ($x_1$) and second ($x_2$) hadron.
In the case of jet production in proton-proton collisions the weights generated by the Monte Carlo program as well as the PDFs can be organised in seven possible initial state combinations of partons: $$\begin{aligned}
\mathrm{gg}: \;\; F^{(0)}(x_{1}, x_{2}; Q^{2}) &=& G_{1}(x_{1})G_{2}(x_{2})\\
\mathrm{qg}: \;\; F^{(1)}(x_{1}, x_{2}; Q^{2}) &=& \left(Q_{1}(x_{1})+
\overline Q_{1}(x_{1})\right) G_{2}(x_{2})\\
\mathrm{gq}: \;\; F^{(2)}(x_{1}, x_{2}; Q^{2}) &=& G_{1}(x_{1})\left(Q_{2}(x_{2})+
\overline Q_{2}(x_{2})\right)\\
\mathrm{qr}: \;\; F^{(3)}(x_{1}, x_{2}; Q^{2}) &=& Q_{1}(x_{1}) Q_{2}(x_{2})
+ \overline Q_{1}(x_{1}) \overline Q_{2}(x_{2}) -D(x_{1}, x_{2})\\
\mathrm{qq}: \;\; F^{(4)}(x_{1}, x_{2}; Q^{2}) &=& D(x_{1}, x_{2})\\
\mathrm{q\bar q}: \;\; F^{(5)}(x_{1}, x_{2}; Q^{2}) &=& \overline D(x_{1}, x_{2})\\
\mathrm{q\bar r}: \;\; F^{(6)}(x_{1}, x_{2}; Q^{2}) &=& Q_{1}(x_{1}) \overline Q_{2}(x_{2})
+ \overline Q_{1}(x_{1}) Q_{2}(x_{2}) -\overline D(x_{1}, x_{2}),\end{aligned}$$ where $g$ denotes gluons, $q$ quarks and $r$ quarks of different flavour $q \neq r$ and we have used the generalized PDFs defined as: $$\begin{aligned}
G_{H}(x) = f_{0/H}(x,Q^{2}), &&
Q_{H}(x) = \sum_{i = 1}^{6} f_{i/H}(x,Q^{2}), \;\;
\overline Q_{H}(x) = \sum_{i = -6}^{-1} f_{i/H}(x,Q^{2}), \nonumber \\
D(x_{1}, x_{2}) &=& \mathop{\sum_{i = -6}^{6}}_{i\neq0} f_{i/H_1}(x_{1},Q^2) f_{i/H_2}(x_{2},Q^{2}), \\
\overline D(x_{1}, x_{2}, \mu^{2}_{F}) &=&
\mathop{\sum_{i = -6}^{6}}_{i\neq0} f_{i/H_1}(x_{1},Q^{2}) f_{-i/H_2}(x_{2},Q^{2}), \nonumber \;\;\end{aligned}$$ where $f_{i/H}$ is the PDF of flavour $i=-6 \dots 6$ for hadron $H$ and $H_1$ ($H_2$) denotes the first or second hadron[^4].
The analogue of eq. \[eq:WfinalxQ\] is then given by: $$\label{eq:WfinalxQ_twohadrons}
W = \sum_p \sum_{l=0}^{6} \sum_{i_{y_1}} \sum_{i_{y_2}} \sum_{i_\tau}
W_{i_{y_1},i_{y_2},i_\tau}^{(p)(l)} \, \left( \frac{\alpha_s\left({Q^2}^{(i_\tau)}\right)}{2\pi}\right)^{p}
F^{(l)}\left(x_1^{(i_{y_1})}, x_2^{(i_{y_1})}, {Q^2}^{(i_\tau)}\right).$$
Including scale depedence in the case of two incoming hadrons
-------------------------------------------------------------
It is again possible to choose arbitrary renormalisation and factorisation scales, specifically for NLO accuracy: $$\begin{aligned}
\label{eq:Wfinalxi_twohadrons}
W(\xi_R, \xi_F) = \sum_{l=0}^{6} \sum_{i_{y_1}} \sum_{i_{y_2}} \sum_{i_\tau}
\left(\frac{\alpha_s\left(\xi_R^2 {Q^2}^{(i_\tau)}\right)\,
}{2\pi}
\right)^{p_{\mathrm{LO}}}
W_{i_{y_1},i_{y_2},i_\tau}^{(p_{\mathrm{LO}})(l)}
F^{(l)}\left(x_1^{(i_{y_1})}, x_2^{(i_{y_1})}, \xi_F^2{Q^2}^{(i_\tau)}\right)
+
\nonumber \\
\left(\frac{\alpha_s\left(\xi_R^2 {Q^2}^{(i_\tau)} \right)\,
}{2\pi}
\right)^{p_{\mathrm{NLO}}}
\left[
\left(
W_{i_{y_1},i_{y_2},i_\tau}^{(p_{\mathrm{NLO}})(l)}
+ 2\pi \beta_0 p_{\mathrm{LO}} \ln \xi_R^2
\,
W_{i_{y_1},i_{y_2},i_\tau}^{(p_{\mathrm{LO}})(l)}
\right)
F^{(l)}\left(x_1^{(i_{y_1})}, x_2^{(i_{y_1})}, \xi_F^2{Q^2}^{(i_\tau)}\right)
\right. \\\left.
- \ln \xi_F^2 \,
W_{i_{y_1},i_{y_2},i_\tau}^{(p_{\mathrm{LO}})(l)}
\left(
F^{(l)}_{q_1 \to P_0\otimes q_1}\left(x_1^{(i_{y_1})}, x_2^{(i_{y_1})},
\xi_F^2{Q^2}^{(i_\tau)}\right) +
F^{(l)}_{q_2 \to P_0\otimes q_2}\left(x_1^{(i_{y_1})}, x_2^{(i_{y_1})},
\xi_F^2{Q^2}^{(i_\tau)}\right)
\right)
\right] \,, \nonumber\end{aligned}$$ where $F^{(l)}_{q_1 \to P_0\otimes q_1}$ is calculated as $F^{(l)}$, but with $q_1$ replaced wtih $P_0 \otimes q_1$, and analogously for $F^{(l)}_{q_2 \to P_0\otimes q_2}$.
Technical implementation
========================
To test the scheme discussed above we use the NLO Monte Carlo program NLOJET++ [@Nagy:2003tz; @*Nagy:2001fj; @*Nagy:2001xb] and the CTEQ6 PDFs [@Pumplin:2002vw]. The grid $W_{i_{y_1},i_{y_2},i_\tau}^{(p)(l)}$ of eq. \[eq:WfinalxQ\_twohadrons\] is filled in a NLOJET++ user module. This module has access to the event weight and parton momenta and it is here that one specifies and calculates the physical observables that are being studied (e.g. jet algorithm).
Having filled the grid we construct the cross-section in a small standalone program which reads the weights from the grid and multiplies them with an arbitrary ${\alpha_s}$ and PDF according to eq. \[eq:WfinalxQ\_twohadrons\]. This program runs very fast (in the order of seconds) and can be called in a PDF fit.
The connection between these two programs is accomplished via a C++ class, which provides methods e.g. for creating and optimising the grid, filling weight events and saving it to disk. The classes are general enough to be extendable for the use with other NLO calculations.
The complete code for the NLOJET++ module, the C++ class and the standalone job is available from the authors. It is still in a development, testing and tuning stage, but help and more ideas are welcome.
The C++ class {#sec:class}
-------------
The main data members of this class are the grids implemented as arrays of three-dimensional ROOT histograms, with each grid point at the bin centers[^5]: $${\rm TH3D[p][l][iobs](x_1,x_2,Q^2)},$$ where the $l$ and $p$ are explained in eq. \[eq:WfinalxQ\_twohadrons\] and $iobs$ denotes the observable bin, e.g. a given $P_T$ range[^6].
The C++ class initialises, stores and fills the grid using the following main methods:
- *Default constructor:* Given the pre-defined kinematic regions of interest, it initializes the grid.
- *Optimizing method:* Since in some bins the weights will be zero over a large kinematic region in $x_1, x_2, Q^2$, the optimising method implements an automated procedure to adapt the grid boundaries for each observable bin. These boundaries are calculated in a first (short) run. In the present implementation, the optimised grid has a fixed number of grid points. Other choices, like a fixed grid spacing, might be implemented in the future.
- *Loading method:* Reads the saved weight grid from a ROOT file
- *Saving method:* Saves the complete grid to a ROOT file, which will be automatically compressed.
The user module for NLOJET++
----------------------------
The user module has to be adapted specifically to the exact definition of the cross-section calculation. If a grid file already exists in the directory where NLOJET++ is started, the grid is not started with the default constructor, but with the optimizing method (see \[sec:class\]). In this way the grid boundaries are optimised for each observable bin. This is necessary to get very fine grid spacings without exceeding the computer memory. The grid is filled at the same place where the standard NLOJET++ histograms are filled. After a certain number of events, the grid is saved in a root-file and the calculation is continued.
The standalone program for constructing the cross-section
---------------------------------------------------------
The standalone program calculates the cross-section in the following way:
1. Load the weight grid from the ROOT file
2. Initialize the PDF interface[^7], load $q(x,Q^2)$ on a helper PDF-grid (to increase the performance)
3. For each observable bin, loop over $i_{y_1},i_{y_2}, i_\tau, l, p$ and calculate $F^{l}(x_1, x_2,Q^2)$ from the appropriate PDFs $q(x,Q^2)$, multiply ${\alpha_s}$ and the weights from the grid and sum over the initial state parton configuration $l$, according to eq. \[eq:WfinalxQ\_twohadrons\].
Results {#sec:results}
=======
We calculate the single inclusive jet cross-section as a function of the jet transverse momentum ($P_T$) for jets within a rapidity of $|y|<0.5$. To define the jets we use the seedless cone jet algorithm as implemented in NLOJET++ using the four-vector recombination scheme and the midpoint algorithm. The cone radius has been put to $R=0.7$, the overlap fraction was set to $f=0.5$. We set the renormalisation and factorization scale to $Q^2=P_{T,max}^2$, where $P_{T,max}$ is the $P_T$ of the highest $P_T$ jet in the required rapidity region[^8].
In our test runs, to be independent from statistical fluctuations (which can be large in particular in the NLO case), we fill in addition to the grid a reference histogram in the standard way according to eq. \[eq:normalint\].
The choice of the grid architecture depends on the required accuracy, on the exact cross-section definition and on the available computer resources. Here, we will just sketch the influence of the grid architecture and the interpolation method on the final result. We will investigate an example where we calculate the inclusive jet cross-section in $N_{\mathrm{obs}} = 100$ bins in the kinematic range $100\, \leq P_T \leq 5000\,\mathrm{GeV}$. In future applications this can serve as guideline for a user to adapt the grid method to his/her specific problem. We believe that the code is transparent and flexible enough to adapt to many applications.
As reference for comparisons of different grid architectures and interpolation methods we use the following:
- *Grid spacing in $y(x)$:* $10^{-5} \leq x_1, x_2 \leq 1.0$ with $N_y=30$
- *Grid spacing in $\tau(Q^2)$:* $100\,\mathrm{GeV} \leq Q \leq 5000\,\mathrm{GeV}$ with $N_\tau=30$
- *Order of interpolation:* $n_y=3,\, n_\tau=3$
The grid boundaries correspond to the user setting for the first run which determines the grid boundaries for each observable bin. In the following we call this grid architecture $30^2$x$30$x$100 (3,3)$. Such a grid takes about $300$ [Mbyte]{} of computer memory. The root-file where the grid is stored has about $50$ [Mbyte]{}.
The result is shown in Fig. \[fig:101obsbins\]a). The reference cross-section is reproduced everywhere to within $0.05\%$. The typical precision is about $0.01\%$. At low and high $P_T$ there is a positive bias of about $0.04\%$. Also shown in Fig. \[fig:101obsbins\]a) are the results obtained with different grid architectures. For a finer $x$ grid ($50^2$x$30$x$100 (3,3)$) the accuracy is further improved (within $0.005\%$) and there is no bias. A finer ($30^2$x$60$x$100 (3,3)$) as well as a coarser ($30^2$x$10$x$100 (3,3)$) binning in $Q^2$ does not improve the precision.
Fig. \[fig:101obsbins\]b) and Fig. \[fig:101obsbins\]c) show for the grid ($30^2$x$30$x$100$) different interpolation methods. With an interpolation of order $n=5$ the precision is $0.01\%$ and the bias at low and high $P_T$ observed for the $n=3$ interpolation disappears. The result is similar to the one obtained with finer $x$-points. Thus by increasing the interpolation order the grid can be kept smaller. An order $n=1$ interpolation gives a systematic negative bias of about $1\%$ becoming even larger towards high $P_T$.
Depending on the available computer resources and the specific problem, the user will have to choose a proper grid architecture. In this context, it is interesting that a very small grid $10^2$x$10$x$100 (5,5)$ that takes only about $10$ [Mbyte]{} computer memory reaches still a precision of $0.5\%$, if an interpolation of order $n=5$ is used (see Fig. \[fig:101obsbins\]d)).
![Ratio between the single inclusive jet cross-section with $100$ $P_T$ bins calculated with the grid technique and the reference cross-section calculated in the standard way. Shown are the standard grid, grids with finer $x$ and $Q^2$ sampling (a) with interpolation of order $1$, $3$ and $5$ (b) (and on a finer scale in c)) and a small grid (d). []{data-label="fig:101obsbins"}](obsbins101.eps "fig:"){width="49.00000%"} ![Ratio between the single inclusive jet cross-section with $100$ $P_T$ bins calculated with the grid technique and the reference cross-section calculated in the standard way. Shown are the standard grid, grids with finer $x$ and $Q^2$ sampling (a) with interpolation of order $1$, $3$ and $5$ (b) (and on a finer scale in c)) and a small grid (d). []{data-label="fig:101obsbins"}](obsbins101_interpol.eps "fig:"){width="49.00000%"} ![Ratio between the single inclusive jet cross-section with $100$ $P_T$ bins calculated with the grid technique and the reference cross-section calculated in the standard way. Shown are the standard grid, grids with finer $x$ and $Q^2$ sampling (a) with interpolation of order $1$, $3$ and $5$ (b) (and on a finer scale in c)) and a small grid (d). []{data-label="fig:101obsbins"}](obsbins101_best.eps "fig:"){width="49.00000%"} ![Ratio between the single inclusive jet cross-section with $100$ $P_T$ bins calculated with the grid technique and the reference cross-section calculated in the standard way. Shown are the standard grid, grids with finer $x$ and $Q^2$ sampling (a) with interpolation of order $1$, $3$ and $5$ (b) (and on a finer scale in c)) and a small grid (d). []{data-label="fig:101obsbins"}](obsbins101_smallgrid.eps "fig:"){width="49.00000%"}
(0,0) ( -450,0)[c)]{} ( -210,0)[d)]{} ( -450,135)[a)]{} ( -210,135)[b)]{}
Conclusions
===========
We have developed a technique to store the perturbative coefficients calculated by an NLO Monte Carlo program on a grid allowing for a-posteriori inclusion of an arbitrary parton density function (PDF) set. We extended a technique that was already successfully used to analyse HERA data to the more demanding case of proton-proton collisions at LHC energies.
The technique can be used to constrain PDF uncertainties, e.g. at high momentum transfers, from data that will be measured at LHC and allows the consistent inclusion of final state observables in global QCD analyses. This will help increase the sensitivity of LHC to find new physics as deviations from the Standard Model predictions.
Even for the large kinematic range for the parton momentum fractions $x_1$ and $x_2$ and of the squared momentum transfer $Q^2$ accessible at LHC, grids of moderate size seem to be sufficient. The single inclusive jet cross-section in the central region $|y|<0.5$ can be calculated with a precision of $0.01\%$ in a realistic example with $100$ bins in the transverse jet energy range $100\, \leq P_T \leq 5000\,\mathrm{GeV}$. In this example, the grid occupies about $300$ [Mbyte]{} computer memory. With smaller grids of order $10$ [Mbyte]{} the reachable accuracy is still $0.5\%$. This is probably sufficient for all practical applications.
Acknowledgment {#acknowledgment .unnumbered}
==============
We would like to thank Z. Nagy, M. H. Seymour, T. Schörner-Sadenius, P. Uwer and M. Wobisch for useful discussions on the grid technique and A. Vogt for discussion on moment-space techniques. We thank Z. Nagy for help and support with NLOJET++. F. Siegert would like to thank CERN for the Summer Student Program.
[^1]: Contribution to the CERN - DESY Workshop 2004/2005, [*HERA and the LHC*]{}.
[^2]: An alternative for the $x$ grid is to use $y = \ln 1/x +
a(1-x)$ with $a$ a parameter that serves to increase the density of points in the large $x$ region.
[^3]: Here, and in the following, renormalisation and factorisation scales have been set equal for simplicity.
[^4]: In the above equation we follow the standard PDG Monte Carlo numbering scheme [@Eidelman:2004wy] where gluons are denoted as $0$, quarks have values from $1$-$6$ and anti-quarks have the corresponding negative values.
[^5]: ROOT histograms are easy to implement, to represent and to manipulate. They are therefore ideal in an early development phase. An additional advantage is the automatic file compression to save space. The overhead of storing some empty bins is largely reduced by optimizing the $x_1$, $x_2$ and $Q^2$ grid boundaries using the NLOJET++ program before final filling. To avoid this residual overhead and to exploit certain symmetries in the grid, a special data class (e.g. a sparse matrix) might be constructed in the future.
[^6]: For the moment we construct a grid for each initial state parton configuration. It will be easy to merge the $qg$ and the $gq$ initial state parton configurations in one grid. In addition, the weights for some of the initial state parton configurations are symmetric in $x_1$ and $x_2$. This could be exploited in future applications to further reduce the grid size.
[^7]: We use the C++ wrapper of the LHAPDF interface [@Whalley:2005nh].
[^8]: Note that beyond LO the $P_{T,max}$ will in general differ from the $P_T$ of the other jets, so when binning an inclusive jet cross section, the $P_T$ of a given jet may not correspond to the renormalisation scale chosen for the event as a whole. For this reason we shall need separate grid dimensions for the jet $P_T$ and for the renormalisation scale. Only in certain moment-space approaches [@Kosower:1997vj] has this requirement so far been efficiently circumvented.
|
---
abstract: 'Conventional intensity cameras recover objects in the direct line-of-sight of the camera, whereas occluded scene parts are considered lost in this process. Non-line-of-sight imaging (NLOS) aims at recovering these occluded objects by analyzing their indirect reflections on visible scene surfaces. Existing NLOS methods temporally probe the indirect light transport to unmix light paths based on their travel time, which mandates specialized instrumentation that suffers from low photon efficiency, high cost, and mechanical scanning. We depart from temporal probing and demonstrate steady-state NLOS imaging using conventional intensity sensors and continuous illumination. Instead of assuming perfectly isotropic scattering, the proposed method exploits directionality in the hidden surface reflectance, resulting in (small) spatial variation of their indirect reflections for varying illumination. To tackle the shape-dependence of these variations, we propose a trainable architecture which learns to map diffuse indirect reflections to scene reflectance using only synthetic training data. Relying on consumer color image sensors, with high fill factor, high quantum efficiency and low read-out noise, we demonstrate high-fidelity color NLOS imaging for scene configurations tackled before with picosecond time resolution.'
author:
- |
Wenzheng Chen$^{1,2}$[^1] Simon Daneau$^{1,3}$ Fahim Mannan$^{1}$ Felix Heide$^{1,4}$\
^1^Algolux ^2^University of Toronto ^3^Université de Montréal ^4^Princeton University
title: 'Steady-state Non-Line-of-Sight Imaging'
---
Introduction
============
Related Work
============
Image Formation Model
=====================
Inverse Indirect Transport for Planar Scenes {#sec:analytic}
=============================================
Learning Inverse Indirect Illumination
======================================
Training Datasets
=================
Evaluation {#sec:eval}
==========
Conclusion
==========
Acknowledgements
================
The authors thank Colin Brosseau for many fruitful discussions and assisting with the experiments. We thank the Vector Institute for supporting Wenzheng Chen.
[^1]: The majority of this work was done while interning at Algolux.
|
---
abstract: 'The IVOA Provenance Data model defines entities, agents and activities as container classes to describe the provenance of datasets, with the executed tasks and responsibilities attached to agents. It also provides a set of classes to describe the activities type and their configuration template, as well as the configuration applied effectively during the execution of a task. Here we highlight lessons learned in the implementation of the CDS ProvHiPS service distributing provenance metadata for the HST HiPS data collections, and for the HST archive original images used to produce the HiPS tiles. ProvHiPS is based on the ProvTAP protocol, the emerging TAP standard for distributing provenance metadata. ProvTAP queries may rapidly become very complex. Various graph representation strategies, including ad hoc solutions, triplestore and SQL CTE have been considered and are discussed shortly.'
author:
- 'Mireille Louys,$^{1,2}$ François Bonnarel,$^2$ Daniel Durand,$^3$ and Anais Egner$^4$'
bibliography:
- 'P2-6.bib'
title: Implementation feedback of the IVOA Provenance data model
---
Introduction: IVOA provenance data model and provenance of CDS HiPS
===================================================================
Datasets used in astronomy are generally the results of a flow of observation and processing steps. Information on this process is generally called “provenance” of the dataset and is stored in various formats and logical organizations. This makes in general the provenance information difficult to compare and use interoperably among different data collections. That’s the main reason why IVOA developed an astronomy-oriented provenance data model during the last years. This formalization not only allows traceability of products but also acknowledgment and contact information, quality and reliability assessment and discovery of datasets by provenance details. At the time of writing the Provenance data model specification is an IVOA proposed recommendation [@2019ProvRec]. HiPS [@fernique2015] defines a new way of organizing image, cube and catalogue data in an all sky and hierarchical way based on HealPIX tessellation of the sky. HiPS datasets are generated from image data collections or catalogues which have their own history. The ProvHiPS service developed at CDS aims at providing provenance information for HiPS stored at CDS back to original raw data when available.
HiPS datasets for HST image collections
=======================================
HiPS are made of hierarchies of tiles containing pixelized information at a given HealPIX order. In the case of image HiPS datasets each tile is generated from a small subset of the original image collection intersecting with the tile. The HiPS format stores inter-nally the progenitor information for each tile in the HiPS tree. The CDS data center publishes HiPS at various wavelengths for the HST image collections. They have been produced from HST drizzled images in collaboration with CADC astronomer Daniel Durand. HST data collections are stored and retrievable from the CADC HST archive through IVOA DAL services. The drizzled images have their own history: they are produced from sets of calibrated images closely related on the sky by a specific type of co-addition called “drizzling”.
A rich tree of related data is then potentially available. We have browsed HiPS tiles metadata and FITS headers of the HST images to extract features relevant in terms of the IVOA Provenance data model to trace historical information and map it into the ProvTAP tables. This resulted in a database containing ten of thousands of entities and activities. Dozen of descriptions of the various kinds of entities have also been produced as well as ten of thousands of configuration parameters for the drizzling and HiPS generation activities.
ProvHiPS implementation
=======================
In order to make such information available we implemented a ProvTAP[^1] service called ProvHiPS. ProvTAP specification is currently an IVOA working draft describing how to map the provenance data model in a TAP service. The heart of it is the Prov-TAP TAP SCHEMA definition providing the list of tables and columns required for storing the provenance metadata information and mapping respectively the classes and attributes of the model. Tables and columns come with datatype, unit, ucd, and utypes consistent with the model. By default a ProvTAP service is queriable via the ADQL language defined in IVOA and provides results in VOTable format.
The CDS ProvHiPS service implements a database containing the provenance information sketched out in the workflow scenario presented in Fig \[fig:workflow\].
Fig. \[fig:query\] shows a 13-joins query tracing the provenance of a single HiPS V HST tile around the target NGC104. The query response is presented in Fig. \[fig:qresp\] as displayed with the TAPHandle application [@2014ASPC..485...15M]. Some drizzled and calibrated images are visualized in Aladin via SAMP messaging as well.
Fig. \[fig:desc\] shows the activity description associated with one of the previous calibration activity. This Activity description provides interoperable typing of the calibration activities as well as a link to the software documentation. More sophisticated user scenarios may include retrieving “siblings” of a given dataset entity using various depths or selecting datasets sharing the same creator agent or generated with similar parameters. The number of joins needed to traverse the provenance graph may increase tremendously. That’s the reason why we experimented various ways of representing and querying graphs on top of relational databases. Former tests with a triplestore architecture has shown promising results [@2019ASPC..523...329]. As published in [@P2_15_adassxxix] and proposed by M.Nullmeier[^2], graphical or Common Table Expressions (CTE) techniques to navigate through graph connections on top of the RDBMS are new solutions to consider. We plan to add such layers on top of our service to improve user-friendliness.
Conclusion
==========
Despite the “complex query” issue, the ProvTAP implementation of ProvHiPS demonstrates it is feasible to map information stored in FITS headers of homogeneous image collections or HiPS metadata into the IVOA PROV DM profile. The scalability of the database allows coping with very large data collections. Retrieval of multiple steps pipelines is easy as long as the appropriate ADQL queries are provided.
We thank the CDS intership program for supporting A. Egner. This work has been partly supported by the ESCAPE project (the European Science Cluster of Astronomy and Particle Physics ESFRI Research Infrastructures) funded by the EU Horizon 2020 research and innovation program under the Grant Agreement n.824064, and also the ASTERICS project under Grant Agreement n.653477.
[^1]: https://wiki.ivoa.net/internal/IVOA/ObservationProvenanceDataModel/ProvTAP.pdf
[^2]: https://www.asterics2020.eu/dokuwiki/lib/exe/fetch.php?media=open:wp4:nullmeier\_tf5\_prov\_custom\_adql.pdf
|
---
abstract: 'We show that the governing equations for two-dimensional gravity water waves with constant non-zero vorticity have a nearly-Hamiltonian structure, which becomes Hamiltonian for steady waves.'
address:
- 'Trinity College, School of Mathematics, Dublin 2, Ireland'
- 'Trinity College, School of Mathematics, Dublin 2, Ireland'
- 'Trinity College, School of Mathematics, Dublin 2, Ireland'
author:
- 'Adrian Constantin, Rossen I. Ivanov and Emil M. Prodanov'
title: 'Nearly-Hamiltonian structure for water waves with constant vorticity'
---
[*Key Words*]{}: Water waves, constant vorticity, Hamiltonian formulation.
[*AMS Subject Classification (2000)*]{}: 35Q35, 37K05, 76B15.
Introduction
============
The mathematical study of water waves was initiated within the framework of linear theory with the work of Airy, Stokes, and their contemporaries in the nineteenth century. Periodic two-dimensional water waves are of special interest since the typical water waves propagating on the surface of the sea (or on a river or lake) present these features. Stokes [@St] noticed that actual water wave characteristics deviate significantly from the predictions of linear theory. This started an extensive study of the nonlinear governing equations for water waves.
A celebrated development in water-wave theory was the discovery by Zakharov [@Z] that the governing equations for two-dimensional irrotational gravity water waves have a Hamiltonian structure - see the discussions in [@BO; @C; @CG]. The aim of this paper is to present a nearly-Hamiltonian formulation for two-dimensional gravity water waves with constant vorticity. For irrotational flows (zero vorticity) we recover Zakharov’s result. Moreover, if we restrict our attention to steady waves, a Hamiltonian structure emerges; we refer to [@GT] for an in-depth discussion of the Hamiltonian structure of steady irrotational water waves. While Zakharov’s ideas were generalized in various directions by several authors - see [@Co0; @KS] for a survey of contributions in this direction, the elegance and simplicity of the nearly-Hamiltonian formulation for flows of constant vorticity makes it conceivable that it might be instrumental in deriving qualitative results for such flows. Related to this possibility, recently [@CS1] stability results for steady water waves with vorticities that depend montonically on the depth were derived from the variational formulation provided in [@CSS]. The reason for considering water waves with constant vorticity is twofold. Firstly, by the Kelvin circulation theorem [@J; @L] a two-dimensional water flow that is initially of constant vorticity remains so at later times and therefore the restriction to such flows is justifiable. Secondly, as already pointed out, the elegance of the nearly-Hamiltonian formulation valid within this setting is mathematically attractive. From the physical point of view we notice that while irrotational flows are appropriate for waves propagating into a region of still water [@L], water waves with vorticity describe wave-current interactions - see the discussions in [@Co; @SCJ; @Th]. Tidal flows are the most prominent example of water flows with constant vorticity [@DP].
Preliminaries
=============
To describe two-dimensional periodic water waves it suffices to consider a cross section of the flow that is perpendicular to the crest line. Choose Cartesian coordinates $(x,y)$ with the $y$-axis pointing vertically upwards and the $x$-axis being the direction of wave propagation. Let $(u(t,x,y),\,v(t,x,y))$ be the velocity field of the flow, let $\{y=0\}$ be the flat bed, and let $\{y=\eta(t,x)\}$ be the water’s free surface.
For gravity water waves the restoring force acting on the water’s free surface is gravity and the effects of surface tension are neglected. Assuming the water density to be constant ($\rho=1$) - this is physically reasonable cf. [@L], we obtain the equation of mass conservation $$u_x+v_y=0.$$ Appropriate for gravity waves is also the assumption of inviscid flow [@L], so that the equation of motion is Euler’s equation $$\begin{cases}
u_t+uu_x+vu_y = -P_x, \\
v_t+uv_x+vv_y = -P_y -g,
\end{cases}$$ where $P(t,x,y)$ is the pressure and $g$ is the gravitational constant of acceleration. The free surface decouples the motion of the water from that of the air so that $$P=P_{atm}\qquad \hbox{on}\quad y=\eta(t,x),$$ must hold, where $P_{atm}$ is the atmospheric pressure [@J]. Since the same particles always form the free surface, we have $$v=\eta_t+u\eta_x\qquad\hbox{on}\quad y=\eta(t,x).$$ On the flat bed the boundary condition $$v=0\qquad\hbox{on}\quad y=0,$$ expresses the fact that water cannot permeate the rigid bed $y=0$. The governing equations for periodic two-dimensional gravity water waves propagating over a flat bed are (1)-(5), with the specification that the periodicity is reflected in the fact that all functions $u,\, v,\,P,\,\eta$ exhibit a periodic dependence in the $x$-variable of, say, period $L>0$. Other than the nonlinear character of the equations, the main difficulty in their analysis lies in the fact that we deal with a free-boundary value problem: the free surface $y=\eta(t,x)$ is not known [*a priori*]{}. Throughout this paper we consider flows of constant vorticity, that is, throughout the fluid the vorticity $$\omega=v_x-u_y$$ is constant ($\omega \in {\mathbb R}$) throughout the fluid domain $$\Omega(t)=\{(x,y) \in {\mathbb R}^2:\ 0<x<L,\ 0<y<\eta(t,x)\},$$ the free surface of which is given by the graph $$S(t)=\{(x,y) \in {\mathbb R}^2:\ 0<x<L,\ y=\eta(t,x)\}$$ restricted to a period cell. Furthermore, we require that $$\int_0^L u(t,x,0)\,dx=0,\qquad t \ge 0.$$ The relevance of this last condition is explained in Section 3.
The nearly-Hamiltonian formulation
==================================
For two-dimensional flows the incompressibility condition (1) ensures the existence of a stream function $\psi(t,x,y)$ determined up to an additive term that depends solely on time by $$u=\psi_y,\quad v=-\psi_x.$$ Since (5) becomes $\psi_x(t,x,0)=0$, we can determine $\psi$ uniquely by setting $\psi=0$ on the flat bed $y=0$, that is, we set $$\psi(t,x_0,y_0)=\int_{0}^{y_0} u(t,x_0,y)\,dy\quad\hbox{for}\quad (x_0,y_0) \in \Omega(t).$$ This explicit formula shows that $\psi$ is $x$-periodic with period $L$. In terms of the stream function, the vorticity $\omega$ is determined from (6) by $$\Delta \psi=-\omega\quad\hbox{in}\quad \Omega(t).$$ Let us now introduce the (generalized) velocity potential $\v(t,x,y)$ via $$u=\v_x-\omega y,\quad v=\v_y.$$ Notice that this is not the standard reduction by the Weyl-Hodge decomposition (see [@CDG; @W0]) since the vector field $W=(-\omega y,\,0)$ is divergence free but does not satisfy the boundary condition $W\cdot n=0$ on a free surface that is not flat ($n$ being the outward normal to the boundary). The value of the (generalized) potential $\v$ at $(t,x_0,y_0)$ can be determined by integrating $\v_x$ along the horizontal segment with endpoints $(0,0)$ and $(x_0,0)$, and subsequently $\v_y$ along the vertical segment joining $(x_0,0)$ to $(x_0,y_0) \in \Omega(t)$: $$\begin{aligned}
\v(t,x_0,y_0) &=& \v(t,0,0)+\int_0^{x_0}\v_x(t,x,0)\,dx+\int_{0}^{y_0} \v_y(t,x_0,y)\,dy \\
&=& \v(t,0,0)+\int_0^{x_0}u(t,x,0)\,dx+\int_{0}^{y_0} v(t,x_0,y)\,dy.\end{aligned}$$ The potential $\v$ is a harmonic function since $$\Delta\v=0\quad\hbox{in}\quad \Omega(t).$$ Notice that, independently of the additive time-dependent term up to which $\v$ is uniquely determined by (11), we have by the above explicit formula for $\v$ that $$\v(t,x_0+L,y_0)-\v(t,x_0,y_0)=\int_{x_0}^{x_0+L} u(t,x,0)\,dx,
\qquad (x_0,y_0) \in \Omega(t).$$ In view of the $x$-periodicity of the function $u$, we see that the right-hand side equals $\displaystyle\int_0^L u(t,x,0)\,dx$. At this point some results on steady water waves (water waves for which the free surface $\eta$, the pressure $P$ and the velocity field $(u,v)$ exhibit an $(x,t)$-dependence of the form $x-ct$, where $c \neq 0$ is the wave speed - that is, in a frame moving at speed $c$ these waves are stationary) are of relevance. In the irrotational case, the existence of steady waves of large amplitude (that is, waves that are not small perturbations of a flat surface) satisfying $\displaystyle\int_0^L u(t,x,0)\,dx=0$ was established in [@AT] - see also the discussions in [@Co2; @T]. For these waves, called [*Stokes waves*]{}, we therefore have that the velocity potential $\v$ is $L$-periodic in the $x$-variable. On the other hand, for flows of constant vorticity $\omega \in {\mathbb R}$, there are steady waves of large amplitude for which $\displaystyle\int_0^L u(t,x,0)\,dx >0$ cf. [@CS0; @CS]. For these waves therefore the (generalized) velocity potential $\v$ is not periodic in the $x$-variable. Thus, while $\v_x$ and $\v_y$ are both $x$-periodic with period $L$, the (generalized) potential $\v$ is not necessarily $L$-periodic in the $x$-variable. The above discussion shows that (7) is the necessary and sufficient condition for $\v$ to be $L$-periodic in the $x$-variable. For steady waves, the relation (7) means that the wave speed is defined as the mean velocity in the moving frame of reference in which the wave is stationary. Imposing (7) for the large-amplitude steady waves with vorticity studied in [@CS; @CE; @CSS; @CS1; @E; @W] simply means that along the continuum of waves constructed in [@CS] the wave speed is not fixed [*a priori*]{}, but varies according to (7). As stated in the Introduction, throughout this paper we consider only flows satisfying (7) so that $\v$ is $L$-periodic in the $x$-variable.
In terms of the functions $\v$ and $\psi$, we can recast the Euler equation (2) in the form $$\nabla \left[ \varphi_t + \frac{1}{2} \,|\nabla \psi|^2 + P + \omega \psi + g y \right] = 0.$$ Thus at each instant $t \ge 0$, $$\displaystyle\left[ \varphi_t +
\frac{1}{2} \,|\nabla \psi|^2 + P + \omega \psi + g y \right]\quad\hbox{is constant throughout}\quad
\Omega(t).$$ This is the generalization for flows of constant vorticity of Bernoulli’s law [@S] for irrotational flows ($\omega=0$). In view of (3), we deduce that $$\displaystyle\left[ \v_t +
\frac{1}{2} \,|\nabla \psi|^2 +\omega \psi + g \eta \right]\quad\hbox{is constant on the
free surface}\quad S(t).$$ Since $\v$ is uniquely determined by (11) up to an arbitrary additive term that is solely time-dependent, we use this freedom to absorb into the definition of $\v$ a suitable time-dependent term so that $$\v_t+\frac{1}{2} \,|\nabla \psi|^2 +\omega \psi + g \eta =0\quad\hbox{on}\quad S(t).$$
Let $\xi$ denote the evaluation of $\v$ at the free surface, $$\xi(t,x)=\v(t,x,\eta(t,x)),\qquad t \ge 0, \ x \in [0,L].$$ Given the constant vorticity $\omega$, the functions $\xi$ and $\eta$, taken to be smooth and $L$-periodic in the $X$-variable, completely determine the motion. Indeed, the function $x \mapsto \eta(t,x)$ fixes the fluid domain $\Omega(t)$, and $\xi(t,\cdot)$ is the appropriate boundary data for a linear elliptic problem of mixed type that determines $\v(t,\cdot,\cdot)$ at time $t \ge 0$. At any given time $t \ge 0$, fix $\eta$ and $\xi$, and let $\v$ be the unique solution of the boundary-value problem $$\begin{cases}
\Delta\v=0\quad\hbox{in}\quad \Omega(t),\\
\v=\xi\quad\hbox{on}\quad S(t),\\
\v_y=0\quad\hbox{on}\quad y=0.
\end{cases}$$ that is $L$-periodic in the $x$-variable. Knowing $\v$, we determine the velocity field $(u,v)$ from (11), the stream function $\psi$ from (10), and the pressure $P$ from Bernoulli’s law (13).
The total energy of the wave motion in a period cell is given by $$H=\iint_{\Omega(t)} \Big\{\frac{u^2+v^2}{2}+gy \Big\}\,dydx.$$ In the above expression, the first term represents the kinetic energy (energy of motion), while the second term is the gravitational potential energy (energy of position). We now show that $H$ is completely determined by the functions $\xi$ and $\eta$. In order to do this, we introduce the [*Hilbert transform*]{}, ${\mathcal T}(\eta)$, defined as follows. At any given time $t \ge 0$, fix $\eta$ and $\xi$, and let $\v$ be the unique solution of the boundary-value problem (16). Since (8) and (11) ensure that $\displaystyle(\psi+\frac{\omega}{2}\, y^2)$ is the harmonic conjugate to $\v$, the Hilbert transform ${\mathcal T}(\eta)\,\xi$ of $\xi$ is given by $$\Big({\mathcal T}(\eta)\,\xi\Big)(x)=\chi(t,x)+\frac{\omega}{2}\,\eta^2(t,x),\qquad x \in [0,L],$$ where $$\chi(t,x)=\psi(t,x,\eta(t,x)),\qquad t \ge 0, \ x \in [0,L].$$ The equations (18)-(19) show that $\chi$ is completely determined by $\eta$ and $\xi$ via $$\chi={\mathcal T}(\eta)\,\xi-\frac{\omega}{2}\,\eta^2.$$ Let $$n=\frac{1}{\sqrt{1+\eta_x^2}}\,\begin{pmatrix}
-\eta_x \\
1
\end{pmatrix},$$ be the outward normal to the free surface $y=\eta(t,x)$. With $dl=\sqrt{1+\eta_x^2}\,dx$, using Green’s identity, we get $$\begin{aligned}
H &=& \frac{1}{2} \iint_{\Omega(t)} |\nabla\psi|^2dydx
+ \frac{1}{2} \int_0^L g \,\eta^2\,dx \nonumber \\
&=& \frac{1}{2} \int_{S(t)} \psi\,[-\psi_x\eta_x+\psi_y]\,dx + \frac{\omega}{2} \iint_{\Omega(t)} \psi\,dydx
+ \frac{1}{2} \int_0^L g \,\eta^2\,dx \end{aligned}$$ in view of (10) and the fact that $\psi=0$ on $y=0$. Another application of Green’s identity for the functions $\psi$ and $y^2$ yields $$\begin{aligned}
&&\iint_{\Omega(t)} \Big(-\,\omega\,y^2\,-\,2 \psi\Big)\,dydx =
\iint_{\Omega(t)} \Big(y^2\,\Delta \psi\,- \psi\,\Delta(y^2)\Big)\,dydx \\
&&\quad = \int_{S(t)} \Big(y^2
\,\frac{\partial \psi}{\partial n} -\frac{\partial y^2}{\partial n}\,\psi\Big)\,dl =
\int_0^L (-\eta_x\psi_x+\psi_y)\,\eta^2\,dx -2\int_0^L \chi\,\eta\,dx\end{aligned}$$ if we take into account (10) and (19). Therefore $$\iint_{\Omega(t)} \psi\,dydx = -\frac{\omega}{6} \int_0^L \eta^3\,dx -
\frac{1}{2} \int_0^L (-\eta_x\psi_x+\psi_y)\,\eta^2\,dx +\int_0^L \chi\,\eta\,dx.$$ Taking into account (8), (11), we further transform the expression (21) for $H$ into $$\frac{1}{2} \int_{S(t)} (\psi-\frac{\omega}{2}\,y^2)\,[\v_y\eta_x+\v_x-\omega\,y]\,dx
+ \frac{1}{2} \int_0^L (g -\frac{\omega^2}{6}\,\eta)\,\eta^2\,dx+\frac{\omega}{2}\int_0^L \chi\,\eta\,dx$$ in view of (22). Since $$\xi_x=\v_x+\v_y\eta_x,\qquad x \in [0,L],$$ and using (18), we obtain $$\begin{aligned}
H(\eta,\xi) &=& \frac{1}{2} \int_0^L \xi_x\,\cdot{\mathcal T}(\eta)\,\xi \,dx
+\, \frac{1}{2} \int_0^L g \,\eta^2\,dx \nonumber \\
&&\qquad -\, \frac{\omega}{2} \int_0^L \xi_x\,\eta^2\,dx
+ \frac{\omega^2}{6}\int_0^L \eta^3\,dx.\end{aligned}$$ We now present the main result of this paper.
[**Theorem 1**]{} [*The governing equations for periodic two-dimensional gravity water waves of constant vorticity $\omega$ are equivalent to the nearly-Hamiltonian system $$\begin{cases}
\dot{\eta} = \displaystyle\frac{\delta H}{\delta \xi}, \vspace{0.3cm}\\
\dot{\xi} = -\,\displaystyle\frac{\delta H}{\delta \eta} - \omega \,\chi
\end{cases}$$ with $y=\eta$ being the free surface, $\xi$ being the evaluation of the (generalized) velocity potential on the free surface, $H=H(\eta,\xi)$ being the total energy of the motion, and $$\chi={\mathcal T}(\eta)\,\xi -\displaystyle\frac{\omega}{2}\,\eta^2$$ being the evaluation of the stream function on the free surface.*]{}
[**Remark**]{} For irrotational flows ($\omega=0$) the system $(24)$ is Hamiltonian: we recover Zakharov’s result [@Z].$\hfill\Box$
Before proceeding with the proof of Theorem 1, let us review briefly the concept of an infinite-dimensional Hamiltonian system - see also [@CG; @O]. An [*infinite-dimensional Hamiltonian system*]{} is a system of partial differential equations of the form $$f_t=J\,\hbox{grad}\,H(f),$$ where $f(t)$ describes a path in a Hilbert space endowed with an inner product $\langle \cdot,
\cdot \rangle$, the [*Hamiltonian function*]{} $H: {\mathcal D} \to {\mathbb R}$ being defined on a dense subspace of the Hilbert space, and with $J$ being a skew-adjoint operator. The gradient in (25) is taken with respect to the inner product $\langle \cdot, \cdot \rangle$ on the Hilbert space. If the operator $J$ is invertible, this set-up defines a [*symplectic structure*]{} on the Hilbert space. The system (24) is nearly-Hamiltonian: one can view $\omega$ as a parameter measuring the deviation of (24) from a Hamiltonian structure of the form (25) with $f=\begin{pmatrix} \eta \\
\xi
\end{pmatrix}$ on the Hilbert space $L^2[0,L] \times L^2[0,L]$, with operator $$J=\begin{pmatrix}
0 & 1\\
-1 & 0
\end{pmatrix},$$ the Hamiltonian function $H$ being given by (23) with $\eta$ and $\xi$ in the dense subspace ${\mathcal D} \subset L^2[0,L]$ of smooth $L$-periodic functions.
[*Proof of Theorem 1.*]{} Let us first compute the $\xi$-gradient of $H$, $\displaystyle\frac{\delta H}{\delta\xi}$. We vary $\xi$ and keep $\eta$ fixed. If $\theta$ is a harmonic function in $\Omega(t)$, $L$-periodic in the $x$-variable and with $\theta_y=0$ on $y=0$, let $\theta_0$ be the evaluation of $\theta$ on $S(t)$. If $\Psi$ is the harmonic conjugate of $\theta$ with $\Psi=0$ on $y=0$, from (18) we get $${\mathcal T}(\eta)\,[\xi]=\chi+\frac{\omega}{2}\,\eta^2,\qquad {\mathcal T}(\eta)\,[\xi+\varepsilon\,\theta_0]=\chi+\varepsilon\,\zeta+\frac{\omega}{2}\,\eta^2,$$ denoting by $\zeta$ the evaluation of $\Psi$ on $S(t)$. Therefore $$\begin{aligned}
\langle \frac{\delta H}{\delta\xi},\, \theta_0 \rangle &=& \lim_{\varepsilon \to 0}
\frac{H(\eta,\xi+\varepsilon \theta_0)-H(\eta,\xi)}{\varepsilon} \nonumber \\
&=& -\frac{1}{2} \int_0^L \Big\{ \zeta_x\,\xi + (\chi_x-\omega\,\eta\,\eta_x)\,\theta_0\,)\,dx \end{aligned}$$ in view of (23) and using the periodicity. Since $$\theta_x=\Psi_y,\qquad \theta_y=-\Psi_x\quad\hbox{throughout}\quad \Omega(t),$$ we obtain $$\begin{aligned}
&&\int_0^L \zeta_x\,\xi\,dx = \int_{S(t)} (\Psi_x+\Psi_y\eta_x)\,\v\,dx =
\int_{S(t)} (-\theta_y+\theta_x\,\eta_x)\,\v\,dx \nonumber \\
&&\quad = -\int_{S(t)} \frac{\partial \theta}{\partial n}\,\v\,dl
= -\int_{S(t)} \theta\,\frac{\partial\v}{\partial n}\,dl =\int_{S(t)} \theta_0 \,(\v_x\eta_x-\v_y)\,dx \end{aligned}$$ using in the next to last step Green’s identity for the harmonic functions $\v$ and $\theta$, which satisfy $\theta_y=\v_y=0$ on $y=0$. From (26)-(27) we infer that $$\langle \frac{\delta H}{\delta\xi},\, \theta_0 \rangle = -\frac{1}{2} \int_{S(t)}
( \v_x\eta_x-\v_y+\chi_x-\omega\,\eta\eta_x )\,\theta_0\,dx= \int_0^L (v-u\,\eta_x)\,\theta_0\,dx,$$ if we take into account (8), (11) and (19). Thus $$\frac{\delta H}{\delta\xi}=v-u\,\eta_x.$$ Since $\dot{\eta}=\eta_t$, we can recast (4) as $$\dot{\eta}=\frac{\delta H}{\delta\xi}.$$
To recover the remaining part of the system (24), we have to compute $\displaystyle\frac{\delta H}{\delta \eta}$. To do this, instead of working with the expression (23), it is simpler to compute the variation in (17). This calculation is not straigthforward due to the fact that there is an implicit nonlinear dependence of $\v$ upon $\eta$. As an example of the intricacies of the calculation, notice that, contrary to a possible first impression, the Hilbert transform ${\mathcal T}(\eta)$ is not skew-adjoint if the surface $y=\eta(t,x)$ is not flat (see [@C; @CG] for a discussion).
Let $$\begin{cases}
\xi_1(t,x)=\v_x(t,x,\eta(t,x)),\\
\xi_2(t,y)=\v_y(t,x,\eta(t,x)),
\end{cases}$$ be the evaluations of the partial derivatives of the potential $\v$ on the free surface $y=\eta(t,x)$. Notice that $\xi_1$ is not to be confused with the function $\xi_x$ since $$\xi_x=\xi_1+\eta_x\xi_2.$$
In terms of the potential $\v$, we can rewrite (17) as $$\begin{aligned}
H(\eta,\v)= - \i\j \, y \, \varphi_x \,dydx &+&
\frac{1}{2} \i\j \omega^2 y^2 \,dydx \nonumber \\
&+& \frac{1}{2} \i\j |\nabla \v|^2 \,dy + \frac{1}{2} \i\j\, g \, \eta^2 \, dx.\end{aligned}$$ We now vary the function $\eta$ describing the free surface by $\d\eta$, keeping $\xi$ fixed. Since $\v$ and $(\psi+\displaystyle{\omega \over 2}\,y)$ are harmonic conjugate functions, by analytic continuation [@B] the function $\v$ has a harmonic extension across the boundary. Varying the domain $\Omega(t)$ to $\Omega_\varepsilon(t)$ by keeping $\xi(t,\cdot)$ fixed means that we solve instead of (16) the problem $$\begin{cases}
\Delta\v_\varepsilon=0\quad\hbox{in}\quad \Omega_\varepsilon(t),\\
\v_\varepsilon=\xi\quad\hbox{on}\quad S_\varepsilon(t),\\
\partial_y\,\v_\varepsilon=0\quad\hbox{on}\quad y=0,
\end{cases}$$ where $$\begin{aligned}
\Omega_\varepsilon(t) &=& \{(x,y) \in {\mathbb R}^2:\ 0<x<L,\ 0<y<\eta(t,x)+\varepsilon\,(\delta\eta)(t,x)\},\\
S_\varepsilon(t) &=& \{(x,y) \in {\mathbb R}^2:\ 0<x<L,\ y=\eta(t,x)+\varepsilon\,(\delta\eta)(t,x)\}.\end{aligned}$$ Since $$\v_\varepsilon(t,x,\eta+\varepsilon\,\delta\eta)=\v(t,x,\eta)=\xi(t,x)$$ we deduce that $$\begin{aligned}
{\v_\varepsilon(t,x,\eta) -\v(t,x,\eta) \over \varepsilon} &=& {\v_\varepsilon(t,x,\eta)-\v_\varepsilon(t,x,\eta+
\varepsilon\,\delta\eta) \over \varepsilon} \\
&=& -\,\delta\eta\,\cdot\, \partial_y\,\v_\varepsilon(t,x,\eta) + O(\varepsilon) \\
&\to& - \,\delta\eta\,\cdot \v_y(t,x,\eta)\quad\hbox{as}\quad \varepsilon \to 0.\end{aligned}$$ Therefore, if $[\d\v]^\#$ is the evaluation of the variation $\d\v$ of $\v$ on $y=\eta(t,x)$, we showed that $$[\d\v]^\#=-\,\xi_2\,\d\eta.$$ Other than a variation $\delta\v$ of $\v$, we have the variation $$\begin{aligned}
\d H & = & - \i \j \omega \, y \, \delta \v_x \, dydx
- \i \omega \, \eta \, \xi_1 \, \delta \eta \, dx
+ \frac{1}{2} \i \omega^2 \eta^2 \, \delta \eta \, dx
\nonumber \\
&&\quad + \i \j (\nabla \varphi) \cdot \nabla \delta \varphi \,dydx +
\frac{1}{2} \i (\xi_1^2+\xi_2^2) \, \delta \eta \, dx + \i g \, \eta \, \delta \eta \, dx \end{aligned}$$ of $H$, since $\d\nabla\v=\nabla\d\v$. Using the formula $$\partial_x \, \j F(x, y) \, dy \, = \j F_x(x, y) \, dy \, + \, F[x, \eta] \,\, \eta_x \, ,$$ the first term on the right-hand side of (31) can be rewritten as $$\begin{aligned}
\i \j \omega \, y \, \delta \varphi_x \, dydx &=& \i \left[ \j \omega \, y \, \delta \varphi \, dy
\right]_x \, dx - \i \omega \, \eta \, [\d\v]^\# \, \eta_x \,dx \nonumber \\
&=& -\,\i \omega\,\eta\,[\d\v]^\#\,\eta_x\,dx=\omega\int_0^L \eta\eta_x\xi_2\d\eta\,dx\end{aligned}$$ by (30) and periodicity. Combining (31)-(32), we obtain $$\begin{aligned}
\delta H &=& \i \Big(- \omega \, \eta \, \eta_x \, \xi_2 \, - \, \omega \, \eta \, \xi_1 \, + \,
\frac{1}{2} \,\omega^2\, \eta^2
+ \, \, \frac{1}{2} (\xi_1^2 +\xi_2^2) \, + \, g \, \eta \Big)\, \delta \eta \, dx \nonumber \\
&&\qquad + \i \j (\nabla \varphi) \cdot \nabla \delta \varphi \,\, dydx \, .\end{aligned}$$ Since $\v$ is harmonic in $\Omega(t)$, applying Green’s identity to the pair of functions $\v$ and $\d\v$, we infer that the last integral in the above expression equals $$\int_{\partial\Omega(t)} \d\v\,\frac{\partial\v}{\partial n}\,dl=-\,
\int_0^L \xi_2\,[\xi_2-\xi_1\,\eta_x]\,\d\eta\,dx$$ in view of (5) and (11), respectively (28) and (30). Therefore (33) becomes $$\d H=\int_0^L \Big( -\omega\,\eta\eta_x\xi_2-\omega\,\eta\xi_1+
\frac{1}{2}\,\omega^2\eta^2+\frac{\xi_1^2+\xi_2^2}{2}+g\,\eta-\xi_2^2
+\xi_1\xi_2\eta_x\Big)\,\d\eta\,dx.$$ Thus $$\frac{\d H}{\d\eta}=-\omega\,\eta\eta_x\xi_2-\omega\,\eta\xi_1+
\frac{1}{2}\,\omega^2\eta^2+\frac{\xi_1^2+\xi_2^2}{2}+g\,\eta-\xi_2^2
+\xi_1\xi_2\eta_x.$$ Combining (14) with (8), (11), (19), and (28), we obtain $$\v_t+\frac{\xi_1^2+\xi_2^2}{2}-\omega\eta\xi_1+\frac{1}{2}\,\omega^2\eta^2
+\omega\,\chi+g\eta=0 \quad\hbox{on}\quad y=\eta(t,x),$$ so that (34) becomes $$\frac{\d H}{\d \eta}=-\v_t-\omega\eta\eta_x\xi_2-\xi_2^2+\eta_x\xi_1\xi_2-\omega\chi.$$ Differentiating (15) with respect to $t$, we get $$\xi_t=\v_t+\xi_2\eta_t\quad\hbox{on}\quad y=\eta(t,x),$$ in view of (28). Furthermore, (4), (11) and (28) yield $$\xi_2=\eta_t+(\xi_1-\omega\eta)\eta_x\quad\hbox{on}\quad y=\eta(t,x).$$ From the previous two relations and (35) we infer $$\frac{\d H}{\d\eta}=-\xi_t-\omega\chi,$$ which completes the proof.$\hfill\Box$
Hamiltonian structure for steady waves
======================================
In this section we consider the case of steady water waves with constant vorticity. If $c \neq 0$ is the speed of the wave, (4) becomes $$-\psi_x=-c\eta_x+\psi_y\eta_x\quad\hbox{on}\quad y=\eta(x-ct),$$ if we use (8). Thus $\partial_x\,\Big[\psi\Big(x-ct,\eta(x-ct)\Big)-\,c\,\eta(x-ct)\Big]=0$ so that in this setting the function $[\psi-cy]$ is constant on the free surface. On the other hand, using (1), we deduce that $$\begin{aligned}
\partial_x \,\Big[ \int_0^\eta (u-c)\,dy\Big] &=& \Big(u(x-ct,\eta(x-ct))\,-c\Big)\,\eta_x(x-ct)
+\int_0^\eta u_x\,dy \\
&=& \Big(u(x-ct,\eta(x-ct))\,-c\Big)\,\eta_x(x-ct)
-\int_0^\eta v_y\,dy \\
&=& \Big(u(x-ct,\eta(x-ct))\,-c\Big)\,\eta_x(x-ct) -v(x-ct,\eta(x-ct))\end{aligned}$$ has to equal zero by (1). Since $\psi_y=u$ throughout the fluid and $\psi=0$ on the flat bed, we have $$\int_0^\eta (u-c)\,dy=(\psi-cy)\Big|_{y=\eta(x-ct)}.$$ The constant value $k$ of the expression on the above left-hand side is the [*relative mass flux*]{} of the flow [@CS]. Since field evidence and laboratory measurements indicate that for steady water waves that are not near the breaking state the relation $u<c$ holds throughout the fluid [@L], we have that $k<0$. These considerations show in view of Theorem 1 that the governing equations for steady water waves with constant vorticity $\omega$ and wave speed $c$ are equivalent to the nearly-Hamiltonian system $$\begin{cases}
\dot{\eta} = \displaystyle\frac{\delta H}{\delta \xi}, \vspace{0.3cm}\\
\dot{\xi} = -\,\displaystyle\frac{\delta H}{\delta \eta} - \omega \,(k+c\eta)
\end{cases}$$ where $k<0$ is the relative mass flux of the flow. From (36) we readily obtain the Hamiltonian formulation for steady water waves with constant vorticity.
[**Theorem 2**]{} [*The governing equations for steady $L$-periodic two-dimensional gravity water waves with constant vorticity $\omega$ are equivalent to the Hamiltonian system $$\begin{cases}
\dot{\eta} = \displaystyle\frac{\delta \hat{H}}{\delta \xi}, \vspace{0.3cm}\\
\dot{\xi} = -\,\displaystyle\frac{\delta \hat{H}}{\delta \eta}\, .
\end{cases}$$ Here $c \neq 0$ is the speed of the wave, $y=\eta$ is the free surface above the flat bed $y=0$, $\xi$ is the evaluation of the (generalized) velocity potential on the free surface, and $$\hat{H}=H +\omega k \int_0^L \eta\,dx+\frac{c\omega}{2}\int_0^L \eta^2\,dx,$$ where $H=H(\eta,\xi)$ is the total energy of the motion.*]{}
[**Remark**]{} For the Hamiltonian system (37) we have ${\mathcal T}(\eta)\xi-c\eta-\displaystyle\frac{\omega}{2}\,\eta^2=k$ in view of (20), since $\xi=k+c\eta$.$\hfill\Box$
.5cm It is a pleasure to thank David Kaup for very useful discussions. We also thank both referees for constructive comments and suggestions. The financial support of Science Foundation Ireland (AC and EMP, grant 04/BRG/M0042) and of the Irish Research Council for Science, Engineering and Technology (RII) is gratefully acknowledged.
[10]{}
C. J. Amick and J. F. Toland, On periodic water-waves and their convergence to solitary waves in the long-wave limit, [*Philos. Trans. Roy. Soc. London Ser. A*]{} [**303**]{} (1981), 633–669.
T. B. Benjamin and P. J. Olver, Hamiltonian structure, symmetries and conservation laws for water waves, [*J. Fluid Mech.*]{} [**125**]{} (1982), 137–185.
R. P. Boas, [*Invitation to Complex Analysis*]{}, McGraw-Hill, Inc., New York, 1987.
J. Cantarella, D. DeTurck, and H. Gluck, Vector calculus and the topology of domains in 3-space, [*Amer. Math. Monthly*]{} [**109**]{} (2002), 409–442.
A. Constantin, A Hamiltonian formulation for free surface water waves with non-vanishing vorticity, [*J. Nonl. Math. Phys.*]{} [**12**]{} (2005), 202–211.
A. Constantin, Wave-current interactions, in [*EQUADIFF 2003*]{}, pp. 207–212, World Sci. Publ., Hackensack, NJ, 2005.
A. Constantin, The trajectories of particles in Stokes waves, [*Inv. Math.*]{}, in print.
A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, [*J. Fluid Mech.*]{} [**498**]{} (2004), 171–181.
A. Constantin and W. Strauss, Exact periodic traveling water waves with vorticity, [*C. R. Math. Acad. Sci. Paris*]{} [**335**]{} (2002), 797–800.
A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, [*Comm. Pure Appl. Math.*]{} [**57**]{} (2004), 481–527.
A. Constantin, D. H. Sattinger, and W. Strauss, Variational formulations for steady water waves with vorticity, [*J. Fluid Mech.*]{} [**548**]{} (2006), 151–163
A. Constantin and W. Strauss, Stability properties of steady water waves with vorticity, [*Comm. Pure Appl. Math*]{}, in print.
W. Craig, Water waves, Hamiltonian systems and Cauchy integrals, in [*Microlocal Analysis and Nonlinear Waves*]{} (Minneapolis, MN, 1988–1989), pp. 37–45, IMA Vol. Math. Appl., 30, Springer, New York, 1991.
W. Craig and M. D. Groves, Hamiltonian long-wave approximations to the water-wave problem, [*Wave Motion*]{} [**19**]{} (1994), 367–389.
A. F. T. DaSilva and D. H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity, [*J. Fluid Mech.*]{} [**195**]{} (1988), 281–302.
M. Ehrnström, Uniqueness for steady water waves with vorticity, [*Int. Math. Res. Not.*]{} [**60**]{} (2005), 3721–3726.
M. D. Groves and J. F. Toland, On variational formulations for steady water waves, [*Arch. Rat. Mech. Anal.*]{} [**137**]{} (1997), 203–226.
R. S. Johnson, [*A Modern Introduction to the Mathematical Theory of Water Waves*]{}, Cambridge University Press, Cambridge, 1997.
B. Kolev and D. H. Sattinger, Variational principles for stationary waves, [*SIAM J. Math. Anal.*]{}, in print.
J. Lighthill, [*Waves in Fluids*]{}, Cambridge University Press, Cambridge, 1978.
P. J. Olver, [*Applications of Lie Groups to Differential Equations*]{}, Springer Verlag, New York, 1986.
J. J. Stoker, [*Water Waves. The Mathematical Theory with Applications*]{}, Interscience Publ., New York, 1957.
G. Stokes, On the theory of oscillatory waves, [*Trans. Cambridge Phil. Soc.*]{} [**8**]{} (1847), 441–455.
C. Swan, I. Cummins, and R. James, An experimental study of two-dimensional surface water waves propagating on depth-varying currents, [*J. Fluid Mech.*]{} [**428**]{} (2001), 273–304.
G. Thomas, Wave-current interactions: an experimental and numerical study, [*J. Fluid Mech.*]{} [**216**]{} (1990), 505–536.
J. F. Toland, Stokes waves, [*Topol. Meth. Nonl. Anal.*]{} [**7**]{} (1996), 1–48.
E. Wahlen, A note on steady gravity waves with vorticity, [*Int. Math. Res. Not.*]{} [**7**]{} (2005), 389–396.
F. W. Warner, [*Foundations of Differentiable Manifolds and Lie Groups*]{}, Springer Verlag, New York, 1983.
V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, [*J. Appl. Mech. Tech. Phys.*]{} [**2**]{} (1968), 190–194.
|
[Simultaneous prediction for independent Poisson processes\
with different durations]{}\
\
[$^{1}$ Department of Mathematical Informatics]{}\
[Graduate School of Information Science and Technology, the University of Tokyo]{}\
[7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, JAPAN]{}\
[komaki@mist.i.u-tokyo.ac.jp]{}\
[$^{2}$ RIKEN Brain Science Institute]{}\
[2-1 Hirosawa, Wako City, Saitama 351-0198, JAPAN]{}
Abstract
Simultaneous predictive densities for independent Poisson observables are investigated. The observed data and the target variables to be predicted are independently distributed according to different Poisson distributions parametrized by the same parameter. The performance of predictive densities is evaluated by the Kullback–Leibler divergence. A class of prior distributions depending on the objective of prediction is introduced. A Bayesian predictive density based on a prior in this class dominates the Bayesian predictive density based on the Jeffreys prior.
Keywords: harmonic time, Jeffreys prior, Kullback–Leibler divergence, predictive density, predictive metric, shrinkage prior
Introduction
============
Suppose that $x_i$ $(i=1,\ldots,d)$ are independently distributed according to the Poisson distribution with mean $r_i \lambda_i$ and that $y_i$ $(i=1,\ldots,d)$ are independently distributed according to the Poisson distribution with mean $s_i \lambda_i$. Then, $$\begin{aligned}
p(x \mid \lambda)
= \prod^d_{i=1} \frac{(r_i \lambda_i)^{x_i}}{x_i!} {\mbox{e}}^{-r_i \lambda_i},
\label{xdist}\end{aligned}$$ and $$\begin{aligned}
p(y \mid \lambda)
= \prod^d_{i=1} \frac{(s_i \lambda_i)^{y_i}}{y_i!} {\mbox{e}}^{-s_i \lambda_i},
\label{ydist}\end{aligned}$$ where $x = (x_1,\ldots,x_d)$ and $y = (y_1,\ldots,y_d)$. Here, $\lambda := (\lambda_1, \ldots, \lambda_d)$ is the unknown parameter, and $r = (r_1,\ldots,r_d)$ and $s = (s_1,\ldots,s_d)$ are known positive constants. The objective is to construct a predictive density $\hat{p}(y ; x)$ for $y$ by using $x$.
The performance of $\hat{p}(y;x)$ is evaluated by the Kullback–Leibler divergence $$D(p(y \mid \lambda), \hat{p}(y;x)) := \sum_y p(y \mid \lambda) \log \frac{p(y \mid \lambda)}{\hat{p}(y;x)}$$ from the true density $p(y \mid \lambda)$ to the predictive density $\hat{p}(y;x)$. The risk function is given by $${{\rm E}}\Bigl[ D(p(y \mid \lambda), \hat{p}(y;x)) \, \Big| \, \lambda \Bigr]
= \sum_x \sum_y p(x \mid \lambda) p(y \mid \lambda) \log \frac{p(y \mid \lambda)}{\hat{p}(y;x)}.$$ It is widely recognized that Bayesian predictive densities $$p_\pi(y \mid x) :=
\frac{\int p(y \mid \lambda) p(x \mid \lambda) \pi(\lambda) {\mbox{d}}\lambda}
{\int p(x \mid \lambda) \pi(\lambda) {\mbox{d}}\lambda},$$ where ${\mbox{d}}\lambda = {\mbox{d}}\lambda_1 \cdots {\mbox{d}}\lambda_d$, constructed by using a prior $\pi$ perform better than plug-in densities $p(y \mid \hat{\lambda})$ constructed by replacing the unknown parameter $\lambda$ by an estimate $\hat{\lambda}(x)$. The choice of $\pi$ becomes important to construct a Bayesian predictive density.
The Jeffreys prior $$\begin{aligned}
\label{jeffreysx}
\pi_\mathrm{J} (\lambda) {\mbox{d}}\lambda_1 \dotsb {\mbox{d}}\lambda_d
\propto \lambda_1^{-\frac{1}{2}} \dotsb \lambda_d^{-\frac{1}{2}} {\mbox{d}}\lambda_1 \dotsb {\mbox{d}}\lambda_d\end{aligned}$$ for $p(x \mid \lambda)$ is proportional to the Jeffreys prior for $p(y \mid \lambda)$ and the volume element prior $\pi_\mathrm{P} (\lambda)$ with respect to the predictive metric discussed in section 4. A natural class of priors including the Jeffreys prior is $$\begin{aligned}
\pi_\beta (\lambda) {\mbox{d}}\lambda_1 \dotsb {\mbox{d}}\lambda_d
:= \lambda_1^{\beta_1-1} \dotsb \lambda_d^{\beta_d-1} {\mbox{d}}\lambda_1 \dotsb {\mbox{d}}\lambda_d,\end{aligned}$$ where $\beta_i > 0$ $(i=1,\ldots,d)$.
We introduce a class of priors defined by $$\pi_{\alpha,\beta,\gamma} (\lambda) {\mbox{d}}\lambda_1 \dotsb {\mbox{d}}\lambda_d :=
\frac{\lambda^{\beta_1-1}_1 \dotsb \lambda^{\beta_d -1}_d}
{(\lambda_1/\gamma_1 + \dotsb + \lambda_d/\gamma_d)^\alpha}
{\mbox{d}}\lambda_1 \dotsb {\mbox{d}}\lambda_d,$$ where $0 \leq \alpha \leq \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} := \sum_i \beta_i$, $\beta_i > 0$, and $\gamma_i > 0$ $(i=1,\ldots,d)$. In the following, a dot as a subscript indicates summation over the corresponding index. Note that $\pi_{\alpha,\beta,\gamma} \propto \pi_{\alpha,\beta,c\gamma}$, where $c>0$ and $c\gamma = (c\gamma_1,\ldots,c\gamma_d)$. The prior $\pi_{\alpha,\beta,\gamma}$ does not depend on $\gamma := (\gamma_1,\ldots,\gamma_d)$ if $\alpha = 0$. If $\alpha > 0$, $\pi_{\alpha, \beta, \gamma}$ puts more weight on parameter values close to $0$ than $\pi_\beta$ does. In this sense, $\pi_{\alpha, \beta, \gamma}$ with $\alpha > 0$ is a shrinkage prior.
There have been several studies for the simple setting $r_1 = r_2 = \cdots = r_d$ and $s_1 = s_2 = \cdots = s_d$. Decision theoretic properties of linear estimators under the Kullback–Leibler loss is studied by [@GY1988AS]. The theory for Bayesian predictive densities for the Poisson model is a generalization of that for Bayesian estimators under the Kullback–Leibler loss. A class of priors $\pi_{\alpha,\beta} := \pi_{\alpha,\beta,\gamma}$ with $\gamma_1 = \cdots = \gamma_d = 1$ is introduced in @Komaki:TheAnnalsOfStatistics:2004. It is shown that the risk of the Bayesian predictive density based on $\pi_{\tilde{\alpha},\beta}$ with $\tilde{\alpha} := \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} -1$ is smaller than the risk of that based on $\pi_\beta$ if $\beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} > 1$. For example, if $d \geq 3$, there exists a Bayesian predictive density that dominates the Bayesian predictive density $p_\mathrm{J}(y \mid x)$ based on the Jeffreys prior because $\beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} = d/2 > 1$. Here, $p_\pi(y \mid x)$ is said to dominate $p_\mathrm{J}(y \mid x)$ if the risk of $p_\pi(y \mid x)$ is not greater than that of $p_\mathrm{J}(y \mid x)$ for all $\lambda$ and the strict inequality holds for at least one point $\lambda$ in the parameter space.
Bayesian predictive densities based on shrinkage priors are discussed by [@komaki2001shrinkage] and [@George:TheAnnalsOfStatistics:2006] for normal models. See also [@George:StatisticalScience:2012] for recent developments of the theory of predictive densities. In practical applications, it often occurs that observed data $x$ and the target variable $y$ to be predicted have different distributions parametrized by the same parameter. Regression models with the same parameter and different explanatory variable values are a typical example. [@Kobayashi:JournalOfMultivariateAnalysis:2008] and [@george2008predictive] showed that shrinkage priors are useful for constructing Bayesian predictive densities for normal linear regression models. [@KBA2014] has studied asymptotic theory for general models other than normal models when $x(i)$ $(i=1,\ldots,N)$ and $y$ have different distributions $p(x \mid \theta)$ and $p(y \mid \theta)$, respectively, with the same parameter $\theta$. However, there has been few studies on nonasymptotic theories of Bayesian predictive densities for non-normal models when the distributions of $x$ and $y$ are different.
In the present paper, we develop finite sample theory for prediction when the data $x$ and the target variable $y$ have different Poisson distributions and , respectively, with the same parameter $\lambda$. The proposed prior depends not only on $r$ corresponding to the data distribution but also on $s$ corresponding to the objective of prediction. Thus, we need to abandon the context invariance of the prior, see e.g. [@dawid1983invariant]. The Bayesian predictive densities studied in the present paper are not represented by using widely known functions such as gamma or beta functions, contrary to the simple setting $r_1 = \cdots = r_d$ and $s_1 = \cdots = s_d$ [@Komaki:TheAnnalsOfStatistics:2004]. However, the predictive densities are represented by introducing a generalization of the Beta function, and the results are proved analytically.
In section 2, we formulate the problem as prediction for time-inhomogeneous Poisson processes and the risk function is represented as an integral with respect to the time. In section 3, we show that a Bayesian predictive density based on a prior in the introduced class $\pi_{\alpha,\beta,\gamma}$ dominates that based on $\pi_\beta$ if $\beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} > 1$. The harmonic time $\tau$ for the time-inhomogeneous Poisson processes is introduced to prove the results. In section 4, we discuss several properties of the proposed prior and the harmonic time $\tau$.
Evaluation of risk
==================
We formulate the problem as prediction for time-inhomogeneous Poisson processes and obtain a useful expression of the risk.
Let $t_i (\tau)$ $(i=1, \dotsb, d)$ be smooth monotonically increasing functions of $\tau \in [0,1]$ satisfying $t_i (0) = r_i$ and $t_i (1) = r_i + s_i$. Let $z_i(\tau)$ $(i=1, \dotsb, d)$ be independent time-inhomogeneous Poisson processes with mean $t_i(\tau) \lambda_i$ and time $\tau$. Then, the density of $z(\tau)$ is $$p(z(\tau) \mid \lambda)
= \prod^d_{i=1} \frac{\{t_i(\tau) \lambda_i\}^{z_i}}{z_i!} {\mbox{e}}^{-t_i(\tau) \lambda_i},$$ where $z(\tau) := (z_1(\tau),\ldots,z_d(\tau))$, and the distributions of $z_i(0)$ and $z_i(1) - z_i(0)$ are identical with those of $x_i$ and $y_i$, respectively. Since $z(0)$ and $z(1)-z(0)$ are independent, prediction of $y$ based on $x$ is equivalent to prediction of $z(1)-z(0)$ based on $z(0)$. We identify $x$ and $y$ with $z(0)$ and $z(1)-z(0)$, respectively.
Let $z_\Delta(\tau) := z(\tau+\Delta) - z(\tau)$. Then, $z_{\Delta=1}(0)$ corresponds to $y$. The density of $z_\Delta(\tau)$ is $$\begin{aligned}
p(z_\Delta(\tau) \mid \lambda) =&
\prod^d_{i=1} \frac{[\{t_i(\tau+\Delta) - t_i(\tau)\} \lambda_i]^{(z_\Delta)_i}}{(z_\Delta)_i!}
{\mbox{e}}^{- \{t_i(\tau+\Delta) - t_i(\tau)\} \lambda_i}.
$$ We designate the prediction of $z_\Delta(\tau)$ in the limit $\Delta \rightarrow 0$ as infinitesimal prediction. The following lemma represents the risk of the original prediction as an integral of the risk of infinitesimal prediction.
\[smallprediction\]
[ 1) ]{} Let $\pi(\lambda)$ be a prior density. Then, $$\begin{aligned}
\frac{\partial}{\partial \Delta} &
{{\rm E}}\Bigl[ D\{p(z_\Delta(\tau) \mid \lambda), p_\pi(z_\Delta(\tau) \mid z(\tau))\} \; \Big| \; \lambda \Bigr]
\bigg|_{\Delta = 0} \notag \\
\label{hyouka-2}
=& \frac{\partial}{\partial \tau} D \{ p(z(\tau) \mid \lambda), p_\pi(z(\tau)) \} \\
\label{hyouka-3}
=& {{\rm E}}\left[
\sum^d_{i=1} \dot{t}_i(\tau)
\biggl\{
\hat{\lambda}^\pi_i(z(\tau),\tau) - \lambda_i
- \lambda_i \log \frac{\hat{\lambda}^\pi_i (z(\tau),\tau)}{\lambda_i} \biggr\}
\; \Bigg| \; \lambda \right],\end{aligned}$$ where $$p_\pi (z(\tau)) := \int p(z(\tau) \mid \lambda) \pi (\lambda) {\mbox{d}}\lambda
= \int \prod^d_{i=1} \frac{\{t_i(\tau) \lambda_i \}^{z_i}}{z_i!}
{\mbox{e}}^{-t_i(\tau) \lambda_i} \pi (\lambda) {\mbox{d}}\lambda,$$ $$\hat{\lambda}_i^{\pi}(z(\tau),\tau) :=
\frac{\int \lambda_i p(z(\tau) \mid \lambda) \pi (\lambda) {\mbox{d}}\lambda}{\int p(z(\tau) \mid \lambda)
\pi (\lambda) {\mbox{d}}\lambda},$$ and $$\dot{t}_i(\tau) := \frac{{\mbox{d}}}{{\mbox{d}}\tau} t(\tau).$$
[ 2) ]{} Let $\pi(\lambda)$ and $\pi'(\lambda)$ be prior densities, and let $p_\pi(y \mid x)$ and $p_{\pi'}(y \mid x)$ be the corresponding Bayesian predictive densities. Then, $$\begin{aligned}
{{\rm E}}\bigl[ & D(p(y \mid \lambda), p_{\pi'} ( y \mid x)) \, \big| \, \lambda \bigr]
- {{\rm E}}\bigl[ D (p(y \mid \lambda), p_{\pi} (y \mid x)) \, \big| \, \lambda \bigr] \notag \\
= & \int^1_0 \frac{\partial}{\partial \Delta} {{\rm E}}\big[ D ( p ( z_\Delta(\tau) \mid \lambda ) , p_{\pi'} ( z_\Delta (\tau) \mid z ( \tau) ) \mid \lambda \big]
\bigg\vert_{\Delta=0}
{\mbox{d}}\tau \notag \\
& \label{lemma1-2-1}
- \int^1_0 \frac{\partial}{\partial \Delta} {{\rm E}}\big[ D ( p ( z_\Delta(\tau) \mid \lambda), p_\pi ( z_\Delta (\tau) \mid z ( \tau) ) \mid \lambda \big]
\bigg\vert_{\Delta=0} {\mbox{d}}\tau \\
= & \int^1_0
{{\rm E}}\left[ \sum_i \dot{t}_i(\tau) \biggl\{
\hat{\lambda}^{\pi'}_i (z(\tau),\tau)
- \lambda_i - \lambda_i \log \frac{\hat{\lambda}^{\pi'}_i (z(\tau),\tau)}{\lambda_i}
\biggr\}
\; \Bigg| \; \lambda \right]
{\mbox{d}}\tau \notag \\
& \label{lemma1-2-2}
- \int^1_0
{{\rm E}}\left[ \sum_i \dot{t}_i(\tau) \biggl\{
\hat{\lambda}^\pi_i(z(\tau),\tau) - \lambda_i - \lambda_i \log \frac{\hat{\lambda}^\pi_i (z(\tau),\tau)}{\lambda_i}
\biggr\}
\; \Bigg| \; \lambda \right]
{\mbox{d}}\tau.\end{aligned}$$
Equation shows that infinitesimal Bayesian prediction based on $\pi$ corresponds to the Bayesian estimator $\hat{\lambda}_\pi$. This fact is a generalization of a result discussed in [@Komaki:JournalOfMultivariateAnalysis:2006] when $r_1=\cdots=r_d$ and $s_1= \cdots =s_d$. By , if $$\begin{aligned}
{{\rm E}}& \left[ \sum_i \dot{t}_i(\tau) \biggl\{
\hat{\lambda}^{\pi'}_i (z(\tau),\tau) - \hat{\lambda}^{\pi}_i (z(\tau),\tau)
- \log \frac{\hat{\lambda}^{\pi'}_i (z(\tau),\tau)}{\hat{\lambda}^{\pi}_i (z(\tau),\tau)}
\biggr\}
\; \Bigg| \; \lambda \right]\end{aligned}$$ is positive for every $\tau \in [0,1]$ and $\lambda$, then the risk of the Bayesian predictive distribution $p_\pi(y \mid x)$ is smaller than that of $p_{\pi'}(y \mid x)$ for every $\lambda$. Intuitively speaking, if the estimators $\hat{\lambda}^{\pi}_i (\cdot,\tau)$ based on $\pi$ is superior in the risk for all $\tau \in [0,1]$, then the Bayesian predictive density $p_\pi(y \mid x)$ is superior in the Kullback–Leibler risk.
Bayesian prediction and estimation
==================================
We introduce a function to represent Bayesian predictive densities and estimators based on $\pi_{\alpha,\beta,\gamma}$.
Suppose that $\gamma \in \mathbb{R}^d$, $\gamma_i > 0 ~(i=1,\ldots,d)$, $x \in \mathbb{R}^d$, $x_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} > 0$, and $0< \alpha < x_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}$. Define $$K(\gamma, x, \alpha) := \int^\infty_0 u^{\alpha-1} \prod^d_{i=1} \frac{1}{(u/\gamma_i + 1)^{x_i}} {\mbox{d}}u.$$
When $\gamma_1 = \dotsb = \gamma_d$, $$\begin{aligned}
K(\gamma, x, \alpha) & = \int^\infty_0 \frac{u^{\alpha-1}}{(u/\gamma_1 + 1)^{{x_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}}}} {\mbox{d}}u
= \gamma_1^\alpha B (x_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} - \alpha, \alpha).\end{aligned}$$ Thus, $K(\gamma, x, \alpha)$ is a generalization of the beta function.
Lemma \[lemma-p\_abc\] below gives explicit forms of Bayesian predictive densities based on $\pi_\beta$ and $\pi_{\alpha,\beta,\gamma}$.
\[lemma-p\_abc\] Suppose that $z_i(\tau)$ $(i=1,\ldots,d)$ are independent time-inhomogeneous Poisson processes with mean $t_i(\tau) \lambda_i$. Let $z_\Delta(\tau) = z(\tau+\Delta) - z(\tau)$, where $\tau \in [0,1)$ and $\Delta \in (0,1-\tau]$.
[ 1) ]{} The Bayesian predictive density based on the prior $\pi_\beta(\lambda) = \lambda_1^{\beta_1-1} \dotsb \lambda_d^{\beta_d-1}$, where $\beta_i > 0$ $(i=1,\ldots,d)$, is given by $$\begin{aligned}
p_\beta & (z_\Delta(\tau) \mid z(\tau)) =
\prod^d_{i=1}
\left\{
\frac{\Gamma (z_i + (z_\Delta)_i + \beta_i)}
{\Gamma (z_i + \beta_i) (z_\Delta)_i!}
\frac{\{t_i(\tau)\}^{z_i + \beta_i}
\{t_i(\tau+\Delta) - t_i(\tau)\}^{(z_\Delta)_i}}
{\{t_i(\tau+\Delta)\}^{z_i + (z_\Delta)_i + \beta_i}}
\right\},\end{aligned}$$ which is a product of negative binomial densities. In particular, when $\tau = 0$ and $\Delta = 1$, $$\begin{aligned}
p_\beta(y \mid x)
= &
\prod_{i=1}^d \left\{ \frac{\Gamma(x_i + y_i + \beta_i)}{\Gamma(x_i + \beta_i) y_i !}
\frac{r_i^{x_i+\beta_i} s_i^{y_i}}{(r_i + s_i)^{x_i + y_i + \beta_i}}
\right\},\end{aligned}$$ where $r_i = t_i(0)$, $r_i+s_i = t_i(1)$, $x=z(1)$, and $y=z_{\Delta=1}(0)$.
[ 2) ]{} The Bayesian predictive density based on the prior $\pi_{\alpha,\beta,\gamma}(\lambda)
= \lambda_1^{\beta_1-1} \dotsb \lambda_d^{\beta_d-1}/
(\lambda_1/\gamma_1 + \cdots + \lambda_d/\gamma_d)^\alpha$, where $0 < \alpha < \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}$, $\beta_i > 0$, and $\gamma_i > 0$ $(i=1,\ldots,d)$, is given by $$\begin{aligned}
p_{\alpha,\beta,\gamma} (z_\Delta(\tau) \mid z(\tau)) =&\
p_\beta (z_\Delta(\tau) \mid z(\tau))
\dfrac{\displaystyle
\int_0^\infty u^{\alpha-1}
\prod^d_{j=1} \frac{1}{\{\frac{u}{t_j(\tau+\Delta) \gamma_j} + 1\}^{z_j + (z_\Delta)_j + \beta_j}} {\mbox{d}}u
}
{\displaystyle
\int_0^\infty u^{\alpha-1}
\prod^d_{j=1} \frac{1}{\{\frac{u}{t_j(\tau) \gamma_j} + 1\}^{z_j + \beta_j}} {\mbox{d}}u
} \\
=&\
p_\beta (z_\Delta(\tau) \mid z(\tau))
\dfrac{K(t(\tau+\Delta) \gamma, z+z_\Delta+\beta, \alpha)}{K(t(\tau) \gamma, z+\beta, \alpha)},\end{aligned}$$ where $t \gamma := (t_1 \gamma_1, t_2 \gamma_2, \ldots, t_d \gamma_d)$.
In particular, when $\tau = 0$ and $\Delta = 1$, $$\begin{aligned}
p_{\alpha,\beta,\gamma} (y \mid x)
=&\ p_\beta(y \mid x)
\dfrac{\displaystyle
\int u^{\alpha-1}
\prod^d_{j=1} \frac{1}{\{\frac{u}{(r_j+s_j)\gamma_j} + 1 \}^{x_j + y_j + \beta_j}} {\mbox{d}}u
}
{\displaystyle
\int u^{\alpha-1}
\prod^d_{j=1} \frac{1}{(\frac{u}{r_j \gamma_j} + 1)^{x_j + \beta_j}} {\mbox{d}}u
} \\
=&\
p_\beta (y \mid x)
\dfrac{K((r+s) \gamma, x+y+\beta, \alpha)}{K(r \gamma, x+\beta, \alpha)},\end{aligned}$$ where $r_i = t_i(0)$, $r_i+s_i = t_i(1)$, $x=z(0)$, $y = z_{\Delta=1}(0)$, $r \gamma := (r_1 \gamma_1, \ldots, r_d \gamma_d)$, and $(r+s) \gamma := ((r_1+s_1) \gamma_1, \ldots, (r_d+s_d) \gamma_d)$.
Lemma \[lemma-estimators\] below gives explicit forms of Bayesian estimators based on $\pi_\beta$ and $\pi_{\alpha,\beta,\gamma}$.
\[lemma-estimators\] Suppose that $z_i(\tau)$ $(i=1,\ldots,d)$ are independently distributed according to the Poisson distribution with mean $t_i(\tau) \lambda_i$.
[ 1) ]{} The posterior mean of $\lambda$ with respect to the observation $z(\tau) = (z_1,\ldots,z_d)$ and the prior $\pi_\beta(\lambda) = \lambda_1^{\beta_1-1} \dotsb \lambda_d^{\beta_d-1}$, where $\beta_i > 0$ $(i=1,\ldots,d)$, is given by $$\begin{aligned}
\hat{\lambda}^{(\beta)}_i(z,\tau) :=& \frac{z_i + \beta_i}{t_i(\tau)}.\end{aligned}$$
[ 2) ]{} The posterior mean of $\lambda$ with respect to the observation $z(\tau) = (z_1,\ldots,z_d)$ and the prior $\pi_{\alpha,\beta,\gamma} = \lambda_1^{\beta_1-1} \dotsb \lambda_d^{\beta_d-1}/
(\lambda_1/\gamma_1 + \cdots + \lambda_d/\gamma_d)^\alpha$, where $0 < \alpha < \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}$, $\beta_i > 0$, and $\gamma_i > 0$ $(i=1,\ldots,d)$, is given by $$\begin{aligned}
\hat{\lambda}_i^{(\alpha,\beta,\gamma)}(z,\tau) :=& \hat{\lambda}^{(\beta)}_i(z,\tau)
\dfrac{\displaystyle
\int u^{\alpha-1}
\prod^d_{j=1} \frac{1}{\left\{\frac{u}{t_j(\tau) \gamma_j} + 1 \right\}^{z_j + \beta_j +\delta_{ij}}} {\mbox{d}}u}
{\displaystyle
\int u^{\alpha-1} \prod^d_{j=1} \frac{1}{\left\{\frac{u}{t_j(\tau) \gamma_j} + 1 \right\}^{z_j + \beta_j}} {\mbox{d}}u} \\
=& \hat{\lambda}^{(\beta)}_i(z,\tau) \frac{K(t\gamma,z+\beta+\delta_i, \alpha)}{K(t\gamma,z+\beta,\alpha)},\end{aligned}$$ where $\delta_{ij}$ is defined to be 1 if $i=j$ and 0 if $i \neq j$, and $\delta_i$ is defined to be the $d$-dimensional vector whose $i$-th element is $1$ and all other elements are $0$.
Let $$\begin{aligned}
\label{coeff}
f_i(t\gamma, z+\beta, \alpha)
:= \frac{K(t\gamma, z+\beta+\delta_i, \alpha)}{K(t\gamma, z+\beta, \alpha)}.\end{aligned}$$ Then, $$\begin{aligned}
\hat{\lambda}_i^{(\alpha,\beta,\gamma)}(z,\tau) =& \hat{\lambda}_i^{(\beta)}(z,\tau) f_i(t(\tau) \gamma, z+\beta, \alpha).\end{aligned}$$ Obviously, $0< f_i(t \gamma, z + \beta,\alpha) < 1$. This inequality is natural because $\pi_{\alpha, \beta, \gamma}$ is a shrinkage prior.
In particular, if $t_1\gamma_1 = \cdots = t_d\gamma_d$, then $$\begin{aligned}
f_i(t \gamma, z+\beta, \alpha)
=&\ \frac{(t_1 \gamma_1)^\alpha B(z_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} + \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} + 1 - \alpha, \alpha)}
{(t_1 \gamma_1)^\alpha B(z_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} + \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} - \alpha, \alpha)}
= \frac{z_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} + \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} - \alpha}{z_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} + \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}},\end{aligned}$$ which does not depend on $t_1 \gamma_1$.
Now, we give the main theorem.
\[maintheorem\] Suppose that $x_i$ and $y_i$ $(i=1,\ldots,d)$ are independently distributed according to the Poisson distributions with mean $r_i \lambda_i$ and $s_i \lambda_i$, respectively. Let $p_\beta (y \mid x)$ be the Bayesian predictive density based on $\pi_\beta(\lambda) = \lambda_1^{\beta_1-1} \dotsb \lambda_d^{\beta_d-1}$. Assume that $\beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} > 1$. Let $\pi^*_\beta(\lambda) := \pi_{\alpha,\beta,\gamma}(\lambda) = \lambda_1^{\beta_1-1} \dotsb \lambda_d^{\beta_d-1}/
(\lambda_1/\gamma_1 + \cdots + \lambda_d/\gamma_d)^\alpha$ with $$\begin{aligned}
\alpha = \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} - 1 ~~ \text{and} ~~~
\gamma_i = \frac{1}{r_i} - \frac{1}{r_i+s_i} ~~~~ (i=1,\ldots,d).\end{aligned}$$ Then, the risk of the Bayesian predictive density $$\begin{aligned}
p_\beta^* (y \mid x)
= p_\beta (y \mid x)
\dfrac{
K \biggl( \dfrac{s}{r}, x+y+\beta, \alpha \biggr)
}
{
K \biggl( \dfrac{s}{r+s}, x+\beta, \alpha \biggr)
}\end{aligned}$$ based on $\pi_\beta^*$, where $$\begin{aligned}
\frac{s}{r} := \left( \frac{s_1}{r_1}, \ldots, \frac{s_d}{r_d} \right) ~~ \text{and} ~~~
\frac{s}{r+s} := \left( \frac{s_1}{r_1 + s_1}, \ldots, \frac{s_d}{r_d + s_d} \right),\end{aligned}$$ is smaller than that of $p_\beta (y \mid x)$ for every $\lambda$.
If $d \geq 3$, there exists a Bayesian predictive density dominating that based on the Jeffreys prior for $p(x \mid \lambda)$ because $\beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} = d/2 > 1$, as in the simple setting with $r_1 = \cdots = r_d$ and $s_1 = \cdots = s_d$ studied in [@Komaki:TheAnnalsOfStatistics:2004]. Note that the prior $\pi^*_\beta$ depends on $r$ and $s$.
Before proving Theorem 1, we prepare Lemmas \[lemma-koukan-0\] and \[lemma-Ksekibun\] below.
\[lemma-koukan-0\] Let $h(x)$ be a real valued function of $x = (x_1,\ldots,x_d) \in \mathbb{N}_0^d$, where $\mathbb{N}_0$ is the set of nonnegative integers. Suppose that $x_i$ $(i=1,\ldots, d)$ are independently distributed according to the Poisson distribution with mean $\lambda_i$. If ${{\rm E}}\bigl[|x_i h(x)| \mid \lambda \bigr] < \infty$, then $$\begin{aligned}
{{\rm E}}[x_i h(x) \mid \lambda] =& {{\rm E}}[\lambda_i h(x + \delta_i) \mid \lambda].\end{aligned}$$
\[lemma-Ksekibun\] Suppose that $\gamma \in \mathbb{R}^d$, $\gamma_i > 0 ~(i=1,\ldots,d)$, $x \in \mathbb{R}^d$, $x_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} > 0$, and $0< \alpha < x_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}$. Then, the following relations hold.
[ 1) ]{} $$\alpha K(\gamma, x, \alpha) = \sum_{i=1}^d \frac{x_i}{\gamma_i} K (\gamma, x + \delta_i, \alpha+1).
\label{lemma-1-0}$$
[ 2) ]{} $$\gamma_i K(\gamma, x, \alpha) = K(\gamma, x + \delta_i, \alpha+1) + \gamma_i K (\gamma, x + \delta_i, \alpha).
\label{lemma-2-0}$$
[ 3) ]{} Let $b = (b_1,b_2,\ldots,b_d) \in \mathbb{R}^d$. Then, $$\begin{aligned}
\sum_{i=1}^d b_i K (\gamma, x+ \delta_i, \alpha)
= & \sum_{i=1}^d \left(\frac{b_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} x_i}{\alpha \gamma_i} - \frac{b_i}{\gamma_i} \right) K(\gamma, x + \delta_i, \alpha + 1) .
\label{lemma-3-0}\end{aligned}$$
Let $$\begin{aligned}
\frac{1}{t_i(\tau)} =& \frac{1}{r_i}(1-\tau) + \frac{1}{r_i+s_i} \tau ~~~ \mbox{for~~} \tau \in [0,1].\end{aligned}$$ Then, $$\begin{aligned}
t_i(\tau) =& r_i \frac{\displaystyle 1 + \frac{s_i}{r_i}}{\displaystyle 1 + \frac{s_i}{r_i}(1-\tau)}\end{aligned}$$ is a smooth monotonically increasing function of $\tau \in [0,1]$ satisfying $t_i(0) = r_i$ and $t_i(1) = r_i + s_i$. Here, $\dot{t}_i/t_i = \gamma_i t_i$ since $\frac{\scriptsize {\mbox{d}}}{\scriptsize {\mbox{d}}\tau} \{1/t_i(\tau)\} = -\dot{t_i}/t_i^2 = -1/r_i + 1/(r_i + s_i) = -\gamma_i$. We call $\tau$ the harmonic time because $\tau$ is the weight of the weighted harmonic mean $t_i(\tau)$ of $r_i$ and $r_i+s_i$.
By Lemma \[lemma-estimators\], the posterior mean of $\lambda$ with respect to $\pi_\beta$ is $$\begin{aligned}
\hat{\lambda}^{(\beta)}_i(z,\tau) =&\ \frac{z_i + \beta_i}{t_i(\tau)},\end{aligned}$$ and the posterior mean $\lambda$ with respect to $\pi_\beta^*$ is $$\begin{aligned}
\hat{\lambda}_i^{(\beta*)}(z,\tau) =&\ \hat{\lambda}_i^{(\beta)}(z,\tau) f_i(\gamma t(\tau),z+\beta,\beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)
= \frac{z_i + \beta_i}{t_i(\tau)} f_i(\gamma t(\tau),z+\beta,\beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1).\end{aligned}$$ Thus, from Lemma \[smallprediction\], it is sufficient to show that $$\begin{aligned}
\sum_i {{\rm E}}\biggl[ &\ \dot{t}_i(\tau) \left\{ \hat{\lambda}_i^{(\beta)}(z(\tau),\tau)
- \hat{\lambda}_i^{(\beta*)}(z(\tau),\tau)
- \lambda_i \log \frac{\hat{\lambda}_i^{(\beta)}(z(\tau),\tau)}{\hat{\lambda}_i^{(\beta*)}(z(\tau),\tau)} \right\}
\, \bigg| \, \lambda \biggr] \notag \\
=& \sum_i {{\rm E}}\biggl[ \dot{t}_i(\tau) \frac{z_i(\tau) + \beta_i}{t_i(\tau)}
\Bigl\{ 1 - f_i(\gamma t(\tau), z(\tau)+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1) \Bigr\} \notag \\
& ~~~~~~~~ + \frac{\dot{t}_i(\tau)}{t_i(\tau)} t_i(\tau) \lambda_i
\log f_i(\gamma t(\tau), z(\tau)+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1) \,\bigg|\, \lambda \biggr]
\label{47-0}\end{aligned}$$ is positive for every $\tau \in [0,1]$ and $\lambda$. Define $\bar{f}_i(\gamma t, z-\delta_i+\beta, \alpha) = f_i(\gamma t, z-\delta_i+\beta, \alpha)$ if $z_i \geq 1$ and $\bar{f}_i(\gamma t, z-\delta_i+\beta, \alpha) = 1$ if $z_i = 0$. Then, by Lemma \[lemma-koukan-0\], is equal to $$\begin{aligned}
{{\rm E}}\biggl[ & \sum_i \frac{\dot{t}_i(\tau)}{t_i(\tau)} (z_i(\tau) + \beta_i)
\Bigl\{ 1 - f_i(\gamma t(\tau), z(\tau)+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1) \Bigr\} \notag \\
&
+ \sum_i \frac{\dot{t}_i(\tau)}{t_i(\tau)} z_i(\tau)
\log \bar{f}_i(\gamma t(\tau), z(\tau)-\delta_i+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1) \, \bigg| \, \lambda \biggr]
\label{54-0}\end{aligned}$$ since $z_i(\tau)$ is independently distributed according to the Poisson distribution with mean $t_i(\tau) \lambda_i$. Note that is the expectation of functions of $z(\tau)$ not depending on $\lambda$. First, we evaluate the first term in the expectation in . By using and , $$\begin{aligned}
1- f_i&(\gamma t, z+\beta,\beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)
= 1 - \frac{K(\gamma t, z + \beta + \delta_i, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)}{K(\gamma t, z + \beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)} \notag \\
=&
1 - \frac{K(\gamma t, z + \beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1) - \dfrac{1}{\gamma_i t_i} K(\gamma t, z + \beta + \delta_i, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}})}
{K(\gamma t, z + \beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)} \notag \\
=& \frac{K(\gamma t, z + \beta + \delta_i, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}})}{\gamma_i t_i K(\gamma t, z + \beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)}.
\label{fK-2}\end{aligned}$$ From $\dot{t}_i/t_i = \gamma_i t_i$ and , we have $$\begin{aligned}
\label{keisu}
\sum_i & \frac{\dot{t}_i}{t_i} (z_i + \beta_i) \{1-f_i(\gamma t, z+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)\}
= \frac{\sum_i (z_i + \beta_i) K(\gamma t, z+\beta+\delta_i, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}})}
{K(\gamma t, z+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)}.\end{aligned}$$ If $z_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} = 0$, then $z_1 = \cdots = z_d = 0$ and $$\begin{aligned}
\sum_i & \frac{\dot{t}_i}{t_i} (z_i + \beta_i) \{1-f_i(\gamma t, z+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)\}
= \frac{\sum_i \beta_i K(\gamma t, \beta+\delta_i, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}})}
{K(\gamma t, \beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)} > 0.\end{aligned}$$ If $z_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} \geq 1$, from , , and , we have $$\begin{aligned}
\sum_i \frac{\dot{t}_i}{t_i} & (z_i + \beta_i) \{1-f_i(\gamma t, z+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)\} \\
=&
\frac{\sum_i \left\{ \dfrac{(z_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} + \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}})(z_i + \beta_i)}{\beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} \gamma_i t_i}
- \dfrac{z_i + \beta_i}{\gamma_i t_i} \right\}
K(\gamma t, z+\beta+\delta_i, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}+1)}
{K(\gamma t, z+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)} \\
=& \frac{\dfrac{z_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}}{\beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}} \sum_i \dfrac{z_i + \beta_i}{\gamma_i t_i}
K(\gamma t, z+\beta+\delta_i, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}+1)}
{K(\gamma t, z+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)} \\
=&
\dfrac{z_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}}{\beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}}
\frac{\beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} K(\gamma t, z+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}})}
{K(\gamma t, z+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)}
= z_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}
\frac{K(\gamma t, z+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}})}
{K(\gamma t, z+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)}.\end{aligned}$$ Next, we evaluate the second term in the expectation in . We have $$\begin{aligned}
\frac{\dot{t}_i}{t} & z_i \log \bar{f}_i(\gamma t, z+\beta-\delta_i, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)
= - \gamma_i t_i z_i \log \left\{\frac{1}{\bar{f}_i(\gamma t, z + \beta -\delta_i, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)} - 1 + 1 \right\}.\end{aligned}$$ From and , if $z_i \geq 1$, $$\begin{aligned}
\frac{1}{\bar{f}_i(\gamma t, z+\beta-\delta_i, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)} -1
=
\frac{K(\gamma t, z+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}})}{\gamma_i t_i K(\gamma t, z+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)}.\end{aligned}$$ Thus, when $z_i \geq 1$, $$\begin{aligned}
\frac{\dot{t}_i}{t} z_i \log \bar{f}_i(\gamma t, z+\beta-\delta_i, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)
=& - \gamma_i t_i z_i
\log \left\{\frac{K(\gamma t, z +\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}})}{\gamma_i t_i K(\gamma t, z +\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)} +1 \right\} \\
>&
- z_i \frac{K(\gamma t, z + \beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}})}{K(\gamma t, z+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)}.\end{aligned}$$ When $z_i = 0$, the equality $$\begin{aligned}
\frac{\dot{t}_i}{t} & z_i \log \bar{f}_i(\gamma t, z+\beta-\delta_i, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)
= - z_i \frac{K(\gamma t, z + \beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}})}{K(\gamma t, z+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)} = 0\end{aligned}$$ obviously holds. Thus, for every $z$, $$\begin{aligned}
\sum_i \frac{\dot{t}_i}{t} & z_i \log \bar{f}_i(\gamma t, z+\beta-\delta_i, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)
\geq
- z_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} \frac{K(\gamma t, z + \beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}})}{K(\gamma t, z+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)}.\end{aligned}$$ The inequality is strict if $z_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}} \geq 1$.
Hence, for every $z \in \mathbb{N}_0^d$, $$\begin{aligned}
\sum_i & \frac{\dot{t}_i}{t_i} (z_i + \beta_i) \{1-f_i(\gamma t, z+\beta, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)\}
+ \sum_i \frac{\dot{t}_i}{t} z_i \log \bar{f}_i(\gamma t, z+\beta-\delta_i, \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}-1)
> 0\end{aligned}$$ Therefore, is greater than $0$ for every $\tau \in [0,1]$ and $\lambda$. Thus, we have proved the desired result.
Relative invariance of the prior along with the harmonic time $\tau$
====================================================================
In this section, $\pi^*_{\beta}$ in Theorem 1 is denoted by $\pi^*_{\beta,r,s}$ to indicate its dependence on $r = (r_1,\ldots,r_d)$ and $s = (s_1,\ldots,s_d)$ explicitly. The prior $\pi^*_{\beta,r,s}$ depends on $r$ and $s$ through $(1/r_1 - 1/(r_1+s_1), \ldots, 1/r_d - 1/(r_d+s_d))$ because $\pi^*_{\beta,r,s} = \pi_{\alpha,\beta,\gamma}$ with $\alpha = \beta_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}$ and $\gamma_i = 1/r_i - 1/(r_i+s_i)$. If there exists a constant $c>0$ such that $$\frac{1}{r_i'} - \frac{1}{r_i'+s_i'}
= c \left(\frac{1}{r_i} - \frac{1}{r_i+s_i}\right)$$ for $i=1,\ldots,d$, then $\pi^*_{\beta,r,s}$ is proportional $\pi^*_{\beta,r',s'}$ because $\pi_{\alpha,\beta,c\gamma} \propto \pi_{\alpha,\beta,\gamma}$.
Consider the harmonic time $\tau \in (-\infty, \min_i (r_i/s_i) + 1)$ satisfying $$\begin{aligned}
\frac{1}{t_i(\tau)}
=& \frac{1}{r_i}(1-\tau) + \frac{1}{r_i+s_i} \tau.\end{aligned}$$ The discussions in previous sections are essentially valid if the time interval $[0,1]$ is extended to $(-\infty, \min_i (r_i/s_i) + 1)$. Suppose that we observe $z(a)$, where $a \in (-\infty, \min_i (r_i/s_i) + 1)$, and predict $z(b) - z(a)$, where $b \in (a, \min_i (r_i/s_i) + 1)$. Since $$\begin{aligned}
\frac{1}{t_i(a)} - \frac{1}{t_i(b)}
=& \left\{\frac{1}{r_i}(1-a) + \frac{1}{r_i+s_i}a \right\}
- \left\{\frac{1}{r_i}(1-b) + \frac{1}{r_i+s_i}b \right\} \\
=& (b - a) \left( \frac{1}{r_i} - \frac{1}{r_i+s_i} \right),\end{aligned}$$ the prior $\pi^*_{\beta,r/(b-a),s/(b-a)}$ for this prediction problem is proportional to the prior $\pi^*_{\beta,r,s}$ for the original prediction problem in which we observe $z(0)$ and predict $z(1) - z(0)$. In this sense, the prior constructed by Theorem 1 is relatively invariant along with the harmonic time $\tau$. This relative invariance corresponds to the fact that the estimators $\hat{\lambda}^{(\beta *)}_i (\cdot,\tau)$ based on $\pi^*_{\beta,r,s}$ is superior in the risk for all $\tau$ and is one reason why the harmonic time $\tau$ is useful to investigate the original prediction problem.
Next, we discuss the relation between the results in previous sections and the asymptotic theory [@KBA2014] for general models when $x(i)$ $(i=1,\ldots,N)$ and $y$ have different distributions $p(x \mid \theta)$ and $p(y \mid \theta)$ with the same parameter $\theta$. The predictive metric ${\mathring{g}_{ij}^{}}$ is defined by $\sum_{k,l} g_{ik} \tilde{g}^{kl} g_{jl}$, where $(g_{ij})$ and $(\tilde{g}_{ij})$ are the Fisher information matrices for $p(x \mid \theta)$ and $p(y \mid \theta)$, respectively, and the $d \times d$ matrix $({\tilde{g}_{}^{ij}})$ is the inverse matrix of $({\tilde{g}_{ij}^{}})$. In the asymptotic theory, the predictive metric ${\mathring{g}_{ij}^{}}$ and the volume element $|{\mathring{g}_{}^{}}|^{1/2} {\mbox{d}}\theta^1 \cdots {\mbox{d}}\theta^d$ of it correspond to the Fisher–Rao metric and the Jeffreys prior, respectively, in the conventional setting.
In the prediction problem for independent time-inhomogeneous Poisson processes with the harmonic time $\tau$, the Fisher information matrix $({g_{ij}^{}})$ for $p(z(\tau) \mid \lambda)$ and the Fisher information matrix $({\tilde{g}_{ij}^{}})$ for $p(z_\Delta(\tau) \mid \lambda)$ are given by $$\begin{aligned}
{g_{ij}^{}}(\lambda;\tau) =
\left\{
\begin{array}{cc}
\displaystyle \frac{t_i(\tau)}{\lambda_i} & (i=j) \\[0.5cm]
0 & (i \neq j)
\end{array}
\right. \end{aligned}$$ and $$\begin{aligned}
{\tilde{g}_{ij}^{}}(\lambda;\tau) =
\left\{
\begin{array}{cc}
\displaystyle \frac{t_i(\tau+\Delta) - t_i(\tau)}{\lambda_i} & (i=j) \\[0.5cm]
0 & (i \neq j)
\end{array}
\right., \end{aligned}$$ respectively. When $\Delta$ is small, ${\tilde{g}_{ii}^{}}(\lambda;\tau) = \dot{t}_i(\tau) \Delta / \lambda_i + \mathrm{o}(\Delta)$. We define the infinitesimal predictive metric by $$\begin{aligned}
\label{poissonpm}
{\mathring{g}_{ij}^{}}(\lambda;\tau) := \lim_{\Delta \rightarrow 0} \Delta \sum_{k,l} {g_{ik}^{}}{\tilde{g}_{}^{ij}}{g_{jl}^{}}
= \left\{
\begin{array}{cc}
\displaystyle \frac{\{t_i(\tau)\}^2}{\dot{t}_i(\tau) \lambda_i}
= \frac{r_i(r_i+s_i)}{\lambda_i} & (i=j) \\[0.5cm]
0 & (i \neq j)
\end{array}
\right.,\end{aligned}$$ which is the limit of the predictive metric as $\Delta \rightarrow 0$. The last equality in is because the relations $\dot{t_i}^2(\tau)/t_i(\tau) = r_i(r_i+s_i)$ $(i=1,\ldots,d)$ holds for the harmonic time $\tau$. The volume element prior based on ${\mathring{g}_{ij}^{}}(\lambda;\tau)$ is defined by $\pi_\text{P}(\lambda;\tau) = |{\mathring{g}_{ij}^{}}(\lambda;\tau)|^{1/2}$ and is proportional to the Jeffreys prior $\pi_\text{J}(\lambda) \propto \prod_i {\lambda_i}^{-1/2}$. Thus, when the harmonic time $\tau$ is adopted, the infinitesimal predictive metric and the volume element prior based on it do not depend on $\tau$. Intuitively speaking, the geometrical structures of infinitesimal prediction are identical for all $\tau$. Hence, there exists a prior superior for infinitesimal predictions for all $\tau$ and the prior is also superior for the original prediction problem. More specifically, the ratio $\pi^*_{\beta,r,s}(\lambda)/\pi_\text{P}(\lambda;\tau)$ does not depend on $\tau$ and is a nonconstant positive superharmonic function with respect to the predictive metric ${\mathring{g}_{ij}^{}}(\lambda;\tau)$ for every $\tau$, see [@KBA2014] for details. This property of the harmonic time $\tau$ is closely related to the relative invariance of the prior $\pi^*_{\beta,r,s}$ along with $\tau$.
Proofs of Lemmas
================
1) First, we prove . We have $$\begin{aligned}
{{\rm E}}\Bigl[ & D\{p(z_\Delta(\tau) \mid \lambda), p_\pi(z_\Delta(\tau) \mid z(\tau))\} \; \Big| \; \lambda \Bigr] =
\sum_{z(\tau), z_\Delta(\tau)} p (z(\tau), z_\Delta(\tau) \mid \lambda)
\log \frac{p(z_\Delta(\tau) \mid \lambda)}{p_{\pi} (z_\Delta(\tau) \mid z(\tau))} \notag \\
=& \sum_{z(\tau), z_\Delta(\tau)} p(z(\tau), z_\Delta(\tau) \mid \lambda) \log p(z(\tau), z_\Delta(\tau) \mid \lambda)
- \sum_{z(\tau)} p(z(\tau) \mid \lambda) \log p (z(\tau) \mid \lambda) \notag \\
& - \sum_{z(\tau), z_\Delta(\tau)} p (z(\tau), z_\Delta(\tau) \mid \lambda) \log p_{\pi} (z(\tau), z_\Delta(\tau))
+ \sum_{z(\tau)} p(z(\tau) \mid \lambda) \log p_{\pi} (z(\tau)).\end{aligned}$$ The conditional density $p(z(\tau) \mid z(\tau+\Delta), \lambda)$ does not depend on $\lambda$ because of the sufficiency of $z(\tau+\Delta) = z(\tau) + z_\Delta(\tau)$. Thus, $$\begin{aligned}
{{\rm E}}\Bigl[ & D\{p(z_\Delta(\tau) \mid \lambda), p_\pi(z_\Delta(\tau) \mid z(\tau))\} \; \Big| \; \lambda \Bigr] \notag \\
= & \sum_{z(\tau), z(\tau+\Delta)} p(z(\tau), z(\tau+\Delta) \mid \lambda)
\log \{p (z(\tau+\Delta) \mid \lambda) p(z(\tau) \mid z(\tau+\Delta)) \} \notag \\
& - \sum_{z(\tau)} p(z(\tau) \mid \lambda) \log p (z(\tau) \mid \lambda) \notag \\
& - \sum_{z(\tau), z(\tau+\Delta)} p (z(\tau), z(\tau+\Delta) \mid \lambda) \log \{p_{\pi} (z(\tau+\Delta))
p(z(\tau) \mid z(\tau+\Delta)) \} \notag \\
& + \sum_{z(\tau)} p(z(\tau) \mid \lambda) \log p_{\pi} (z(\tau)) \notag \\
= & \sum_{z(\tau+\Delta)} p(z(\tau+\Delta) \mid \lambda) \log p (z(\tau+\Delta) \mid \lambda)
- \sum_{z(\tau)} p(z(\tau) \mid \lambda) \log p (z(\tau) \mid \lambda) \notag \\
& - \sum_{z(\tau+\Delta)} p (z(\tau+\Delta) \mid \lambda) \log p_{\pi} (z(\tau+\Delta))
+ \sum_{z(\tau)} p(z(\tau) \mid \lambda) \log p_{\pi} (z(\tau)).
\label{eq:diff}\end{aligned}$$ Therefore, we have $$\begin{aligned}
\frac{\partial}{\partial \Delta} &
{{\rm E}}\Bigl[ D\{p(z_\Delta(\tau) \mid \lambda), p_\pi(z_\Delta(\tau) \mid z(\tau))\} \; \Big| \; \lambda \Bigr]
\bigg|_{\Delta = 0} \notag \\
= &
\frac{\partial}{\partial \tau} \sum_z p (z(\tau)\mid\lambda) \log \frac{p(z(\tau) \mid \lambda)}{p_\pi (z(\tau))}
= \frac{\partial}{\partial \tau} D \{ p(z(\tau) \mid \lambda), p_\pi(z(\tau)) \}
\label{18-1}\end{aligned}$$ because ${{\rm E}}\Bigl[ D\{p(z_\Delta(\tau) \mid \lambda), p_\pi(z_\Delta(\tau) \mid z(\tau))\} \; \Big| \; \lambda \Bigr] = 0$ when $\Delta = 0$. Next, we prove . We have $$\begin{aligned}
\frac{\partial }{\partial \tau} & p (z(\tau) \mid \lambda)
= \frac{{\mbox{d}}}{{\mbox{d}}\tau} \prod^d_{i=1} \frac{ \{ t_i(\tau) \lambda_i \}^{z_i}} {z_i!} {\mbox{e}}^{ - t_i (\tau) \lambda_i}
\notag \\
= & \sum^d_{j=1} \left[ \prod^d_{i=1} z_j \frac{ \{ t_i(\tau) \lambda_i \}^{z_i - \delta_{ij}} }{z_i!}
\dot{t}_j (\tau) \lambda_j {\mbox{e}}^{ -t_i (\tau) \lambda_i}
- \prod^d_{i=1} \frac{ \{ t_i(\tau) \lambda_i \} ^{z_i} } {z_i!}
\dot{t}_j (\tau) \lambda_j {\mbox{e}}^{ -t_i (\tau) \lambda_i} \right]
\notag \\
= & \sum^d_{j=1} \left[ \prod^d_{i=1} z_j \frac{\dot{t}_j(\tau)}{t_j(\tau)} \frac{ \{ t_i(\tau) \lambda_i \}^{z_i}}{z_i!}
{\mbox{e}}^{ -t_i (\tau) \lambda_i }
- \prod^d_{i=1} \frac{\dot{t}_j(\tau)}{t_j(\tau)} t_j(\tau) \lambda_j \frac{ \{ t_i ( \tau) \lambda_i \}^{z_i}}{z_i!}
{\mbox{e}}^{-t_i(\tau) \lambda_i}\right]
\notag \\
= & \sum^d_{j=1} \frac{\dot{t}_j(\tau)}{t_j(\tau)} \{z_j - t_j(\tau) \lambda_j\} p(z(\tau) \mid \lambda).
\label{eq-bibun}\end{aligned}$$ Similarly, $$\begin{aligned}
\frac{\partial }{\partial \tau} p_\pi (z (\tau))
= & \sum^d_{j=1} \frac{\dot{t}_j(\tau)}{t_j(\tau)} \{z_j - t_j(\tau) \hat{\lambda}^\pi_j(z,\tau)\}
p_\pi (z (\tau)).
\label{eq-bibundouyou}\end{aligned}$$ From Lemma \[lemma-koukan-0\], $$\begin{aligned}
\sum_z & \sum^{d}_{j=1} \{z_j - t_j (\tau) \lambda_j \} p (z(\tau) \mid \lambda) \log p_{\pi} (z(\tau)) \notag \\
= & \sum_z \sum^{d}_{j=1} t_j (\tau) \lambda_j p ( z (\tau) \mid \lambda )
\log \frac { p_\pi ( z (\tau) + \delta_j ) }{ p_\pi ( z (\tau) ) }.
\label{13-A}\end{aligned}$$ Since $$\begin{aligned}
p_\pi (z (\tau) + \delta_j)
=& \int \prod^{d}_{i=1} \frac{ \{ t_i (\tau) \lambda_i \}^{z_i + \delta_{ij}} }{(z_i + \delta_{ij}) !}
{\mbox{e}}^{ - t_i \lambda_i } \pi ( \lambda) {\mbox{d}}\lambda \\
=& \int \frac{ t_j ( \tau) \lambda_j}{z_j + 1} \prod^{{\mbox{d}}}_{i=1}
\frac{ \{t_i(\tau) \lambda_i \}^{z_i} }{z_i!} {\mbox{e}}^{-t_i(\tau) \lambda_i} \pi (\lambda) {\mbox{d}}\lambda,\end{aligned}$$ we have $$\begin{aligned}
\frac{p_\pi (z(\tau) + \delta_j )}{ p_\pi (z(\tau))} = \frac{t_j(\tau) \hat{\lambda}^\pi_j(z,\tau)}{z_j + 1}.
\label{13-B}\end{aligned}$$ From , , , , and Lemma \[lemma-koukan-0\], $$\begin{aligned}
\frac{\partial}{\partial \tau} & \sum_z p (z(\tau)\mid\lambda) \log p_\pi (z(\tau)) \notag \\
=& \sum_z \left\{\frac{\partial}{\partial \tau} p(z(\tau) \mid \lambda) \right\} \log p_\pi (z(\tau)) +
\sum_z p(z(\tau)\mid\lambda) \frac{\frac{\partial}{\partial \tau} p_\pi (z(\tau))}{p_\pi (z(\tau))} \notag \\
=& \sum_z \sum^d_{j=1} \frac{\dot{t}_j(\tau)}{t_j(\tau)}
t_j (\tau) \lambda_j p ( z (\tau) \mid \lambda )
\log \frac { p_\pi ( z (\tau) + \delta_j ) }{ p_\pi ( z (\tau) ) } \notag \\
&+ \sum_z \sum^d_{j=1} p(z(\tau)\mid\lambda) \frac{\dot{t}_j(\tau)}{t_j(\tau)}
\{z_j - t_j(\tau) \hat{\lambda}^\pi_j(z,\tau)\} \notag \\
=& \sum_z \sum^d_{j=1} p ( z (\tau) \mid \lambda ) \dot{t}_j(\tau)
\lambda_j \log \frac{t_j (\tau) \hat{\lambda}^\pi_j (z,\tau)}{z_j + 1}
+ \sum_z \sum^d_{j=1} p(z(\tau)\mid\lambda) \dot{t}_j(\tau) \{\lambda_j - \hat{\lambda}^\pi_j(z,\tau)\}.\end{aligned}$$ Similarly we have, $$\begin{aligned}
\frac{\partial}{\partial \tau} & \sum_z p (z(\tau)\mid\lambda) \log p (z(\tau) \mid \lambda)
= \sum_z \sum^d_{j=1} p ( z (\tau) \mid \lambda ) \dot{t}_j(\tau)
\lambda_j \log \frac{t_j (\tau) \lambda_j}{z_j + 1}. \notag\end{aligned}$$ Thus, $$\begin{aligned}
\frac{\partial}{\partial \tau} & \sum_z p (z(\tau)\mid\lambda) \log \frac{p(z(\tau) \mid \lambda)}{p_\pi (z(\tau))} \notag \\
=& \sum_z p ( z (\tau) \mid \lambda )
\sum^d_{j=1} \dot{t}_j(\tau) \lambda_j
\left\{
\frac{\hat{\lambda}_j^\pi(z,\tau)}{\lambda_j} - 1 -\log \frac{\hat{\lambda}^\pi_j(z,\tau)}{\lambda_j}
\right\}.\end{aligned}$$ 2) From , we have $$\begin{aligned}
{{\rm E}}\bigl[ & D(p(y \mid \lambda), p_{\pi'} ( y \mid x)) \, \big| \, \lambda \bigr]
- {{\rm E}}\bigl[ D (p(y \mid \lambda), p_{\pi} (y \mid x)) \, \big| \, \lambda \bigr] \notag \\
=& {{\rm E}}\Bigl[ D \bigl\{ p(z_{\Delta=1}(0) \mid \lambda), p_{\pi'} (z_{\Delta=1}(0) \mid z(0)) \bigr\} \, \Big| \, \lambda \Bigr] \notag \\
& - {{\rm E}}\Bigl[ D \bigl\{ p(z_{\Delta=1}(0) \mid \lambda), p_{\pi} (z_{\Delta=1}(0) \mid z(0)) \bigr\} \, \Big| \, \lambda \Bigr] \notag \\
= & \int^1_0 \frac{\partial}{\partial \tau} \sum_{z(\tau)} p(z(\tau) \mid \lambda ) \log p_\pi (z(\tau)) {\mbox{d}}\tau
- \int^1_0 \frac{\partial}{\partial \tau} \sum_{z(\tau)} p(z(\tau) \mid \lambda) \log p_{\pi'} (z(\tau)) {\mbox{d}}\tau.\end{aligned}$$ Thus, we obtain the desired results and from and , respectively.
1) Let $z_i = z_i(\tau)$ and $z'_i = (z_\Delta)_i(\tau)$. Then, we have $$\begin{aligned}
\int p(z &\mid \lambda) \pi_\beta(\lambda) {\mbox{d}}\lambda
= \int \prod^d_{i=1} \frac{\{t_i(\tau)\}^{z_i} \lambda_i^{z_i + \beta_i - 1}}{z_i!}
{\mbox{e}}^{-t_i(\tau) \lambda_i} {\mbox{d}}\lambda_1 \dotsb {\mbox{d}}\lambda_d \\
=& \prod^d_{i=1} \frac{\{t_i(\tau)\}^{z_i} \Gamma(z_i + \beta_i)}
{z_i! t_i(\tau)^{z_i + \beta_i}}\end{aligned}$$ and $$\begin{aligned}
\int p(z,&\ z'\mid \lambda) \pi_\beta(\lambda) {\mbox{d}}\lambda \\
=& \int \prod^d_{i=1} \frac{\{t_i(\tau)\}^{z_i}
\{t_i(\tau+\Delta) - t_i(\tau)\}^{z'_i}
\lambda_i^{z_i + z'_i + \beta_i - 1}}{z_i! z'_i!}
{\mbox{e}}^{- t_i(\tau+\Delta) \lambda_i}
{\mbox{d}}\lambda_1 \dotsb {\mbox{d}}\lambda_d \\
=& \prod^d_{i=1}
\frac{\{t_i(\tau)\}^{z_i} \{t_i(\tau+\Delta) - t_i(\tau)\}^{z'_i}
\Gamma(z_i + z'_i + \beta_i)}
{z_i! z'_i! \{t_i(\tau+\Delta)\}^{z_i + z'_i + \beta_i}}.\end{aligned}$$ From $p_\beta(z' \mid z) = p_\beta(z, z')/p_\beta(z)$, we have the desired result.
2) If $\gamma_i > 0$ $(i=1,\ldots,d)$ and $\alpha > 0$, $$\begin{aligned}
\int_0^\infty u^{\alpha-1} \exp \left(- u \sum_{i=1}^d \frac{\lambda_i}{\gamma_i} \right) {\mbox{d}}u
= \frac{\Gamma (\alpha)}{(\sum_{i=1}^d \frac{\lambda_i}{\gamma_i})^\alpha}.\end{aligned}$$ Thus, $$\begin{aligned}
\pi_{\alpha,\beta,\gamma}(\lambda) =& \frac{\prod_{i=1}^d \lambda_i^{\beta_i-1}}{\Gamma (\alpha)}
\int_0^\infty u^{\alpha-1} \exp \left(- u \sum_{j=1}^d \frac{\lambda_j}{\gamma_j} \right) {\mbox{d}}u.
\label{priorintegral} \end{aligned}$$ Therefore, since $$\begin{aligned}
\Gamma(\alpha) & p_{\alpha,\beta,\gamma}(z)
= \Gamma(\alpha) \int p(z \mid \lambda) \pi_{\alpha,\beta,\gamma}(\lambda) {\mbox{d}}\lambda \\
=& \int \prod^d_{i=1} \frac{\{t_i(\tau)\}^{z_i} \lambda_i^{z_i +\beta_i-1}}{z_i!} {\mbox{e}}^{-t_i(\tau) \lambda_i}
\int^\infty_0 u^{\alpha-1} \exp \biggl( - u \sum_j \frac{\lambda_j}{\gamma_j} \biggr) {\mbox{d}}u
{\mbox{d}}\lambda_1 \dotsb {\mbox{d}}\lambda_d \\
=& \int_0^\infty u^{\alpha-1} \int \prod^d_{i=1} \frac{\{t_i(\tau)\}^{z_i} \lambda_i^{z_i + \beta_i - 1}}{z_i!}
{\mbox{e}}^{-\{\frac{u}{\gamma_i} + t_i(\tau)\}\lambda_i} {\mbox{d}}\lambda_1 \dotsb {\mbox{d}}\lambda_d {\mbox{d}}u \\
=& \int_0^\infty u^{\alpha-1} \prod^d_{i=1} \frac{\{t_i(\tau)\}^{z_i} \Gamma(z_i + \beta_i)}
{z_i! \{\frac{u}{\gamma_i} + t_i(\tau)\}^{z_i + \beta_i}} {\mbox{d}}u \\
=& \left[ \prod^d_{i=1} \frac{\{t_i(\tau)\}^{z_i} \Gamma (z_i + \beta_i)}{z_i! \{t_i(\tau)\}^{z_j + \beta_i}} \right]
\int_0^\infty u^{\alpha-1}
\prod^d_{j=1} \frac{1}{\{\frac{u}{t_i(\tau) \gamma_i} + 1\}^{z_j + \beta_i}} {\mbox{d}}u\end{aligned}$$ and $$\begin{aligned}
\Gamma(\alpha) & p_{\alpha,\beta,\gamma}(z,z')
= \Gamma(\alpha) \int p(z,z' \mid \lambda) \pi_{\alpha,\beta,\gamma}(\lambda) {\mbox{d}}\lambda \\
=& \int \prod^d_{i=1} \frac{
\{t_i(\tau)\}^{z_i}
\{t_i(\tau+\Delta) - t_i(\tau)\}^{z'_i}
\lambda_i^{z_i + z_i' + \beta_i - 1}}{z_i!z'_i!} {\mbox{e}}^{-t_i(\tau + \Delta) \lambda_i} \\
& \times \int^\infty_0 u^{\alpha-1} \exp \biggl(- u \sum_j \frac{\lambda_j}{\gamma_j} \biggr) {\mbox{d}}u
{\mbox{d}}\lambda_1 \dotsb {\mbox{d}}\lambda_d \\
=& \int_0^\infty u^{\alpha-1} \int \prod^d_{i=1} \frac{
\{t_i(\tau)\}^{z_i}
\{t_i(\tau+\Delta) - t_i(\tau)\}^{z'_i}
\lambda_i^{z_i + z'_i + \beta_i - 1}}{z_i! z'_i!}
{\mbox{e}}^{- \{\frac{u}{\gamma_i} + t_i(\tau+\Delta)\}\lambda_i}
{\mbox{d}}\lambda_1 \dotsb {\mbox{d}}\lambda_d {\mbox{d}}u \\
=& \int_0^\infty u^{\alpha-1} \prod^d_{i=1}
\frac{\{t_i(\tau)\}^{z_i} \{t_i(\tau+\Delta) - t_i(\tau)\}^{z'_i}
\Gamma(z_i + z'_i + \beta_i)}
{z_i! z'_i! \{\frac{u}{\gamma_i} + t_i(\tau+\Delta)\}^{z_i + z'_i + \beta_i}} {\mbox{d}}u \\
=& \left[
\prod^d_{i=1}
\frac{\{t_i(\tau)\}^{z_i} \{t_i(\tau+\Delta) - t_i(\tau)\}^{z'_i}
\Gamma (z_i + z'_i + \beta_i)}{z_i! z'_i! \{t_i(\tau+\Delta)\}^{z_i + z'_i + \beta_i}} \right]
\int_0^\infty u^{\alpha-1} \prod^d_{j=1}
\frac{1}{\{\frac{u}{t_j(\tau+\Delta) \gamma_j} + 1\}^{z_j + z'_j + \beta_j}} {\mbox{d}}u,\end{aligned}$$ we obtain the desired result from $p_{\alpha,\beta,\gamma} (z' \mid z) = p_{\alpha,\beta,\gamma}(z,z')/p_{\alpha,\beta,\gamma}(z)$.
1) The posterior mean of $\lambda_i$ with respect to $\pi_\beta$ is given by $$\begin{aligned}
\hat{\lambda}_i^{(\beta)} &:= \frac{\int \lambda_i p(z(\tau) \mid \lambda) \pi_\beta(\lambda) {\mbox{d}}\lambda}
{\int p(z(\tau) \mid \lambda) \pi_\beta(\lambda) {\mbox{d}}\lambda}
= \frac{\displaystyle \int \lambda_i \prod^d_{j=1} \frac{\lambda_j^{z_j + \beta_i - 1}}{z_j!}
{\mbox{e}}^{-t_i(\tau) \lambda_j} {\mbox{d}}\lambda_1 \dotsb {\mbox{d}}\lambda_d}
{\displaystyle \int \prod^d_{k=1} \frac{\lambda_k^{z_k + \beta_k - 1}}{z_k!}
{\mbox{e}}^{-t_k(\tau) \lambda_k} {\mbox{d}}\lambda_1 \dotsb {\mbox{d}}\lambda_d} \\
&= \frac{\displaystyle \frac{\Gamma \left(z_i + \beta_i + 1 \right)}{t_i(\tau)^{z_i + \beta_i + 1}}
\prod_{j \neq i} \frac{\Gamma \left(z_j + \beta_i \right)}{t_k(\tau)^{z_k + \beta_i}}}
{\displaystyle \prod^d_{k=1} \frac{\Gamma(z_k + \beta_k)}{t_k(\tau)^{z_k + \beta_k}}}
= \frac{z_i + \beta_i}{t_i(\tau)}.\end{aligned}$$
2) By using , we have $$\begin{aligned}
\Gamma(\alpha) & \int p(z(\tau) \mid \lambda) \pi_{\alpha,\beta,\gamma}(\lambda) {\mbox{d}}\lambda \\
=& \int \prod^d_{i=1} \frac{\lambda_i^{z_i + \beta_i - 1}}{z_i!} {\mbox{e}}^{-t_i(\tau) \lambda_i}
\int^\infty_0 u^{\alpha-1} \exp
\biggl(- u \sum_j \frac{\lambda_j}{\gamma_j} \biggr) {\mbox{d}}u
{\mbox{d}}\lambda_1 \dotsb {\mbox{d}}\lambda_d \\
=& \int_0^\infty u^{\alpha-1} \int \prod^d_{i=1} \frac{\lambda_i^{z_i + \beta_i - 1}}{z_i!}
{\mbox{e}}^{- \{t_i(\tau) + \frac{u}{\gamma_i}\}\lambda_i}
{\mbox{d}}\lambda_1 \dotsb {\mbox{d}}\lambda_d {\mbox{d}}u \\
=& \left\{\prod^d_{i=1} \frac{\Gamma (z_i + \beta_i)}{z_i!} \right\} \int_0^\infty u^{\alpha-1}
\prod^d_{j=1} \frac{1}{\left\{t_j(\tau) + \frac{u}{\gamma_j}\right\}^{z_j + \beta_j}} {\mbox{d}}u,\end{aligned}$$ and $$\begin{aligned}
\Gamma(\alpha) & \int \lambda_i p(z(\tau) \mid \lambda) \pi_{\alpha,\beta,\gamma}(\lambda) {\mbox{d}}\lambda \\
=& \int \lambda_i \left( \prod^d_{j=1} \frac{\lambda_j^{z_j + \beta_j - 1}}{z_j!} {\mbox{e}}^{-t_j(\tau) \lambda_j} \right)
\int^\infty_0 u^{\alpha-1} \exp \biggl(- u \sum_k \frac{\lambda_k}{\gamma_k} \biggr) {\mbox{d}}u
{\mbox{d}}\lambda_1 \dotsb {\mbox{d}}\lambda_d \\
= & \frac{\Gamma \left(z_i + \beta_i + 1 \right)}{{z_i!}}
\left\{\prod_{j \neq i} \frac{\Gamma \left(z_j + \beta_j \right)}{{z_j!}} \right\} \\
& \times \int_0^\infty u^{\alpha-1}
\frac{1}{\left\{ t_i(\tau) + \frac{u}{\gamma_i} \right\}^{z_i + \beta_i + 1}}
\left[ \prod_{k \neq i} \frac{1}{\left\{ t_k(\tau) + \frac{u}{\gamma_k} \right\}^{z_k + \beta_k}} \right]
{\mbox{d}}u.\end{aligned}$$ Thus, the posterior mean of $\lambda$ with respect to $\pi_{\alpha,\beta,\gamma}$ is given by $$\begin{aligned}
\hat{\lambda}_i^{(\alpha,\beta,\gamma)} :=&
\frac{\displaystyle \int_0^\infty \lambda_i p(z(\tau) \mid \lambda) \pi_{\alpha,\beta,\gamma}(\lambda) {\mbox{d}}\lambda}
{\displaystyle \int_0^\infty p(z(\tau) \mid \lambda) \pi_{\alpha,\beta,\gamma}(\lambda) {\mbox{d}}\lambda}
= \frac{z_i + \beta_i}{t_i(\tau)}
\frac{\displaystyle
\int_0^\infty \displaystyle
u^{\alpha-1} \prod^d_{j=1} \dfrac{1}{\left\{\frac{u}{t_j(\tau) \gamma_j} + 1 \right\}^{z_j + \beta_j +\delta_{ij}}} {\mbox{d}}u}
{\displaystyle
\int_0^\infty \displaystyle
u^{\alpha-1} \prod^d_{j=1} \dfrac{1}{\left\{\frac{u}{t_j(\tau) \gamma_j} + 1 \right\}^{z_j + \beta_j}} {\mbox{d}}u}.\end{aligned}$$
We have $$\begin{aligned}
{{\rm E}}[x_i h(x) \mid \lambda] =&
\sum_x \prod^d_{j=1} \frac{\lambda_j^{x_j}}{x_j!} {\mbox{e}}^{-\lambda_j} x_i h (x)
= \sum_x \prod^d_{j=1} \lambda_i \frac{\lambda ^{x_j}_j}{x_j!} {\mbox{e}}^{-\lambda_j} h(x + \delta_i) \\
=& {{\rm E}}[\lambda_i h(x + \delta_i) \mid \lambda].\end{aligned}$$
1) By partial integration, $$\begin{aligned}
K (\gamma,&\ x, \alpha) = \int^\infty_0 u^{\alpha-1} \prod^d_{i=1} \frac{1}{(u/\gamma_i + 1)^{x_i}} {\mbox{d}}u \\
=& \left[ \frac{u^\alpha}{\alpha} \prod^d_{i=1} \frac{1}{(u/\gamma_i + 1)^{x_i}} \right]^\infty_0
+ \int^\infty_0 \frac{u^\alpha}{\alpha} \sum^d_{i=1} \left\{\prod_{j \neq i} \frac{1}{(u/\gamma_j + 1)^{x_j}} \right\}
\frac{x_i/\gamma_i}{(u/\gamma_i + 1)^{x_i +1}} {\mbox{d}}u \\
=& \frac{1}{\alpha} \sum_i \frac{x_i}{\gamma_i} K(\gamma, x + \delta_i, \alpha+1).\end{aligned}$$ 2) We have $$\begin{aligned}
K (\gamma, x + \delta_i, \alpha+1)
= & \int_0^\infty u^\alpha \Bigl\{\prod_j \frac{1}{(u/\gamma_j + 1)^{x_j}} \Bigr\} \frac{1}{u/\gamma_i + 1} {\mbox{d}}u \\
= & \int_0^\infty u^{\alpha-1} \Bigl\{\prod_j \frac{1}{(u/\gamma_j + 1)^{x_j}} \Bigr\} \frac{1}{u/\gamma_i + 1}
\gamma_i (u/\gamma_i + 1 - 1) {\mbox{d}}u \\
= & \gamma_i K (\gamma, x, \alpha) - \gamma_i K(\gamma, x + \delta_i, \alpha).\end{aligned}$$ 3) From , we have $$\begin{aligned}
\sum_i b_i K (\gamma, x+ \delta_i, \alpha)
=& \sum_i b_i \left\{ K (\gamma, x, \alpha) - \frac{1}{\gamma_i} K (\gamma, x + \delta_i, \alpha+1) \right\}
\notag \\
= & \frac{b_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}}{\alpha} \alpha K(\gamma, x, \alpha) - \sum_i \frac{b_i}{\gamma_i} K (\gamma, x + \delta_i, \alpha+1).\end{aligned}$$ By using , $$\begin{aligned}
\sum_i b_i K (\gamma, x+ \delta_i, \alpha)
=&
\frac{b_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}}{\alpha} \sum_i \frac{x_i}{\gamma_i} K (\gamma, x + \delta_i, \alpha+1)
- \sum_i \frac{b_i}{\gamma_i} K (\gamma, x + \delta_i, \alpha+1) \\
= & \sum_i \left(\frac{b_{{\raisebox{-0.25ex}{\scalebox{1.4}{${\cdot}$}}}}}{\alpha} \frac{x_i}{\gamma_i} - \frac{b_i}{\gamma_i} \right) K(\gamma, x + \delta_i, \alpha+1) .\end{aligned}$$
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was partially supported by Grant-in-Aid for Scientific Research (23300104, 23650144) and by the Aihara Project, the FIRST program from JSPS, initiated by CSTP.
[10]{} natexlab\#1[\#1]{}
Dawid, A. P. (1983). Invariant prior distributions. In Kotz, S., Johnson, N. L., Read, C. B. (Eds.), *Encyclopedia of Statistical Sciences, vol. 4.* Wiley-Interscience, New York, 228–236.
George, E. I., Liang, F. & Xu, X. (2006). Improved minimax predictive densities under Kullback–Leibler loss. *Annals of Statistics* **34**, 78–91.
George, E. I., Liang, F. & Xu, X. (2012). From minimax shrinkage estimation to minimax shrinkage prediction. *Statistical Science* **27**, 82–94.
George, E. I. & Xu, X. (2008). Predictive density estimation for multiple regression. *Econometric Theory* **24**, 528–544.
Ghosh, M. & Yang, M.-C. (1988). Simultaneous estimation of Poisson means under entropy loss. *Annals of Statistics* **16**, 278–291.
Kobayashi, K. & Komaki, F. (2008). Bayesian shrinkage prediction for the regression problem. *Journal of Multivariate Analysis* **99**, 1888–1905.
Komaki, F. (2001). A shrinkage predictive distribution for multivariate normal observables. *Biometrika* **88**, 859–864.
Komaki, F. (2004). Simultaneous prediction of independent Poisson observables. *Annals of Statistics* **32**, 1744–1769.
Komaki, F. (2006). A class of proper priors for Bayesian simultaneous prediction of independent Poisson observables. *Journal of Multivariate Analysis* **97**, 1815–1828.
Komaki, F. (2013). Asymptotic properties of Bayesian predictive densities when the distributions of data and target variables are different, submitted.
|
---
abstract: 'We present a language model consisting of a collection of costed bidirectional finite state automata associated with the head words of phrases. The model is suitable for incremental application of lexical associations in a dynamic programming search for optimal dependency tree derivations. We also present a model and algorithm for machine translation involving optimal “tiling” of a dependency tree with entries of a costed bilingual lexicon. Experimental results are reported comparing methods for assigning cost functions to these models. We conclude with a discussion of the adequacy of annotated linguistic strings as representations for machine translation.'
author:
- |
Hiyan Alshawi\
AT&T Research\
600 Mountain Avenue, Murray Hill, NJ 07974, USA\
hiyan@research.att.com
title: |
Head Automata and Bilingual Tiling:\
Translation with Minimal Representations
---
Introduction
============
Until the advent of statistical methods in the mainstream of natural language processing, syntactic and semantic representations were becoming progressively more complex. This trend is now reversing itself, in part because statistical methods reduce the burden of detailed modeling required by constraint-based grammars, and in part because statistical models for converting natural language into complex syntactic or semantic representations is not well understood at present. At the same time, lexically centered views of language have continued to increase in popularity. We can see this in lexicalized grammatical theories, head-driven parsing and generation, and statistical disambiguation based on lexical associations.
These themes — simple representations, statistical modeling, and lexicalism — form the basis for the models and algorithms described in the bulk of this paper. The primary purpose is to build effective mechanisms for machine translation, the oldest and still the most commonplace application of non-superficial natural language processing. A secondary motivation is to test the extent to which a non-trivial language processing task can be carried out without complex semantic representations.
In Section \[sec:headauto\] we present reversible mono-lingual models consisting of collections of simple automata associated with the heads of phrases. These [*head automata*]{} are applied by an algorithm with admissible incremental pruning based on semantic association costs, providing a practical solution to the problem of combinatoric disambiguation (Church and Patil 1982). The model is intended to combine the lexical sensitivity of N-gram models (Jelinek et al. 1992) and the structural properties of statistical context free grammars (Booth 1969) without the computational overhead of statistical lexicalized tree-adjoining grammars (Schabes 1992, Resnik 1992). The quantitative dependency model described here grew out of the model presented in Alshawi 1996a. An alternative model based on transducer versions of the automata is described in Alshawi 1996b.
For translation, we use a model for mapping dependency graphs written by the source language head automata. This model is coded entirely as a bilingual lexicon, with associated cost parameters. The transfer algorithm described in Section \[sec:transfer\] searches for the lowest cost ‘tiling’ of the target dependency graph with entries from the bilingual lexicon. Dynamic programming is again used to make exhaustive search tractable, avoiding the combinatoric explosion of shake-and-bake translation (Whitelock 1992, Brew 1992).
In Section \[sec:costs\] we present a general framework for associating costs with the solutions of search processes, pointing out some benefits of cost functions other than log likelihood, including an error-minimization cost function for unsupervised training of the parameters in our translation application. Section \[sec:experiment\] briefly describes an English-Chinese translator employing the models and algorithms. We also present experimental results comparing the performance of different cost assignment methods.
Finally, we return to the more general discussion of representations for machine translation and other natural language processing tasks, arguing the case for simple representations close to natural language itself.
Head Automata Language Models {#sec:headauto}
=============================
Lexical and Dependency Parameters
---------------------------------
Head automata mono-lingual language models consist of a [*lexicon*]{}, in which each entry is a pair $\langle w,m \rangle$ of a word $w$ from a vocabulary $V$ and a head automaton $m$ (defined below), and a [*parameter table*]{} giving an assignment of costs to events in a generative process involving the automata.
We first describe the model in terms of the familiar paradigm of a generative statistical model, presenting the parameters as conditional probabilities. This gives us a stochastic version of dependency grammar (Hudson 1984).
Each derivation in the generative statistical model produces an [*ordered dependency tree*]{}, that is, a tree in which nodes dominate ordered sequences of left and right subtrees and in which the nodes have labels taken from the vocabulary $V$ and the arcs have labels taken from a set $R$ of relation symbols. When a node with label $w$ immediately dominates a node with label $w'$ via an arc with label $r$, we say that $w'$ is an [*r-dependent*]{} of the [*head*]{} $w$. The interpretation of this directed arc is that relation $r$ holds between particular instances of $w$ and $w'$. (A word may have several or no $r$-dependents for a particular relation $r$.) A recursive left-parent-right traversal of the nodes of an ordered dependency tree for a derivation yields the word string for the derivation.
A head automaton $m$ of a lexical entry $\langle w, m \rangle$ defines possible ordered local trees immediately dominated by $w$ in derivations. Model parameters for head automata, together with dependency parameters and lexical parameters, give a probability distribution for derivations.
A [*dependency parameter*]{}
> $P(\downarrow,w'|w,r')$
is the probability, given a head $w$ with a dependent arc with label $r'$, that $w'$ is the $r'$-dependent for this arc.
A [*lexical parameter*]{}
> $P(m,q|r,\downarrow,w)$
is the probability that a local tree immediately dominated by an $r$-dependent $w$ is derived by starting in state $q$ of some automaton $m$ in a lexical entry $\langle w,m \rangle$. The model also includes lexical parameters
> $P(w,m,q|\rhd)$
for the probability that $w$ is the head word for an entire derivation initiated from state $q$ of automaton $m$.
Head Automata
-------------
A head automaton is a weighted finite state machine that writes (or accepts) a pair of sequences of relation symbols from $R$:
> $(\langle r_1 \cdots r_k \rangle, \langle r_{k+1} \cdots r_n \rangle)$.
These correspond to the relations between a head word and the sequences of dependent phrases to its left and right (see Figure \[fig:automaton\]). The machine consists of a finite set $q_0, \cdots, q_s$ of states and an [*action table*]{} specifying the finite cost (non-zero probability) actions the automaton can undergo.
(3324,1740)(664,-1654) (1876,-1636)[(0,0)\[lb\][$w_k$]{}]{} (741,-1636)[(0,0)\[lb\][$w_1$]{}]{} (3176,-1186)[(0,0)\[lb\][. . .]{}]{} (1201,-1186)[(0,0)\[lb\][. . .]{}]{} (716,-1211)[(0,0)\[lb\][$r_1$]{}]{} (2526,-1636)[(0,0)\[lb\][$w_{k+1}$]{}]{} (3176,-1561)[(0,0)\[lb\][. . .]{}]{} (1201,-1561)[(0,0)\[lb\][. . .]{}]{} (2251, 14)[(0,0)\[lb\][$w$]{}]{} (2251,-361)[(0,0)\[lb\][$m$]{}]{} (3751,-1636)[(0,0)\[lb\][$w_{n}$]{}]{} (3726,-1211)[(0,0)\[lb\][$r_n$]{}]{} (2551,-1336)[(1425,300)]{} (676,-1336)[(1425,300)]{} (1801,-1036)[( 0,-1)[300]{}]{} (976,-1036)[( 0,-1)[300]{}]{} (676,-1336)[(1425,300)]{} (2951,-1036)[( 0,-1)[300]{}]{} (2601,-1211)[(0,0)\[lb\][$r_{k+1}$]{}]{} (1851,-1211)[(0,0)\[lb\][$r_k$]{}]{} (2161,-481)[(-2,-1)[900]{}]{} (2485,-499)[( 4,-3)[600]{}]{} (2101,-136)[( 0,-1)[225]{}]{} (2101,-361)[( 1,-1)[225]{}]{} (2326,-586)[( 1, 1)[225]{}]{} (2551,-361)[( 0, 1)[225]{}]{} (2551,-136)[(-1, 0)[450]{}]{} (2101,-136)[( 0, 1)[ 0]{}]{} (3676,-1036)[( 0,-1)[300]{}]{}
There are three types of action for an automaton $m$: left transitions, right transitions, and stop actions. These actions, together with associated probabilistic model parameters, are as follows.
- Left transition: if in state $q_{i-1}$, $m$ can write a symbol $r$ onto the right end of the current left sequence and enter state $q_i$ with probability $P(\leftarrow,q_i,r|q_{i-1},m)$.
- Right transition: if in state $q_{i-1}$, $m$ can write a symbol $r$ onto the left end of the current right sequence and enter state $q_i$ with probability $P(\rightarrow,q_i,r|q_{i-1},m)$.
- Stop: if in state $q$, $m$ can stop with probability $P(\Box|q,m)$, at which point the sequences are considered complete.
For a consistent probabilistic model, the probabilities of all transitions and stop actions from a state $q$ must sum to unity. Any state of a head automaton can be an initial state, the probability of a particular initial state in a derivation being specified by lexical parameters. A derivation of a pair of symbol sequence thus corresponds to the selection of an initial state, a sequence of zero or more transitions (writing the symbols) and a stop action. The probability, given an initial state $q$, that automaton $m$ will a generate a pair of sequences, i.e.
> $P(\langle r_1 \cdots r_k \rangle, \langle r_{k+1} \cdots r_n \rangle | m, q)$
is the product of the probabilities of the actions taken to generate the sequences. The case of zero transitions will yield empty sequences, corresponding to a leaf node of the dependency tree.
From a linguistic perspective, head automata allow for a compact, graded, notion of lexical subcategorization (Gazdar et al. 1985) and the linear order of a head and its dependent phrases. Lexical parameters can control the saturation of a lexical item (for example a verb that is both transitive and intransitive) by starting the same automaton in different states. Head automata can also be used to code a grammar in which states of an automaton for word $w$ corresponds to X-bar levels (Jackendoff 1977) for phrases headed by $w$.
Head automata are formally more powerful than finite state automata that accept regular languages in the following sense. Each head automaton defines a formal language with alphabet $R$ whose strings are the concatenation of the left and right sequence pairs written by the automaton. The class of languages defined in this way clearly includes all regular languages, since strings of a regular language can be generated, for example, by a head automaton that only writes a left sequence. Head automata can also accept some non-regular languages requiring coordination of the left and right sequences, for example the language $a^nb^n$ (requiring two states), and the language of palindromes over a finite alphabet.
Derivation Probability
----------------------
Let the probability of generating an ordered dependency subtree $D$ headed by an $r$-dependent word $w$ be $P(D|w,r)$. The recursive process of generating this subtree proceeds as follows:
1. Select an initial state $q$ of an automaton $m$ for $w$ with lexical probability $P(m,q|r,\downarrow,w)$.
2. Run the automaton $m_0$ with initial state $q$ to generate a pair of relation sequences with probability $P(\langle r_1 \cdots r_k \rangle,\langle r_{k+1} \cdots r_n \rangle|m,q)$.
3. For each relation $r_i$ in these sequences, select a dependent word $w_i$ with dependency probability $P(\downarrow,w_i|w,r_i)$.
4. For each dependent $w_i$, recursively generate a subtree with probability $P(D_i|w_i,r_i)$.
We can now express the probability $P(D_0)$ for an entire ordered dependency tree derivation $D_0$ headed by a word $w_0$ as
> $P(D_0) = \\
> \hspace*{10mm} P(w_0,m_0,q_0|\rhd) \\
> \hspace*{10mm} P(\langle r_1 \cdots r_k \rangle,\langle r_{k+1} \cdots r_n \rangle|m_0,q_0)\\
> \hspace*{10mm} \prod_{1 \leq i \leq n}
> P(\downarrow,w_i|w_0,r_i) P(D_i|w_i,r_i).$
In the translation application we search for the highest probability derivation (or more generally, the N-highest probability derivations). For other purposes, the probability of strings may be of more interest. The probability of a string according to the model is the sum of the probabilities of derivations of ordered dependency trees yielding the string.
In practice, the number of parameters in a head automaton language model is dominated by the dependency parameters, that is, $O(|V|^2|R|)$ parameters. This puts the size of the model somewhere in between 2-gram and 3-gram model. The similarly motivated link grammar model (Lafferty, Sleator and Temperley 1992) has $O(|V|^3)$ parameters. Unlike simple N-gram models, head automata models yield an interesting distribution of sentence lengths. For example, the average sentence length for Monte-Carlo generation with our probabilistic head automata model for ATIS was 10.6 words (the average was 9.7 words for the corpus it was trained on).
Analysis and Generation
=======================
Analysis
--------
Head automaton models admit efficient lexically driven analysis (parsing) algorithms in which partial analyses are costed incrementally as they are constructed. Put in terms of the traditional parsing issues in natural language understanding, “semantic" associations coded as dependency parameters are applied at each parsing step allowing semantically suboptimal analyses to be eliminated, so the analysis with the best semantic score can be identified without scoring an exponential number of syntactic parses. Since the model is lexical, linguistic constructions headed by lexical items not present in the input are not involved in the search the way they are with typical top-down or predictive parsing strategies.
We will sketch an algorithm for finding the lowest cost ordered dependency tree derivation for an input string in polynomial time in the length of the string. In our experimental system we use a more general version of the algorithm to allow input in the form of word lattices.
The algorithm is a bottom-up tabular parser (Younger 1967, Early 1970) in which constituents are constructed “head-outwards” (Kay 1989, Sata and Stock 1989). Since we are analyzing bottom-up with generative model automata, the algorithm ‘runs’ the automata backwards. Edges in the parsing lattice (or “chart”) are tuples representing partial or complete phrases headed by a word $w$ from position $i$ to position $j$ in the string:
> $\langle w,t,i,j,m,q,c \rangle$.
Here $m$ is the head automaton for $w$ in this derivation; the automaton is in state $q$; $t$ is the dependency tree constructed so far, and $c$ is the cost of the partial derivation. We will use the notation $C(x|y)$ for the cost of a model event with probability $P(x|y)$; the assignment of costs to events is discussed in Section \[sec:costs\].
[*Initialization:* ]{} For each word $w$ in the input between positions $i$ and $j$, the lattice is initialized with phrases
> $\langle w,\{\},i,j,m,q_f,c_f \rangle$
for any lexical entry $\langle w,m \rangle$ and any final state $q_f$ of the automaton $m$ in the entry. A final state is one for which the stop action cost $c_f = C(\Box|q_f,m)$ is finite.
[*Transitions:* ]{} Phrases are combined bottom-up to form progressively larger phrases. There are two types of combination corresponding to left and right transitions of the automaton for the word acting as the head in the combination. We will specify left combination; right combination is the mirror image of left combination. If the lattice contains two phrases abutting at position $k$ in the string:
> $\langle w_1,t_1,i,k,m_1,q_1,c_1 \rangle$\
> $\langle w_2,t_2,k,j,m_2,q_2,c_2 \rangle$,
and the parameter table contains the following finite costs parameters (a left $r$-transition of $m_2$, a lexical parameter for $w_1$, and an $r$-dependency parameter):
> $c_3=C(\leftarrow,q_2,r|q'_2,m_2)$\
> $c_4=C(m_1,q_1|r,\downarrow,w_1)$\
> $c_5=C(\downarrow,w_1|w_2,r)$,
then build a new phrase headed by $w_2$ with a tree $t_2'$ formed by adding $t_1$ to $t_2$ as an $r$-dependent of $w_2$:
> $\langle w_2,t_2',i,j,m_2,q_2',c_1+c_2+c_3+c_4+c_5 \rangle$.
When no more combinations are possible, for each phrase spanning the entire input we add the appropriate start of derivation cost to these phrases and select the one with the lowest total cost.
[*Pruning:* ]{} The dynamic programming condition for pruning suboptimal partial analyses is as follows. Whenever there are two phrases
> $p = \langle w,t,i,j,m,q,c \rangle$\
> $p' = \langle w,t',i,j,m,q,c' \rangle$,
and $c'$ is greater than $c$, then we can remove $p'$ because for any derivation involving $p'$ that spans the entire string, there will be a lower cost derivation involving $p$. This pruning condition is effective at curbing a combinatorial explosion arising from, for example, prepositional phrase attachment ambiguities (coded in the alternative trees $t$ and $t'$).
The worst case asymptotic time complexity of the analysis algorithm is $O(\mbox{min}(n^2,|V|^2) n^3)$, where $n$ is the length of an input string and $|V|$ is the size of the vocabulary. This limit can be derived in a similar way to cubic time tabular recognition algorithms for context free grammars (Younger 1967) with the grammar related term being replaced by the term $\mbox{min}(n^2,|V|^2)$ since the words of the input sentence also act as categories in the head automata model. In this context “recognition” refers to checking that the input string can be generated from the grammar. Note that our algorithm is for analysis (in the sense of finding the best derivation) which, in general, is a higher time complexity problem than recognition.
Generation
----------
By generation here we mean determining the lowest cost linear surface ordering for the dependents of each word in an [*unordered*]{} dependency structure resulting from the transfer mapping described in Section \[sec:transfer\]. In general, the output of transfer is a dependency graph and the task of the generator involves a search for a backbone dependency tree for the graph, if necessary by adding dependency edges to join up unconnected components of the graph. For each graph component, the main steps of the search process, described non-deterministically, are
1. Select a node with word label $w$ having a finite start of derivation cost $C(w,m,q|\rhd)$.
2. Execute a path through the head automaton $m$ starting at state $q$ and ending at state $q'$ with a finite stop action cost $C(\Box|q',m)$. When making a transition with relation $r_i$ in the path, select a graph edge with label $r_i$ from $w$ to some previously unvisited node $w_i$ with finite dependency cost $C(\downarrow,w_i|w,r_i)$. Include the cost of the transition (e.g. $C(\rightarrow,q_i,r_i|q_{i-1},m)$) in the running total for this derivation.
3. For each dependent node $w_i$, select a lexical entry with cost $C(m_i,q_i|r_i,\downarrow,w_i)$, and recursively apply the machine $m_i$ from state $q_i$ as in step 2.
4. Perform a left-parent-right traversal of the nodes of the resulting dependency tree, yielding a target string.
The target string resulting from the lowest cost tree that includes all nodes in the graph is selected as the translation target string. The independence assumptions implicit in head automata models mean that we can select lowest cost orderings of local dependency trees, below a given relation $r$, independently in the search for the lowest cost derivation.
When the generator is used as part of the translation system, the dependency parameter costs are not, in fact, applied by the generator. Instead, because these parameters are independent of surface order, they are applied earlier by the transfer component, influencing the choice of structure passed to the generator.
Transfer Maps {#sec:transfer}
=============
Transfer Model Bilingual Lexicon
--------------------------------
The transfer model defines possible mappings, with associated costs, of dependency trees with source-language word node labels into ones with target-language word labels. Unlike the head automata monolingual models, the transfer model operates with unordered dependency trees, that is, it treats the dependents of a word as an unordered bag. The model is general enough to cover the common translation problems discussed in the literature (e.g. Lindop and Tsujii 1991 and Dorr 1994) including many-to-many word mapping, argument switching, and head switching.
A transfer model consists of a bilingual lexicon and a transfer parameter table. The model uses [*dependency tree fragments*]{}, which are the same as unordered dependency trees except that some nodes may not have word labels. In the [*bilingual lexicon*]{}, an entry for a source word $w_i$ (see top portion of Figure \[fig:tile\]) has the form
> $\langle w_i,H_i,n_i,G_i,f_i \rangle$
where $H_i$ is a source language tree fragment, $n_i$ (the [*primary node*]{}) is a distinguished node of $H_i$ with label $w_i$, $G_i$ is a target tree fragment, and $f_i$ is a [*mapping function*]{}, i.e. a (possibly partial) function from the nodes of $H_i$ to the nodes of $G_i$.
The [*transfer parameter table*]{} specifies costs for the application of transfer entries. In a context-independent model, each entry has a single cost parameter. In context-dependent transfer models, the cost function takes into account the identities of the labels of the arcs and nodes dominating $w_i$ in the source graph. (Context dependence is discussed further in Section \[sec:costs\].) The set of transfer parameters may also include costs for the [*null transfer entries*]{} for $w_i$, for use in derivations in which $w_i$ is translated by the entry for another word $v$. For example, the entry for $v$ might be for translating an idiom involving $w_i$ as a modifier.
Each entry in the bilingual lexicon specifies a way of mapping part of a dependency tree, specifically that part “matching” (as explained below) the source fragment of the entry, into part of a target graph, as indicated by the target fragment. Entry mapping functions specify how the set of target fragments for deriving a translation are to be combined: whenever an entry is applied, a global node-mapping function is extended to include the entry mapping function.
Matching, Tiling, and Derivation
--------------------------------
Transfer mapping takes a source dependency tree $S$ from analysis and produces a minimum cost derivation of a target graph $T$ and a (possibly partial) function $f$ from source nodes to target nodes. In fact, the transfer model is applicable to certain types of source dependency graphs that are more general than trees, although the version of the head automata model described here only produces trees.
We will say that a tree fragment $H$ [*matches*]{} an unordered dependency tree $S$ if there is a function $g$ (a [*matching function*]{}) from the nodes of $H$ to the nodes of $S$ such that
- $g$ is a total one-one function;
- if a node $n$ of $H$ has a label, and that label is word $w$, then the word label for $g(n)$ is also $w$;
- for every arc in $H$ with label $r$ from node $n_1$ to node $n_2$, there is an arc with label $r$ from $g(n_1)$ to $g(n_2)$.
Unlike first order unification, this definition of matching is not commutative and is not deterministic in that there may be multiple matching functions for applying a bilingual entry to an input source tree. A particular match of an entry against a dependency tree can be represented by the matching function $g$, a set of arcs $A$ in $S$, and the (possibly context dependent) cost $c$ of applying the entry.
A [*tiling*]{} of a source graph with respect to a transfer model is a set of entry matches
> $\{\langle E_1, g_1, A_1, c_1 \rangle, \cdots, \langle E_k, g_k, A_k, c_k \rangle\}$
which is such that
- $k$ is the number of nodes in the source tree $S$.
- Each $E_i$, $1 \leq i \leq k$, is a bilingual entry $\langle w_i,H_i,n_i,G_i,f_i \rangle$ matching $S$ with function $g_i$ (see Figure \[fig:tile\]) and arcs $A_i$.
- For primary nodes $n_i$ and $n_j$ of two distinct entries $E_i$ and $E_j$, $g_i(n_i)$ and $g_j(n_j)$ are distinct.
- The sets of edges $A_i$ form a partition of the edges of $S$.
- The images $g_i(L_i)$ form a partition of the nodes of $S$, where $L_i$ is the set of [*labeled*]{} source nodes in the source fragment $H_i$ of $E_i$.
- $c_i$ is the cost of the match specified by the parameter table.
(4839,3270)(964,-3148) (4741,-526)[( 5,-3)[375]{}]{} (5116,-751)[( 1, 0)[675]{}]{} (5791,-751)[(-1, 1)[600]{}]{} (5191,-151)[(-6,-5)[450]{}]{} (2101,-1261)[(-3,-5)[1125]{}]{} (976,-3136)[( 1, 0)[2925]{}]{} (3901,-3136)[(-1, 1)[1800]{}]{} (5099,-2224)[(-1, 3)[225]{}]{} (4874,-1549)[( 6, 1)[450]{}]{} (2101,-211)[( 0,-1)[1875]{}]{} (2251,-136)[( 1, 0)[2400]{}]{} (4651,-136)[( 1, 0)[450]{}]{} (2551,-586)[(-1, 1)[450]{}]{} (4651,-2611)[( 5,-3)[375]{}]{} (5026,-2836)[( 1, 0)[675]{}]{} (5701,-2836)[(-1, 1)[600]{}]{} (5101,-2236)[(-6,-5)[450]{}]{} (4501,-811)[(0,0)\[lb\][$G_i$]{}]{} (1501,-811)[(0,0)\[lb\][$H_i$]{}]{} (4576,-2986)[(0,0)\[lb\][$T$]{}]{} (1276,-2911)[(0,0)\[lb\][$S$]{}]{} (2176,-961)[(0,0)\[lb\][$g_i$]{}]{} (3376, 14)[(0,0)\[lb\][$f_i$]{}]{} (3526,-2086)[(0,0)\[lb\][$f$]{}]{} (1801,-586)[( 1, 0)[750]{}]{} (5101,-2236) (5176,-136) (2101,-136) (2101,-2236) (2101,-2236)[(-2,-3)[300]{}]{} (2101,-136)[(-2,-3)[300]{}]{} (5296,-1502)[(-1,-3)[225]{}]{} (2101,-2236)[( 1, 0)[2400]{}]{} (4501,-2236)[( 1, 0)[450]{}]{} (2101,-2236)[( 1, 1)[450]{}]{} (2101,-2236)[(-2, 3)[300]{}]{} (2551,-2686)[(-1, 1)[450]{}]{} (1801,-2686)[( 1, 0)[750]{}]{}
A tiling of $S$ yields a costed derivation of a target dependency graph $T$ as follows:
- The cost of the derivation is the sum of the costs $c_i$ for each match in the tiling.
- The nodes and arcs of $T$ are composed of the nodes and arcs of the target fragments $G_i$ for the entries $E_i$.
- Let $f_i$ and $f_j$ be the mapping functions for entries $E_i$ and $E_j$. For any node $n$ of $S$ for which target nodes $f_i(g_i^{-1}(n))$ and $f_j(g_i^{-1}(n))$ are defined, these two nodes are identified as a single node $f(n)$ in $T$.
The merging of target fragment nodes in the last condition has the effect of joining the target fragments in a consistent fashion. The node mapping function $f$ for the entire tree thus has a different role from the alignment function in the IBM statistical translation model (Brown et al. 1990, 1993); the role of the latter includes the linear ordering of words in the target string. In our approach, target word order is handled exclusively by the target monolingual model.
Transfer Algorithm
------------------
The main transfer search is preceded by a bilingual lexicon matching phase. This leads to greater efficiency as it avoids repeating matching operations during the search phase, and it allows a static analysis of the matching entries and source tree to identify subtrees for which the search phase can safely prune out suboptimal partial translations.
#### Transfer Configurations {#transfer-configurations .unnumbered}
In order to apply target language model relation costs incrementally, we need to distinguish between complete and incomplete arcs: an arc is complete if both its nodes have labels, otherwise it is incomplete. The output of the lexicon matching phrase, and the partial derivations manipulated by the search phase are both in the form of [*transfer configurations*]{}
> $\langle S, R, T, P, f, c, I \rangle$
where $S$ is the set of source nodes and arcs consumed so far in the derivation, $R$ the remaining source nodes and arcs, $f$ the mapping function built so far, $T$ the set of nodes and complete arcs of the target graph, $P$ the set of incomplete target arcs, $c$ the partial derivation cost, and $I$ a set of source nodes for which entries have yet to be applied.
#### Lexical matching phase {#lexical-matching-phase .unnumbered}
The algorithm for lexical matching has a similar control structure to standard unification algorithms, except that it can result in multiple matches. We omit the details. The lexicon matching phase returns, for each source node $i$, a set of [*runtime entries*]{}. There is one runtime entry for each successful match and possibly a null entry for the node if the word label for $i$ is included in successful matches for other entries. Runtime entries are transfer configurations of the form
> $\langle H_i, \phi, G_i, P_i, f_i, c_i, \{i\} \rangle$
in which $H_i$ is the source fragment for the entry with each node replaced by its image under the applicable matching function; $G_i$ the target fragment for the entry, except for the incomplete arcs $P_i$ of this fragment; $f_i$ the composition of mapping function for the entry with the inverse of the matching function; $c_i$ the cost of applying the entry in the context of its match with the source graph plus the cost in the target model of the arcs in $G_i$.
#### Transfer Search {#transfer-search .unnumbered}
Before the transfer search proper, the resulting runtime entries together with the source graph are analyzed to determine [*decomposition nodes*]{}. A decomposition node $n$ is a source tree node for which it is safe to prune suboptimal translations of the subtree dominated by $n$. Specifically, it is checked that $n$ is the root node of all source fragments $H_n$ of runtime entries in which both $n$ [*and*]{} its node label are included, and that $f_n(n)$ is not dominated by (i.e. not reachable via directed arcs from) another node in the target graph $G_n$ of such entries.
Transfer search maintains a set $M$ of active runtime entries. Initially, this is the set of runtime entries resulting from the lexicon matching phase. Overall search control is as follows:
1. Determine the set of decomposition nodes.
2. Sort the decomposition nodes into a list $D$ such that if $n_1$ dominates $n_2$ in $S$ then $n_2$ precedes $n_1$ in $D$.
3. If $D$ is empty, apply the subtree transfer search (given below) to $S$, return the lowest cost solution, and stop.
4. Remove the first decomposition node $n$ from $D$ and apply the subtree transfer search to the subtree $S'$ dominated by $n$, to yield solutions\
$\langle S',\phi,T',\phi,f',c',\phi \rangle$.
5. Partition these solutions into subsets with the same word label for the node $f'(n)$, and select the solution with lowest cost $c'$ from each subset.
6. Remove from $M$ the set of runtime entries for nodes in $S'$.
7. For each selected subtree solution, add to $M$ a new runtime entry $\langle S', \phi, T', f', c', \{n\} \rangle$.
8. Repeat from step 3.
The subtree transfer search maintains a queue $Q$ of configurations corresponding to partial derivations for translating the subtree. Control follows a standard non-deterministic search paradigm:
1. Initialize $Q$ to contain a single configuration\
$\langle \phi, R_0, \phi, \phi, \phi, 0, I_0 \rangle$ with the input subtree $R_0$ and the set of nodes $I_0$ in $R_0$.
2. If $Q$ is empty, return the lowest cost solution found and stop.
3. Remove a configuration $\langle S, R, T, P, f, c, I \rangle$ from the queue.
4. If $R$ is empty, add the configuration to the set of subtree solutions.
5. Select a node $i$ from $I$.
6. For each runtime entry $\langle H_i, \phi, G_i, P_i, f_i, c_i, \{i\} \rangle$ for $i$, if $H_i$ is a subgraph of $R$, add to $Q$ a configuration $\langle S \cup H_i, R-H_i, T \cup G_i \cup G', P \cup P_i-G', f \cup f_i, c+c_i+c_{G'}, I-\{i\} \rangle$, where $G'$ is the set of newly completed arcs (those in $P \cup P_i$ with both node labels in $T \cup G_i \cup P \cup P_i$) and $c_{G'}$ is the cost of the arcs $G'$ in the target language model.
7. For any source node $n$ for which $f(n)$ and $f_i(n)$ are both defined, merge these two target nodes.
8. Repeat from step 2.
Keeping the arcs $P$ separate in the configuration allows efficient incremental application of target dependency costs $c_{G'}$ during the search, so these costs are taken into account in the pruning step of the overall search control. This way we can keep the benefits of monolingual/bilingual modularity (Isabelle and Macklovitch 1986) without the computational overhead of transfer-and-filter (Alshawi et al. 1992).
It is possible to apply the subtree search directly to the whole graph starting with the initial runtime entries from lexical matching. However, this would result in an exponential search, specifically a search tree with a branching factor of the order of the number of matching entries per input word. Fortunately, long sentences typically have several decomposition nodes, such as the heads of noun phrases, so the search as described is factored into manageable components.
Cost Functions {#sec:costs}
==============
Costed Search Processes
-----------------------
The head automata model and transfer model were originally conceived as probabilistic models. In order to take advantage of more of the information available in our training data, we experimented with cost functions that make use of incorrect translations as negative examples and also to treat the correctness of a translation hypothesis as a matter of degree.
To experiment with different models, we implemented a general mechanism for associating costs to solutions of a search process. Here, a search process is conceptualized as a non-deterministic computation that takes a single input string, undergoes a sequence of state transitions in a non-deterministic fashion, then outputs a solution string. Process states are distinct from, but may include, head automaton states.
A cost function for a search process is a real valued function defined on a pair of equivalence classes of process states. The first element of the pair, a [*context*]{} $c$, is an equivalence class of states before transitions. The second element, an [*event*]{} $e$, is an equivalence class of states after transitions. (The equivalence relations for contexts and events may be different.) We refer to an event-context pair as a [*choice*]{}, for which we use the notation
> $(e | c)$
borrowed from the special case of conditional probabilities. The cost of a derivation of a solution by the process is taken to be the sum of costs of choices involved in the derivation.
We represent events and contexts by finite sequences of symbols (typically words or relation symbols in the translation application). We write
> $C(a_1 \cdots a_n | b_1 \cdots b_k)$
for the cost of the event represented by $\langle a_1 \cdots a_n \rangle$ in the context represented by$\langle b_1 \cdots b_k \rangle$.
“Backed off” costs can be computed by averaging over larger equivalence classes (represented by shorter sequences in which positions are eliminated systematically). A similar smoothing technique has been applied to the specific case of prepositional phrase attachment by Collins and Brooks (1995). We have used backed off costs in the translation application for the various cost functions described below. Although this resulted in some improvement in testing, so far the improvement has not been statistically significant.
Model Cost Functions
--------------------
Taken together, the events, contexts, and cost function constitute a [*process cost model*]{}, or simply a [*model*]{}. The cost function specifies the [*model parameters*]{}; the other components are the [*model structure*]{}.
We have experimented with a number of model types, including the following.
[*Probabilistic model:* ]{} In this model we assume a probability distribution on the possible events for a context, that is,
> $\sum_{e} P(e | c) = 1$.
The cost parameters of the model are defined as:
> $C(e | c) = -\ln(P(e | c))$.
Given a set of solutions from executions of a process, let $n^+(e|c)$ be the number of times choice $(e|c)$ was taken leading to acceptable solutions (e.g. correct translations) and $n^+(c)$ be the number of times context $c$ was encountered for these solutions. We can then estimate the probabilistic model costs with
> $C(e | c) \approx \ln(n^+(c)) - \ln(n^+(e | c))$.
[*Discriminative model:* ]{} The costs in this model are likelihood ratios comparing positive and negative solutions, for example correct and incorrect translations. (See Dunning 1993 on the application of likelihood ratios in computational linguistics.) Let $n^-(e|c)$ be the count for choice $(e|c)$ leading to negative solutions. The cost function for the discriminative model is estimated as
> $C(e | c) \approx \ln(n^-(e | c)) - \ln(n^+(e | c))$.
[*Mean distance model:* ]{} In the mean distance model, we make use of some measure of goodness of a solution $t_s$ for some input $s$ by comparing it against an ideal solution $\hat{t}_s$ for $s$ with a distance metric $h$:
> $h(t_s,\hat{t}_s) \mapsto d$
in which $d$ is a non-negative real number. A parameter for choice $(e|c)$ in the distance model
> $C(e|c) = E_h(e|c)$
is the mean value of $h(t_s,\hat{t}_s)$ for solutions $t_s$ produced by derivations including the choice $(e|c)$.
[*Normalized distance model:* ]{} The mean distance model does not use the constraint that a particular choice faced by a process is always a choice between events with the same context. It is also somewhat sensitive to peculiarities of the distance function $h$. With the same assumptions we made for the mean distance model, let
> $E_h(c)$
be the average of $h(t_s,\hat{t}_s)$ for solutions derived from sequences of choices including the context $c$. The cost parameter for $(e|c)$ in the normalized distance model is
> $C(e | c) = \frac{E_h(e|c)}{E_h(c)}$,
that is, the ratio of the expected distance for derivations involving the choice and the expected distance for all derivations involving the context for that choice.
#### Reflexive Training {#reflexive-training .unnumbered}
If we have a manually translated corpus, we can apply the mean and normalized distance models to translation by taking the ideal solution $\hat{t}_s$ for translating a source string $s$ to be the manual translation for $s$. In the absence of good metrics for comparing translations, we employ a heuristic string distance metric to compare word selection and word order in $t_s$ and $\hat{t}_s$.
In order to train the model parameters without a manually translated corpus, we use a “reflexive” training method (similar in spirit to the “wake-sleep” algorithm, Hinton et al. 1995). In this method, our search process translates a source sentence $s$ to $t_s$ in the target language and then translates $t_s$ back to a source language sentence $s'$. The original sentence $s$ can then act as the ideal solution of the overall process. For this training method to be effective, we need a reasonably good initial model, i.e. one for which the distance $h(s,s')$ is inversely correlated with the probability that $t_s$ is a good translation of $s$.
Experimental System {#sec:experiment}
===================
We have built an experimental translation system using the monolingual and translation models described in this paper. The system translates sentences in the ATIS domain (Hirschman et al. 1993) between English and Mandarin Chinese. The translator is in fact a subsystem of a speech translation prototype, though the experiments we describe here are for transcribed spoken utterances. (We informally refer to the transcribed utterances as sentences.) The average time taken for translation of sentences (of unrestricted length) from the ATIS corpus was around 1.7 seconds with approximately 0.4 seconds being taken by the analysis algorithm and 0.7 seconds by the transfer algorithm.
English and Chinese lexicons of around 1200 and 1000 words respectively were constructed. Altogether, the entries in these lexicons made reference to around 200 structurally distinct head automata. The transfer lexicon contained around 3500 paired graph fragments, most of which were used in both transfer directions. With this model structure, we tried a number of methods for assigning cost functions. The nature of the training methods and their corresponding cost functions meant that different amounts of training data could be used, as discussed further below.
The methods make use of a supervised training set and an unsupervised training set, both sets being chosen at random from the 20,000 or so ATIS sentences available to us. The supervised training set comprised around 1950 sentences. A subcollection of 1150 of these sentences were translated by the system, and the resulting translations manually classified as ‘good’ (800 translations) or ‘bad’ (350 translations). The remaining 800 supervised training set sentences were hand-tagged for prepositional attachment points. (Prepositional phrase attachment is a major cause of ambiguity in the ATIS corpus, and moreover can affect English-Chinese translation, see Chen and Chen 1992.) The attachment information was used to generate additional negative and positive counts for dependency choices. The unsupervised training set consisted of approximately 13,000 sentences; it was used for automatic training (as described under ‘Reflexive Training’ above) by translating the sentences into Chinese and back to English.
A. [*Qualitative Baseline:* ]{} In this model, all choices were assigned the same cost except for irregular events (such as unknown words or partial analyses) which were all assigned a high penalty cost. This model gives an indication of performance based solely on model structure.
B. [*Probabilistic:* ]{} Counts for choices leading to good translations for sentences of the supervised training corpus, together with counts from the manually assigned attachment points, were used to compute negated log probability costs.
C. [*Discriminative:* ]{} The positive counts as in the probabilistic method, together with corresponding negative counts from bad translations or incorrect attachment choices, were used to compute log likelihood ratio costs.
D. [*Normalized Distance:* ]{} In this fully automatic method, normalized distance costs were computed from reflexive translation of the sentences in the unsupervised training corpus. The translation runs were carried out with parameters from method A.
E. [*Bootstrapped Normalized Distance:* ]{} The same as method D except that the system used to carry out the reflexive translation was running with parameters from method C.
Table \[tab:methods\] shows the results of evaluating the performance of these models for translating 200 unrestricted length ATIS sentences into Chinese. This was a previously unseen test set not included in any of the training sets. Two measures of translation acceptability are shown, as judged by a Chinese speaker. (In separate experiments, we verified that the judgments of this speaker were near the average of five Chinese speakers). The first measure, “meaning and grammar”, gives the percentage of sentence translations judged to preserve meaning without the introduction of grammatical errors. For the second measure, “meaning preservation”, grammatical errors were allowed if they did not interfere with meaning (in the sense of misleading the hearer).
-------- ------------- ------------------
Method Meaning and Meaning
Grammar (%) Preservation (%)
A 29 71
D 37 71
B 46 82
C 52 83
E 54 83
-------- ------------- ------------------
: Translation performance of different cost assignment methods
\[tab:methods\]
In the table, we have grouped together methods A and D for which the parameters were derived without human supervision effort, and methods B, C, and E which depended on the same amount of human supervision effort. This means that side by side comparison of these methods has practical relevance, even though the methods exploited different amounts of data. In the case of E, the supervision effort was used only as an oracle during training, not directly in the cost computations.
We can see from Table \[tab:methods\] that the choice of method affected translation quality (meaning and grammar) more than it affected preservation of meaning. A possible explanation is that the model structure was adequate for most lexical choice decisions because of the relatively low degree of polysemy in the ATIS corpus. For the stricter measure, the differences were statistically significant, according to the sign test at the 5% significance level, for the following comparisons: C and E each outperformed B and D, and B and D each outperformed A.
Language Processing and Semantic Representations
================================================
The translation system we have described employs only simple representations of sentences and phrases. Apart from the words themselves, the only symbols used are the dependency relations $R$. In our experimental system, these relation symbols are themselves natural language words, although this is not a necessary property of our models. Information coded explicitly in sentence representations by word senses and feature constraints in our previous work (Alshawi 1992) is implicit in the models used to derive the dependency trees and translations. In particular, dependency parameters and context-dependent transfer parameters give rise to an implicit, graded notion of word sense.
For language-centered applications like translation or summarization, for which we have a large body of examples of the desired behavior, we can think of the task in terms of the formal problem of modeling a relation between strings based on examples of that relation. By taking this viewpoint, we seem to be ignoring the intuition that most interesting natural language processing tasks (translation, summarization, interfaces) are semantic in nature. It is therefore tempting to conclude that an adequate treatment of these tasks requires the manipulation of artificial semantic representation languages with well-understood formal denotations. While the intuition seems reasonable, the conclusion might be too strong in that it rules out the possibility that natural language itself is adequate for manipulating semantic denotations. After all, this is the primary function of natural language.
The main justification for artificial semantic representation languages is that they are unambiguous by design. This may not be as critical, or useful, as it might first appear. While it is true that natural language is ambiguous and under-specified out of context, this uncertainty is greatly reduced by context to the point where further resolution (e.g. full scoping) is irrelevant to the task, or even the intended meaning. The fact that translation is insensitive to many ambiguities motivated the use of unresolved quasi-logical form for transfer (Alshawi et al. 1992).
To the extent that contextual resolution is necessary, context may be provided by the state of the language processor rather than complex semantic representations. Local context may include the state of local processing components (such as our head automata) for capturing grammatical constraints, or the identity of other words in a phrase for capturing sense distinctions. For larger scale context, I have argued elsewhere (Alshawi 1987) that memory activation patterns resulting from the process of carrying out an understanding task can act as global context without explicit representations of discourse. Under this view, the challenge is how to exploit context in performing a task rather than how to map natural language phrases to expressions of a formalism for coding meaning independently of context or intended use.
There is now greater understanding of the formal semantics of under-specified and ambiguous representations. In Alshawi 1995, I provide a denotational semantics for a simple under-specified language and argue for extending this treatment to a formal semantics of natural language strings as expressions of an under-specified representation. In this paradigm, ordered dependency trees can be viewed as natural language strings annotated so that some of the implicit relations are more explicit. A milder form of this kind of annotation is a bracketed natural language string. We are not advocating an approach in which linguistic structure is ignored (as it is in the IBM translator described by Brown et al. 1990), but rather one in which the syntactic and semantic structure of a string is implicit in the way it is processed by an interpreter.
One important advantage of using representations that are close to natural language itself is that it reduces the degrees of freedom in specifying language and task models, making these models easier to acquire automatically. With these considerations in mind, we have started to experiment with a version of the translator described here with even simpler representations and for which the model structure, not just the parameters, can be acquired automatically.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work on cost functions and training methods was carried out jointly with Adam Buchsbaum who also customized the English model to ATIS and integrated the translator into our speech translation prototype. Jishen He constructed the Chinese ATIS language model and bilingual lexicon and identified many problems with early versions of the transfer component. I am also grateful for advice and help from Don Hindle, Fernando Pereira, Chi-Lin Shih, Richard Sproat, and Bin Wu.
References {#references .unnumbered}
==========
Alshawi, H. 1996a. “Qualitative and Quantitative Models of Speech Translation”. In [*The Balancing Act: Combining Symbolic and Statistical Approaches to Language*]{}, edited by P. Resnik and J. Klavans, The MIT Press, Cambridge, Massachusetts.
Alshawi, H. 1996b. “Head Automata for Speech Translation”. In [*Proceedings of ICSLP 96, the Fourth International Conference on Spoken Language Processing*]{}, Philadelphia, Pennsylvania.
Alshawi, H. 1995. “Underspecified First Order Logics”. In [*Semantic Ambiguity and Underspecification*]{}, edited by K. van Deemter and S. Peters, CSLI Publications, Stanford, California.
Alshawi, H. 1992. [*The Core Language Engine*]{}. MIT Press, Cambridge, Massachusetts.
Alshawi, H. 1987. [*Memory and Context for Language Interpretation*]{}. Cambridge University Press, Cambridge, England.
Alshawi, H., D. Carter, B. Gamback and M. Rayner. 1992. “Swedish-English QLF Translation”. In [*The Core Language Engine*]{}, edited by H. Alshawi, The MIT Press, Cambridge, Massachusetts.
Booth, T. 1969. “Probabilistic Representation of Formal Languages”. [*Tenth Annual IEEE Symposium on Switching and Automata Theory*]{}.
Brew, C. 1992. “Letting the Cat out of the Bag: Generation for Shake-and-Bake MT”. Proceedings of COLING92, the International Conference on Computational Linguistics, Nantes, France.
Brown, P., J. Cocke, S. Della Pietra, V. Della Pietra, F. Jelinek, J. Lafferty, R. Mercer and P. Rossin. 1990. “A Statistical Approach to Machine Translation”. [*Computational Linguistics*]{} 16:79–85.
Brown, P.F., S.A. Della Pietra, V.J. Della Pietra, and R.L. Mercer. 1993. “The Mathematics of Statistical Machine Translation: Parameter Estimation”. [*Computational Linguistics*]{} 19:263–312.
Chen, K.H. and H. H. Chen. 1992. “Attachment and Transfer of Prepositional Phrases with Constraint Propagation”. [*Computer Processing of Chinese and Oriental Languages*]{}, Vol. 6, No. 2, 123–142.
Church K. and R. Patil. 1982. “Coping with Syntactic Ambiguity or How to Put the Block in the Box on the Table”. [*Computational Linguistics*]{} 8:139–149.
Collins, M. and J. Brooks. 1995. “Prepositional Phrase Attachment through a Backed-Off Model.” [*Proceedings of the Third Workshop on Very Large Corpora*]{}, Cambridge, Massachusetts, ACL, 27–38.
Dorr, B.J. 1994. “Machine Translation Divergences: A Formal Description and Proposed Solution”. [*Computational Linguistics*]{} 20:597–634.
Dunning, T. 1993. “Accurate Methods for Statistics of Surprise and Coincidence.” [*Computational Linguistics*]{}. 19:61–74.
Early, J. 1970. “An Efficient Context-Free Parsing Algorithm”. [*Communications of the ACM*]{} 14: 453–60.
Gazdar, G., E. Klein, G.K. Pullum, and I.A.Sag. 1985. [*Generalised Phrase Structure Grammar*]{}. Blackwell, Oxford.
Hinton, G.E., P. Dayan, B.J. Frey and R.M. Neal. 1995. “The ‘Wake-Sleep’ Algorithm for Unsupervised Neural Networks”. [*Science*]{} 268:1158–1161.
Hudson, R.A. 1984. [*Word Grammar*]{}. Blackwell, Oxford.
Hirschman, L., M. Bates, D. Dahl, W. Fisher, J. Garofolo, D. Pallett, K. Hunicke-Smith, P. Price, A. Rudnicky, and E. Tzoukermann. 1993. “Multi-Site Data Collection and Evaluation in Spoken Language Understanding”. In [*Proceedings of the Human Language Technology Workshop*]{}, Morgan Kaufmann, San Francisco, 19–24.
Isabelle, P. and E. Macklovitch. 1986. “Transfer and MT Modularity”, [*Eleventh International Conference on Computational Linguistics*]{}, Bonn, Germany, 115–117.
Jackendoff, R.S. 1977. [*X-bar Syntax: A Study of Phrase Structure*]{}. MIT Press, Cambridge, Massachusetts.
Jelinek, F., R.L. Mercer and S. Roukos. 1992. “Principles of Lexical Language Modeling for Speech Recognition”. In [*Advances in Speech Signal Processing*]{}, edited by S. Furui and M.M. Sondhi. Marcel Dekker, New York.
Lafferty, J., D. Sleator and D. Temperley. 1992. “Grammatical Trigrams: A Probabilistic Model of Link Grammar”. In [*Proceedings of the 1992 AAAI Fall Symposium on Probabilistic Approaches to Natural Language*]{}, 89-97.
Kay, M. 1989. “Head Driven Parsing”. In [*Proceedings of the Workshop on Parsing Technologies*]{}, Pittsburg, 1989.
Lindop, J. and J. Tsujii. 1991. “Complex Transfer in MT: A Survey of Examples”. Technical Report 91/5, Centre for Computational Linguistics, UMIST, Manchester, UK.
Resnik, P. 1992. “Probabilistic Tree-Adjoining Grammar as a Framework for Statistical Natural Language Processing”. In [*Proceedings of COLING-92*]{}, Nantes, France, 418-424.
Sata, G. and O. Stock. 1989. “Head-Driven Bidirectional Parsing”. In [*Proceedings of the Workshop on Parsing Technologies*]{}, Pittsburg, 1989.
Schabes, Y. 1992. “Stochastic Lexicalized Tree-Adjoining Grammars”. In [*Proceedings of COLING-92*]{}, Nantes, France, 426-432.
Whitelock, P.J. 1992. “Shake-and-Bake Translation”. Proceedings of COLING92, the International Conference on Computational Linguistics, Nantes, France.
Younger, D. 1967. Recognition and Parsing of Context-Free Languages in Time $n^3$. [*Information and Control*]{}, 10, 189–208.
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---
abstract: |
We measure the branching fraction for the charmless semi-inclusive process $B\to \etapr X_s$, where the meson has a momentum in the range 2.0 to 2.7 in the center-of-mass frame and $X_s$ represents a system comprising a kaon and zero to four pions. We find $\mathcal{B}(B\to \etapr
X_s)=(3.9\pm0.8\stat\pm0.5\syst\pm0.8{\ensuremath{\mathrm{(model)}}\xspace})\times 10^{-4}$. We also obtain the $X_s$ mass distribution and find that it tends to favor models predicting high masses.
title: '[**Study of high momentum production in $B\to\etapr X_s$** ]{}'
---
-PUB-[03]{}/[038]{}\
SLAC-PUB-[10295]{}\
authors\_oct2003
The production of high momentum mesons in $B$ meson decays is expected to be dominated by the $B\to\etapr X_s$ process, where $X_s$ is a strange hadronic system, generated by the $b\to sg^*$ transition as depicted in Fig. \[fig:diagrams\](a-c). Figure \[fig:diagrams\](d) shows the color-suppressed modes $\Bzb\to\etapr
D^{(*)0}$, which are significant sources of background and which have been measured for the first time recently [@ref:colsup]. Contributions from $b
\rightarrow u$ transitions and other sources of $\etapr$ are expected to be negligible [@ref:BaBarEtaPPi].
The large $B \to \etapr X_s$ branching fraction measured by the CLEO collaboration [@ref:FirstCleo], prompted intense theoretical activity, which focused the special character of the $\etapr$ meson as receiving much of its mass from the QCD anomaly . A later measurement by CLEO confirmed the large production, measuring $\mathcal{B}(B\to\etapr X_{nc})=(4.6\pm1.1\stat\pm0.4\syst\pm0.5{\rm (bkg)})\times10^{-4}$ [@ref:LastCleo], where $X_{nc}$ denotes a charmless recoiling hadronic system.\
The rate for $ B \to \etapr X_s$ and especially the fully background-subtracted distribution of the mass of $X_s$ can provide important clues to the dynamics of weak decays and to the structure of the isosinglet pseudoscalar mesons.\
We present results for the branching fraction $\mathcal{B}(B\to\etapr X_s)$ and the mass spectrum of $X_s$. The signal is analyzed for momentumbetween $2.0$ and $2.7~\gevc$ in the CM to suppress background coming from $b\to c\to \etapr$ cascades such as $B\to D_sX$ with $D_s\to\etapr X$, $B\to DX$ with $D\to\etapr X$, $B\to\Lambda_cX$ with $\Lambda_c\to\etapr X$. Our analysis is based on data collected with the detector [@BABARNIM] at the PEP-II asymmetric $e^+e^-$ collider located at the Stanford Linear Accelerator Center. An integrated luminosity of 81.4 , corresponding to 88.4 million pairs, was recorded at the resonance (on-resonance) and 9.6 were recorded 40 $\mev$ below this resonance (off-resonance), for continuum background studies.\
Two tracking devices are used for the detection of charged particles: a silicon vertex tracker consisting of five layers of double-sided silicon microstrip detectors, and a 40-layer central drift chamber, both operating in the 1.5 T magnetic field of a superconducting solenoid. Photons and electrons are detected by a CsI(Tl) electromagnetic calorimeter. Charged-particle identification is provided by the average energy loss ($dE/dx$) in the tracking devices, and by an internally reflecting ring-imaging Cherenkov detector covering the central region.\
We select events by requiring at least four charged tracks and a value of the ratio of the second to zeroth Fox-Wolfram moment [@ref:fox] less than 0.5.\
We form a $B$ candidate by combining an $\etapr\to\eta\pi^+\pi^-$, where the $\eta$ decays into $\gamma\gamma$, with a $K^+$ or a that is reconstructed in the $\pi^+\pi^-$ channel, and up to four pions, of which at most one is a $\pi^0$, leading to 16 possible channels [@ref:chargeconj]:
$B^+\to\etapr K^+(+\pi^0)~~~~~~~~~~~~B^0\to\etapr\KS(+\pi^0)$ $B^+\to\etapr K^+\pi^+\pi^-(+\pi^0)~~~~B^0\to\etapr\KS\pi^+\pi^-(+\pi^0)$\
$B^+\to\etapr\KS\pi^+(+\pi^0)~~~~~~~~~B^0\to\etapr K^+\pi^-(+\pi^0)$\
$B^+\to\etapr\KS\pi^+\pi^+\pi^-(+\pi^0)~B^0\to\etapr
K^+\pi^-\pi^+\pi^-(+\pi^0)$
The mass of the $\eta\to\gamma\gamma$, $\KS\to\pi^+\pi^-$ and $\pi^0\to\gamma\gamma$ candidates are required to lie within 3$\sigma$ ($\sigma=16,3$ and $6~\mevcc$ respectively) of their known values and are then kinematically constrained to their nominal masses.\
To identify the $s$ quark in the $X_s$ system, we require a or a track consistent with a charged kaon. The charged-kaon selection has been optimized to reduce background from $B\to\etapr\pi,~\etapr\rho$, and $\etapr a_1$ decays. For the , we require the angle $\alpha$ between the momentum of the candidate and its flight direction to be less than 0.05 radians, as it peaks at zero for true particles.\
We require candidates for $B\to\etapr X_s$ to be consistent with a $B$ decay, based on the beam-energy-substituted mass, $\mes=\sqrt{(s/2+\mathbf{p}_0.\mathbf{p}_B)^2/E_0^2-\mathbf{p}_B^2}$ and the energy difference, $\Delta E=E_B^*-\sqrt{s}/2$, where $E$ and $\mathbf{p}$ denote the energy and momentum of the particles, the subscripts $0$ and $B$ refer to the initial and the $B$ candidate, respectively, the asterisk denotes the rest frame, and $\sqrt{s}$ is the $e^+e^-$ center-of-mass energy. In addition, the cosine of the angle between the thrust axis of the $B$ candidate and that of the rest of the event in the center-of-mass frame ($\cos\theta^*_T$) is used to remove continuum background, which is peaked near $|\cos\theta^*_T|=1$, while signal events are uniformly distributed. We require $\mes>5.265~\gevcc$, $|\Delta E|<0.1~\gev$, and $|\cos\theta^*_T|<0.8$. For each event, we select the candidate with the smallest $\chi^2$, with $\chi^2$ defined by
$\chi^2=(\mes-M_B)^2/\sigma^2(\mes)+(\Delta E)^2/\sigma^2(\Delta
E)$,
where $M_B$ is the $B$-meson mass and where where the resolutions $\sigma(\mes)=3~\mevcc$ and $\sigma(\Delta E)=25~\mev$ are obtained from Monte Carlo simulation. The remaining continuum background is subtracted with the use of off-resonance data.\
The background contribution from color-suppressed modes $\Bzb\to\etapr D^{(*)0}$ is estimated from a Monte Carlo simulation which uses our measurement of its branching fraction, $\mathcal{B}(\Bzb\to\etapr
D^{(*)0})=(1.7\pm0.4\stat\pm0.2\syst)\times 10^{-4}$ [@ref:colsup].
To determine efficiencies, we model the signal using a combination of the two-body mode $B\to\etapr K$ and, for $X_s$ masses above the $K\pi$ threshold, a non-resonant derived from the theoretical predictions [@ref:Atw; @ref:Hou; @ref:Fritzsch], which are based on the anomalous $\etapr$-gluon-gluon coupling and which favor high-mass $X_s$ systems. The fraction of the two-body mode is constrained in the simulation model to be between 10% and 15% [@ref:newBaBarEtaPK; @ref:BelleEtaPK]. When not forming a $K$ meson, the $X_s$ fragments into $s\bar{q}$ and $s\bar{q}g$ ($q=u,d$). We find that the overall efficiency is $(6.0\pm0.2)$% for the $K^{\pm}$ modes and $(4.7\pm0.1)\%$ for the modes, including the branching fraction $\mathcal{B}(\KS\to\pi^+\pi^-)$.\
The branching fraction of $B\to\etapr X_s$ is computed through a fit to the number of signal events, with momentum between $2.0$ and $2.7~\gevc$, both for on-resonance and off-resonance data. To parameterize the background, we use a Gaussian function for the signal and a second order polynomial. For the fit of the off-resonance data sample, we constrain the mass and width of the to the values obtained with on-resonance data. Figure \[fig:etaPrFits\] shows the fits of the $\eta\pi\pi$ invariant mass distributions for the $K^{\pm}$ and modes. The fitted yields are reported in Table \[Ta:summary\].\
$K^{\pm}$ modes modes
--------------- ----------------- ----------------
$Y_{\rm ON}$ $577.0\pm34.0$ $367.0\pm34.0$
$Y_{\rm OFF}$ $18.9\pm8.5$ $21.7\pm8.4$
$Y_{\rm CS}$ $63.6\pm 11.4$ $26.9\pm4.5$
$Y$ $353.1\pm 80.5$ $156.1\pm79.1$
: Results of the fits for $K^{\pm}$ and modes. Yields for on-resonance data ($Y_{\rm ON}$), off-resonance data ($Y_{\rm OFF}$), expectation from color-suppressed background ($Y_{\rm CS}$) and on-resonance data after background subtraction ($Y$) are given. A luminosity scale factor, $f=8.48$ , is applied to the off-resonance yield.[]{data-label="Ta:summary"}
The semi-inclusive branching fraction is computed by performing a weighted average of the results obtained for the $K^{\pm}$ and modes. The detection efficiencies are corrected to account for the and $\eta$ branching fractions to the channel we observe. For the modes, we convert the result so it corresponds to $K^0$ and ${\overline K}^0$. The final state $X_s$ includes both $K^+$- and $K^0$-tagged decays. Assuming that their branching fractions are equal, we obtain $\mathcal{B}(B\to\etapr X_s)=(3.9\pm0.8\stat\pm0.5\syst\pm0.8{\ensuremath{\mathrm{(model)}}\xspace})\times 10^{-4}$. We obtain the systematic error by combining the sources listed in Table \[Ta:systExcl\].
Source $K^{\pm}$ syst (%) syst (%)
--------------------------------------------------- -------------------- ----------
Tracking 3.4 3.3
$\eta,\pi^0$ detection 7.0 8.2
$K/\KS$ ID 2.5 4.3
$\mathcal{B}(\etapr\to\eta_{\gamma\gamma}\pi\pi)$ 3.4 3.4
$N_{\BB}$ 1.1 1.1
MC sample size 3.0 3.0
$\etapr D^{(*)0}$ subtraction 3.0 2.9
Total 12.1 13.5
Model 20 20
: Contribution of different sources to the systematic error for modes with a $K^{\pm}$ or .[]{data-label="Ta:systExcl"}
The largest uncertainty arises from our model of the $X_s$ system. To estimate that uncertainty, we use an alternative model which consists of a combination of resonant modes: $\etapr K$, $\etapr K^*(892)$, $\etapr K_1(1270)$, $\etapr
K_1(1400)$, $\etapr K^*(1410)$, $\etapr K_2^*(1430)$, $\etapr
K_3^*(1780)$, and $\etapr K_4^*(2045)$. The variability of the efficiency and our knowledge of the resonant sector lead us to assign a 20% systematic uncertainty. Other systematic uncertainties include track reconstruction efficiency, reconstruction efficiencies of $\pi^0\to\gamma\gamma$, $\eta\to\gamma\gamma$, and $\KS\to\pi^+\pi^-$ candidates, charged-kaon identification efficiency, secondary branching fractions, number of events ($N_{\BB}$), the size of our Monte-Carlo sample, and subtraction of the background from $\Bzb\to \etapr D^{(*)0}$.\
To explore the $X_s$ mass distribution, we select $B$ candidates for which the mass of the is within three standard deviations of the known value, and subtract the continuum contribution by using on-resonance data in the sideband $5.200<\mes<5.265~\gevcc$. The continuum background scaling factor ($\mathcal{A}$), from the sideband to signal regions, is computed from off-resonance data to be $0.591\pm0.118$. The resulting mass distributions are shown in Fig. \[Fi:RawMXs\] for all $B$ modes and separately for the $B^0$ modes. The peak at $m(X_s)\simeq500 \mevcc$ corresponds to the two body mode $B\to
\etapr K$.
\(a) (b)
To obtain the full $X_s$ spectrum, we fit the mass distribution in bins of $X_s$ mass. The efficiency, averaged over the charged and neutral kaons, as a function of $m(X_s)$, is shown in Fig. \[Fi:effmxs\]. The correction for the feed-across between bins is included in the efficiencies.
According to simulations, the $X_s$ system is correctly reconstructed for 85% (60%) of the candidates in the region $m(X_s)<1.5~\gevcc$ ($m(X_s)>1.5~\gevcc$). For correctly reconstructed events, the experimental resolution varies from 5 to 15 $\mevcc$ for low and high masses, respectively. In the case of misreconstructed events, the resolution ranges from 100 to 150 $\mevcc$. Table \[Ta:YieldMxs\] shows the fitted yields for the raw signal, the sideband region, the expected color-suppressed background, and the yield after full background subtraction, as a function of $m(X_s)$.
$m(X_s)$ range $Y_{ON}$ $Y_{SB}$ $Y_{CS}$ $Y$
---------------- -------------- -------------- -------------- ----------------
$[0.4,0.6]$ $200\pm15$ $46.1\pm8.8$ — $172.8\pm15.9$
$[0.6,1.2]$ $120 \pm 14$ $100 \pm 13$ — $60.9\pm16.0$
$[1.2,1.5]$ $114\pm15$ $112\pm14$ $1.1\pm0.3$ $46.7\pm17.1$
$[1.5,1.8]$ $150\pm18$ $163\pm17$ $7.7\pm1.6$ $46.0\pm20.7$
$[1.8,2.0]$ $140\pm17$ $93\pm15$ $47.4\pm9.6$ $37.6\pm21.4$
$[2.0,2.3]$ $149\pm20$ $142\pm18$ $26.2\pm4.5$ $38.9\pm23.1$
$[2.3,2.5]$ $80\pm14$ $70\pm14$ $4.9\pm0.9$ $33.7\pm16.3$
: Fitted yields for on-resonance data and color-suppressed background for different $m(X_s)$ ranges in . The sideband yields ($Y_{SB}$) must be corrected by the sideband to signal region scaling factor (see text) before subtraction.[]{data-label="Ta:YieldMxs"}
The branching fraction as a function of $m(X_s)$, obtained from the fully background-subtracted yield (Table \[Ta:YieldMxs\]), is shown in Fig. \[Fi:BFVsMxs\].
\(a) (b)
We compare data and simulation by forming a difference. The probability for the nonresonant $X_s$ model (Fig. \[Fi:BFVsMxs\](a)) to fit the data is 61% while it is close to $\sim10^{-7}$ for the equal mixture of resonances (Fig. \[Fi:BFVsMxs\](b)). We find improved agreement with the resonant model if the weights of $K_3^*$ and $K_4^*$ are increased by a factor of 1.5, leading to a probability of 2%.\
As a consistency check of the method, we measure the two-body decay modes ($X_s=K^{\pm},\KS$), and find $171.0\pm14.0$ and $27.1\pm5.6$ events in on-resonance data for $\etapr K^{\pm}$ and $\etapr\KS$ respectively, and no signal events for both channels in off-resonance data, leading to the branching fractions $\mathcal{B}(B^{\pm}\to\eta^{\prime}K^{\pm})=(6.9\pm0.6\stat)\times 10^{-5}$ and $\mathcal{B}(B^0\to\eta^{\prime}K^0)=(5.6\pm1.2\stat)\times 10^{-5}$. These values are fully compatible with what has been measured by recent exclusive analyses [@ref:newBaBarEtaPK; @ref:BelleEtaPK].\
In summary, we have measured the branching fraction, $\mathcal{B}(B\to\etapr X_s)=(3.9\pm0.8\stat\pm0.5\syst\pm0.8{\ensuremath{\mathrm{(model)}}\xspace})\times 10^{-4}$, for $2.0<p^*(\etapr)<2.7~\gevc$. We have also derived the $m(X_s)$ spectrum and found that the data tends to confirm models predicting a peak at high masses and seems to disfavor predictions based only on the diagram of Fig. \[fig:diagrams\](a,b) for which $m(X_s)$ peaks near 1.4-1.5 [@ref:Datta].\
Among the various theoretical conjectures to explain this production, an $\etapr gg$ coupling due to the QCD anomaly has been widely suggested as a likely explanation. However, the $\etapr gg$ form factor initially proposed [@ref:Atw] is disfavored by recent studies of the inclusive production $\Upsilon(1S)\to\etapr X$ [@ref:KaganNew; @ref:CLEOUps1]. A recently updated approach [@ref:Fritzsch] exploiting the same gluon anomaly could in principle account for the observed branching fraction and the $m(X_s)$ spectrum.
We are grateful for the excellent luminosity and machine conditions provided by our 2 colleagues. The collaborating institutions wish to thank SLAC for its support and kind hospitality. This work is supported by DOE and NSF (USA), NSERC (Canada), IHEP (China), CEA and CNRS-IN2P3 (France), BMBF (Germany), INFN (Italy), NFR (Norway), MIST (Russia), and PPARC (United Kingdom). Individuals have received support from the Swiss NSF, A. P. Sloan Foundation, Research Corporation, and Alexander von Humboldt Foundation.
[99]{} Collaboration, B. Aubert [*et al.*]{}, hep-ex/0310028, submitted to . Collaboration, B. Aubert [*et al.*]{}, hep-ex/0308015, hep-ex/0311016. CLEO Collaboration, , 1786 (1998). D. Atwood and A. Soni, Phys. Lett. B [**405**]{}, 150 (1997). W.S. Hou and B. Tseng, , 434 (1998). H. Fritzsch and Y-F. Zhou, , 034015 (2003). In this paper, the Fermi motion of the $b$ quark is added to a model developed earlier in H. Fritzsch, Phys. Lett. B [**415**]{}, 83 (1997). This leads to a $X_s$ spectrum favoring high masses, comparable to the one predicted by the two previous references. F. Yuan and K.T. Chao, , 2495 (1997). CLEO Collaboration, G.Bonvicini [*et al.*]{}, , 011101 (2003). Collaboration, B. Aubert [*et al.*]{}, , 1 (2002). G.C. Fox and S. Wolfram, , 1581 (1978). Throughout this paper, whenever a mode is given, the charge conjugate state is also implied. A.Datta [*et al.*]{}, Phys. Lett. B [**419**]{}, 369 (1998). Collaboration, B. Aubert [*et al.*]{}, , 161801 (2003). Belle Collaboration, K. Abe [*et al.*]{}, Phys. Lett. B [**517**]{}, 309 (2001). A.L. Kagan, AIP Conf. Proc. 618, 310 (2002). CLEO Collaboration, M. Artuso [*et al.*]{}, , 052003 (2003).
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abstract: 'An effective action is obtained of a Bose gas in the bulk separated into two regions by a strong external potential depending on the single coordinate. The main attention is focused on the relaxation of the difference between phases of the weakly coupling condensates of the different bulk domains separated from each other by the external potential. The cases of low and high temperatures are considered.'
address: 'R.S.C. ”Kurchatov Institute” Kurchatov sq. 46, Moscow 123182, Russia.'
author:
- 'V. S. Babichenko'
title: Low Temperature Relaxation of the Phase In an Inhomogeneous Bose Gas
---
\[theorem\][Acknowledgement]{} \[theorem\][Algorithm]{} \[theorem\][Axiom]{} \[theorem\][Claim]{} \[theorem\][Conclusion]{} \[theorem\][Condition]{} \[theorem\][Conjecture]{} \[theorem\][Corollary]{} \[theorem\][Criterion]{} \[theorem\][Definition]{} \[theorem\][Example]{} \[theorem\][Exercise]{} \[theorem\][Lemma]{} \[theorem\][Notation]{} \[theorem\][Problem]{} \[theorem\][Proposition]{} \[theorem\][Remark]{} \[theorem\][Solution]{} \[theorem\][Summary]{}
The experimental realization of the Bose-Einstein condensation at ultralow temperatures in atomic vapors \[1\] provides an example of the systems in which the approximation of the weakly non-ideal gas is well applicable. This fact is connected with the small density of particles in the systems concerned. The realization of such systems gives the possibility of the experimental investigations of the macroscopic quantum phenomenons of different types. Recently, the manifestations of the macroscopic quantum phase and its behavior are studied actively \[2-5\], \[8\].
The investigation of the kinetic phenomena due to the relaxation of the order parameter is of a doubtless interest for the study of macroscopic quantum phenomena. In the present work we study the spatial relaxation of the phase of the order parameter in the inhomogeneous Bose gas with the weak coupling between different spatial regions due to a barrier tunneling.
The system concerned in the present work is a Bose gas of the small density in the bulk separated into two regions by a strong external potential $%
U\left( \overrightarrow{r}\right) $ which depends on the x coordinate and does not depend on $\overrightarrow{r}_{\perp }=\left( y,z\right) $, i.e., $%
U\left( \overrightarrow{r}\right) =U\left( x\right) $. For simplicity, we assume that the external potential has a rectangular shape $U\left( x\right)
=U_{0}$ for $-d<x<d$ and $U\left( x\right) =0$ beyond this region of the x coordinate. The height of the external potential $U_{0}$ is supposed to be the largest energy parameter in the system, in particular, $U_{0}>>\mu $ where $\mu $ is the chemical potential $\mu =n\lambda $, $\lambda $ is the scattering amplitude of Bose particles, and $n$ is the density of the Bose gas. Due to this assumption the interaction between particles of the Bose gas in the region of the influence of the external potential, i.e., in the region $-d<x<d$, can be neglected.
The left and the right domains of the bulk are supposed to have the same temperature $T$ smaller than the Bose-condensation temperature $T_{c}$. We assume that the densities of the right and the left domains of the bulk have the values $n_{1}=n-\frac{1}{2}\Delta n$ and $n_{2}=n+\frac{1}{2}\Delta n$ correspondingly and the difference between densities $\Delta n$ is much smaller than the average density n. We will consider both the case of the nonzero $\Delta n$ and the case of $\Delta n=0$. At the same time, in these two cases the phases of the left and the right Bose condensates are supposed to be different at the initial time moment. This assumption means that at the initial time moment there is a nonzero current from one side of the bulk to another. This non-equilibrium initial state will relax to the equilibrium state having the same densities and the same phases of the condensates for both sides of the bulk if $\Delta n=0$ or if $\Delta n$ can change in time, and will relax to the stationary state if the density difference $\Delta n$ keeps constant and nonzero. This relaxation process is studied in the present work.
Note that in the case of superconductors the similar initial non-equilibrium state results in the Josephson oscillations with a small damping \[6\], \[7\]. The essential distinction of the Bose gas consisting of neutral atoms from superconductors in which the Cooper pairs represent the charged objects is the presence of the gapless excitation spectrum. The presence of low energy excitations results in the change of the character of the initial state relaxation making the relaxation essentially faster. In the case of the homogeneous in y-z plane difference between the phases of the condensates the excitations radiated during the relaxation process have the one-dimensional character. The one-dimensional character of the radiated excitations results in a divergency of the small momentum correlators of these excitations. Hence, the consideration of the relaxation process to the second order in the tunneling amplitude is not correct in contrast to the case of superconductors \[7\] and requires more accurate analysis \[9\], \[14\], \[15\].
The effective action for the difference of phases.
==================================================
Below the approach to the problem of the phase difference relaxation without using the perturbation theory in the tunneling amplitude is developed. For this purpose the effective action for the difference between phases of the Bose field at the right-hand and the left-hand boundaries of the potential barrier is derived. This effective action is obtained by the integration over the bulk components of the Bose field with the fixed values of this field at the potential barrier boundaries. The obtained effective action contains the relaxation part which is proportional to the first power of the frequency of the phase field. For the small frequencies the relaxation part is found to be much larger than the usual kinetic part of the phase dynamics which is proportional to the second power of the frequency. In order to describe the phase difference relaxation process the consideration is developed in the framework of the Schwinger-Keldysh technique \[10\], \[11\] which is very convenient for the description of quantum kinetic processes.
The generating functional of the system can be written in the form $Z=\int
D\psi D\overline{\psi }e\exp \left\{ i\left( S_{U}+S_{j}\right) \right\} $. The action $S_{U}$ of the inhomogeneous non-equilibrium Bose gas is given by
$$S_{U}=\oint dt\int d^{3}r\left\{ \overline{\psi }\left[ i\partial _{t}+\mu
\left( x\right) -U\left( x\right) \right] \psi -\frac{1}{2}\left(
\overrightarrow{\nabla }\overline{\psi }\right) \left( \overrightarrow{%
\nabla }\psi \right) -\frac{\lambda }{2}\left( \overline{\psi }\psi \right)
^{2}\right\}$$ and the term $S_{j}$ reads as $S_{j}=\oint dt\int d^{3}r\left( \overline{%
\psi }j+\overline{j}\psi \right) $, where $j$ and $\overline{j}$ are the infinitely small sources. The integration over the time t in the action is realized over the time reversing contour and is denoted as $\oint dt$. The Planck constant $\hbar $ and mass m of a Bose particle are set equal to unity $\hbar =m=1$. The chemical potential $\mu \left( x\right) $ is supposed to have the constant value $\mu _{1}=\mu -\frac{1}{2}\Delta \mu $ for the coordinate xd and $\mu _{2}=\mu +\frac{1}{2}\Delta \mu $ for x-d where the chemical potential $\mu $ obeys the equality $\mu =\lambda n$ and the chemical potential difference obeys the equality $\Delta \mu
=\lambda \Delta n$. The small value of $\Delta n$ results in the small value of the chemical potential difference $\Delta \mu <<\mu $.
Due to the large magnitude of potential $U_{0}$ in the region x$\in \left[
-d,d\right] $ the interaction between Bose particles can be neglected and the integral for Z over the fields $\psi ,\overline{\psi }$ takes the Gaussian form in this region of the x-coordinate. In this connection the integral for Z over the fields $\psi \left( x,\overrightarrow{r}_{\perp
};t\right) ,\overline{\psi }\left( x,\overrightarrow{r}_{\perp };t\right) $ for x$\in \left[ -d,d\right] $ with the fixed magnitudes of these fields at the boundary of the region of the nonzero external potential can be calculated. Moreover, the characteristic scale of the $\psi $-field variation in the time and in the $\overrightarrow{r}_{\perp }$ space is supposed to be much larger than $1/U_{0}$ and $1/\sqrt{U_{0}}$, respectively. It is these fields that describe the slow relaxation process possessing the characteristic frequency proportional to the small tunneling coefficient. Due to this assumption the tunneling amplitude of the $\psi $ fields has the local character for the time variable t and space variables $%
\overrightarrow{r}_{\perp }$. The magnitudes of the $\psi $-fields at the boundary of the region of the nonzero external potential are denoted as $%
\psi \left( d,\overrightarrow{r}_{\perp },t\right) =\psi _{s1}\left(
\overrightarrow{r}_{\perp },t\right) $; $\psi \left( -d,\overrightarrow{r}%
_{\perp },t\right) =\psi _{s2}\left( \overrightarrow{r}_{\perp },t\right) $. The calculation of the integral for Z over the $\psi $-fields in the region x$\in \left[ -d,d\right] $ gives the action in the form of the sum of four terms $S=S_{vol}+S_{surf}^{\left( \rho \right) }+S_{surf}^{\left(
tunn\right) }+S_{j}$. The part of the action $S_{vol}$ is
$$S_{vol}=\oint dt\int\limits_{U\left( x\right) =0}dx\int d^{2}r_{\perp
}\left\{ \overline{\psi }\left( i\partial _{t}+\mu \right) \psi -\frac{1}{2}%
\left( \overrightarrow{\nabla }\overline{\psi }\right) \left(
\overrightarrow{\nabla }\psi \right) -\frac{\lambda }{2}\left( \overline{%
\psi }\psi \right) ^{2}\right\} \eqnum{1}$$
where the region of the integration over the x-coordinate is the sum of the regions $\left( -L,-d\right) \cup \left( d,L\right) $. The parts of the action $S_{surf}^{\left( \rho \right) }$ and $S_{surf}^{\left( tunn\right) }$ can be represented in the form \[17\] $\ $
$$S_{surf}^{\left( \rho \right) }=-\frac{\varkappa }{2}\coth \left( 2\varkappa
d\right) \oint dt\int d^{2}r_{\perp }\left\{ \mid \psi _{s1}\left(
\overrightarrow{r}_{\perp },t\right) \mid ^{2}+\mid \psi _{s2}\left(
\overrightarrow{r}_{\perp },t\right) \mid ^{2}\right\} \eqnum{2}$$
$$S_{surf}^{\left( tunn\right) }=\frac{\varkappa }{2\sinh \left( 2\varkappa
d\right) }\oint dt\int d^{2}r_{\perp }\left\{ \overline{\psi }_{s1}\left(
\overrightarrow{r}_{\perp },t\right) \psi _{s2}\left( \overrightarrow{r}%
_{\perp },t\right) e^{i\Delta \mu t}+\overline{\psi }_{s2}\left(
\overrightarrow{r}_{\perp },t\right) \psi _{s1}\left( \overrightarrow{r}%
_{\perp },t\right) e^{-i\Delta \mu t}\right\} \eqnum{3}$$
where the magnitude $\varkappa $ is $\varkappa =\sqrt{2U_{0}}$, the index 1 denotes the right-hand side of the bulk and the index 2 denotes the left-hand side of the bulk. The term $S_{j}$ can be written as $S_{j}=\oint
dt\sum\limits_{\alpha =1,2}\left( \overline{\psi }_{s\alpha }j+\overline{j}%
\psi _{s\alpha }\right) $ where $j$ and $\overline{j}$ are the infinitely small sources. Later on we assume that the width d of the potential barrier is sufficiency large so that the inequality $\varkappa d>>1$ takes place and, thus, $\sinh \left( 2\varkappa d\right) \thickapprox \cosh \left(
2\varkappa d\right) \thickapprox \frac{1}{2}e^{2\varkappa d}>>1$. The term $%
S_{surf}^{\left( tunn\right) }$ describes the tunneling between the right-hand and left-hand sides of the bulk. From the form of this tunneling part of the action it can easily be seen that the tunneling amplitude for the condensate particles and for the low energy non-condensate particles has the same nonzero magnitude in contrast with the work \[8\].
Our goal is the calculation of the functional integral for Z over the $\psi $-fields in the bulk, i.e., in the region x$\in \left( -L,-d\right) \cup
\left( d,L\right) $ with the fixed values at the potential barrier boundary, and, thus, obtain the effective action for the fields at the boundary $\psi
_{s1}$, $\psi _{s2}$.
Below the onedimensional character of the system and its excitations is supposed. In this supposition the considering fields depend on the x coordinate only. Later on, it is convenient to introduce the dimensionless space and time coordinates via the following change of these variables $%
x\rightarrow \xi x;$ $t\rightarrow t/\mu $. Thus, the system of units which we use later measures the quantities of the dimension of length in units of $\xi $ and the quantities of the dimension of energy in units of $%
\mu $. Moreover, we translate the x coordinate for the right-hand side of the bulk on the value -d and for the left-hand side on +d, so as the right-hand and left-hand boundaries of the potential barrier take the zero coordinates x=0.
It is convenient to represent the fields $\psi $ in the modulus-phase form $%
\psi =Re^{i\varphi }$; $\overline{\psi }=Re^{-i\varphi }$. Below the modulus $R$ of the field $\psi $ is represented as the sum of the saddle-point configuration $F\left( x\right) $ and the fluctuations of the modulus field $\rho $ which characterize the density fluctuations
$$R\left( x,t\right) =F\left( x\right) +\rho \left( x,t\right) \eqnum{4}$$
The saddle-point configuration $F\left( x\right) $ obeys the equation
$$\left( 1+\frac{1}{2}\nabla _{x}^{2}-F^{2}\left( x\right) \right) F\left(
x\right) =\varkappa F\left( x\right) \delta \left( x\right) \eqnum{5}$$
and has the form $F\left( x\right) =\tanh \left( \mid x\mid +X_{0}\right) $, and the constant $X_{0}$ is the small value $X_{0}=1/\varkappa $. The fluctuations $\rho $ can be represented as the sum of two summands
$$\rho \left( x,t\right) =\delta \rho \left( x,t\right) +\frac{\nabla
_{x}F\left( x\right) }{\nabla _{x}F\left( 0\right) }\rho _{s}\left( t\right)
\eqnum{6}$$
where the fluctuations $\delta \rho \left( x,t\right) $ obey the zero boundary conditions $\delta \rho \left( 0,t\right) =0$. Note, that the second summand of (6) is the zero mode of the Hamiltonian $\widehat{H}_{1}=-%
\frac{1}{2}\nabla _{x}^{2}-1+3F^{2}$ which describe the dynamics of the fluctuations $\rho \left( x,t\right) $. To describe the slow process of the tunneling through the potential barrier we are interesting for the fields $%
\delta \rho $ and $\varphi $ which are slow varying values at the space scale $\xi $ and the time scale $1/\mu $. For the derivation of the slow field effective action we substitute the field $\psi $ into the action (1-3) in the modulus-phase representation where the field $R\left( x,t\right) $ has the form (4) and where $\rho $ is determined by (6). At the same time, in certain terms of the action due to the slowness of $\delta \rho $ and $%
\varphi $ the function like $F^{2}\left( x\right) $ can be replaced by the function $1-\delta \left( x\right) $, where $\delta \left( x\right) $ is the $\delta $-function. Integrating over the fields $\delta \rho \left(
x,t\right) $ and $\rho _{s}\left( t\right) $ in the generation functional we obtain the effective action for the slow phase field
$$S^{\left( slow\right) }\left[ \varphi \right] =K\oint dt\int\limits_{U\left(
x\right) =0}dx\left\{ \frac{1}{2}\left( \partial _{t}\varphi \right) ^{2}-%
\frac{1}{2}\left( \nabla _{x}\varphi \right) ^{2}+\left[ \frac{1}{2}\left(
\partial _{t}\varphi _{s}\right) ^{2}+\gamma \cos \left( \Delta \varphi
_{s}\left( t\right) +\Delta \mu t\right) \right] \delta \left( x\right)
\right\}$$
where $\gamma =\frac{2}{\varkappa }\exp \left( -2\varkappa d\right) $, the value $\Delta \varphi _{s}\left( t\right) $ is the difference of phases at the right-hand and the left-hand boundaries of the potential barrier $\Delta
\varphi _{s}\left( t\right) =\varphi _{s2}\left( t\right) -\varphi
_{s1}\left( t\right) $, the constant $K$ is $K=n\xi S_{\perp }>>1$ and $%
\delta \left( x\right) $ is the Dirac $\delta $-function. The value $%
S_{\perp }$ is the square of the bulk section which is supposed to obey the inequality $S_{\perp }\lesssim \xi ^{2}$ to ensure the 1D character of the system. The dimensionless constant $K=n\xi S_{\perp }$ has the large value $%
K>>1$ due to the large value of the parameter $n\xi ^{3}>>1$. The last inequality is ensured by the application of the gas approximation.
The last step to the derivation of the effective action for the phase difference $\Delta \varphi _{s}\left( t\right) $ is the integration in Z over the bulk components of the phase $\varphi \left( x,t\right) $ with the fixing boundary values $\varphi _{s}\left( t\right) $. For this purpose the field $\varphi \left( x,t\right) $ is represented in the form of the Fourier expansion $\varphi \left( x,t\right) =\sum\limits_{k}\varphi _{k}\left(
t\right) e^{ikx}$ with the Fourier components $\varphi _{k}$ obeying the condition $\varphi _{s}\left( t\right) =\sum\limits_{k}\varphi _{k}\left(
t\right) $ where the coordinate of the right-hand boundary of the potential barrier we choose as x=0. The integration over $\varphi _{k}$ gives
$$S_{eff}\left[ \varphi _{s}\right] =\oint dtdt^{\prime }\left\{ \frac{1}{2}%
\left( \partial _{t}\varphi _{s}\right) ^{2}\delta \left( t-t^{\prime
}\right) +\frac{1}{2}\varphi _{s}\left( t\right) \left[ <\widehat{D}>\right]
_{t,t^{\prime }}^{-1}\varphi _{s}\left( t^{\prime }\right) +\gamma \cos
\left( \Delta \varphi _{s}\left( t\right) -\Delta \mu t\right) \delta \left(
t-t^{\prime }\right) \right\} \eqnum{8}$$
where we denote $<\widehat{D}>=\int \frac{dk}{2\pi }\widehat{D}_{k}$. In the frequency-momentum representation in the ”triangular” form \[10\], \[11\] at the temperature T the phonon propagator $\widehat{D}\left( \omega ,k\right) $ can be written
$$\widehat{D}\left( \omega ,k\right) =\left(
\begin{array}{cc}
0 & D^{A}\left( \omega ,k\right) \\
D^{R}\left( \omega ,k\right) & D^{K}\left( \omega ,k\right)
\end{array}
\right) \eqnum{9}$$
where $D^{R,A}\left( \omega ,k\right) =\left[ \left( \omega \pm i\delta
\right) ^{2}-k^{2}\right] ^{-1}$ and $D^{K}\left( \omega ,k\right) =\coth
\left( \frac{\omega }{2T}\right) \left( D^{R}-D^{A}\right) $. The simple calculations of $<\widehat{D}>_{\omega }$ give
$$<\widehat{D}>_{\omega }=\frac{1}{2}\left(
\begin{array}{cc}
0 & <D^{A}>_{\omega } \\
<D^{R}>_{\omega } & <D^{K}>_{\omega }
\end{array}
\right) \text{\ } \eqnum{10}$$
where $<D^{R}>_{\omega }=\frac{1}{i\left( \omega +i\delta \right) }$; $%
<D^{A}>_{\omega }=\frac{1}{-i\left( \omega -i\delta \right) }$; $\
<D^{K}>_{\omega }=2\coth \left( \frac{\omega }{2T}\right) \frac{1}{i\omega }$.
The effective action for small frequencies and the Fokker-Planck equation.
==========================================================================
Introducing the half-sum and the difference of the fields instead of the fields $\varphi _{s1}$ and $\varphi _{s2}$ we obtain the effective action for the difference of phases at the boundary of the potential barrier. In the ”triangle” representation it can be written as
$$S\left[ \varphi \right] =2K\int dt\left\{ \varphi \left[ \frac{1}{2}\widehat{%
\omega }^{2}+i\widehat{\omega }\right] \Phi +\Phi \left[ \frac{1}{2}\widehat{%
\omega }^{2}-i\widehat{\omega }\right] \varphi +i\varphi \left[ 2\widehat{%
\omega }\coth \left( \frac{\widehat{\omega }}{2T}\right) \right] \varphi -V%
\left[ \Phi ,\varphi \right] \right\} \eqnum{11}$$
where $\widehat{\omega }=i\partial _{t}$, the function $V\left[ \Phi
,\varphi \right] $ is $V\left[ \Phi ,\varphi \right] =\gamma \sin \left(
\Phi +\Delta \mu t\right) \sin \left( \varphi \right) $. In this representation the fields $\Phi $ and $\varphi $ are expressed via the fields belonging to the upper and the lower time contour branches as $\Phi =%
\frac{1}{2}\left( \varphi _{+}+\varphi _{-}\right) $ and $\varphi =\frac{1}{2%
}\left( \varphi _{+}-\varphi _{-}\right) $. Note that as usually in the ”triangle” representation of the Keldysh-Schwinger technique the field $%
\Phi $ describes the kinetics of the phase difference and the field $\varphi
$ describes the quantum noise of this value.
The fields $\Phi $ and $\varphi $ can be fragmented into fast and slow components. The slow fields are the fields having the frequencies $\omega $ obeying the inequality $\omega <\omega _{0}$ and the fast fields are the fields with the frequencies $\omega >\omega _{0}$ where $\gamma <<\omega
_{0}<<1$. The part of the action for the fast fields is obtained by the expansion of the potential $V$ up to the second order of the fast components $\delta \Phi $, $\delta \varphi $ and has the form
$$S^{\left( fast\right) }\left[ \delta \varphi ,\delta \Phi \right] =2K\int
dt\left\{
\begin{array}{c}
\delta \varphi \left[ \frac{1}{2}\widehat{\omega }^{2}+i\widehat{\omega }%
\right] \delta \Phi +\delta \Phi \left[ \frac{1}{2}\widehat{\omega }^{2}-i%
\widehat{\omega }\right] \delta \varphi +i\delta \varphi \left[ 2\widehat{%
\omega }\coth \left( \frac{\widehat{\omega }}{2T}\right) \right] \delta
\varphi - \\
-V_{\varphi \Phi }^{\prime \prime }\delta \varphi \delta \Phi -\frac{1}{2}%
V_{\Phi \Phi }^{\prime \prime }\left( \delta \Phi \delta \Phi +\delta
\varphi \delta \varphi \right)
\end{array}
\right\} \eqnum{12}$$
$$V_{\varphi \Phi }^{\prime \prime }=\gamma \cos \left( \Phi +\Delta \mu
t\right) \cos \left( \varphi \right) \text{; \ \ \ }V_{\Phi \Phi }^{\prime
\prime }=V_{\varphi \varphi }^{\prime \prime }=-\gamma \sin \left( \Phi
+\Delta \mu t\right) \sin \left( \varphi \right) \text{\ }$$ where $\delta \varphi ,\delta \Phi $ are the fast components of fields and $%
\varphi ,\Phi $ are the slow components. The part of the action for the slow fields can be written as
$$S^{\left( slow\right) }\left[ \varphi \right] =2K\int dt\left\{ -2\varphi
\partial _{t}\Phi -V\left[ \Phi ,\varphi \right] +i\varphi \left[ 2\widehat{%
\omega }\coth \left( \frac{\widehat{\omega }}{2T}\right) \right] \varphi
\right\} \eqnum{13}$$
Due to the large value of the constant $K>>1$, so as the value $\frac{1}{K}%
\ln \frac{1}{\gamma }$ obeys the inequality $\frac{1}{K}\ln \frac{1}{\gamma }%
<<1$, the action $S^{\left( fast\right) }\left[ \delta \varphi ,\delta \Phi %
\right] $ can be approximated as
$$S^{\left( fast\right) }\left[ \delta \varphi ,\delta \Phi \right] =2K\int
dt\left\{
\begin{array}{c}
\delta \varphi \left[ \frac{1}{2}\widehat{\omega }^{2}+i\widehat{\omega }%
\right] \delta \Phi +\delta \Phi \left[ \frac{1}{2}\widehat{\omega }^{2}-i%
\widehat{\omega }\right] \delta \varphi +i\delta \varphi \left[ 2\widehat{%
\omega }\coth \left( \frac{\widehat{\omega }}{2T}\right) \right] \delta
\varphi - \\
-V_{\varphi \Phi }^{\prime \prime }<\delta \varphi \delta \Phi >_{0}-\frac{1%
}{2}V_{\Phi \Phi }^{\prime \prime }<\delta \Phi \delta \Phi >_{0}-\frac{1}{2}%
V_{\varphi \varphi }^{\prime \prime }<\delta \varphi \delta \varphi >_{0}
\end{array}
\right\} \eqnum{14}$$
where
$$\begin{aligned}
&<&\delta \varphi _{t}\delta \varphi _{t}>_{0}=0\text{; }<\delta \Phi
_{t}\delta \Phi _{t}>_{0}=\frac{i}{2K}\int\limits_{\mid \omega \mid >\omega
_{0}}\frac{d\omega }{2\pi }D^{\left( K\right) }\left( \omega \right) =\frac{1%
}{4\pi K}\ln \left( \frac{1}{\omega _{0}}\right) \text{; } \eqnum{15} \\
&<&\delta \varphi _{t}\delta \Phi _{t}>_{0}=i\int\limits_{\mid \omega \mid
>\omega _{0}}\frac{d\omega }{2\pi }D_{0}^{\left( R\right) }\left( \omega
\right) =\frac{i}{8K} \nonumber\end{aligned}$$
The correlator $\widehat{D}_{0}$ is defined by the equality $\widehat{D}%
_{0}=\left( \frac{\omega ^{2}}{2}\widehat{\sigma }_{x}+\frac{1}{2}<\widehat{D%
}>^{-1}\right) ^{-1}$, where $\widehat{\sigma }_{x}$ is the $\widehat{\sigma
}_{x}$- Pauli matrix. Thus, taking into account the renormalizations resulting from the fast fluctuations the effective action for the slow fields can be written as
$$S^{\left( slow\right) }\left[ \varphi ,\Phi \right] =2K\int dt\left\{
-2\varphi \left( \partial _{t}\Phi \right) -V\left[ \Phi ,\varphi \right]
+i\varphi \left[ 2\widehat{\omega }\coth \left( \frac{\widehat{\omega }}{2T}%
\right) \right] \varphi -\frac{i}{8K}V_{\varphi \Phi }^{\prime \prime
}\right\} \eqnum{16}$$
In this expression the renormalizations proportional to the correlators $%
<\delta \Phi _{t}\delta \Phi _{t}>_{0}$ give the small renormalization of the constant $\gamma $ in the potential $V$. They can be neglected due to the smallness of the value $\frac{1}{K}\ln \left( \frac{1}{\omega _{0}}%
\right) <<1$. On the other side the renormalizations proportional to the correlators $<\delta \varphi _{t}\delta \Phi _{t}>_{0}$ are taken into account because they can become comparable with the relaxation term $\varphi %
\left[ 2\omega \coth \left( \frac{\omega }{2T}\right) \right] \varphi $ in the case of the low temperatures and small frequencies $T\thicksim \frac{1}{K%
}$, $\omega \thicksim \frac{1}{K}$.
It is convenient to shift the field $\Phi \rightarrow \Phi -\Delta \mu t$. As a result of this shift the action (16) takes the form
$$S^{\left( slow\right) }\left[ \varphi ,\Phi \right] =2K\int dt\left\{
-2\varphi \left( \partial _{t}\Phi \right) -V^{\left( 0\right) }\left[ \Phi
,\varphi \right] +i\varphi \left[ 2\widehat{\omega }\coth \left( \frac{%
\widehat{\omega }}{2T}\right) \right] \varphi -\frac{i}{8K}V_{\varphi \Phi
}^{\left( 0\right) \prime \prime }\right\}$$ where $V^{\left( 0\right) }\left[ \Phi ,\varphi \right] =\gamma \sin \Phi
\sin \left( \varphi \right) -v_{0}\varphi $, at that $v_{0}=2\Delta \mu $. Taking into account the smallness of the quantum noise $\varphi $ due to the large value of the constant $K>>1$ the action can be expanded in the powers of the field $\varphi $. This expansion gives
$$S_{eff}^{\left( slow\right) }\left[ \varphi ,\Phi \right] =2K\int dt\left\{
-2\varphi \left( \partial _{t}\Phi \right) +i\varphi \left[ 2\widehat{\omega
}\coth \left( \frac{\widehat{\omega }}{2T}\right) +\frac{1}{16K}\gamma \cos
\Phi \right] \varphi -A\left( \Phi \right) \varphi -i\frac{1}{8K}A^{\prime
}\left( \Phi \right) \right\} \eqnum{17}$$
where
$$A\left( \Phi \right) =\gamma \sin \Phi -v_{0}\text{; \ \ \ }A^{\prime
}\left( \Phi \right) =\gamma \cos \Phi \text{\ } \eqnum{18}$$
The first correction to the renormalized amplitude of the potential $V\left[
\Phi ,\varphi \right] $, i.e., to the constant $\gamma $ with respect to the value $1/K$ has the form $\gamma ^{\ast }=\gamma \left( 1-\frac{1}{4\pi K}%
\ln \left( \frac{K}{\gamma }\right) \right) $. Integrating over $\varphi $ in the generation functional Z we obtain
$$S_{eff}^{\left( slow\right) }\left[ \Phi \right] =-\frac{1}{2}Sp\ln \left(
\widehat{\Gamma }\right) +2K\int dtdt^{\prime }\left\{ \frac{i}{4}\left[
-2\left( \partial _{t}\Phi \right) -A_{t}\right] \widehat{\Gamma }%
_{t,t^{\prime }}\left[ -2\left( \partial _{t^{\prime }}\Phi \right)
-A_{t^{\prime }}\right] -i\frac{1}{8K}A_{t}^{\prime }\delta \left(
t-t^{\prime }\right) \right\}$$ where the correlator $\widehat{\Gamma }_{t,t^{\prime }}$ is
$$\widehat{\Gamma }=\left[ 2\widehat{\omega }\coth \left( \frac{\widehat{%
\omega }}{2T}\right) +\frac{1}{16K}\gamma ^{\ast }\cos \Phi \right]
^{-1}\approx \left[ 2\widehat{\omega }\coth \left( \frac{\widehat{\omega }}{%
2T}\right) +\frac{1}{16K}\gamma ^{\ast }\right] ^{-1} \eqnum{19}$$
In the approximation of the local in time action we have
$$S_{eff}^{\left( slow\right) }\left[ \Phi \right] =i2K\int dt\left\{ \frac{1}{%
4}\left[ 2\left( \partial _{t}\Phi _{t}\right) +A\left( \Phi _{t}\right) %
\right] ^{2}\widehat{\Gamma }_{0}-\frac{1}{8K}A^{\prime }\left( \Phi
_{t}\right) \right\} \eqnum{20}$$
$$\widehat{\Gamma }_{0}=\frac{1}{4T+\frac{1}{16K}\gamma ^{\ast }}=\frac{1}{%
4T^{\ast }} \eqnum{21}$$
Finally, the action can be represented in the form
$$S_{eff}^{\left( slow\right) }\left[ \Phi \right] =i2K\int dt\left\{ \frac{M}{%
2}\left( \partial _{t}\Phi \right) ^{2}+\frac{M}{8}A^{2}-\frac{1}{8K}%
A^{\prime }\right\} \eqnum{22}$$
where $M=2\widehat{\Gamma }_{0}=\frac{1}{2T^{\ast }}$. Note that the generation potential Z with the action $S_{eff}^{\left( slow\right) }\left[
\Phi \right] $ (22) has the form corresponding to the quantum mechanics with the ”imaginary time” in which the role of the ”imaginary time” plays the usual real time. This action corresponds to the action which is obtained for the classical random process describing by the Langevin equation with the white noise \[14\] and having the form of the supersymmentric quantum mechanics. The Schredinger equation and the Hamiltonian corresponding to the action $S_{eff}^{\left( slow\right) }\left[ \Phi \right] $ are
$$-\frac{1}{2K}\partial _{t}\Psi =\widehat{H}_{FP}\Psi \eqnum{23}$$
$$\widehat{H}_{FP}=-\frac{1}{\left( 2K\right) ^{2}}\frac{d^{2}}{2Md^{2}\Phi }+%
\frac{M}{8}A^{2}-\frac{1}{8K}A^{\prime } \eqnum{24}$$
This Hamiltonian $\widehat{H}_{FP}$ can be represented in the factorized form
$\widehat{H}_{FP}=-\frac{1}{2M\left( 2K\right) ^{2}}\left( \frac{d}{d\Phi }%
-MKA\right) \left( \frac{d}{d\Phi }+MKA\right) $. The transformation $\Psi
=\exp \left\{ MKV\left( \Phi \right) \right\} P\left( \Phi \right) $, where $%
V\left( \Phi \right) =\gamma ^{\ast }\left( 1-\cos \Phi \right) -v_{0}\Phi $ and $A=\frac{d}{d\Phi }\left( V\left( \Phi \right) \right) $ transfer the Hamiltonian (18) to the form of the Fokker-Planck operator $\widehat{L}%
_{FP}=-\frac{1}{2M\left( 2K\right) ^{2}}\frac{d}{d\Phi }\left( \frac{d}{%
d\Phi }+2MKA\right) $. The stationary solution of the corresponding Fokker-Planck equation in the case of the zero velocity $v_{0}=0$ has the form $P\left( \Phi \right) =C_{0}\exp \left( -\frac{KV\left( \Phi \right) }{%
T^{\ast }}\right) $, where $V\left( \Phi \right) =\gamma ^{\ast }\left(
1-\cos \Phi \right) $ and the constant $C_{0}$ is $C_{0}=\left( \frac{%
K\gamma ^{\ast }}{2\pi T^{\ast }}\right) ^{\frac{1}{2}}$. In the case of the nonzero velocity $v_{0}$ the stationary solution of the Fokker-Planck equation is determined by the equation
$$\left( \frac{d}{d\Phi }+2MKA\right) P\left( \Phi \right) =j$$ and can be found taking into account the condition of the periodicity of the probability distribution $P\left( 0\right) =P\left( \Phi \right) $ and the normalization condition $\int\limits_{0}^{2\pi }d\Phi P\left( \Phi \right)
=1 $ \[13\], \[18\]. The constant j is found from the normalization condition for $P\left( \Phi \right) $ and for the case of the small $\Delta n$ we obtain $j=\frac{v_{0}}{\gamma ^{\ast }}\exp \left\{ -2\left( \frac{K\gamma
^{\ast }}{T^{\ast }}\right) \right\} $. The probability of the phase slip process is determined by the value $\left( \partial _{t}<\Phi >\right) $ $%
\sim j$. For the zero temperature and small velocities $v_{0}$, so as $%
v_{0}=\Delta \mu <<T^{\ast }$, the value $T^{\ast }$ can be considered as the effective temperature and for the large time t, when the action can be considered in the local in time form, determines the diffusion coefficient. In the opposite case $v_{0}=\Delta \mu >>T^{\ast }$ the diffusion coefficient is defined by the value $\widehat{\Gamma }_{t,t}=\ln \left(
\frac{1}{v_{0}}\right) $ and this value should be substituted instead of the value $\left( \gamma ^{\ast }/T^{\ast }\right) $. This can be seen from the expression (20) for $\widehat{\Gamma }$ with taking into account that in the case of the large value of $\Delta \mu $ compared with $T^{\ast }$ the frequencies $\omega $ should be cut of for the small values by the chemical potential difference $\Delta \mu $. Note, that the result for the probability of the phase slip $\left( 1/\tau _{ps}\right) \thicksim
v_{0}\exp \left\{ -2K\ln \left( \frac{1}{v_{0}}\right) \right\} $ coincides with the result of the works \[15-17\].
The Fokker-Planck equation describing the relaxation process of the difference of phases has been obtained both for high temperatures and for low temperatures. For high temperatures $T\gtrsim \gamma $ the obtained Fokker-Planck equation is equal to the classical Langevin equation with the white noise with the correlator proportional to 1/T \[18\]. For the low temperatures $T<<\gamma $ and the small density difference $\Delta n$, so as the inequality $v_{0}<<1/K$ takes place, the relaxation kinetics of the difference of phases is described by the Fokker-Planck equation with the effective temperature $T^{\ast }$ and the diffusion coefficient determined by both the usual temperature and the quantum fluctuations of the Bose gas. For the small temperatures and the large value of the density difference $%
\Delta n$, so as $v_{0}>>\gamma /K$, the diffusion coefficient is determined by the value $v_{0}$. In the last case the result is analogues to the result of the works \[15-17\].
The author thanks S. Burmistrov, A. Kozlov and Yu. Kagan for helpful discussions. This work was supported by the Russian Foundation for Basic Research, by the Netherlands Organization for Scientific Research (NWO) and by INTAS -2001-2344.
\[1\] M. H. Anderson, J. H. Ensher, M. B. Matthews, C. E. Weiman, and E. A. Cornell, Science 269, 198 (1995); K. B. Davis et al., Phys. Rev. Lett. 75, 3969 (1995); C. C. Bradley, C. A. Sacklet, J. T. Tollett, and R. G. Hulet, Phys. Rev. Lett. 75, 1687 (1995).
\[2\] D. S. Hall, M. R. Matthews, C. E. Weiman, and E. A. Cornell, Phys. Rev. Lett. 81, 1543 (1998).
\[3\] F. Dalfovo, L. P. Pitaevskii, and S. Stringari, Phys. Rev. A 54, 4213 (1996).
\[4\] A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoi, Phys. Rev. Lett. 79, 4950 (1997).
\[5\] I. Zapata, F. Sols, and Leggett, Phys. Rev. A 57, R28 (1998).
\[6\] B. D. Josephson, Phys. Lett. 1, 251 (1962).
\[7\] V. Ambegaokar, U. Eckern, and G. Schen, Phys. Rev. Lett. 48, 1745 (1982),U. Eckern, and G. Schen, V. Ambegaokar, Phys. Rev. B 30, 6419 (1984).
\[8\] F. Meier, and W. Zwerger, cond-mat/ 9904147; F. Meier, and W. Zwerger, Phys. Rev. A 64, 033610 (2001).
\[9\] V. S. Babichenko, cond-mat/ 0109248.
\[10\] L. V. Keldysh, JETP, 47, 1515, (1964).
\[11\] J. Schwinger, J. Math. Phys. 2, 407 (1961).
\[12\] R. P. Feinman, A. R. Hibbs, ”Quantum Mechanics and Path Integrals”, N.Y., 1965.
\[13\] H. Risken, Z. Phys., 251, 231 (1972); K. Kawasaki, Progr. Theor. Phys. 51, 1064 (1974); R. Graham Z. Phys. B40, 149, (1980).
\[14\] M. V. Feigelman, A. M. Tsvelik, JETP, 83, 1430 (1982).
\[15\] H. P. Buchler, V. B. Geshkenbein, and G. Blatter, Phys. Rev. Lett. 87, 100403 (1982).
\[16\] C. L. Kane and M. R. A. Fisher, Phys. Rev. Lett. 68, 1220 (1992).
\[17\] A. Schmid, Phys.Rev.Lett. 51,1506 (1983); S. A. Bulgadaev JETP Lett. 39, 315 (1984); S. E. Korshunov JETP Lett. 65, 1025 (1987).
\[18\] V. Ambegaokar and B. I. Halperin, Phys. Rev. Lett. 22, 1364 (1969).
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abstract: 'Most existing caching solutions for wireless networks rest on assumption that the file popularity distribution is perfectly known. In this paper, we consider optimal caching designs for perfect, imperfect and unknown file popularity distributions in large-scale multi-tier wireless network. First, in the case of perfect file popularity distribution, we formulate , which is nonconvex. We an efficient parallel iterative algorithm to obtain a stationary point parallel successive convex approximation (SCA). Then, in the case of imperfect file popularity distribution, we formulate To solve this challenging robust optimization problem, we transform it to an equivalent complementary geometric programming (CGP), and an efficient iterative algorithm a stationary point SCA. To the best of our knowledge, this is the first work explicitly considering the estimation error of file popularity distribution in the optimization of caching design. Next, in the case of unknown file popularity distribution, we formulate is a challenging nonconvex stochastic optimization problem, we develop an efficient iterative algorithm to obtain a stationary point stochastic parallel SCA. As far as we know, this is the first work considering stochastic optimization in a large-scale wireless network. Finally, by numerical results, we show that the proposed solutions achieve gains over existing schemes in all three cases,'
author:
-
title: 'Optimal Caching Designs for Perfect, Imperfect and Unknown File Popularity Distributions in Large-Scale Multi-Tier Wireless Networks'
---
Cache, multi-tier wireless network, stochastic geometry, robust optimization, stochastic optimization, complementary geometric programming
Introduction
============
The rapid proliferation of smart mobile devices has triggered an unprecedented growth of the global mobile data traffic. , has been proposed as an effective way to support the dramatic traffic growth, by reducing the distance between popular contents and requesters, and alleviating the backhaul load.
Caching in single-tier wireless networks has been actively studied [@TassiulasIT13; @ICC15Giovanidis; @cui2016analysis]. Specifically, in [@TassiulasIT13], the authors consider the optimal caching design and transmission strategy to minimize the required link capacity in a square grid wireless network. In [@ICC15Giovanidis] and [@cui2016analysis], the authors consider random caching at BSs, analyze and optimize the hit probability [@ICC15Giovanidis] and the successful transmission probability (STP) [@cui2016analysis] in large-scale wireless networks that capture the stochastic nature of geographic locations of BSs and users.[^1] As caching can successfully alleviate the urgent backhaul requirement for small cells, a significant amount of research effort has been devoted to optimal caching design in multi-tier wireless networks [@femtocaching13; @cui2017analysis; @li2016optimization; @wen2016cache; @Wang2017Joint]. For instance, in [@femtocaching13], the authors consider coded and uncoded caching at small BSs to minimize the expected downloading time in a macro cell with multiple small BSs . In [@cui2017analysis; @li2016optimization; @wen2016cache; @Wang2017Joint], the authors consider hybrid caching [@cui2017analysis] and random caching [@li2016optimization; @wen2016cache; @Wang2017Joint] in large-scale multi-tier wireless networks, and focus on the analysis and optimization of the STP. Specifically, in our previous work [@cui2017analysis], we obtain optimal hybrid caching design for a two-tier wireless network. In [@li2016optimization] and [@wen2016cache], the authors obtain optimal caching design for a multi-tier network in the case of uniform signal-to-interference ratio (SIR) threshold for all users. In the general case of arbitrary SIR thresholds for users, the optimization problem is nonconvex, and in [@wen2016cache], an optimal caching solution of a simplified convex problem is used as a sub-optimal solution of the original nonconvex problem. In our previous work [@Wang2017Joint], a stationary point of the nonconvex problem is obtained only for a two-tier wireless network. Note that most existing works on caching [@cooperative; @wanli; @TassiulasIT13; @ICC15Giovanidis; @cui2016analysis; @femtocaching13; @cui2017analysis; @li2016optimization; @wen2016cache; @Wang2017Joint; @centralali; @maddah2015decentralized] assume that the file popularity distribution is perfectly known. In practice, however, such an assumption cannot be reasonably justified [@Tatar2014].
Some recent works consider caching design in the case the file popularity distribution is not known and only instantaneous file requests from users can be observed[@bacstuug2015transfer; @Leconte; @Bharath; @Chenyang; @Trend; @cachingmimoLiu15; @blasco2014learning; @song; @reinforcement; @JSTSP18]. These works generally fall into two categories. One category adopts two-stage methods for caching design in the case of unknown file popularity distribution [@bacstuug2015transfer; @Bharath; @Chenyang; @Trend; @Leconte]. Specifically, in the first stage, the file popularity distribution is estimated based on historical file requests, via various learning approaches; in the second stage, caching schemes are proposed based on the estimated popularity distribution. To be specific, in [@bacstuug2015transfer] , the file popularity distribution is estimated using transfer learning [@bacstuug2015transfer] Based on the estimated file popularity distribution, the authors consider caching the most popular files, and analyze the backhaul offloading [@bacstuug2015transfer] In [@Bharath], the request frequency for each file obtained from historical file requests is considered as the popularity of the file, and the performance gap for minimizing the offloading time caused by the estimation error is analyzed. In [@Chenyang], the file popularity distribution is estimated by learning user preferences via probabilistic latent semantic analysis, and a greedy algorithm is proposed to obtain a caching solution of the offloading probability maximization problem (under the estimated file popularity distribution) with performance guarantee. Note that these works [@bacstuug2015transfer; @Bharath; @Chenyang; @Trend; @Leconte] consider either simple caching design without performance guarantee, such as caching the most popular files at each BS [@bacstuug2015transfer; @Trend], or optimization-based caching design obtained by optimizing simple performance metrics that may not fully reflect natures of wireless networks (such as fading [@Bharath] and stochastic locations of BSs and users [@Chenyang; @Leconte]). In addition, note that [@bacstuug2015transfer; @Bharath; @Chenyang; @Trend; @Leconte] fail to consider estimation errors in designing caching schemes.
The other category adopts single-stage methods for caching design [@cachingmimoLiu15; @blasco2014learning; @song; @reinforcement; @JSTSP18] in the case of unknown file popularity, where caching solutions are gradually updated while accumulating file requests using stochastic optimization [@cachingmimoLiu15] or reinforcement learning [@blasco2014learning; @song; @reinforcement; @JSTSP18]. Compared with two-stage methods, Specifically, in [@cachingmimoLiu15], the authors optimize power control and caching for video streaming in a multi-cell multi-user MIMO network using techniques for stochastic optimization. In [@blasco2014learning] and [@song], the authors formulate the optimal caching design problem for a single-cell wireless network [@blasco2014learning] and the optimal cooperative caching design problem for a multi-cell wireless network [@song], and develop low-complexity algorithms to obtain approximate solutions using results for multi-armed bandit problems. In [@reinforcement] and [@JSTSP18], the authors formulate the dynamic optimal caching design problem for a single-cell wireless network [@reinforcement] and a multi-cell wireless network [@JSTSP18], and develop approximate solutions using Q-learning techniques. In addition, the caching solutions in [@cachingmimoLiu15; @blasco2014learning; @song; @JSTSP18; @reinforcement] are for one BS [@blasco2014learning; @reinforcement] or a single-tier of BSs [@cachingmimoLiu15; @song; @JSTSP18].
Therefore, further studies are to optimize caching design in multi-tier wireless networks when perfect file popularity distribution is not known. In this paper, we consider optimal caching design in three cases, i.e., the case of perfect file popularity distribution (where the file popularity distribution has been estimated, and the estimation error is negligible), the case of imperfect file popularity distribution (where the file popularity distribution has been estimated, and a deterministic bound of the estimation error is known) and the case of unknown file popularity distribution (where there is no prior information of the file popularity distribution, but instantaneous file requests from some users can be observed over time), in a large-scale multi-tier wireless network. Our main contributions are summarized below.
- In the case of perfect file popularity distribution, we formulate the STP maximization problem, which is nonconvex with a complicated objective function. We an efficient parallel iterative algorithm to obtain a stationary point parallel successive convex approximation (SCA) [@razaviyayn2014parallel]. Specifically, by carefully an approximation function for each tier, we obtain closed-form optimal solutions the approximate all tiers at each iteration, and hence can significantly reduce the complexity and improve the convergence speed of the iterative algorithm.
- In the case of imperfect file popularity distribution, we formulate the worst-case STP (over all possible values of the true file popularity distribution) maximization problem. is a challenging problem which does not lie in the category of convex-concave games that can be easily solved. We transform it to an equivalent complementary geometric programming (CGP) and an efficient iterative algorithm a stationary point using SCA [@4275017]. Note that this case corresponds to that considered in the second stage of the two-stage methods proposed in [@bacstuug2015transfer; @Bharath; @Chenyang; @Trend; @Leconte]. But the essential difference is that we explicitly consider the estimation error of file popularity distribution in the optimization of caching design.
- In the case of unknown file popularity distribution, we formulate the stochastic STP (i.e., the STP in the stochastic form) maximization problem. is a challenging nonconvex stochastic optimization problem, we develop an efficient parallel iterative algorithm to obtain a stationary point stochastic parallel SCA [@7412752]. Specifically, by carefully an approximation function for each tier, we obtain closed-form optimal solutions the approximate all tiers at each iteration, make full use of instantaneous file requests from users at each slot, and significantly improve the convergence speed of the iterative algorithm. Note that this case corresponds to that considered in the single-stage methods proposed in [@cachingmimoLiu15; @blasco2014learning; @song; @reinforcement; @JSTSP18]. The key difference is that we consider stochastic optimization in a large-scale wireless network which captures the channel fading, interference and stochastic nature of wireless networks.
- Finally, by numerical simulations, we show the convergence of the proposed solutions. We also show that the proposed solutions achieve gains over existing schemes in all three cases.
System Model {#sec:netmodel}
============
Network Model
-------------
In this part, we first elaborate on the network model which extends those in [@li2016optimization; @wen2016cache; @Wang2017Joint] in the sense that besides perfect file popularity distribution, it also models imperfect and unknown file popularity distributions. We consider a large-scale $M$-tier network consisting of $M$ tiers of BSs, where $M\geq2$,[^2] as shown in Fig. \[fig:system\]. The locations of the BSs in tier $m$ are spatially distributed as an independent homogeneous Poisson point process (PPP), denoted as $\Phi_{m}$, with density $\lambda_{m}$, for all $m\in\mathcal{M}\triangleq\{1,2,\dots,M\}$. The locations of the users are also distributed as an independent homogeneous PPP $\Phi_{u}$. Consider a discrete-time system with time being slotted. Let $t\in\{1,2,\dots\}$ denote the slot index. Each BS in tier $m$ has one transmit antenna with transmission power $P_m$. Each user has one receive antenna. All BSs are operating on the same frequency band Both path loss and small-scale fading are considered: for path loss, a transmitted signal from either tier with distance $D$ is attenuated by a factor $D^{-\alpha}$, where $\alpha>2$ is the path loss exponent [@cui2017analysis; @li2016optimization; @wen2016cache; @Wang2017Joint]; for small-scale fading, at each slot, Rayleigh fading channels are adopted. Since a multi-tier network is primarily interference-limited, we ignore the thermal noise for simplicity[@wen2016cache].
Let $\mathcal N\triangleq \{1,2,\cdots, N\}$ denote the set of $N$ files in the $M$-tier network. For ease of illustration, as in [@femtocaching13; @cui2017analysis; @li2016optimization; @wen2016cache; @Wang2017Joint; @bacstuug2015transfer], assume that all files have the same size.[^3] At each slot, a user requests at most one file at random. For ease of analysis, assume $\pi_s(t)$, $s\in\Phi_u$, $t\in\{1,2,\dots\}$ are i.i.d. with respect to $s$ and $t$ [@cui2017analysis; @li2016optimization; @wen2016cache; @Wang2017Joint; @bacstuug2015transfer].[^4] Thus, $\mathbf a\triangleq (a_n)_{n\in \mathcal N}$ represents the file popularity distribution, which usually evolves at a slower timescale. In this paper, we consider the following three cases of file popularity distribution.\
**Perfect file popularity distribution:** In this case, we assume that the file popularity distribution has been estimated by some learning methods, and the estimation error is negligible. That is, the exact value of $\mathbf{a}$ is known.\
**Imperfect file popularity distribution:** In this case, we assume that the file popularity distribution has been estimated by some learning methods, and a deterministic bound of the estimation error is known. Note that this case corresponds to that considered in the second stage of the two-stage methods proposed in [@bacstuug2015transfer; @Bharath; @Chenyang]. Let $\widehat{\mathbf a}\triangleq(\widehat a_n)_{n\in\mathcal N}$ denote the estimated file popularity distribution, and let $\boldsymbol\Delta\triangleq(\Delta_n)_{n\in\mathcal N}$ denote the estimation error. Assume $\sum_{n\in\mathcal N}\widehat a_n=1$, $\sum_{n\in\mathcal N}\Delta_n=0$, and $|\Delta_n|\leq\varepsilon_n$ for some known $\varepsilon_n>0$, for all $n\in\mathcal N$, which are usually satisfied for effective learning methods. The (true) file popularity distribution is given by $\mathbf a=\widehat{\mathbf a}+\boldsymbol\Delta$, and satisfies $\mathbf a\in\mathcal A \triangleq\left\{ (x_n)_{n\in\mathcal{N}} \ \middle|\ \underline a_n\leq x_n\leq \overline a_n, n\in\mathcal{N}, \sum_{n\in\mathcal{N}}x_n=1 \right\}$, where $\underline a_n\triangleq\max\{\widehat a_n-\varepsilon_n,0\}$, and $\overline a_n\triangleq\min\{\widehat a_n+\varepsilon_n,1\}$ for all $n\in\mathcal N$.\
**Unknown file popularity distribution:** In this case, we assume that there is no prior information of the file popularity distribution $\mathbf{a}$, but the file requests from the users in some set $\mathcal U\subseteq\Phi_u$ can be observed over time. Here $\mathcal U$ can represent the set of users located a BS or a cluster of BSs.[^5] Note that this case corresponds to that considered in the single-stage methods proposed in [@cachingmimoLiu15; @blasco2014learning; @song; @reinforcement; @JSTSP18].
The $M$-tier network consists of cache-enabled BSs. In tier $m$, each BS is equipped with a cache of size $K_m<N$ (in number of files) to store $K_m$ different popular files out of $N$. We say every $K_m$ different files form a combination. Thus, there are in total $\binom{N}{K_m}$ different combinations, each with $K_m$ different files. Let $\mathcal I_m$ denote the set of indices for the combinations. More detailed descriptions of $\mathcal{I}_m$ and $\mathcal{I}_{m,n}$ can be found in [@cui2016analysis; @cui2017analysis; @Wang2017Joint].
![[]{data-label="fig:system"}](system.eps){width="12cm"}
Caching and User Association
----------------------------
To provide high spatial file diversity, we consider random caching in the cache-enabled $M$-tier network [@li2016optimization; @wen2016cache; @Wang2017Joint], as illustrated in Fig. \[fig:system\]. In particular, each BS in tier $m$ stores $K_m$ different files with certain probability [@li2016optimization; @wen2016cache; @Wang2017Joint]. The probability that combination $ i\in\mathcal I_m $ is stored each BS tier $m$ is $ p_{m,i} $, where $p_{m,i}$ satisfies $$\begin{aligned}
&0\leq p_{m,i}\leq1, \ m\in\mathcal{M}, \quad i\in \mathcal I_m,\label{eqn:p-interval}\\
&\sum_{i\in \mathcal I_m}p_{m,i}=1,\quad m\in\mathcal{M}.\label{eqn:p-sum1}\end{aligned}$$ A random caching design is specified by the caching distributions Let $\mathcal I_{m,n}\subset\mathcal I_m$ denote the set of indices for the $\binom{N-1}{K_m-1}$ combinations containing file $n$. Based on $\mathbf{p}$, we also define the probability that file $n$ is stored at a BS in tier $m$, i.e., $$\begin{aligned}
T_{m,n} \triangleq \sum_{i\in\mathcal {I}_{m,n}}p_{m,i}, \quad m\in\mathcal{M},\ n \in \mathcal N. \label{eqn:T-def}\end{aligned}$$ From [@ICC15Giovanidis; @cui2016analysis; @cui2017analysis; @Wang2017Joint], we know that the constraints on $ \mathbf p$ in , and can be equivalently rewritten as the following constraints on $\mathbf T\triangleq(T_{m,n})_{m\in\mathcal M,n\in\mathcal N}$: $$\begin{aligned}
&0\leq T_{m,n}\leq 1,\quad m\in\mathcal M,\ n\in\mathcal N, \label{eqn:T1}\\
&\sum\limits_{n\in\mathcal N}T_{m,n}=K_m,\quad m\in\mathcal M. \label{eqn:T2}\end{aligned}$$ The constraints in are due to the fact that each file combination in tier $m$ contains $K_m$ different files and the sum of the caching probabilities all file combinations is one. The details can be found in [@cui2016analysis]. For any $\mathbf T$ satisfying and , one corresponding can be easily obtained using the method in [@ICC15Giovanidis].[^6] If a file is stored in a tier, a user requesting the file is associated with the BS which provides the maximum long-term average received power among all BSs storing the file [@li2016optimization; @wen2016cache; @Wang2017Joint]. Otherwise, the user will be served through other service mechanisms [@li2016optimization; @wen2016cache; @Wang2017Joint], the investigation of which is beyond the scope of this paper.[^7] The probability that an arbitrary user requesting file $n$ is associated with tier $m$ is given [@li2016optimization; @wen2016cache; @Wang2017Joint]: $$\begin{aligned}
&A_{m,n}(\mathbf T)
=\frac{\lambda_mT_{m,n}}{\lambda_m T_{m,n} + \sum_{l\in\mathcal{M}\backslash\{m\}}\lambda_{l}T_{l,n}\left(\frac{P_l}{P_m}\right)^{ \frac{2}{\alpha} }}, \quadm\in\mathcal M,n\in\mathcal N. \label{eqn:user-association-prob}\end{aligned}$$
Performance Metrics {#Sec:perf}
-------------------
In this paper, we study w.l.o.g. the performance of a typical user $u_0$, which is located at the origin. Suppose $u_0$ requests file $n$ and is associated with tier $m$. Let $\ell_0\in\Phi_{m}$ denote the index of the serving BS of $u_0$. We denote $D_{m',\ell,0}$ and $h_{m',\ell,0}{\stackrel{d}{\sim}}\mathcal{CN}\left(0,1\right)$ as the distance and the small-scale channel between BS $\ell\in\Phi_{m'}$ and $u_{0}$, respectively. For analytical tractability, as in [@wen2016cache; @Wang2017Joint], we assume all BSs are active for serving their own users.[^8] In this case, the SIR of $u_{0}$, denoted by ${\rm SIR}_{m,n,0}$, is given by [@Wang2017Joint]: $$\begin{aligned}
&{\rm SIR}_{m,n,0}=\frac{{D_{m,\ell_0,0}^{-\alpha}}\left|h_{m,\ell_0,0}\right|^{2}}{\sum\limits_{\ell\in\Phi_{m}\backslash \{\ell_0\}}D_{m,\ell,0}^{-\alpha}\left|h_{m,\ell,0}\right|^{2}+\sum\limits_{j\in\mathcal{M}\backslash \{m\}}\sum\limits_{\ell\in\Phi_{j}}D_{j,\ell,0}^{-\alpha}\left|h_{j,\ell,0}\right|^{2}\frac{P_{j}}{P_{m}} }.\label{eqn:SIR}\end{aligned}$$ Note that the distribution of ${\rm SIR}_{m,n,0}$ is affected by $\mathbf T$. We assume that file $n$ delivered from tier $m$ can be decoded correctly at $u_0$ if ${\rm SIR_{m,n,0}\geq\tau_{m}}$, where $\tau_{m}$ represents a threshold for tier $m$ [@wen2016cache]. Requesters are mostly concerned about whether their desired files can be successfully received. In the following, we introduce the performance metrics in the three cases of file popularity distribution.\
**Perfect file popularity distribution:** When the exact value of $\mathbf a$ is known, we adopt the probability that a randomly requested file by $u_0$ is successfully transmitted, called the STP [@Wang2017Joint]: $$\begin{aligned}
q\left(\mathbf a,\mathbf T\right)\triangleq& \sum_{m\in\mathcal M}\sum_{n\in\mathcal N} a_n A_{m,n}\left(\mathbf T\right){\rm Pr}\left[{\rm SIR}_{m,n,0}\geq\tau_{m}\right]\nonumber\\
=&\sum_{m\in\mathcal M}\sum_{n\in\mathcal N}{\frac{a_n T_{m,n}}{\sum_{l\in\mathcal{M}}{\theta_{l,m}T_{l,n}}+\eta_{m}}}, \label{eqn:STP}\end{aligned}$$ as the performance metric,[^9] where $\theta_{l,m}$ and $\eta_{m}$ are given by: $$\begin{aligned}
\theta_{l,m}=\frac{2\lambda_{l}}{\alpha\lambda_m}\left(\frac{P_l}{P_m}\tau_{m} \right)^{\frac{2}{\alpha}}\left(B'\left(\frac{2}{\alpha},1-\frac{2}{\alpha}, \frac{1}{1+\tau_{m}} \right)-B\left(\frac{2}{\alpha},1-\frac{2}{\alpha}\right)\right)+\frac{\lambda_{l}}{\lambda_m}\left(\frac{P_l}{P_m}\right)^{\frac{2}{\alpha}},\label{eqn:theta}\end{aligned}$$ $$\eta_{m}=\sum_{l\in\mathcal{M}}\frac{2\lambda_l}{\alpha\lambda_m}\left(\frac{P_l}{P_m}\tau_{m}\right)^{\frac{2}{\alpha}}B\left(\frac{2}{\alpha},1-\frac{2}{\alpha}\right),\label{eqn:eta}$$ respectively. Note that $q(\mathbf a,\mathbf T)$ is a linear function of $\mathbf a$ and a nonconcave function of $\mathbf T$.\
**Imperfect file popularity distribution:** When the exact value of $\mathbf a$ is not known except that it falls within a known set $\mathcal A$, we adopt the worst-case STP: $$\begin{aligned}
q_{\text{wt}}(\mathcal{A},\mathbf{T})\triangleq \min_{\mathbf a\in\mathcal{A}} q(\mathbf{a},\mathbf{T}),\label{eqn:wtSTP}\end{aligned}$$ as the performance metric.[^10]\
**Unknown file popularity distribution:** When there is no prior information of $\mathbf{a}$, but $\{\pi_s(t):s\in\mathcal U\}$ can be obtained at each slot $t$ for some $\mathcal U\subseteq\Phi_u$, we adopt the STP in the stochastic form, called the stochastic STP: $$\begin{aligned}
q_{\text{st}}(\mathbf a,\mathbf T)\triangleq\mathbb{E}\left[q(\boldsymbol\xi,\mathbf T)\right], \label{eqn:stSTP}\end{aligned}$$ as the performance metric, where with $\mathbf I[\cdot]$ denoting the indicator function, $\boldsymbol\xi\triangleq(\xi_n)_{n\in\mathcal N}$, Note that $q_{\text{st}}(\mathbf a,\mathbf T)=q(\mathbf{a},\mathbf{T})$ as for all $n\in\mathcal N$ and $t\in\{1,2,\dots\}$.
\[fig:problem\]
In Section \[Sec:pe\], Section \[Sec:ro\] and Section \[Sec:sto\], we shall maximize the STP, the worst-case STP and the stochastic STP in the cases of perfect, imperfect and unknown file popularity distribution, respectively, as shown in Fig \[fig:problem\].[^11]
Performance Optimization for Perfect File Popularity Distribution {#Sec:pe}
=================================================================
In this section, we consider the case of perfect file popularity distribution. In this case, we would like to maximize the STP, by optimizing the caching probabilities. We formulate the optimal random caching design problem as follows.
\[prob:rand\] $$\begin{aligned}
q^*(\mathbf a) \triangleq \max_{\mathbf{T}}\quad &q\left(\mathbf a,\mathbf{T}\right)\nonumber\\
\text{s.t.} \quad &\eqref{eqn:T1},\eqref{eqn:T2},\nonumber
\end{aligned}$$ where $ q\left(\mathbf a,\mathbf{T}\right) $ is given by .
Problem \[prob:rand\] is equivalent to Problem 0 in [@wen2016cache]. It is non-convex (as the objective function is non-convex in $\mathbf T$), and in [@wen2016cache] a suboptimal solution of it is obtained by solving an approximate convex problem. In the following, we extend the technique in [@Wang2017Joint] for the case of $M=2$ to the case of $M\geq2$, and develop an efficient parallel iterative algorithm to obtain a stationary point of Problem \[prob:rand\] using parallel SCA. Different from the cyclic computation mechanism in [@Wang2017Joint], the parallel computation mechanism here can speed up the computation, especially for large $M$. Specifically, this algorithm updates the caching probabilities of the $M$ tiers, i.e., $\mathbf{T}_m\triangleq(T_{m,n})_{n\in\mathcal{N}}$, $m\in\mathcal M$, at each iteration in a parallel manner, by maximizing $M$ approximate functions of $q\left(\mathbf a,\mathbf{T}\right)$.
For notation convenience, define $$\begin{aligned}
q_{j}\left(\mathbf a,\mathbf{T}_m,\mathbf{T}_{-m}\right)\triangleq\sum_{n\in\mathcal{N}}{\frac{a_n T_{j,n}}{\sum_{l\in\mathcal{M}}{\theta_{l,j}T_{l,n}}+\eta_{j}}},\label{eqn:qm}\end{aligned}$$ where $\mathbf T_{-m}\triangleq\left(\mathbf T_j\right)_{j\in\mathcal M,j\neq m}$. Note that $q(\mathbf a,\mathbf T)$ can be rewritten as: $$\begin{aligned}
q\left(\mathbf a,\mathbf{T}_m,\mathbf{T}_{-m}\right)\triangleq\sum_{j\in\mathcal M}q_{j}\left(\mathbf a,\mathbf{T}_m,\mathbf{T}_{-m}\right).\nonumber\end{aligned}$$ Let $\mathbf{T}_m^{(k)}$ denote the caching probabilities of tier $m$ obtained at iteration $k$, and denote $\mathbf{T}^{(k)}\triangleq\big(\mathbf{T}_m^{(k)}\big)_{m\in\mathbf{M}}$. At iteration $k+1$, choose $$\begin{aligned}
&h_m\big(\mathbf{a},\mathbf{T}_m,\mathbf{T}^{(k)}\big)\triangleq\nonumber\\
&q_{m}\left(\mathbf{a},\mathbf{T}_m,\mathbf{T}_{-m}^{(k)}\right)+\sum_{j\in\mathcal{M},j\neq m}\left(q_{j}\left(\mathbf{a},\mathbf{T}_m^{(k)},\mathbf{T}_{-m}^{(k)}\right)
-\sum_{n\in\mathcal{N}}\frac{a_n\theta_{m,j}T_{j,n}^{(k)}\left(T_{m,n}-T_{m,n}^{(k)}\right)}{\left({\sum_{l\in\mathcal{M}}{\theta_{l,j}T_{l,n}^{(k)}}+\eta_{j}}\right)^2}\right), \label{eqn:randap}\end{aligned}$$ as an approximation function of $q(\mathbf a,\mathbf{T})$ for updating $\mathbf T_m$. Note that the strongly concave component function of $q\big(\mathbf{a},\mathbf{T}_m,\mathbf{T}_{-m}^{(k)}\big)$, i.e., $q_{m}\big(\mathbf{a},\mathbf{T}_m,\mathbf{T}_{-m}^{(k)}\big)$, is left unchanged, and the other nonconcave (actually convex) component functions, i.e., $q_{j}\big(\mathbf{a},\mathbf{T}_m,\mathbf{T}_{-m}^{(k)}\big)$, $j\in\mathcal{M}$, $j\neq m$, are linearized at $\mathbf{T}_m=\mathbf{T}_m^{(k)}$. This choice of the approximate function is beneficial from several aspects[@Wang2017Joint]. Firstly, it can guarantee the convergence of the algorithm to a stationary point of Problem \[prob:rand\], which will be shown in Theorem \[thm:conv-rand\]. Secondly, it usually leads to fast convergence of the algorithm by exploiting the partial concavity of the objective function, which will be shown in Fig. \[fig:simulation-convergence\]. Thirdly, it yields a closed-form optimal solution of the optimization problem for each tier at each iteration, which will be shown in Lemma \[lem:closed-formrand\], and hence a low-complexity algorithm.
Specifically, at iteration $k$, we first solve the following problem for each tier $m\in\mathcal M$ separately, in a parallel manner.
\[prob:randsp\] $$\begin{aligned}
\overline{\mathbf{T}}_m^{(k)} \triangleq \mathop{\arg\max}_{\mathbf{T}_m}\ &h_m\big(,\mathbf{a},\mathbf{T}_m,\mathbf{T}^{(k-1)}\big) \nonumber\\
\text{s.t.} \quad &0\leq T_{m,n}\leq 1,\quad n\in\mathcal N, \label{eqn:Tm1}\\
&\sum\limits_{n\in\mathcal N}T_{m,n}=K_m. \label{eqn:Tm2}\end{aligned}$$
Problem \[prob:randsp\] is a convex optimization problem and Slater’s condition is satisfied, implying that strong duality holds. Based on KKT conditions, we can obtain closed-form optimal solution of Problem \[prob:randsp\].
\[lem:closed-formrand\] For all $m\in\mathcal{M}$, the optimal solution of Problem \[prob:randsp\] is given by $$\begin{aligned}
&\overline T_{m,n}^{(k)}=
\left[\frac{1}{\theta_{m,m}}\sqrt{\frac{a_n\left(\sum\limits_{l\in\mathcal{M},l\neq m}\theta_{l,m}T_{l,n}^{(k-1)}+\eta_{m}\right)}{ \nu_m^{*({k})} + \sum\limits_{j\in\mathcal{M},j\neq m}\frac{a_n\theta_{m,j}T_{j,n}^{(k-1)}}{\left({\sum\limits_{l\in\mathcal{M}}{\theta_{l,j}T_{l,n}^{(k-1)}}+\eta_{j}}\right)^2} }} - \frac{\sum\limits_{l\in\mathcal{M},l\neq m}\theta_{l,m}T_{l,n}^{(k-1)}+\eta_{m}}{\theta_{m,m}}\right]^1_0, \label{eqn:BSUM-opt-sol}\end{aligned}$$ where $[x]^1_0\triangleq\min\left\{\max\{x,0\},1\right\}$ and $\nu_m^{*({k})}$ is the Lagrange multiplier that satisfies $\sum_{n\in\mathcal{N}}\overline T_{m,n}^{(k)}=K_m$.
Then, we update the caching probabilities of tier $m$ by: $$\begin{aligned}
&\mathbf{T}_m^{(k)}=(1-\gamma^{(k)})\mathbf{T}_m^{(k-1)}+\gamma^{(k)}\overline{\mathbf{T}}_m^{(k)},\label{eqn:updateTm}\end{aligned}$$ where $\gamma^{(k)}$ is a positive diminishing stepsize satisfying $$\begin{aligned}
&\gamma^{(k)}>0,\quad \lim_{k\to\infty}\gamma^{(k)}=0,\quad \sum_{k=1}^\infty\gamma^{(k)}=\infty,\quad \sum_{k=1}^\infty\left(\gamma^{(k)}\right)^2<\infty. \label{eqn:gamma}\end{aligned}$$ Finally, the details of the proposed parallel iterative algorithm are summarized in Algorithm \[alg:rand\]. Based on [@razaviyayn2014parallel Theorem 1], we can show the following result.
\[thm:conv-rand\] If the stepsize $\{\gamma^{(k)}\}$ satisfies , then every limit point of $\{\mathbf{T}^{(k)}\}$ generated by Algorithm \[alg:rand\] is a stationary point of Problem 1.
Please refer to Appendix A.
**initialization**: choose any feasible solution $\mathbf{T}^{(0)}$ of Problem \[prob:rand\] as the initial point, and set $k=1$.\
**repeat** for all $m\in\mathcal{M}$, compute $\overline{\mathbf{T}}_m^{(k)}$ according to , and update $\mathbf{T}_m^{(k)}$ according to . set $k=k+1$. **until** some convergence criteria is met.
\[alg:rand\]
Robust Optimization for Imperfect File Popularity Distribution {#Sec:ro}
==============================================================
In this section, we consider the case of imperfect file popularity distribution. In this case, we would like to maximize the worst-case STP, by optimizing the caching probabilities. We formulate the robust optimal random caching design problem as follows.
\[prob:ro\] $$\begin{aligned}
q_{\text{wt}}^* \triangleq \max_{\mathbf{T}}\ &\underbrace{\min_{\mathbf a\in\mathcal{A}} q(\mathbf{a},\mathbf{T})}_{= q_{\text{wt}}(\mathcal{A},\mathbf{T})}\nonumber\\
\text{s.t.} \quad &\eqref{eqn:T1},\eqref{eqn:T2},\nonumber\end{aligned}$$ where $q(\mathbf{a},\mathbf{T})$ is given by .
Problem \[prob:ro\] is a challenging maximin problem, which does not lie in the category of convex-concave games that can be easily solved (as $q(\mathbf a,\mathbf T)$ is a nonconcave function of $\mathbf T$). In the following, we solve it in two steps.
Firstly, we transform the maximin problem in Problem \[prob:ro\] to an equivalent maximization problem. As the inner problem $\min_{\mathbf{a}\in\mathcal{A}}q\left(\mathbf a,\mathbf{T}\right)$ is a linear programming (LP) with respect to $\mathbf{a}$ and strong duality holds for LP, the inner problem $\min_{\mathbf{a}\in\mathcal{A}}q\left(\mathbf a,\mathbf{T}\right)$ shares the same optimal value with its dual problem. Thus, we can transform Problem \[prob:ro\] to the following equivalent maximization problem by replacing the inner problem with its dual problem.
\[prob:dualmax\] $$\begin{aligned}
q_{\text{wt}}^{\star} \triangleq \max_{\mathbf{T},\boldsymbol{\lambda}\succeq\mathbf0,\boldsymbol{\mu}\succeq\mathbf0,{\nu}} \quad &\sum_{n\in\mathcal{N}}{(\lambda_n \underline a_n-\mu_n \overline a_n)}-\nu\nonumber\\
\text{s.t.} \quad &\eqref{eqn:T1},\eqref{eqn:T2},\nonumber\\
&\sum_{m\in\mathcal{M}}{\frac{T_{m,n}}{\sum_{l\in\mathcal{M}}{\theta_{l,m}T_{l,n}}+\eta_{m}}}+\mu_n-\lambda_n+\nu=0,\quad n\in\mathcal{N},\label{eqn:dual1}
\end{aligned}$$ where $\boldsymbol{\lambda}\triangleq(\lambda_n)_{n\in\mathcal{N}}$ and $\boldsymbol{\mu}\triangleq(\mu_n)_{n\in\mathcal{N}}$.
Note that $\boldsymbol{\lambda}$, $\boldsymbol{\mu}$ and $\nu$ are dual variables for the dual problem of the inner problem, corresponding to $a_n\geq\underline a_n$, $a_n\leq\overline a_n$, $n\in\mathcal N$, and $\sum_{n\in\mathcal N}a_n=1$, respectively.
\[lem:roeq\] Problem \[prob:ro\] and Problem \[prob:dualmax\] have the same optimal value and optimal caching probabilities.
Please refer to Appendix B.
Based on Lemma \[lem:roeq\], we can solve Problem \[prob:dualmax\] instead of Problem \[prob:ro\]. Problem \[prob:dualmax\] is nonconvex, as the constraints in are nonconvex. In what follows, we show how to obtain a stationary point of Problem \[prob:dualmax\] using SCA. We first rewrite $\nu$ as $\nu_1-\nu_2$ with $\nu_1,\nu_2>0$, and define new $\mathbf{x}\triangleq(x_{m,n})_{m\in\mathcal{M},n\in\mathcal{N}}$:[^12] $$\begin{aligned}
x_{m,n}=\sum_{l=1}^M \theta_{l,m} T_{l,n}+ \eta_{m}, \quad m\in\mathcal{M}, n\in\mathcal{N}. \label{eqn:xeq}\end{aligned}$$ We also introduce a new variable $y>0$ which serves as a lower bound of the objective function of Problem \[prob:dualmax\]: $$\begin{aligned}
y\leq\sum_{n\in\mathcal{N}}{(\lambda_n \underline a_n-\mu_n \overline a_n)}-\nu. \label{eqn:yneq}\end{aligned}$$ Therefore, Problem \[prob:dualmax\] can be equivalently transformed to the following problem.[^13]
\[prob:roep\] $$\begin{aligned}
\max_{\substack{\mathbf{T},\mathbf{x},\boldsymbol{\lambda},\boldsymbol{\mu}\succ\mathbf 0\\ {\nu_1},{\nu_2},y>0}} \quad &y \nonumber\\
\text{s.t.} \quad
&\frac{y+\sum_{n\in\mathcal{N}}{\mu_n \overline a_n}+\nu_1}{\sum_{n\in\mathcal{N}}{\lambda_n \underline a_n}+\nu_2}\leq1, \label{eqn:c1}\\
&\frac{\lambda_n+\nu_2}{\sum_{m\in\mathcal{M}} {T_{m,n}}{x_{m,n}^{-1}}+\mu_n+\nu_1}\leq1, \quad n\in\mathcal{N},\label{eqn:c2}\\
&\frac{\sum_{l\in\mathcal{M}} \theta_{l,m} T_{l,n} + \eta_{m}}{x_{m,n}}\leq1, \quad m\in\mathcal{M},\ n\in\mathcal{N}, \label{eqn:x1}\\
&T_{m,n}\leq 1,\quad m\in\mathcal M,\ n\in\mathcal N, \label{eqn:T3}\\
&\sum\limits_{n\in\mathcal N}T_{m,n}\leq K_m,\quad m\in\mathcal M. \label{eqn:T4}\end{aligned}$$
Note that the inequality constraints in , and are active at any optimal solution of Problem \[prob:roep\], and hence can replace the equality constraints in , and , respectively. In Problem \[prob:roep\], a monomial subject to upper bounds on posynomials (i.e., , and ) and upper bounds on the ratios of posynomials (i.e., and ). Thus, Problem \[prob:roep\] is a CGP, and can be solved by the method proposed in [@4275017], which is based on SCA. The main idea is to solve a sequence of successively refined geometric programmings (GPs), each of which is obtained by approximating the denominators of the ratios of posynomials in and with monomials. Specifically, at iteration $k$, update $\big(\mathbf{T}^{(k)},\mathbf{x}^{(k)},\boldsymbol{\lambda}^{(k)},\boldsymbol{\mu}^{(k)},{\nu_1}^{(k)},{\nu_2}^{(k)},y^{(k)}\big)$ by solving the following approximate GP of Problem \[prob:roep\], which is parameterized by $\big(\mathbf{T}^{(k-1)},\mathbf{x}^{(k-1)},\boldsymbol{\lambda}^{(k-1)},\boldsymbol{\mu}^{(k-1)},{\nu_1}^{(k-1)},{\nu_2}^{(k-1)}\big)$ obtained at iteration $k-1$.
\[prob:roap\] $$\begin{aligned}
&\hspace{-32mm}\big(\mathbf{T}^{(k)},\mathbf{x}^{(k)},\boldsymbol{\lambda}^{(k)},\boldsymbol{\mu}^{(k)},{\nu_1}^{(k)},{\nu_2}^{(k)},y^{(k)}\big)\triangleq\mathop{\arg\max}_{\substack{\mathbf{T},\mathbf{x},\boldsymbol{\lambda},\boldsymbol{\mu}\succ\mathbf 0\\ {\nu_1},{\nu_2},y>0}} \quad y \nonumber\\
\text{s.t.} \quad &\eqref{eqn:x1},\eqref{eqn:T3},\eqref{eqn:T4},\nonumber\\
&\frac{y+\sum_{n=1}^N{\mu_n \overline a_n}+\nu_1}{\prod_{n\in\mathcal{N}}{\left(\frac{\lambda_n \underline a_n}{\sigma_n^{(k)}}\right)^{\sigma_n^{(k)}}}\left(\frac{\nu_2}{\gamma_1^{(k)}}\right)^{\gamma_1^{(k)}}}\leq1, \\
&\frac{\lambda_n+\nu_2}{\prod_{m\in\mathcal{M}} \left(\frac{T_{m,n}x_{m,n}^{-1}}{\beta_{m,n}^{(k)}}\right)^{\beta_{m,n}^{(k)}}\left(\frac{\mu_n}{\gamma_{2,n}^{(k)}}\right)^{\gamma_{2,n}^{(k)}}\left(\frac{\nu_1}{\gamma_{3,n}^{(k)}}\right)^{\gamma_{3,n}^{(k)}}}\leq1,\quad n\in\mathcal{N},
\end{aligned}$$ where $$\begin{aligned}
\sigma_{n}^{(k)}\triangleq\ &\frac{\lambda_n^{(k-1)} \underline a_n}{\sum_{n\in\mathcal{N}}{\lambda_n^{(k-1)} \underline a_n}+\nu_2^{(k-1)}},\nonumber\\
\beta_{m,n}^{(k)}\triangleq\ &\frac{T_{m,n}^{(k-1)}\left(x_{m,n}^{(k-1)}\right)^{-1}}{\sum_{m\in\mathcal{M}} {T_{m,n}^{(k-1)}}{\left(x_{m,n}^{(k-1)}\right)^{-1}}+\mu_n^{(k-1)}+\nu_1^{(k-1)}},\nonumber\\
\gamma_{1}^{(k)}\triangleq\ &\frac{\nu_2^{(k-1)}}{\sum_{n\in\mathcal{N}}{\lambda_n^{(k-1)} \underline a_n}+\nu_2^{(k-1)}},\nonumber\\
\gamma_{2,n}^{(k)}\triangleq\ &\frac{\mu_n^{(k-1)}}{\sum_{m\in\mathcal{M}} {T_{m,n}^{(k-1)}}{\left(x_{m,n}^{(k-1)}\right)^{-1}}+\mu_n^{(k-1)}+\nu_1^{(k-1)}},\nonumber\\
\gamma_{3,n}^{(k)}\triangleq\ &\frac{\nu_1^{(k-1)}}{\sum_{m\in\mathcal{M}} {T_{m,n}^{(k-1)}}{\left(x_{m,n}^{(k-1)}\right)^{-1}}+\mu_n^{(k-1)}+\nu_1^{(k-1)}}.\nonumber\end{aligned}$$
Problem \[prob:roap\] is a standard GP, which can be readily transformed into a convex problem and solved The details for solving Problem \[prob:roep\] are summarized in Algorithm \[alg:ro\]. By the convergence result in [@4275017 Proposition 3], and by comparing the KKT conditions of Problem \[prob:dualmax\] and Problem \[prob:roep\], we have the following result.
\[thm:conv-ro\] $\left(\mathbf{T}^{(k)}\right.$, $\mathbf{x}^{(k)}$, $\boldsymbol{\lambda}^{(k)}$, $\boldsymbol{\mu}^{(k)}$, ${\nu_1}^{(k)}$, ${\nu_2}^{(k)}$, $\left.y^{(k)}\right)$ obtained by Algorithm \[alg:ro\] converges to a stationary point of Problem \[prob:roep\], as $k\to \infty$. Furthermore, the limit point of $\left\{\left(\mathbf{T}^{(k)}\right.\right.$, $\boldsymbol{\lambda}^{(k)}$, $\boldsymbol{\mu}^{(k)}$, $\left.\left.{\nu_1}^{(k)}-{\nu_2}^{(k)}\right)\right\}$ is a stationary point of Problem \[prob:dualmax\].
Please refer to Appendix C.
**initialization**: choose any feasible solution $\big(\mathbf{T}^{(0)}$, $\mathbf{x}^{(0)}$, $\boldsymbol{\lambda}^{(0)}$, $\boldsymbol{\mu}^{(0)}$, ${\nu_1}^{(0)}$, ${\nu_2}^{(0)}$, $y^{(0)}\big)$ of Problem \[prob:roep\] as the initial point, and set $k=1$.\
**repeat** compute $\big(\mathbf{T}^{(k)}$, $\mathbf{x}^{(k)}$, $\boldsymbol{\lambda}^{(k)}$, $\boldsymbol{\mu}^{(k)}$, ${\nu_1}^{(k)}$, ${\nu_2}^{(k)}$, $y^{(k)}\big)$ by transforming Problem \[prob:roap\] into a GP\
in convex form, and solving it with standard convex optimization techniques. set $k=k+1$. **until** some convergence criteria is met.
\[alg:ro\]
Stochastic Optimization for Unknown File Popularity Distribution {#Sec:sto}
================================================================
In this section, we consider the case of unknown file popularity distribution. In this case, we would like to maximize the stochastic STP, by optimizing the caching probabilities. We formulate the stochastic optimal random caching design problem as follows.
\[prob:sto\] $$\begin{aligned}
\max_{\mathbf{T}}\quad &\underbrace{\mathbb{E}\left[q(\boldsymbol\xi,\mathbf T)\right]}_{=q_\text{st}(\mathbf a,\mathbf{T})}\nonumber\\
\text{s.t.} \quad &\eqref{eqn:T1},\eqref{eqn:T2},\nonumber
\end{aligned}$$ where $q\left(\mathbf a,\mathbf T\right)$ is given by .
Problem \[prob:sto\] is a nonconvex stochastic optimization problem, which is more challenging than a convex one. In the following, we develop an efficient parallel iterative algorithm to obtain a stationary point of Problem \[prob:sto\], using stochastic parallel SCA [@7412752]. Similarly, the parallel computation mechanism here can speed up the computation, especially for large $M$. Specifically, this algorithm updates the caching probabilities of the $M$ tiers, i.e., $\mathbf{T}_m\triangleq(T_{m,n})_{n\in\mathcal{N}}$, $m\in\mathcal M$, at each slot $t$ in a parallel manner, by maximizing $M$ approximate functions of $q_{st}(\mathbf a,\mathbf{T})$.
Let $\mathbf{T}_m^{(t)}$ denote the caching probabilities of tier $m$ obtained at slot $t$, and denote $\mathbf{T}^{(t)}\triangleq\big(\mathbf{T}_m^{(t)}\big)_{m\in\mathbf{M}}$. At slot $t$, choose $$\begin{aligned}
&\widehat{h}_{m}\left(\boldsymbol\xi^{(t)},\mathbf{T}_m,\mathbf{T}^{(t-1)}\right)=\rho^{(t)}\Bigg(q_{m}\left(\boldsymbol\xi^{(t)},\mathbf{T}_m,\mathbf{T}_{-m}^{(t-1)}\right)-\nonumber\\
&\sum_{j\in\mathcal{M},j\neq m}\sum_{n\in\mathcal{N}}\frac{\xi^{(t)}_n\theta_{m,j,K_j}T_{j,n}^{(t-1)}\left(T_{m,n}-T_{m,n}^{(t-1)}\right)}{\left({\sum_{l\in\mathcal{M}}{\theta_{l,j,K_j}T_{l,n}^{(t-1)}}+\eta_{j,K_j}}\right)^2}\Bigg)
+(1-\rho^{(t)})\sum_{n\in\mathcal N}\left({T}_{m,n}-{T}_{m,n}^{(t-1)}\right){{f}^{(t-1)}_{m,n}}\label{eqn:hhat}\end{aligned}$$ as an approximation function of $q_{st}(\mathbf a,\mathbf{T})$ for updating $\mathbf T_m$. Here, [^14] $\boldsymbol\xi^{(t)}\triangleq(\xi^{(t)}_n)_{n\in\mathcal N}$, $\rho^{(t)}$ is a positive diminishing stepsize satisfying $$\begin{aligned}
\rho^{(t)}>0,\quad \lim_{t\to\infty}\rho^{(t)}=0,\quad \sum_{t=1}^\infty\rho^{(t)}=\infty,\quad \sum_{t=1}^\infty\left(\rho^{(t)}\right)^2<\infty,\label{eqn:rho}\end{aligned}$$ and $f^{(t)}_{m,n}$ is given by $$\begin{aligned}
&f^{(t)}_{m,n}=(1-\rho^{(t)})f^{(t-1)}_{m,n}+\rho^{(t)}\left(\frac{\xi_n}{{\sum_{l\in\mathcal{M}}{\theta_{l,m}T_{l,n}^{(t)}}+\eta_{m}}}-\sum_{j\in\mathcal{M}}\frac{\xi_n\theta_{m,j}T_{j,n}^{(t)}}{\left({\sum_{l\in\mathcal{M}}{\theta_{l,j}T_{l,n}^{(t)}}+\eta_{j}}\right)^2}\right),\label{eqn:ft}\end{aligned}$$ where $f^{(0)}_{m,n}=0$ , $m\in\mathcal M$, $n\in\mathcal N$.
Note that the strongly concave component function of $q\big(\boldsymbol\xi^{(t)},\mathbf{T}_m,\mathbf{T}_{-m}^{(t-1)}\big)$, i.e., $q_{m}\big(\boldsymbol\xi^{(t)},\mathbf{T}_m,$ $\mathbf{T}_{-m}^{(t-1)}\big)$, is left unchanged, and the other nonconcave (actually convex) component functions, i.e., $q_{j}\big(\boldsymbol\xi^{(t)},\mathbf{T}_m,\mathbf{T}_{-m}^{(t-1)}\big)$, $j\in\mathcal{M}$, $j\neq m$, are linearized at $\mathbf{T}_m=\mathbf{T}_m^{(t-1)}$. In addition, note that the of $a_n$ at each slot $t$, i.e., $\xi^{(t)}_n$, becomes more accurate as ${\sum_{s\in\mathcal U}\mathbf I\left[\pi^{(t)}_s\neq0\right]}$ increases, and the of $\nabla_{{T}_{m,n}}q_\text{st}\big(\mathbf a,\mathbf{T}^{(t)}\big)$ based on accumulated instantaneous file requests $\{\pi_s(t):s\in\mathcal U\}$, $t\in\{1,2,\dots,t-1\}$, i.e., ${f}^{(t)}_{m,n}$, becomes more accurate as $t$ increases. This choice of the approximate function, $\widehat{h}_{m}\left(\boldsymbol\xi^{(t)},\mathbf{T}_m,\mathbf{T}^{(t-1)}\right)$, given in , is beneficial for similar reasons as in the case of perfect file popularity distribution.
Specifically, at slot $t$, we first solve the following problem for each tier $m\in\mathcal M$ separately, in a parallel manner.
\[prob:stoap\] $$\begin{aligned}
\widehat{\mathbf{T}}_m^{(t)}\triangleq\mathop{\arg\max}_{\mathbf{T}_m} \quad &\widehat{h}_{m}\left(\boldsymbol\xi^{(t)},\mathbf{T}_m,\mathbf{T}^{(t-1)}\right) \nonumber\\*
\text{s.t.} \quad &\eqref{eqn:Tm1},\eqref{eqn:Tm2} \nonumber\end{aligned}$$
Problem \[prob:stoap\] is a convex optimization problem and Slater’s condition is satisfied, implying that strong duality holds. Based on KKT conditions, we can obtain closed-form optimal solution of Problem \[prob:stoap\].
\[lem:closed-formsto\] For all $m\in\mathcal{M}$, the optimal solution of Problem \[prob:stoap\] is given by: $$\begin{aligned}
&\widehat T_{m,n}^{(t)}=
\left[\frac{1}{\theta_{m,m}}\sqrt{\frac{\rho^{(t)}\xi^{(t)}_n\left(\sum\limits_{l\neq m,l\in\mathcal{M}}\theta_{l,m}T_{l,n}^{(t-1)}+\eta_{m}\right)}{ \nu_m^{*({t})} + \rho^{(t)}\sum\limits_{\substack{j\neq m\\j\in\mathcal{M}}}\frac{\xi^{(t)}_n\theta_{m,j}T_{j,n}^{(t-1)}}{\left({\sum\limits_{l\in\mathcal{M}}{\theta_{l,j}T_{l,n}^{(t-1)}}+\eta_{j}}\right)^2}-(1-\rho^{(t)})f^{(t-1)}_{m,n} }} - \frac{\sum\limits_{\substack{l\neq m\\l\in\mathcal{M}}}\theta_{l,m}T_{l,n}^{(t-1)}+\eta_{m}}{\theta_{m,m}}\right]^1_0, \label{eqn:closed-formsto}\end{aligned}$$ where $[x]^1_0\triangleq\min\left\{\max\{x,0\},1\right\}$ and $\nu_m^{*({t})}$ is the Lagrange multiplier that satisfies $\sum_{n\in\mathcal{N}}\widehat T_{m,n}^{(t)}=K_m$.
Then, we update the caching probabilities of tier $m$ by $$\begin{aligned}
&\mathbf{T}_m^{(t)}=(1-\omega^{(t)})\mathbf{T}_m^{(t-1)}+\omega^{(t)}\widehat{\mathbf{T}}_m^{(t)},\label{eqn:recTk}\end{aligned}$$ where $\omega^{(t)}$ is a positive diminishing stepsize satisfying $$\begin{aligned}
\omega^{(t)}=0,\quad \lim_{t\to\infty}\omega^{(t)}=0,\quad \sum_{t=1}^\infty\omega^{(t)}=\infty,\quad \sum_{t=1}^\infty\left(\omega^{(t)}\right)^2<\infty,\quad \lim_{t\to\infty}\frac{\omega^{(t)}}{\rho^{(t)}}=0. \label{eqn:omega}\end{aligned}$$ Finally, the details of the proposed stochastic parallel iterative algorithm are summarized in Algorithm \[alg:sto\].[^15] Based on [@7412752 Theorem 1], we can show the following result.
\[thm:conv-sto\] If the stepsizes $\{\rho^{(t)}\}$ and $\{\omega^{(t)}\}$ satisfy and , respectively, then every limit point of $\{\mathbf{T}^{(t)}\}$ generated by Algorithm \[alg:sto\] is almost surely a stationary point of Problem \[prob:sto\].
Please refer to Appendix D.
\[alg:sto\] **initialization**: choose any feasible solution $\mathbf{T}^{(0)}$ of Problem \[prob:sto\] as the initial point, and set $t=1$.\
**repeat** observing $\{\pi_s(t):s\in\mathcal U\}$, and for all $m\in\mathcal{M}$, compute $\widehat{\mathbf{T}}_m^{(t)}$ according to , and\
update $\mathbf{T}_m^{(t)}$ according to . set $t=t+1$. **until** some convergence criteria is met.
Numerical Results {#Sec:simu}
=================
In this section, we show the convergence of the proposed algorithms and compare the proposed algorithms with baseline schemes. We choose $M=3$, $N = 500$,[^16] $\alpha = 3$, $\lambda_1=3.2\times10^{-7}$, $\lambda_2=8\times10^{-6}$, $\lambda_3=8 \times10^{-4}$, $P_1=10^3P_3$ and $P_2=10^{1.4}P_3$. For the case of perfect file popularity distribution, we assume the file popularity follows Zipf distribution, i.e., $a_n = \frac{n^{-\gamma}}{\sum_{n\in\mathcal N}n^{-\gamma}}$, $n\in\mathcal N$, where $\gamma$ is the Zipf exponent. For the case of imperfect file popularity distribution, we assume the estimated file popularity is a Zipf distribution, i.e., $\widehat a_n = \frac{n^{-\widehat\gamma}}{\sum_{n\in\mathcal N}n^{-\widehat\gamma}}$, $n\in\mathcal N$, where $\widehat\gamma$ is the Zipf exponent, and $\varepsilon_n=\epsilon a_n$, $n\in\mathcal N$ for some $\epsilon\in(0,1)$. For the case of unknown file popularity distribution, we assume that file requests follow the Zipf distribution, i.e., $a_n = \frac{n^{-\gamma}}{\sum_{n\in\mathcal N}n^{-\gamma}}$, $n\in\mathcal N$, as in the case of perfect file popularity.
Convergence
-----------
First, we show the convergence of the proposed algorithms. Fig. \[fig:simulation-convergence\] illustrates the STP, worst-case STP and stochastic STP versus the number of iterations. In addition, Algorithm \[alg:sto\] with a larger $U$ converges faster, as more observations for file requests can be used to approximate the file popularity distribution and the gradient of the stochastic STP.
\[fig:simulation-convergence\]
Performance
-----------
Next, we compare the STP of Algorithm \[alg:rand\] (obtained using 20 iterations), the STP of Algorithm \[alg:ro\] (obtained using 30 iterations) and the STP of Algorithm \[alg:sto\] with (obtained using iterations (slots)) with those of three baseline schemes. Baseline 1 (most popular) refers to the design in which each BS in tier $m$ stores the $K_m$ most popular files [@bacstuug2015transfer] according to ${\mathbf a}$ in the case of perfect file popularity distribution, $\widehat{\mathbf a}$ in the case of imperfect file popularity distribution, and the estimated file popularity $\widetilde{\mathbf a}\triangleq(\widetilde a_n)_{n\in\mathcal N}$ in the case of unknown file popularity distribution, where Baseline 2 (i.i.d. file popularity) refers to the design in which each BS in tier $m$ randomly stores $K_m$ files, in an i.i.d. manner [@Bharath] with file $n$ being selected with probabilities ${a}_n$ in the case of perfect file popularity distribution, $\widehat{a}_n$ in the case of imperfect file popularity distribution, and $\widetilde{a}_n$ in the case of unknown file popularity distribution. [^17]
\[fig:simulation-performance-pe\]
\[fig:simulation-performance-ro\]
\[fig:simulation-performance-sto\]
Fig. \[fig:simulation-pe-size\], Fig. \[fig:simulation-ro-size\] and Fig. \[fig:simulation-sto-size\] illustrate the STP, the worst-case STP and the stochastic STP versus the cache sizes, respectively. From Fig. \[fig:simulation-pe-size\], Fig. \[fig:simulation-ro-size\] and Fig. \[fig:simulation-sto-size\], we can observe that as the cache sizes $K_m$, $m\in\mathcal M$ increase, the STP of each scheme increases. This is because as $K_m$, $m\in\mathcal M$ increase, each BS can cache more files, and the probability that a randomly requested file is cached at a nearby BS increases. In addition, note that for each scheme, its STP in Fig. \[fig:simulation-pe-size\] is very close to its stochastic STP in Fig. \[fig:simulation-sto-size\], since under the same file popularity distribution, its stochastic STP converges to its STP as the number of goes to infinity.
Fig. \[fig:simulation-pe-zipf\], Fig. \[fig:simulation-ro-zipf\] and Fig. \[fig:simulation-sto-zipf\] illustrate the STP, the worst-case STP and the stochastic STP versus the Zipf exponents, respectively. From Fig. \[fig:simulation-pe-zipf\], Fig. \[fig:simulation-ro-zipf\] and Fig. \[fig:simulation-sto-zipf\], we can observe that as the Zipf exponent $\gamma$ $(\widehat\gamma)$ increases, the STP of each scheme increases. This is because when $\widehat\gamma$ ($\gamma$) increases, the tail of the Zipf distribution becomes small, and hence, the probability that a randomly requested file is cached at a nearby BS increases for each scheme. Similarly, note that for each scheme, its STP in Fig. \[fig:simulation-pe-zipf\] is very close to its stochastic STP in Fig. \[fig:simulation-sto-zipf\].
Fig. \[fig:simulation-ro-error\] illustrates the worst-case STP versus the error bound parameter $\epsilon$ in the case of imperfect file popularity. From Fig. \[fig:simulation-ro-error\], we can see that as $\epsilon$ increases, the worst-case STP of each scheme decreases. The reason is that as $\epsilon$ increases, the possible estimation error increases, and the worst-case STP decreases. In addition, we can observe that the decrease rates of the worst-case STPs of all schemes are different, as the estimation error has different impacts on these schemes.
Finally, from Fig \[fig:simulation-performance-pe\], Fig \[fig:simulation-performance-ro\] and Fig \[fig:simulation-performance-sto\], we can observe that our proposed solutions outperform the baseline schemes in all three cases. This is because in each case, the properties of the file popularity distribution are appropriately captured when formulating the optimal caching design problem, and a stationary point of the challenging optimization problem is obtained using the proposed algorithm.
Conclusion
==========
In this paper, we considered optimal caching designs for perfect, imperfect and unknown file popularity distributions in large-scale multi-tier wireless networks. First, in the case of perfect file popularity distribution, we formulate , which is nonconvex. We an efficient parallel iterative algorithm to obtain a stationary point. Then, in the case of imperfect file popularity distribution, we formulate is a challenging robust optimization problem, we an efficient iterative algorithm a stationary point. Next, in the case of unknown file popularity distribution, we formulate is a challenging nonconvex stochastic optimization problem, we develop an efficient iterative algorithm to obtain a stationary point. Finally, by numerical results, we show that the proposed solutions achieve gains over existing schemes in all three cases.
Appendix A: Proof of Theorem \[thm:conv-rand\] {#appendix-a-proof-of-theoremthmconv-rand .unnumbered}
==============================================
We show that the assumptions in [@razaviyayn2014parallel Theorem 1] are satisfied.
- It is clear that $h_m\big(\mathbf{a},\mathbf{T}_m,\mathbf{T}^{(k)}\big)$ is continuously differentiable for any given $\mathbf{a}$ and $\mathbf{T}^{(k)}$. It remains to show that $h_m\big(\mathbf{a},\mathbf{T}_m,\mathbf{T}^{(k)}\big)$ is strongly convex in $\mathbf T_m$. Note that the Hessian of $h_m\big(\mathbf{a},\mathbf{T}_m,\mathbf{T}^{(k)}\big)$ is $\boldsymbol\nabla^2_{\mathbf T_m}h_m\big(\mathbf{a},\mathbf{T}_m,\mathbf{T}^{(k)}\big)\triangleq\Big(\frac{\partial^2 h_m\big(\mathbf{a},\mathbf{T}_m,\mathbf{T}^{(k)}\big)}{\partial T_{m,n}\partial T_{m,n'}}\Big)_{n\in\mathcal N,n'\in\mathcal N}$, where $$\begin{aligned}
\frac{\partial^2 h_m\big(\mathbf{a},\mathbf{T}_m,\mathbf{T}^{(k)}\big)}{\partial T_{m,n}\partial T_{m,n'}}=
\begin{cases}
{\frac{-2 a_n \eta_m \theta_{m,m}}{\left({\theta_{m,m}T_{m,n}}+\sum_{l\in\mathcal{M},l\neq m}{\theta_{l,m}T^{(k)}_{l,n}}+\eta_{m}\right)^3}},\quad &n'= n,\\
0,\quad &n'\neq n.
\end{cases}\end{aligned}$$ Thus, $\boldsymbol\nabla^2_{\mathbf T_m}h_m\big(\mathbf{a},\mathbf{T}_m,\mathbf{T}^{(k)}\big)\preceq-b\mathbf E$, where $b\triangleq\min\limits_{n\in\mathcal N}{\frac{2 a_n \eta_m \theta_{m,m}}{\left({\theta_{m,m}}+\sum_{l\in\mathcal{M},l\neq m}{\theta_{l,m}T^{(k)}_{l,n}}+\eta_{m}\right)^3}}$, and $\mathbf E$ is the identity matrix. That is, $h_m\big(\mathbf{a},\mathbf{T}_m,\mathbf{T}^{(k)}\big)$ is uniformly strongly concave with constant $b>0$. Thus, the first assumption of [@razaviyayn2014parallel Theorem 1] is satisfied.
- By , we have $\nabla_{\mathbf{T}_{m}}{h_m\big(\mathbf{a},\mathbf{T}_m,\mathbf{T}^{(k)}\big)}=\nabla_{\mathbf{T}_{m}}{q\big(\mathbf a,\mathbf T\big)}$ for all $m\in\mathcal M$. Thus, the second assumption of [@razaviyayn2014parallel Theorem 1] is satisfied.
- It is clear that $h_m\big(\mathbf{a},\mathbf{T}_m,\mathbf{T}^{(k)}\big)$ is smooth on the constraint set determined by and for any given $\mathbf{a}$ and $\mathbf{T}_m$, and hence its derivative is Lipschitz continuous. Thus, the third assumption of [@razaviyayn2014parallel Theorem 1] is satisfied.
Therefore, Theorem \[thm:conv-sto\] readily follows from [@razaviyayn2014parallel Theorem 1].
Appendix B: Proof of Lemma \[lem:roeq\] {#appendix-b-proof-of-lemmalemroeq .unnumbered}
=======================================
Firstly, the inner problem of Problem \[prob:ro\], $\min_{\mathbf{a}\in\mathcal{A}}q_{\infty}\left(\mathbf a,\mathbf{T}\right)$, can be rewritten as follows.
\[prob:inter\] $$\begin{aligned}
\min_{\mathbf a} \quad &q(\mathbf a,\mathbf T)\nonumber\\
\text{s.t.} \quad
&\underline a_n\leq a_n,\quad n\in\mathcal{N},\label{eqn:alowbound}\\
&a_n\leq\overline a_n,\quad n\in\mathcal{N},\label{eqn:aupbound}\\
&\sum_{n\in\mathcal N}a_n=1. \label{eqn:asum}
\end{aligned}$$
It is clear that Problem \[prob:inter\] is an LP with respect to $\mathbf a$ for any given $\mathbf T$, satisfying and . From [@boyd2004convex pp.225], we obtain its dual problem as below.
\[prob:interdual\] $$\begin{aligned}
\max_{\boldsymbol{\lambda},\boldsymbol{\mu},{\nu}} \quad &\sum_{n\in\mathcal{N}}{(\lambda_n \underline a_n-\mu_n \overline a_n)}-\nu\nonumber\\
\text{s.t.} \quad &\eqref{eqn:dual1},\nonumber\\
&\lambda_n\geq 0, \quad n\in\mathcal{N},\label{eqn:dual20}\\
&\mu_n\geq 0, \quad n\in\mathcal{N},\label{eqn:dual30}
\end{aligned}$$
where $\boldsymbol{\lambda}$, $\boldsymbol{\mu}$ and $\nu$ are the Lagrange multipliers corresponding to , and , respectively.
As strong duality holds for LP, the dual problem in Problem \[prob:interdual\] and the primal problem in Problem \[prob:inter\] share the same optimal value, which can be viewed as a function of $\mathbf T$. Thus, the maximin problem in Problem \[prob:ro\] can be equivalently converted to the maximization problem in Problem \[prob:dualmax\], by replacing the inner LP problem in Problem \[prob:inter\] with its dual problem in Problem \[prob:interdual\].
Appendix C: Proof of Theorem \[thm:conv-ro\] {#appendix-c-proof-of-theoremthmconv-ro .unnumbered}
============================================
By [@4275017 Proposition 3], it can be easily shown that Algorithm \[alg:ro\] converges to a stationary point of Problem \[prob:roep\]. Let $\left(\mathbf{T}^{\star}\right.$, $\mathbf{x}^{\star}$, $\boldsymbol{\lambda}^{\star}$, $\boldsymbol{\mu}^{\star}$, ${\nu_1}^{\star}$, $\left.{\nu_2}^{\star}, y^{\star}\right)$ denote a stationary point of Problem \[prob:roep\], which satisfies the KKT conditions of Problem \[prob:roep\]. It remains to show that $\left(\mathbf{T}^{\star}\right.$, $\boldsymbol{\lambda}^{\star}$, $\boldsymbol{\mu}^{\star}$, $\left.{\nu_1}^{\star}-{\nu_2}^{\star}\right)$ is a stationary point of Problem \[prob:dualmax\], which satisfies the KKT conditions of Problem \[prob:dualmax\].
First, we derive the KKT conditions of Problem \[prob:roep\] based on its equivalent form given as follows.
\[prob:roep2\] $$\begin{aligned}
\min_{{\mathbf{T},\mathbf{x},\boldsymbol{\lambda},\boldsymbol{\mu}, {\nu_1},{\nu_2},y}} \quad &-y \nonumber\\
\text{s.t.} \quad &\eqref{eqn:dual20},\eqref{eqn:dual30},\nonumber\\
&y-\sum_{n\in\mathcal{N}}{(\lambda_n \underline a_n-\mu_n \overline a_n)}+(\nu_1-\nu_2)\leq0\label{eqn:provprob1}\\
&\lambda_n-\mu_n-(\nu_1-\nu_2)-\sum_{m\in\mathcal{M}} {T_{m,n}}{x_{m,n}^{-1}}\leq0, \quad n\in\mathcal{N},\label{eqn:provprob2}\\
&\left({\sum_{l\in\mathcal{M}} \theta_{l,m} T_{l,n} + \eta_{m}}\right){x^{-1}_{m,n}}\leq1, \quad m\in\mathcal{M},\ n\in\mathcal{N}, \label{eqn:provprob3}\\
&T_{m,n}\geq 0,\quad m\in\mathcal M,\ n\in\mathcal N, \label{eqn:provprob4}\\
&T_{m,n}\leq 1,\quad m\in\mathcal M,\ n\in\mathcal N, \label{eqn:provprob5}\\
&\sum\limits_{n\in\mathcal N}T_{m,n}\leq K_m,\quad m\in\mathcal M. \label{eqn:provprob6}\end{aligned}$$
The Lagrangian function of Problem \[prob:roep2\] is given by: $$\begin{aligned}
&\mathcal{L}\left(\mathbf{T},\mathbf{x}, \boldsymbol{\lambda}, \boldsymbol{\mu}, {\nu_1}, {\nu_2}, y, \widetilde{\boldsymbol\nu}_1, \widetilde{\boldsymbol\nu}_2, \widetilde\lambda_0, \widetilde{\boldsymbol\lambda}, \widetilde{\boldsymbol\mu}, \widetilde{\boldsymbol\kappa}_1, \widetilde{\boldsymbol\kappa}_2, \widetilde{\boldsymbol\iota}\right)=
-y+\sum_{n\in\mathcal N}\widetilde\nu_{1,n}(-\lambda_n)+\sum_{n\in\mathcal N}\widetilde\nu_{2,n}(-\mu_n)\nonumber\\
&+\widetilde\lambda_0\left(y-\sum_{n\in\mathcal{N}}{(\lambda_n \underline a_n-\mu_n \overline a_n)}+(\nu_1-\nu_2)\right)
+\sum_{n\in\mathcal N}\widetilde\lambda_n\left(\lambda_n-\mu_n-(\nu_1-\nu_2)-\sum_{m\in\mathcal{M}} {T_{m,n}}{x_{m,n}^{-1}}\right)\nonumber\\
&+\sum_{m\in\mathcal M}\sum_{n\in\mathcal N}\widetilde\mu_{m,n}\left(\left({\sum_{l\in\mathcal{M}} \theta_{l,m} T_{l,n} + \eta_{m}}\right){x^{-1}_{m,n}}-1\right)
+\sum_{m\in\mathcal M}\widetilde\iota_m\left(\sum\limits_{n\in\mathcal N}T_{m,n}-K_m\right)\nonumber\\
&+\sum_{m\in\mathcal M}\sum_{n\in\mathcal N}\widetilde\kappa_{1,m,n}(-T_{m,n})
+\sum_{m\in\mathcal M}\sum_{n\in\mathcal N}\widetilde\kappa_{2,m,n}(T_{m,n}-1), \label{eqn:lag}\end{aligned}$$ where $\widetilde{\boldsymbol\nu}_1\triangleq(\widetilde\nu_{1,n})_{n\in\mathcal N}$, $\widetilde{\boldsymbol\nu}_2\triangleq(\widetilde\nu_{2,n})_{n\in\mathcal N}$, $\widetilde\lambda_0$, $\widetilde{\boldsymbol\lambda}\triangleq(\widetilde\lambda_{n})_{n\in\mathcal N}$, $\widetilde{\boldsymbol\mu}\triangleq(\widetilde\mu_{m,n})_{m\in\mathcal M,n\in\mathcal N}$, $\widetilde{\boldsymbol\kappa}_1\triangleq(\widetilde\kappa_{1,m,n})_{m\in\mathcal M,n\in\mathcal N}$, $\widetilde{\boldsymbol\kappa}_2\triangleq(\widetilde\kappa_{2,m,n})_{m\in\mathcal M,n\in\mathcal N}$ and $\widetilde{\boldsymbol\iota}\triangleq(\widetilde\iota_{m})_{m\in\mathcal M}$ are the Lagrange multipliers corresponding to , , , , , , and , respectively.
Next, as $\left(\mathbf{T}^{\star}\right.$, $\mathbf{x}^{\star}$, $\boldsymbol{\lambda}^{\star}$, $\boldsymbol{\mu}^{\star}$, ${\nu_1}^{\star}$, ${\nu_2}^{\star}$, $\left.y^{\star}\right)$ is a stationary point of Problem \[prob:roep\], there exists a dual point $\left(\widetilde{\boldsymbol\nu}_1^{\star}\right.$, $\widetilde{\boldsymbol\nu}_2^{\star}$, $\widetilde\lambda_0^{\star}$, $\widetilde{\boldsymbol\lambda}^{\star}$, $\widetilde{\boldsymbol\mu}^{\star}$, $\widetilde{\boldsymbol\kappa}_1^{\star}$, $\widetilde{\boldsymbol\kappa}_2^{\star}$, $\left.\widetilde{\boldsymbol\iota}^{\star}\right)$ together with $\left(\mathbf{T}^{\star}\right.$, $\mathbf{x}^{\star}$, $\boldsymbol{\lambda}^{\star}$, $\boldsymbol{\mu}^{\star}$, ${\nu_1}^{\star}$, ${\nu_2}^{\star}$, $\left.y^{\star}\right)$ satisfies the following KKT conditions:
- Feasibility: The primal point $\left(\mathbf{T}^{\star}\right.$, $\mathbf{x}^{\star}$, $\boldsymbol{\lambda}^{\star}$, $\boldsymbol{\mu}^{\star}$, ${\nu_1}^{\star}$, $\left.{\nu_2}^{\star}, y^{\star}\right)$ satisfies the primal constraints in , , , , , , and , and the dual point $\left(\widetilde{\boldsymbol\nu}_1^{\star}\right.$, $\widetilde{\boldsymbol\nu}_2^{\star}$, $\widetilde\lambda_0^{\star}$, $\widetilde{\boldsymbol\lambda}^{\star}$, $\widetilde{\boldsymbol\mu}^{\star}$, $\widetilde{\boldsymbol\kappa}_1^{\star}$, $\widetilde{\boldsymbol\kappa}_2^{\star}$, $\left.\widetilde{\boldsymbol\iota}^{\star}\right)$ satisfies the dual constraints. $$\begin{aligned}
\widetilde\nu_{1,n}^{\star}, \widetilde\nu_{2,n}^{\star}, \widetilde\lambda_0^{\star}, \widetilde\lambda_n^{\star}, \widetilde\mu_{m,n}^{\star}, \widetilde\kappa_{1,m,n}^{\star}, \widetilde\kappa_{2,m,n}^{\star}, \widetilde\iota_{m}^{\star}\geq0, \quad m\in\mathcal M,\ n\in\mathcal N. \label{eqn:dual-feasible}\end{aligned}$$
- Complementary slackness: $\left(\mathbf{T}^{\star}\right.$, $\mathbf{x}^{\star}$, $\boldsymbol{\lambda}^{\star}$, $\boldsymbol{\mu}^{\star}$, ${\nu_1}^{\star}$, $\left.{\nu_2}^{\star}, y^{\star}\right)$ and $\left(\widetilde{\boldsymbol\nu}_1^{\star}\right.$, $\widetilde{\boldsymbol\nu}_2^{\star}$, $\widetilde\lambda_0^{\star}$, $\widetilde{\boldsymbol\lambda}^{\star}$, $\widetilde{\boldsymbol\mu}^{\star}$, $\widetilde{\boldsymbol\kappa}_1^{\star}$, $\widetilde{\boldsymbol\kappa}_2^{\star}$, $\left.\widetilde{\boldsymbol\iota}^{\star}\right)$ satisfy: $$\begin{aligned}
&\widetilde\nu_{1,n}^{\star}\lambda_n^{\star}=0,\quad {n\in\mathcal N},\label{eqn:lambdacs}\\
&\widetilde\nu_{2,n}^{\star}\mu_n^{\star}=0,\quad {n\in\mathcal N},\label{eqn:mucs}\\
&\widetilde\lambda_0^{\star}\left(y^{\star}-\sum_{n\in\mathcal{N}}{(\lambda_n^{\star} \underline a_n-\mu_n^{\star} \overline a_n)}+(\nu_1^{\star}-\nu_2^{\star})\right)=0,\\
&\widetilde\lambda_n^{\star}\left(\lambda_n^{\star}-\mu_n^{\star}-(\nu_1^{\star}-\nu_2^{\star})-\sum_{m\in\mathcal{M}} {T_{m,n}^{\star}}({x_{m,n}^{\star}})^{-1}\right)=0,\quad {n\in\mathcal N},\label{eqn:slack4}\\
&\widetilde\mu_{m,n}^{\star}\left(\left({\sum_{l\in\mathcal{M}} \theta_{l,m} T_{l,n}^{\star} + \eta_{m}}\right)({x_{m,n}^{\star}})^{-1}-1\right)=0,\quad{m\in\mathcal M},\ {n\in\mathcal N},\label{eqn:slack5}\\
&\widetilde\kappa_{1,m,n}^{\star}T_{m,n}^{\star}=0,\quad {m\in\mathcal M},\ {n\in\mathcal N},\label{eqn:k0}\\
&\widetilde\kappa_{2,m,n}^{\star}(T_{m,n}^{\star}-1)=0,\quad {m\in\mathcal M},\ {n\in\mathcal N},\label{eqn:k1}\\
&\widetilde\iota_m^{\star}\left(\sum\limits_{n\in\mathcal N}T_{m,n}^{\star}-K_m\right)=0,\quad {m\in\mathcal M}.\label{eqn:slack8}\end{aligned}$$
- Zero derivative: $$\begin{aligned}
&\frac{\partial\mathcal{L}\left(\mathbf{T}^{\star},\mathbf{x}^{\star}, \boldsymbol{\lambda}^{\star}, \boldsymbol{\mu}^{\star}, {\nu_1}^{\star}, {\nu_2}^{\star}, y^{\star}, \widetilde{\boldsymbol\nu}_1^{\star}, \widetilde{\boldsymbol\nu}_2^{\star}, \widetilde\lambda_0^{\star}, \widetilde{\boldsymbol\lambda}^{\star}, \widetilde{\boldsymbol\mu}^{\star}, \widetilde{\boldsymbol\kappa}_1^{\star}, \widetilde{\boldsymbol\kappa}_2^{\star}, \widetilde{\boldsymbol\iota}^{\star}\right)}{\partial y}=\widetilde\lambda_0^{\star}-1=0,\label{eqn:deri-1}\\
&\frac{\partial\mathcal{L}\left(\mathbf{T}^{\star},\mathbf{x}^{\star}, \boldsymbol{\lambda}^{\star}, \boldsymbol{\mu}^{\star}, {\nu_1}^{\star}, {\nu_2}^{\star}, y^{\star}, \widetilde{\boldsymbol\nu}_1^{\star}, \widetilde{\boldsymbol\nu}_2^{\star}, \widetilde\lambda_0^{\star}, \widetilde{\boldsymbol\lambda}^{\star}, \widetilde{\boldsymbol\mu}^{\star}, \widetilde{\boldsymbol\kappa}_1^{\star}, \widetilde{\boldsymbol\kappa}_2^{\star}, \widetilde{\boldsymbol\iota}^{\star}\right)}{\partial\lambda_n}\nonumber\\
&=-\widetilde\nu_{1,n}^{\star}-\widetilde\lambda_0^{\star}\underline a_n+\widetilde\lambda_n^{\star}=0,\quad {n\in\mathcal N},\label{eqn:deri-2}\\
&\frac{\partial\mathcal{L}\left(\mathbf{T}^{\star},\mathbf{x}^{\star}, \boldsymbol{\lambda}^{\star}, \boldsymbol{\mu}^{\star}, {\nu_1}^{\star}, {\nu_2}^{\star}, y^{\star}, \widetilde{\boldsymbol\nu}_1^{\star}, \widetilde{\boldsymbol\nu}_2^{\star}, \widetilde\lambda_0^{\star}, \widetilde{\boldsymbol\lambda}^{\star}, \widetilde{\boldsymbol\mu}^{\star}, \widetilde{\boldsymbol\kappa}_1^{\star}, \widetilde{\boldsymbol\kappa}_2^{\star}, \widetilde{\boldsymbol\iota}^{\star}\right)}{\partial \mu_n}\nonumber\\
&=-\widetilde\nu_{2,n}^{\star}+\widetilde\lambda_0^{\star}\overline a_n-\widetilde\lambda_n^{\star}=0,\quad {n\in\mathcal N},\label{eqn:deri-3}\\
&\frac{\partial\mathcal{L}\left(\mathbf{T}^{\star},\mathbf{x}^{\star}, \boldsymbol{\lambda}^{\star}, \boldsymbol{\mu}^{\star}, {\nu_1}^{\star}, {\nu_2}^{\star}, y^{\star}, \widetilde{\boldsymbol\nu}_1^{\star}, \widetilde{\boldsymbol\nu}_2^{\star}, \widetilde\lambda_0^{\star}, \widetilde{\boldsymbol\lambda}^{\star}, \widetilde{\boldsymbol\mu}^{\star}, \widetilde{\boldsymbol\kappa}_1^{\star}, \widetilde{\boldsymbol\kappa}_2^{\star}, \widetilde{\boldsymbol\iota}^{\star}\right)}{\partial \nu_1}=\widetilde\lambda_0^{\star}-\sum_{n\in\mathcal N}\widetilde\lambda_n^{\star}=0,\label{eqn:deri-4}\\
&\frac{\partial\mathcal{L}\left(\mathbf{T}^{\star},\mathbf{x}^{\star}, \boldsymbol{\lambda}^{\star}, \boldsymbol{\mu}^{\star}, {\nu_1}^{\star}, {\nu_2}^{\star}, y^{\star}, \widetilde{\boldsymbol\nu}_1^{\star}, \widetilde{\boldsymbol\nu}_2^{\star}, \widetilde\lambda_0^{\star}, \widetilde{\boldsymbol\lambda}^{\star}, \widetilde{\boldsymbol\mu}^{\star}, \widetilde{\boldsymbol\kappa}_1^{\star}, \widetilde{\boldsymbol\kappa}_2^{\star}, \widetilde{\boldsymbol\iota}^{\star}\right)}{\partial x_{m,n}}\nonumber\\
&=-\frac{\widetilde\lambda_n^{\star} T_{m,n}^{\star}}{x_{m,n}^{\star}}+\frac{\widetilde\mu_{m,n}^{\star}}{x_{m.n}^{\star}}{\left({\sum_{l\in\mathcal{M}} \theta_{l,m} T_{l,n}^{\star} + \eta_{m}}\right)}=0,\quad {m\in\mathcal M},\ {n\in\mathcal N},\label{eqn:deri-5}\\
&\frac{\partial\mathcal{L}\left(\mathbf{T}^{\star},\mathbf{x}^{\star}, \boldsymbol{\lambda}^{\star}, \boldsymbol{\mu}^{\star}, {\nu_1}^{\star}, {\nu_2}^{\star}, y^{\star}, \widetilde{\boldsymbol\nu}_1^{\star}, \widetilde{\boldsymbol\nu}_2^{\star}, \widetilde\lambda_0^{\star}, \widetilde{\boldsymbol\lambda}^{\star}, \widetilde{\boldsymbol\mu}^{\star}, \widetilde{\boldsymbol\kappa}_1^{\star}, \widetilde{\boldsymbol\kappa}_2^{\star}, \widetilde{\boldsymbol\iota}^{\star}\right)}{\partial T_{m,n}}\nonumber\\
&=-\widetilde\lambda_n^{\star} ({x_{m,n}^{\star}})^{-1}+\sum_{l\in\mathcal M}\widetilde\mu_{l,n}^{\star}\theta_{m,l}({x_{l,n}^{\star}})^{-1}-\widetilde\kappa_{1,m,n}^{\star}+\widetilde\kappa_{2,m,n}^{\star}+\widetilde\iota_m^{\star}=0,\quad {m\in\mathcal M},\ {n\in\mathcal N}.\label{eqn:deri-6}\end{aligned}$$
Then, based on the KKT conditions of Problem \[prob:roep\], we show that the primal point $\left(\mathbf{T}^{\star}\right.$, $\mathbf{x}^{\star}$, $\boldsymbol{\lambda}^{\star}$, $\boldsymbol{\mu}^{\star}$, $\left.{\nu_1}^{\star}-{\nu_2}^{\star}\right)$ and the dual point $\left(\widetilde{\boldsymbol\nu}_1^{\star}\right.$, $\widetilde{\boldsymbol\nu}_2^{\star}$, $\widetilde{\boldsymbol\lambda}^{\star}$, $\widetilde{\boldsymbol\kappa}_1^{\star}$, $\widetilde{\boldsymbol\kappa}_2^{\star}$, $\left.\widetilde{\boldsymbol\iota}^{\star}\right)$ satisfy the KKT conditions of Problem \[prob:dualmax\], implying that $\left(\mathbf{T}^{\star}\right.$, $\boldsymbol{\lambda}^{\star}$, $\boldsymbol{\mu}^{\star}$, $\left.{\nu_1}^{\star}-{\nu_2}^{\star}\right)$ is a stationary point of Problem \[prob:dualmax\]. Note that the Lagrangian function of Problem \[prob:dualmax\] is given by: $$\begin{aligned}
&\mathcal{L}\left(\mathbf{T},\boldsymbol{\lambda}, \boldsymbol{\mu}, {\nu}, \widetilde{\boldsymbol\nu}_1, \widetilde{\boldsymbol\nu}_2, \widetilde{\boldsymbol\lambda}, \widetilde{\boldsymbol\kappa}_1, \widetilde{\boldsymbol\kappa}_2, \widetilde{\boldsymbol\iota}\right)=-\sum_{n\in\mathcal{N}}{(\lambda_n \underline a_n-\mu_n \overline a_n)}+\nu\nonumber\\
&+\sum_{n\in\mathcal N}\widetilde\nu_{1,n}(-\lambda_n)+\sum_{n\in\mathcal N}\widetilde\nu_{2,n}(-\mu_n)
+\sum_{n\in\mathcal N}\widetilde\lambda_n\left(\lambda_n-\mu_n-\nu-\sum_{m\in\mathcal{M}}{\frac{T_{m,n}}{\left({\sum_{l\in\mathcal{M}} \theta_{l,m} T_{l,n} + \eta_{m}}\right)}}\right)\nonumber\\
&+\sum_{m\in\mathcal M}\sum_{n\in\mathcal N}\widetilde\kappa_{1,m,n}(-T_{m,n})+\sum_{m\in\mathcal M}\sum_{n\in\mathcal N}\widetilde\kappa_{2,m,n}(T_{m,n}-1)+\sum_{m\in\mathcal M}\widetilde\iota_m\left(\sum\limits_{n\in\mathcal N}T_{m,n}-K_m\right). \label{eqn:lag2}\end{aligned}$$ Now, we prove that the primal point $\left(\mathbf{T}^{\star}\right.$, $\mathbf{x}^{\star}$, $\boldsymbol{\lambda}^{\star}$, $\boldsymbol{\mu}^{\star}$, $\left.{\nu_1}^{\star}-{\nu_2}^{\star}\right)$ and the dual point $\left(\widetilde{\boldsymbol\nu}_1^{\star}\right.$, $\widetilde{\boldsymbol\nu}_2^{\star}$, $\widetilde{\boldsymbol\lambda}^{\star}$, $\widetilde{\boldsymbol\kappa}_1^{\star}$, $\widetilde{\boldsymbol\kappa}_2^{\star}$, $\left.\widetilde{\boldsymbol\iota}^{\star}\right)$ satisfy the following KKT conditions.
- Complementary slackness: It is clear that $\left(\mathbf{T}^{\star}\right.$, $\mathbf{x}^{\star}$, $\boldsymbol{\lambda}^{\star}$, $\boldsymbol{\mu}^{\star}$, $\left.{\nu_1}^{\star}-{\nu_2}^{\star}\right)$ and $\left(\widetilde{\boldsymbol\nu}_1^{\star}\right.$, $\widetilde{\boldsymbol\nu}_2^{\star}$, $\widetilde{\boldsymbol\lambda}^{\star}$, $\widetilde{\boldsymbol\kappa}_1^{\star}$, $\widetilde{\boldsymbol\kappa}_2^{\star}$, $\left.\widetilde{\boldsymbol\iota}^{\star}\right)$ satisfy , , and .
- Zero derivative: Substituting $\widetilde\lambda_0^{\star}=1$ into , and , we have: $$\begin{aligned}
&\frac{\partial\mathcal{L}\left(\mathbf{T}^{\star},\boldsymbol{\lambda}^{\star}, \boldsymbol{\mu}^{\star}, {\nu}^{\star}, \widetilde{\boldsymbol\nu}_1^{\star}, \widetilde{\boldsymbol\nu}_2^{\star}, \widetilde{\boldsymbol\lambda}^{\star}, \widetilde{\boldsymbol\kappa}_1^{\star}, \widetilde{\boldsymbol\kappa}_2^{\star}, \widetilde{\boldsymbol\iota}^{\star}\right)}{\partial\lambda_n}=-\widetilde\nu_{1,n}^{\star}-\underline a_n+\widetilde\lambda_n^{\star}=0,\quad {n\in\mathcal N},\label{eqn:deri-10}\\
&\frac{\partial\mathcal{L}\left(\mathbf{T}^{\star},\boldsymbol{\lambda}^{\star}, \boldsymbol{\mu}^{\star}, {\nu}^{\star}, \widetilde{\boldsymbol\nu}_1^{\star}, \widetilde{\boldsymbol\nu}_2^{\star}, \widetilde{\boldsymbol\lambda}^{\star}, \widetilde{\boldsymbol\kappa}_1^{\star}, \widetilde{\boldsymbol\kappa}_2^{\star}, \widetilde{\boldsymbol\iota}^{\star}\right)}{\partial \mu_n}=-\widetilde\nu_{2,n}^{\star}+\overline a_n-\widetilde\lambda_n^{\star}=0,\quad {n\in\mathcal N}.\label{eqn:deri-20}\\
&\frac{\partial\mathcal{L}\left(\mathbf{T}^{\star},\boldsymbol{\lambda}^{\star}, \boldsymbol{\mu}^{\star}, {\nu}^{\star}, \widetilde{\boldsymbol\nu}_1^{\star}, \widetilde{\boldsymbol\nu}_2^{\star}, \widetilde{\boldsymbol\lambda}^{\star}, \widetilde{\boldsymbol\kappa}_1^{\star}, \widetilde{\boldsymbol\kappa}_2^{\star}, \widetilde{\boldsymbol\iota}^{\star}\right)}{\partial \nu}=1-\sum_{n\in\mathcal N}\widetilde\lambda_n^{\star}=0. \label{eqn:deri-30}\end{aligned}$$ Substitute $\widetilde\lambda_0^{\star}=1$ into , we have $\widetilde\lambda_n^{\star}=\underline a_n+\widetilde\nu_{1,n}^{\star}>0$. By $\widetilde\lambda_n^{\star}>0$ and , we have $$\begin{aligned}
\widetilde\mu_{m,n}^{\star}=\frac{\widetilde\lambda_n T_{m,n}^{\star}}{\left({\sum_{l\in\mathcal{M}} \theta_{l,m} T_{l,n}^{\star} + \eta_{m}}\right)}>0.\label{eqn:mueq}\end{aligned}$$ In addtion, by $\widetilde\mu_{m,n}^{\star}>0$ and , we have $$\begin{aligned}
{x^{\star}_{m,n}}=\left({\sum_{l\in\mathcal{M}} \theta_{l,m} T_{l,n}^{\star} + \eta_{m}}\right).\label{eqn:xeq}\end{aligned}$$ Substituting and into , we have: $$\begin{aligned}
&\frac{\partial\mathcal{L}\left(\mathbf{T}^{\star},\boldsymbol{\lambda}^{\star}, \boldsymbol{\mu}^{\star}, {\nu}^{\star}, \widetilde{\boldsymbol\nu}_1^{\star}, \widetilde{\boldsymbol\nu}_2^{\star}, \widetilde{\boldsymbol\lambda}^{\star}, \widetilde{\boldsymbol\kappa}_1^{\star}, \widetilde{\boldsymbol\kappa}_2^{\star}, \widetilde{\boldsymbol\iota}^{\star}\right)}{\partial T_{m,n}}\nonumber\\
&=-\widetilde\lambda_n^{\star}\nabla_{T_{m,n}}\frac{T_{m,n}^{\star}}{\left({\sum_{l\in\mathcal{M}} \theta_{l,m} T_{l,n}^{\star} + \eta_{m}}\right)}-\widetilde\kappa_{1,m,n}^{\star}+\widetilde\kappa_{2,m,n}^{\star}+\widetilde\iota_m^{\star}=0,\ {m\in\mathcal M},\ {n\in\mathcal N}.\label{eqn:deri-40}\end{aligned}$$ Note that , , and are the zero derivative conditions of Problem \[prob:dualmax\].
- Feasibility: Given , , , and , to show that $\left(\mathbf{T}^{\star}\right.$, $\mathbf{x}^{\star}$, $\boldsymbol{\lambda}^{\star}$, $\boldsymbol{\mu}^{\star}$, $\left.{\nu_1}^{\star}-{\nu_2}^{\star}\right)$ and $\left(\widetilde{\boldsymbol\nu}_1^{\star}\right.$, $\widetilde{\boldsymbol\nu}_2^{\star}$, $\widetilde{\boldsymbol\lambda}^{\star}$, $\widetilde{\boldsymbol\kappa}_1^{\star}$, $\widetilde{\boldsymbol\kappa}_2^{\star}$, $\left.\widetilde{\boldsymbol\iota}^{\star}\right)$ satisfy the feasibility conditions of Problem \[prob:dualmax\], it remains to show: $$\begin{aligned}
&\sum_{m\in\mathcal{M}}{\frac{T_{m,n}^\star}{\sum_{l\in\mathcal{M}}{\theta_{l,m}T_{l,n}^\star}+\eta_{m}}}+\mu_n^\star-\lambda_n^\star+\nu^\star_1-\nu^\star_2=0,\quad n\in\mathcal{N},\label{eqn:dual100}\\
&\sum\limits_{n\in\mathcal N}T_{m,n}^\star=K_m,\quad m\in\mathcal M. \label{eqn:T200}\end{aligned}$$ Note that can be easily shown by $\widetilde\lambda_n^{\star}>0$, and . Now, we show as below. If $\widetilde\kappa_{2,m,n}^{\star}\neq0$ for all $n\in\mathcal N$, by , we have $T^{\star}_{m,n}=1$ for all $n\in\mathcal N$, implying $\sum_{n\in\mathcal N}T^{\star}_{m,n}=N$, which is contradictory with $\sum_{n\in\mathcal N}T^{\star}_{m,n}\leq K_m$. Thus, there exists $n\in\mathcal N$, such that $\widetilde\kappa_{2,m,n}^{\star}=0$. Substituting $\widetilde\kappa_{2,m,n}^{\star}=0$ into $\eqref{eqn:deri-40}$, we have: $$\begin{aligned}
\widetilde\iota_m^{\star}=\widetilde\lambda_n^{\star}\nabla_{T_{m,n}}\frac{T_{m,n}^{\star}}{\left({\sum_{l\in\mathcal{M}} \theta_{l,m} T_{l,n}^{\star} + \eta_{m}}\right)}+\widetilde\kappa_{1,m,n}^{\star}>0. \nonumber\end{aligned}$$ By $\widetilde\iota_m^{\star}>0$ and , we have .
Therefore, $\left(\mathbf{T}^{\star}\right.$, $\mathbf{x}^{\star}$, $\boldsymbol{\lambda}^{\star}$, $\boldsymbol{\mu}^{\star}$, $\left.{\nu_1}^{\star}-{\nu_2}^{\star}\right)$ and $\left(\widetilde{\boldsymbol\nu}_1^{\star}\right.$, $\widetilde{\boldsymbol\nu}_2^{\star}$, $\widetilde{\boldsymbol\lambda}^{\star}$, $\widetilde{\boldsymbol\kappa}_1^{\star}$, $\widetilde{\boldsymbol\kappa}_2^{\star}$, $\left.\widetilde{\boldsymbol\iota}^{\star}\right)$ satisfy the KKT conditions of Problem \[prob:dualmax\].
Appendix D: Proof of Theorem \[thm:conv-sto\] {#appendix-d-proof-of-theoremthmconv-sto .unnumbered}
=============================================
We show that the assumptions in [@7412752 Theorem 1] are satisfied.
- The constraint set of Problem \[prob:sto\] determined by and is compact and convex. Thus, Assumption a) in [@7412752 Theorem 1] is satisfied.
- It is clear that $q_m(\boldsymbol\xi,\mathbf T)$ is smooth on the constraint set of Problem \[prob:sto\] for any given $\boldsymbol\xi$, and hence it is continuously differentiable and its derivative is Lipschitz continuous. Thus, Assumption b) in [@7412752 Theorem 1] is satisfied.
- Random variables $\boldsymbol\xi^0,\boldsymbol\xi^1,\dots$ are bounded and identically distributed. Thus, Assumption c) in [@7412752 Theorem 1] is satisfied.
Therefore, Theorem \[thm:conv-sto\] readily follows from [@7412752 Theorem 1].
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
Z. [Chen]{}, J. [Lee]{}, T. Q. S. [Quek]{}, and M. [Kountouris]{}, “Cooperative caching and transmission design in cluster-centric small cell networks,” *[IEEE]{} Trans. Wireless Commun.*, vol. 16, no. 5, pp. 3401–3415, May 2017.
W. [Wen]{}, Y. [Cui]{}, F. [Zheng]{}, S. [Jin]{}, and Y. [Jiang]{}, “Random caching based cooperative transmission in heterogeneous wireless networks,” *[IEEE]{} Trans. Commun.*, vol. 66, no. 7, pp. 2809–2825, July 2018.
S. Gitzenis, G. S. Paschos, and L. Tassiulas, “Asymptotic laws for joint content replication and delivery in wireless networks,” *[IEEE]{} Trans. Inf. Theory*, vol. 59, no. 5, pp. 2760–2776, May 2013.
B. Blaszczyszyn and A. Giovanidis, “Optimal geographic caching in cellular networks,” in *IEEE Int. Conf. on Commun. (ICC)*, June 2015, pp. 1–6.
Y. Cui, D. Jiang, and Y. Wu, “Analysis and optimization of caching and multicasting in large-scale cache-enabled wireless networks,” *IEEE Trans. Wireless Commun.*, vol. 15, no. 7, pp. 5101–5112, Jul 2016.
K. Shanmugam, N. Golrezaei, A. Dimakis, A. Molisch, and G. Caire, “Femtocaching: Wireless content delivery through distributed caching helpers,” *[IEEE]{} Trans. Inf. Theory*, vol. 59, no. 12, Dec 2013.
Y. Cui and D. Jiang, “Analysis and optimization of caching and multicasting in large-scale cache-enabled heterogeneous wireless networks,” *IEEE Trans. Wireless Commun.*, vol. 16, no. 1, pp. 250–264, Jan 2017.
K. Li, C. Yang, Z. Chen, and M. Tao, “Optimization and analysis of probabilistic caching in $n$ -tier heterogeneous networks,” *[IEEE]{} Trans. Wireless Commun.*, vol. 17, no. 2, pp. 1283–1297, Feb 2018.
J. Wen, K. Huang, S. Yang, and V. O. K. Li, “Cache-enabled heterogeneous cellular networks: Optimal tier-level content placement,” *[IEEE]{} Trans. Wireless Commun.*, vol. 16, no. 9, pp. 5939–5952, Sept 2017.
Y. Cui, Z. Wang, Y. Yang, F. Yang, L. Ding, and L. Qian, “Joint and competitive caching designs in large-scale multi-tier wireless multicasting networks,” *[IEEE]{} Trans. Commun.*, vol. 66, no. 7, pp. 3108–3121, July 2018.
M. A. [Maddah-Ali]{} and U. [Niesen]{}, “Fundamental limits of caching,” *[IEEE]{} Trans. Inf. Theory*, vol. 60, no. 5, pp. 2856–2867, May 2014.
M. A. Maddah-Ali and U. Niesen, “Decentralized coded caching attains order-optimal memory-rate tradeoff,” *[IEEE/ACM]{} Trans. Netw.*, vol. 23, no. 4, pp. 1029–1040, 2015.
J. Andrews, F. Baccelli, and R. Ganti, “A tractable approach to coverage and rate in cellular networks,” *[IEEE]{} Trans. Commun.*, vol. 59, no. 11, pp. 3122–3134, November 2011.
A. Tatar, M. D. De Amorim, S. Fdida, and P. Antoniadis, “A survey on predicting the popularity of web content,” *Journal of Internet Services and Applications*, vol. 5, no. 1, p. 8, Aug 2014.
E. Ba[ş]{}tu[ğ]{}, M. Bennis, and M. Debbah, “A transfer learning approach for cache-enabled wireless networks,” in *2015 13th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt)*, May 2015, pp. 161–166.
M. [Leconte]{}, G. [Paschos]{}, L. [Gkatzikis]{}, M. [Draief]{}, S. [Vassilaras]{}, and S. [Chouvardas]{}, “Placing dynamic content in caches with small population,” in *IEEE INFOCOM 2016 - The 35th Annual IEEE International Conference on Computer Communications*, April 2016, pp. 1–9.
B. N. Bharath, K. G. Nagananda, and H. V. Poor, “A learning-based approach to caching in heterogenous small cell networks,” *[IEEE]{} Trans. Commun.*, vol. 64, no. 4, pp. 1674–1686, April 2016.
B. Chen and C. Yang, “Caching policy for cache-enabled d2d communications by learning user preference,” *[IEEE]{} Trans. Commun.*, pp. 1–1, 2018.
S. [Li]{}, J. [Xu]{}, M. [van der Schaar]{}, and W. [Li]{}, “Trend-aware video caching through online learning,” *IEEE Transactions on Multimedia*, vol. 18, no. 12, pp. 2503–2516, Dec 2016.
A. Liu and V. Lau, “Exploiting base station caching in [MIMO]{} cellular networks: Opportunistic cooperation for video streaming,” *[IEEE]{} Trans. Signal Process.*, vol. 63, no. 1, pp. 57–69, Jan 2015.
P. Blasco and D. G[ü]{}nd[ü]{}z, “Learning-based optimization of cache content in a small cell base station,” in *IEEE Int. Conf. on Commun. (ICC)*, June 2014, pp. 1897–1903.
J. Song, M. Sheng, T. Q. S. Quek, C. Xu, and X. Wang, “Learning-based content caching and sharing for wireless networks,” *[IEEE]{} Trans. Commun.*, vol. 65, no. 10, pp. 4309–4324, Oct 2017.
C. Zhong, M. C. Gursoy, and S. Velipasalar, “A deep reinforcement learning-based framework for content caching,” in *2018 52nd CISS*, March 2018, pp. 1–6.
A. Sadeghi, F. Sheikholeslami, and G. B. Giannakis, “Optimal and scalable caching for 5g using reinforcement learning of space-time popularities,” *IEEE Journal of Selected Topics in Signal Processing*, vol. 12, no. 1, pp. 180–190, Feb 2018.
M. Razaviyayn, M. Hong, Z.-Q. Luo, and J.-S. Pang, “Parallel successive convex approximation for nonsmooth nonconvex optimization,” in *Advances in Neural Information Processing Systems*, 2014, pp. 1440–1448.
M. Chiang, C. W. Tan, D. P. Palomar, D. O’neill, and D. Julian, “Power control by geometric programming,” *[IEEE]{} Trans. Wireless Commun.*, vol. 6, no. 7, pp. 2640–2651, July 2007.
Y. Yang, G. Scutari, D. P. Palomar, and M. Pesavento, “A parallel decomposition method for nonconvex stochastic multi-agent optimization problems,” *[IEEE]{} Trans. Signal Process.*, vol. 64, no. 11, pp. 2949–2964, June 2016.
S. Boyd and L. Vandenberghe, *Convex optimization*.1em plus 0.5em minus 0.4emCambridge university press, 2004.
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[^3]: The results in this paper can be easily extended to the case of different file sizes by considering file combinations of the same total size, but formed by files of possibly different sizes [@Wang2017Joint].
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[^8]: This assumption corresponds to the worst-case interference strength for the typical user. The performance obtained under this assumption provides a lower bound on the performance of the practical network where some void BSs may be shut down.
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[^10]: Note that one of the major techniques for designing systems that are robust against modeling uncertainties is to optimize the worst-case performance.
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[^12]: Note that $\eta_{m}>0$, as $B\left(\frac{2}{\alpha},1-\frac{2}{\alpha}\right)>0$, and in most practical cases, $\theta_{l,m}>0$ [@Wang2017Joint]. Thus, in the rest of this paper, we consider the case where $x_{m,n}>0$ for all $m\in\mathcal{M}$ and $n\in\mathcal{N}$.
[^13]: For ease of analysis, in Problem \[prob:roep\], we consider $\mathbf T\succ0$ instead of $\mathbf T\succeq0$, which does not change the optimal value or affect the numerical solution.
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abstract: 'Interstellar dust contains a component which reveals its presence by emitting a broad, unstructured band of light in the 540 to 950 nm wavelength range, referred to as Extended Red Emission (ERE). The presence of interstellar dust and ultraviolet photons are two necessary conditions for ERE to occur. This is the basis for suggestions which attribute ERE to an interstellar dust component capable of photoluminescence. In this study, we have collected all published ERE observations with absolute-calibrated spectra for interstellar environments, where the density of ultraviolet photons can be estimated reliably. In each case, we determined the band-integrated ERE intensity, the wavelength of peak emission in the ERE band, and the efficiency with which absorbed ultraviolet photons are contributing to the ERE. The data show that radiation is not only driving the ERE, as expected for a photoluminescence process, but is modifying the ERE carrier as manifested by a systematic increase in the ERE band’s peak wavelength and a general decrease in the photon conversion efficiency with increasing densities of the prevailing exciting radiation. The overall spectral characteristics of the ERE and the observed high quantum efficiency of the ERE process are currently best matched by the recently proposed silicon nanoparticle (SNP) model. Using the experimentally established fact that ionization of semiconductor nanoparticles quenches their photoluminescence, we proceeded to test the SNP model by developing a quantitative model for the excitation and ionization equilibrium of SNPs under interstellar conditions for a wide range of radiation field densities. With a single adjustable parameter, the cross section for photoionization, the model reproduces the observations of ERE intensity and ERE efficiency remarkably well. The assumption that about 50% of the ERE carriers are neutral under radiation conditions encountered in the diffuse interstellar medium leads to a prediction of the ionization cross section of SNPs of average diameter of 3.5 nm for single-photon ionization of $ \leq 3.4 \cdot 10^{-15}$ $\rm cm^{2}$. The shift of the ERE band’s peak wavelength toward larger values with increasing radiation density requires a change of the size distribution of the actively luminescing ERE carriers through a gradual removal of the smaller particles by size-dependent photofragmentation. We propose that heat-assisted Coulomb decay of metastable, multiply charged SNPs is such a process, which will remove selectively the smaller components of an existing SNP size distribution.'
author:
- 'Tracy L. Smith & Adolf N. Witt'
title: The Photophysics of the Carrier of Extended Red Emission
---
Introduction
============
Extended Red Emission (ERE) is an interstellar photoluminescence phenomenon, which is widely observed in astrophysical environments where interstellar dust is exposed to ultraviolet photons (Gordon et al. 1998, and references therein). The emission appears in the form of a broad, unstructured band, limited in extent to the 540 - 950 nm spectral range. The pronounced variability of the peak wavelength of this band (610 - 880 nm) and the correlated variability of the FWHM of the band (60 - 180 nm) are two defining characteristics of the ERE (Darbon et al. 1999). Recent summaries of the observational data concerning the ERE have been given by Witt et al. (1998), Smith (2000) and by Ledoux et al. (2001), which should be consulted for the numerous observational references.
The peak intensity of the ERE in different environments, ranging from the high-Galactic-latitude diffuse interstellar medium (ISM) of the Milky Way Galaxy to HII regions, is generally proportional to the density of the illuminating radiation field (Gordon et al. 1998). This is expected for a photon-driven process involving an interstellar dust mixture, in which the ERE carrier represents an approximately constant fraction of the dust mass. However, a high degree of spatial variability of the ERE intensity, e.g. the presence of bright ERE filaments contrasted by locations without any detectable ERE within the same individual objects, is also part of the ERE phenomenon (e.g. Witt & Malin 1989; Witt & Boroson 1990). While faintest on an absolute scale in the diffuse ISM of the Galaxy, the ERE, there, is comparatively intense relative to the density of the prevailing radiation field (Gordon et al. 1998; Szomoru & Guhathakurta 1998). The correlation of ERE intensity with HI column density at high galactic latitudes has led to an estimate of a lower limit to the quantum yield of the ERE process of $10 \pm 3$ % (Gordon et al. 1998). This lower limit was arrived at with the assumption that the ERE carrier is the sole source of absorption of interstellar photons in the 90 - 550 nm spectral range. Given that almost certainly other dust components contribute to this absorption, the intrinsic quantum efficiency of the ERE carrier must exceed the lower-limit estimate in inverse proportion to the fraction of photons actually absorbed by it.
The realization that the ERE carrier is both a luminescing agent of high intrinsic quantum efficiency ($ \gg$ 10%) as well as a major contributor to the UV/optical absorption by interstellar grains had a profound impact on the discussion over the identification of the ERE carrier’s nature. If we estimate the intrinsic quantum yield to be about 50%, the ERE carrier intercepts about 20% of the photons absorbed by interstellar dust in the 90 - 550 nm range. In view of this high cross section, the ERE carrier is restricted chemically to the small number of relatively abundant but highly depleted elements making up most of the mass of interstellar grains, i.e. C, Fe, Si, Mg, plus O and H. Even without these new restrictions, many of the previously proposed ERE carrier candidates, which are laboratory analog materials and models, had encountered serious difficulties. Proposed candidates either lacked the ability to match the wide range of ERE spectral variations or they implied the presence of additional emission features or significant continuum luminescence shortward of 540 nm, which is not in agreement with observations. Witt et al. (1998) and Ledoux et al.(2001) present detailed discussions of proposed ERE carrier candidates and the shortcomings of most.
With the high intrinsic quantum efficiency added to the observational constraints, the discussion over viable ERE carrier candidates currently has narrowed to a choice between crystalline silicon nanoparticles (Ledoux et al. 1998; Witt et al. 1998; Ledoux et al. 2001) and aromatic hydrocarbon clusters (AHC) (Seahra & Duley 1999). The silicon nanoparticle (SNP) model derives its strength from the extensive agreement between laboratory-derived experimental data on SNP luminescence properties (e.g. Ledoux et al. 2000) and the astronomical ERE data, while relevant laboratory data for the AHCs currently do not exist for the size range of particles proposed by Seahra and Duley (1999). The AHC model, thus, is based upon theoretical extrapolations from smaller hydrocarbon molecules. Particular weaknesses of the AHC model include the relative constancy of the wavelength of its predicted emission peak (Seahra & Duley 1999, Fig. 2), its prediction of secondary features not confirmed by observation (Gordon et al. 2000), and a suggested correlation between the strength of absorption in the 217 nm UV extinction band and the intensity of the ERE, again at variance with observation.
By contrast, the SNP model can explain the variable peak wavelength of the ERE band through the size dependence of photoluminescence by oxygen-passivated silicon nanocrystals residing within the quantum-confinement regime (Delerue et al. 1993; Ledoux et al. 2000; Takeoka et al. 2000). Furthermore, the quantum efficiency of individual well-passivated SNPs has been found experimentally to be near 100% (Credo et al. 1999; Ledoux et al. 2001). The absorption coefficient per unit volume of nanocrystalline silicon (Theiss 1997; Li & Draine 2001) is about an order of magnitude larger than that of average interstellar dust (Huffman 1977), so that the cosmic abundance of Si seems sufficient to explain the observed ERE intensities (Witt et al. 1998) and the required associated UV/optical absorption cross section. Ledoux et al. (2001) have estimated that the mass of interstellar SNPs would be about 2% of the mass of all interstellar dust, if the intrinsic quantum efficiency of interstellar SNPs is 50%. This would require about 20% of the interstellar silicon in the case that the relative abundance of silicon is given by solar abundances, but close to 40%, if B-star abundances prevail (Snow & Witt 1996). Furthermore, recent observations (Gordon et al. 2000) have revealed tentative evidence for the existence of a second, related luminescence band at 1.15 $\mu$m in a bright ERE filament in the reflection nebula NGC 7023, matching in wavelength and width a similar band seen in SNPs at low temperatures in the laboratory (Meyer et al. 1993; Takeoka et al. 2000). An increasingly compelling case can thus be made that SNPs may well be the carriers of the ERE, because their experimentally established properties meet the constraints imposed by ERE observations and cosmic abundances. However, new constraints based upon the expected infrared emission characteristics of SNPs (Li & Draine 2001) have been proposed recently, which, when combined with suitably sensitive observations, may soon place severe limits on the possible interstellar abundance of oxygen-passivated SNPs or of pure-silicon SNPs.
Both the AHC and the SNP models, just as earlier ERE models based on hydrogenated amorphous carbon solids, such as HAC (Duley 1985) or QCC (Sakata et al. 1992), rely on across-bandgap recombination of electrons and holes in a semiconductor structure for the source of emission in the ERE band. The width of the band, in all instances, is attributed to a distribution of sizes of the luminescent entities with size-dependent bandgaps, while the peak wavelength of the band is related to the bandgap of the dominant particle size. In this paper we plan to examine whether the photophysics of such semiconductor nanoparticles under astrophysical conditions can account for the characteristics of the ERE seen in a wide range of astrophysical environments. We will be guided by experimental results found for SNPs and other semiconductor nanoparticles, and we will use experimentally determined parameters for SNPs, where available, as input into numerical calculations.
First, however, we begin by characterizing the environments, where ERE has been observed. We compute the UV-radiation densities for these environments and compile data for the wavelengths of peak ERE intensities and band-integrated ERE intensities. We proceed by estimating ERE quantum yields for locations other than the diffuse ISM. Then, starting from a strong base of experimentally determined properties of SNPs, we develop a model for the photophysics of the ERE carrier in interstellar environments. This model must explain, in a self-consistent manner, the observed variations of the ERE intensity, the ERE quantum efficiency, and the ERE peak wavelength as a function of the density and hardness of the local radiation fields. This model will identify the photoionization of the ERE carrier as the physical process which controls the fraction of carrier particles capable of luminescing in a given environment, which itself is characterized by the density and spectrum of the prevailing photons and by the density and temperature of the free electrons. Finally, this model must lend itself to making specific predictions both with respect to future ERE observations as well as to yet not-measured carrier properties.
In Section 2 of this paper, we accumulate the available observational data, which provide information on ERE intensities, ERE quantum efficiencies, and ERE-band peak wavelengths in different sources. Critical elements will be the estimates of the UV radiation densities and the lower limits to the ERE quantum efficiencies, which will be determined from information about the spectral types and luminosities of the illuminating stars, the projected distances of the observed regions from the sources in question, and the observed ratios of ERE intensity to scattered light intensity over the same wavelength band. In Section 3 we introduce evidence showing photoionization to be a possible process controlling the ERE; we discuss the ionization mechanisms and their respective rates, and we present the relevant rate equations and their solutions. Section 4 contains the discussion where observational data and model predictions will be compared. This is followed by a summary in Section 5.
The Intensity, Peak Wavelength, and Efficiency of ERE
=====================================================
Data and Observations
---------------------
We restricted our data sample to objects and environments where the sources of illumination are well-known with respect to spectral type, distance, and luminosity, thus permitting a reliable estimate of the density of the exciting radiation field. Furthermore, we focused on objects where the dust compositions are derived from average interstellar dust rather than being the result of local dust formation. Thus, we excluded existing ERE observations of planetary nebulae (Furton & Witt 1990, 1992) and limited ourselves to reflection nebulae, HII regions, dark nebulae, and the diffuse, high-$\mid$b$\mid$ Galactic cirrus, all in the Milky Way Galaxy.
Twenty-one of the twenty-two reflection nebulae (RN) included here were observed by Witt & Boroson (1990) during a spectroscopic survey of RN, wherein the ERE band-integrated intensity and the peak wavelength of the ERE band were measured. For the Orion Nebula, these quantities were taken from the observations of Perrin & Sivan (1992). The data for the high-Galactic-latitude dark nebula Lynds 1780 (L 1780) were taken from the spectrophotometric observations of Mattila (1979). One observation of an ERE peak wavelength for the Red Rectangle was obtained from Rouan et al. (1995) and one peak ERE wavelength point for the Bubble Nebula (NGC 7635) was taken from Sivan & Perrin (1993). The ERE peak wavelength measurement for a high-Galactic-latitude cirrus came from Szomoru & Guhathakurta (1998) and the efficiency and band-integrated intensity of ERE in the ISM were taken from Gordon et al. (1998).
Dependence of ERE Properties on Radiation Field
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Gordon et al. (1998) demonstrated that in a multitude of ERE sources the ERE band-integrated intensities and the intensities of the scattered light in the same wavelength band are roughly proportional to each other over several orders of magnitude in intensity. The scattered light intensity is determined by the column density of scattering grains and the density of the illuminating radiation in the wavelength range of the ERE band. This finding, therefore, suggests that, on average, the ERE carriers represent a fixed fraction of the dust present and that the closely related illuminating radiation field at wavelengths shorter than the ERE band is the source of ERE excitation. To quantify this relationship, an estimate of the densities of the exciting radiation fields is essential.
An analysis of ERE intensity variations in reflection nebulae in relation to the degree of internal reddening as a function of position within these nebulae by Witt & Schild (1985) and the non-detection of ERE in reflection nebulae illuminated by stars cooler than 7000 K (Darbon et al. 1999; Witt & Rogers 1991) suggest that the excitation of the ERE carrier is dominated by photons in the wavelength range shortward of 250 nm. We estimated the relevant radiation field density in the diffuse ISM, $U_{isrf}$, by integrating over the flux values of Mathis et al. (1983) from 91 to 250 nm at a distance of 10 kpc from the Galactic Center. All other radiation field densities will be normalized to this reference value, $U_{isrf}$ = 9.7$\cdot 10^{-14}$ ergs $\rm cm^{-3}$.
Then, for nebulae with known illuminating stars, we calculated the radiation field density at the top of the illuminating star’s atmosphere, $U_{star}$, using the star’s temperature and the appropriate atmosphere model (Kurucz et al. 1974) for the wavelength range from 91 to 250 nm. This radiation field density was then corrected for geometric dilution,
$$\begin{aligned}
U_{obs} & = & \frac{U_{star}}{2}(\frac{r}{D})^{2},\end{aligned}$$
where $U_{obs}$ is the radiation field density at the point of ERE observation, *r* is the stellar radius and *D* is the projected distance of the ERE observation from the star in the plane of the sky. The radiation field strength for each point of observation was then expressed in units of $U_{isrf}$,
$$\begin{aligned}
U & = & \frac{U_{obs}}{U_{isrf}}.\end{aligned}$$
Since the values of *D* are measured in the plane of the sky, they are lower limits to the true distances of the observed positions from the stars, and *U* is an upper limit to the actual radiation field strength. Figure 1 shows a plot of the accumulated ERE intensity values versus the associated densities of the radiation field within specific and among different RN, the Orion Nebula, L 1780, the diffuse ISM, and the Red Rectangle. The diffuse ISM point is an average of many lines of sight (Gordon et al. 1998). The downward-pointing error bars on some of the RN points indicate upper limit determinations for the ERE intensity by Witt & Boroson (1990). The Red Rectangle is included here to indicate its exceptional ERE brightness, but we note that the dust in this bi-polar proto-planetary nebula is most likely of local origin and thus is not representative of the typical interstellar dust mixture. Figure 1 confirms that the different ERE intensities in the diffuse ISM, in RN and in the Orion Nebula are a result of scaling the density of the exciting radiation fields, as expected for a photoluminescence process. Specifically, we expect the ERE intensity to exhibit a proportionality given by
$$\begin{aligned}
I(ERE) & \propto & U \cdot \eta \cdot (1 - exp(-\tau_{ERE})),\end{aligned}$$
where $\eta$ is the ERE photon conversion efficiency, defined in the following section, and where $\tau_{ERE}$ is the effective optical depth for dust absorption at the wavelength of the ERE. The examination of Figure 1 reveals that the detected ERE intensities range over four orders of magnitude, while the corresponding exciting radiation field densities vary by about six orders of magnitude. Given that the values of $\tau_{ERE}$ are comparable in the different ERE sources, Figure 1 provides the first qualitatative evidence for substantial variations in the ERE efficiency among different environments. In particular, the Red Rectangle, illuminated by a relatively cool B 9.5 III star, must derive its exceptional ERE intensity, when compared with the more typical RN, from a combination of high intrinsic efficiency and relative overabundance of ERE carriers, most likely produced in the local mass outflow.
Figure 2 shows the relationship between ERE-band peak wavelength, $\lambda_{p}$, and radiation field density. Again, due to the projection effect, the radiation field strengths calculated for the RN, Orion Nebula, and the Red Rectangle are strict upper limits. Figure 2 indicates that the longest peak wavelengths for the ERE band are found in high radiation density environments, and the shortest are found in the lowest radiation density environments, as represented by the high-Galactic-latitude cirrus. The high-$\mid$b$\mid$ dark nebula L 1780 appears to take on an outlier role. We will comment on the likely explanation for this effect in Sect. 4.3.
The Photon Conversion Efficiency of the ERE Process
---------------------------------------------------
One can estimate a lower limit to the ERE conversion efficiency, if both the number of emitted ERE photons and the number of UV/optical photons absorbed within the same environment are known. It is assumed that the photons absorbed by dust which can possibly lead to the emission of an ERE photon are in the wavelength range between 91 nm (ionization of H) and 540 nm (edge of ERE band), although 250 nm is probably a more realistic upper limit of this range (see Sect. 2.2). The efficiency determined in this manner is a lower limit only, because other dust components, which are not contributing to the ERE, may participate in the photon absorption and the ERE excitation may occur primarily in the short-wavelength portion of the assumed wavelength window of exciting radiation.
Spectroscopic observations of reflection nebulae show the ERE band superimposed on a scattered light continuum. The scattered intensity beneath the ERE band, $I_{sca}$(band), is a direct probe of the number of illuminating photons within the ERE band and the number of dust grains present in the observed volume giving rise to the ERE. Given the knowledge of (1) the illuminating star’s spectral energy distribution, which relates the number of illuminating and scattered photons in the ERE band to the number of photons capable of exciting the ERE, (2) the wavelength dependence of extinction and (3) the wavelength dependence of the dust albedo, the number of photons absorbed between 91 nm and 540 nm, $N_{abs}$(91-540 nm), can be estimated. The photoluminescence efficiency can then be found as
$$\begin{aligned}
\eta & = & \left(\frac{N_{ERE}}{N_{sca}(\rm band)}\right)\left(\frac{N_{sca}(\rm band)}
{N_{abs}(91-540 \rm nm)}\right)\end{aligned}$$
where $N_{ERE}/N_{sca}$(band) is the ratio of the number of ERE photons and scattered photons, each integrated over all solid angles. This ratio is not observationally accessible to a spatially fixed observer. Instead, we measure the intensity ratio $I_{ERE}/I_{sca}$(band), which depends strongly on the range of the dominant scattering angles in a given nebula, defined by the unknown nebular geometry and the direction to the observer. The direction dependence of the intensity ratio is caused by the fact that the scattered light intensity is subject to a highly asymmetric phase function, while the ERE is expected to be emitted isotropically. This expectation is supported by experimental evidence, showing colloidal SNPs to exhibit no polarization memory (Koch et al. 1996). Multiple phonon-exciton interactions during the long radiative lifetime of SNPs as well as the rapid rotation of free-flying SNPs effectively destroy all directional relations between exciting photons and emitted ERE photons. This assessment is confirmed by the observed ERE band intensities and the corresponding scattered light intensities seen in RN (Witt & Boroson 1990, Fig. 1). For a given scattered light intensity, the ERE intensities in RN vary by about one order of magnitude at most, while for a given ERE intensity the corresponding scattered light intensities vary by nearly two orders of magnitudes. We hope to overcome this basic problem by averaging over multiple observations in a given RN and by averaging the data for a relatively large sample of RN representing a range of geometries, and with this reservation set $N_{ERE}/N_{sca}$(band) = $I_{ERE}/I_{sca}$(band).
The ratio of photons scattered within the wavelength range of the ERE band to photons absorbed within the wavelength range 91 - 540 nm, $N_{sca}$(band)/$N_{abs}$(91-540 nm), was determined with the help of a Monte Carlo radiative transfer code. This code included the local dust opacity law, characterized by the value of the ratio of total to selective extinction, $R_{V}$, the wavelength dependence of the dust albedo and phase function, and the spectral energy distribution of the exciting star, characterized by its effective temperature. The principal source of uncertainty, again, is the unknown geometry of the RN, which can be different in each instance. For simplicity, we adopted a spherical geometry with a centrally embedded star. The optical depth of the radius (1 pc) was taken to be 0.5 at the ERE wavelength and the density of the nebulae was assumed uniform. The absolute surface brightness and radial surface brightness distribution produced with such models agree well with measured surface brightness data for bright reflection nebulae (Witt & Schild 1986). In the model output, we examined the ratio $N_{sca}(\rm band)/N_{abs}(91-540$ $\rm nm)$ at an intermediate angular offset corresponding to 0.3 of the projected radius of the nebula. This arrangement covers a large range of scattering angles but still favors forward-scattering, appropriate for a collection of objects which include many of the RN with the highest-known surface brightnesses. Calculations were performed for two values of the UV albedo, 0.4 and 0.6, to correspond to limiting cases of low $R_{V}$/strong 218 nm absortion feature and high $R_{V}$/weak 218 nm absorption feature, respectively. The dust albedo in the ERE band was set at 0.6, consistent with numerous observational determinations (Witt & Gordon 2000). The phase function asymmetry was assumed to increase with decreasing wavelength, e.g., g = 0.6 at the ERE wavelength vs. g = 0.7 at the wavelength of the exciting UV radiation. An example of the model results for the ratio
$$\begin{aligned}
\frac{I(ERE)}{I_{sca}(\rm band)} & = & \eta \cdot \frac{N_{abs}(91-540 \rm nm)}{N_{sca}(\rm band)}\end{aligned}$$
is shown for $\eta$=0.01 as a contour plot in Figure 3, for $a_{UV}$=0.6, 2.5$\le R_{V} \le 5.0$, and $1\cdot 10^{4} \rm K \le T_{eff} \le 3.5\cdot 10^{4} \rm K$, corresponding to stars of approximate spectral types A 0 to O 7. Figure 3 illustrates the fact that $I_{ERE}$/$I_{sca}$ can change easily over a factor of 30 to 40 for the same value of the quantum efficiency $\eta$, depending upon the values of $R_{V}$ and $T_{eff}$. Correct estimates for the nebular opacity law and the spectral type of the illuminating stars are, therefore, of critical importance. Table 1 shows the values we adopted for the nebulae with ERE detections and well-determined upper limits.
Object Star Sp. T. $R_{V}$ $I_{ERE}$/$I_{sca}$(band) $\eta$(%)
------------ -------------- ------------ --------------------------- ------------
NGC 1333 B8 V 4.7 0.07 - 0.16 1.8 - 4.0
NGC 1788 B9 V 4 $<$ 0.03 $<$ 1.04
NGC 1999 A0 $\sim$ 3.6 0.06 2.0
NGC 2068 B2 II - III 4 $<$ 0.03 $<$ 0.16
NGC 2023 B1.5 V 4.11 0.02 - 0.42 0.04 - 1.2
NGC 2071 B5 V 4.5 $<$ 0.1 $<$ 1.2
NGC 2247 B3 Pe 3.25 0.08 0.4
NGC 2327 B1 3.1 - 4.5 0.06 0.1
NGC 7023 B3 Ve 3.2 0.04 - 0.15 0.2 - 0.6
IC 59 B0 V 3.5 $<$ 0.16 $<$ 0.19
IC 63 B0 IVe $\sim$ 3.5 0.68 4.7
IC 348 B5 V 3.5 $<$ 0.05 $<$ 0.4
IC 426 B8 p $\sim$ 4 0.21 7.9
IC 435 B5 V 5.3 $<$ 0.01 $<$ 0.13
IC 5076 B8 Ia 3.2 $<$ 0.09 $<$ 2.48
CED 167 B6 V 3.2 $<$ 0.01 $<$ 0.13
CED 201 B9.5 V 3.42 0.02 0.5
Maja B7 III 3.2 $<$ 0.02 $<$ 0.32
Merope B6 IV 3.6 $<$ 0.01 $<$ 0.01
VdB 132 B3 V 4 $<$ 0.02 $<$ 0.08
Orion Neb. 06, 07, 2xB0 5.5 0.20 - 0.64 0.4 - 1.3
L1780 ISRF 3.1 0.6 13
ISM ISRF 3.1 0.2 - 2 10
: Data Related to Efficiency Estimates \[tbl-1\]
The uncertainties introduced by the range in UV dust albedos depend on the temperature of the illuminating stars and range from 15% to 25% for stars ranging from late B to late O spectral type. The uncertainties go in the direction of increasing the efficiency for a given ratio $I_{ERE}/I_{sca}(band)$ for a higher UV albedo. While the overall uncertainties of the derived efficiencies for individual RN observations are difficult to assess for reasons discussed above, we estimate the uncertainty of the efficiency of the $\emph{average}$ of all RN detections to be about a factor of two.
The estimated efficiencies, as defined above, are plotted in Figure 4 as a function of the UV radiation density. The highest efficiencies found are associated with the lowest radiation densities. Also, among reflection nebulae, illuminated by stars between spectral types A 0 and B 0, we note a significant decline of the ERE efficiency by about two orders of magnitude as the radiation density increases by a similar factor. The Orion Nebula data do not follow this general trend, a fact that needs to be explained by our model (*c.f.* Sect. 3.4). We do not include the Red Rectangle data in this graph, because the dust mixture in this object is not derived from interstellar dust but is locally produced. We will discuss the efficiency of the ERE carriers in the Red Rectangle below.
The Red Rectangle
-----------------
The Red Rectangle occupies a special role in the problem of understanding the ERE. Given the relatively cool illuminating star in the object, the exceptionally high ERE intensity suggests that the ERE carriers are present here with an unusually high relative abundance, most likely as a result of local production. By employing the same method as used for RN, we estimate an efficiency for the photon conversion of $\sim$ 20%, the highest value seen in any object. The principal source of uncertainty here is the unusual opacity law characterizing the matter in the Red Rectangle (Sitko et al. 1981), which is unlike that of any interstellar or nebular dust. There is only a very weak indication of absorption at the interstellar extinction band centered at 217.5 nm, but massive absorption shortward of 180 nm is present instead. The ERE is strongest along the X-shaped walls of the bi-polar outflow cone (Schmidt & Witt 1991), while the unidentified infrared band (UIB) at 3.3 $\mu$m is found to be weak to non-existent along the walls or interface areas, but very strong in the dust ring surrounding the central binary star (Kerr et al. 1999). Additionally, all of the remaining UIBs at 6.2, 7.7, 8.6 and 11.3 $\mu$m have been observed in the Red Rectangle (Russell et al. 1978). Waters et al. (1998) have used observations from the Infrared Space Observatory (ISO) to confirm the existence of crystalline silicates in a circum-binary disk in the nebula, which suggests a mixed chemistry for this object.
The Photoionization of Semiconductor Nanoparticles Under Astrophysical Conditions
=================================================================================
The Silicon Nanoparticle Model
------------------------------
Before discussing the photoionization of semiconductor nanoparticles in general, it is essential to briefly summarize the main characteristics of the SNP model as representative of a possible carrier of the ERE. Silicon is an indirect-bandgap semiconductor material with a bandgap energy of 1.17 eV at 0 K. The indirect-bandgap nature implies that an excited electron located in the conduction band needs to undergo a change in momentum state before it can recombine with a hole in the valence band. This process, which involves electron-phonon interactions, has a long time constant ($\sim 10^{-5}$ - $10^{-3}$ s; Takeoka et al. 2000). Consequently, in the absence of spatial confinement and with an abundance of non-radiative recombination routes available in bulk silicon, photoluminescence in bulk crystalline silicon is a highly unlikely process. Several important changes occur when a transition to nanocrystals of silicon is considered (Freedhoff & Marchetti 1997). First, size-dependent quantum confinement of the electron-hole exciton causes the bandgap to widen, shifting it from 1.17 eV to values consistent with the wavelength range of the ERE band (Ledoux et al. 2000). Second, the spatial confinement of the electron forces its wavefunction to spread in momentum space in accordance with the Heisenberg uncertainty principle, greatly enhancing the probability of radiative recombination with the hole in the valence band without phonon collisions. Thus, efficient photoluminescence at energies much higher than 1.17 eV becomes possible. However, this occurs only, provided that all dangling Si bonds at the surface of the nanocrystal are passivated, e.g., by hydrogen or oxygen atoms. Observational constraints suggest that interstellar Si nanocrystals would be passivated by oxygen (Witt et al. 1998). For a range of diameters from 1 nm to 7 nm, the peak energy of photoluminescence of SNPs varies between 1.9 eV and 1.2 eV (Takeoka et al. 2000), in agreement with the wavelength range over which ERE has been observed in astronomical sources. And third, the small size of the nanocrystal makes it much easier to arrive at a defect-free structure. The presence of defects would otherwise provide for radiationless recombination routes, which would be detrimental to a high quantum efficiency. As a consequence, the resulting well-passivated, nanocrystalline system is a particle capable of photoluminescence with essentially 100% efficiency, emitting a luminescence photon upon each excitation by a photon with an energy exceeding that of the bandgap of the particle in question (Credo et al. 1998). An appropriate size distribution of such particles can then account for the observed interstellar luminescence seen as ERE, provided suitable shorter-wavelength photons are present to cause the excitation.
Secondary, but equally important, aspects of the SNP model concern a plausible mode of formation under astrophysical conditions and adequate cosmic abundance of silicon to explain the observed ERE intensities. Witt et al. (1998) suggested that the formation of interstellar SNPs could occur as a result of the nucleation of SiO molecules in oxygen-rich stellar outflows (Castro-Carrizo et al. 2001; Roche et al. 1991; Hirano et al. 2001; Garray et al. 2000; Zhang et al. 2000; Rietmeijer et al. 1999; Gail & Sedlmayr 1999). To lead to SNPs, the nucleation of SiO must be followed by annealing and phase separation into an elemental silicon phase in the core and a passivating mantle of $\rm SiO_{2}$. In fact, the $\rm SiO_{2}$ phase is thermodynamically more favorable for solid silicon oxides. Recent experiments involving the evaporation and recondensation of SiO lend strong support to this suggestion (Rinnert et al. 1998, 1999; Murakami et al. 1998). The nucleation of SiO molecules is a proposed first stage of the formation of silicate grains (Gail & Sedlmayr 1999), although the theoretical understanding of this formation process is still far from complete. Given the high mass absorption coefficient of nanocrystalline silicon (Theiss 1997) and the high quantum efficiency of SNPs, it has been estimated that the mass of interstellar SNPs amounts to only 1% to 2% of the interstellar dust mass (Ledoux et al. 2001), while silicon contributes about 8% of the total mass of interstellar solids condensed from a gas of solar composition (Whittet 1992, p.52).
The absorption of interstellar photons by SNPs occurs primarily in the mid- and far-ultraviolet (Zubko et al. 1999; Theiss 1997), which is consistent with astronomical assessments of the excitation of the ERE (Witt & Schild 1985; Darbon et al. 1999). The expected oxide coating of SNPs not only is essential for effective passivation, assuring a high overall photoluminescence efficiency, it also contributes to the interstellar Si-O stretch absorption band at 9.7 $\mu$m and to a possibly prominent emission feature at 20 $\mu$m (Li & Draine 2001). Also, the oxide coating assures that even the smallest SNPs do not luminesce shortward of 540 nm by creating surface-related electronic states within the growing band gap of such particles (Wolkin et al. 1999). Finally, given that only about 25% of the energy of an absorbed typical ultraviolet photon is used to power the ERE, the bulk of the absorbed energy goes towards heating the SNPs (Duley 1992; Li & Draine 2001). As a consequence of the small size of the SNPs, temperature fluctuations may raise the temperatures of the particles temporarily into the range from 50 K to 300 K (Li & Draine 2001), causing them to contribute to thermally driven band and continuum dust emission, mainly in the 15 - 60 $\mu$m wavelength region.
Photoionization Control of ERE Carrier Photoluminescence
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The observational data summarized in Sect. 2 strongly suggest that the ultraviolet photon field present in different ERE sources not only powers the ERE (Figure 1), but also controls the size distribution of luminescing particles (Figure 2) and the fraction of absorbing particles capable of luminescing (Figure 4). Two mechanisms must be considered: (i) partial, size-dependent photodestruction and (ii), size-dependent quenching of the photoluminescence by photoionization. Typical SNPs with spectra matching those of the ERE have between 200 and 6000 silicon atoms and their instantaneous photodestruction by thermal heating is a very unlikely process. The maximum temperatures of SNPs in typical ERE sources predicted by Li & Draine (2001) are in the 300 K - 500 K range for SNPs with diameters of 2nm, which is too low for thermal evaporation. An alternative process is fragmentation by Coulomb explosions of multiply charged SNPs. Such fragmentation, mostly by ejection of singly-charged monomers and dimers, has been seen experimentally in multiply-ionized, free silicon nanoparticles by Ehbrecht & Huisken (1999) and Bescos et al. (2000). It is not known, how the presence of an oxide shell affects this disintegration process. However, there is considerable experimental evidence that single ionization of semiconductor nanocrystals, including SNPs, effectively quenches their photoluminescence (Nirmal & Brus 1999; Banin et al. 1999; Efros & Rosen 1997; Nirmal et al. 1996; Kharchenko & Rosen 1996; Wang et al. 2001). If an ionized nanocrystal is photoexcited, the photogenerated electron-hole pair recombines by transferring its energy to the strongly Coulomb-coupled second hole via a non-radiative Auger decay process (Nirmal & Brus 1999). Partial ionization of SNPs is expected under conditions existing in the diffuse ISM and a higher degree of ionization, including multiple ionization, is expected in regions of higher radiation density (Bakes & Tielens 1994; Weingartner & Draine 2001). The threshold for single-photon ionization of free silicon clusters with about 200 silicon atoms has been found experimentally to lie near 5.1 eV (Fuke et al. 1993). We expect the ionization to be more difficult, if not requiring a higher energy, when a passivating oxide shell is present, but no experimental data appear to exist for this case. The energy per exciting photon of most laboratory photoluminescence experiments is less than the ionization threshold value. Consequently, the ionization process usually seen under laboratory conditions is Auger ionization (Chepic et al. 1990; Kovalev et al. 2000b), which arises when a second electron is excited into the conduction band before the first electron has completed its recombination with its hole and when the combined excitation energies of the two electrons exceed the ionization threshold. The strong Coulomb interaction between the two excited electrons causes one to recombine with a hole while the other leaves the system. The Auger process is both highly efficient in quantum-confined nanocrystals, and particularly important in SNPs as a result of the long photoluminescence lifetime of singly-excited electrons residing in the conduction band, which ranges up to msec in duration, but is dependent on size and temperature of the SNPs (Takeoka et al. 2000).
Under interstellar conditions, the photon spectrum extends to 13.6 eV, and photoionization by single photons must clearly be the dominant process in the diffuse ISM, given the high degree of dilution of the interstellar radiation field. As we will show below, despite the fact that the rate of the two-photon Auger ionization process increases with the square of the radiation field density, the two ionization processes are not likely to reach comparable rates in any of the radiation environments, where ERE is observed.
As shown in Figure 4, the observed lower limit of the ERE quantum yield in the diffuse ISM is about $10 \pm 3$ % (Gordon et al. 1998). After making allowance for other grain components contributing to the absorption of UV/optical photons without contributing to the ERE, we may estimate that the intrinsic quantum efficiency of the ERE carrier particles could be near 50% (Witt et al. 1998). If photoionization controls the ability of the ERE carrier to luminesce, the intrinsic quantum efficiency then is proportional to the non-ionized fraction of the carrier particles. Thus, we find that the degree of ionization of the ERE carrier particles in the diffuse ISM must be fairly low, not exceeding 50%. The degree of ionization could be less, if a certain fraction of the carriers is otherwise defected and thus unable to luminesce, because the structure of such defected nanoparticles permits radiation-less recombination of electrons and holes. These expectations regarding the ionization state of nm-sized ERE carrier particles are in good agreement with the predictions of likely charge states for particles of 3 nm diameter in the cold, diffuse ISM by Bakes & Tielens (1994) and Weingartner & Draine (2001)
The expected low degree of ionization in interstellar space places a strict upper limit on the ionization cross section of SNPs for single-photon ionization, assuming that SNPs cause the ERE. We will calculate this cross section below. Since no measurements exist for this cross section, our calculation will represent a specific prediction made by the SNP model, which later laboratory experiments may test.
The Photoionization Equilibrium Rate Equations and their Solutions
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Let us assume that interstellar SNPs are perfectly passivated and that the only way to quench their photoluminescence is by photoionization. We further assume the absorption characteristics of SNPs to be unchanged, whether or not the latter are ionized. Then, under interstellar conditions, the photoluminescence properties of SNPs are controlled by the probabilities of occupation of four distinct states. The first is $P_{0}$, the probability that a SNP is in the neutral ground state, with all electrons in the valence band. It is seen easily that $P_{0}$ measures the intrinsic photoluminescence quantum efficiency of the total SNP ensemble, consisting of neutral and ionized particles. The second essential probability is $P_{1}$, which measures the fraction of SNPs in the neutral, excited state in which one electron is momentarily in the conduction band. Since the decay from this state almost invariably is via the emission of a photoluminescence photon, $P_{1}$ is directly proportional to the ERE intensity produced by the SNPs. A third state, measured by the probability $P_{2}$, represents neutral SNPs in which a second photoexcitation has occurred before the first excitation had enough time to decay. The most likely result of the strong Coulomb interaction of the two electrons is Auger ionization on a timescale of typically $\tau_{A} \sim 10^{-10}$ s (Delerue et al. 1998), rendering the SNP unable to luminesce. However, under astrophysical conditions this probabilty is expected to be very small. The fourth state of importance is the ionized state, measured by the probability $P_{+}$. SNPs can enter this state via either one of the two ionization processes. We will consider only the once-ionized state here, although multiply-ionized SNPs are likely in environments with high photon densities and comparatively low densities of free electrons. We ignore the possibility of negatively charged SNPs, because the rate of photoionization of negatively charged SNPs is expected to be large compared to the electron capture rate of a neutral SNP, given the radiation and electron densities in ERE-emitting environments. Our model accounts for the presence of higher positive charge states by increasing the value of $\tau$, the lifetime of the ionized state. We will discuss the role of multiple ionization in SNPs in Section 3.5 of this paper.
We can formally express the time rate of change of the four probabilities $P_{0}, P_{1}, P_{2}$, and $P_{+}$, by the following rate equations (Chepic et al. 1990): $$\begin{aligned}
\dot{P_{0}} = -W_{1} \cdot P_{0} - \frac{P_{0}}{\tau_{i}} + \frac{P_{1}}{\tau_{1}}\end{aligned}$$ $$\begin{aligned}
\dot{P_{1}} = W_{1} \cdot P_{0} + \frac{P_{2}}{\tau_{2}} - \frac{P_{1}}{\tau_{1}} - W_{2} \cdot P_{1} +
\frac{P_{+}}{\tau}\end{aligned}$$ $$\begin{aligned}
\dot{P_{2}} = W_{2} \cdot P_{1} - \frac{P_{2}}{\tau_{2}} - \frac{P_{2}}{\tau_{A}}\end{aligned}$$ $$\begin{aligned}
\dot{P_{+}} = \frac{P_{2}}{\tau_{A}} + \frac{P_{0}}{\tau_{i}} - \frac{P_{+}}{\tau}\end{aligned}$$ and, consistent with our assumptions, the following normalization: $$\begin{aligned}
P_{0} + P_{1} + P_{2} + P_{+} = 1.0\end{aligned}$$ To solve these equations for any given environment, we assume steady state: $$\begin{aligned}
\dot{P_{0}} = \dot{P_{1}} = \dot{P_{2}} = \dot{P_{+}} = 0\end{aligned}$$
The factors $W_{1}$ and $W_{2}$, respectively, represent the rates of photoexcitation of SNPs in the neutral ground state and the neutral, once-excited state by the prevailing radiation field, such that the combined energy of the two photons causing the excitations exceeds the ionization potential of the SNPs. For our subsequent calculations we will assume $W_{1} = W_{2}$, which is approximately correct (Efros & Rosen 1997), given the ionization potential (Fuke et al. 1993) and typical bandgaps of SNPs. The lifetimes appearing in Equations (6) through (9) are: $\tau_{1}$, the radiative lifetime of the singly-excited state; $\tau_{2}$, the radiative lifetime of the doubly-excited state; $\tau_{A}$, the lifetime of the doubly-excited state against Auger ionization; $\tau_{i}$, the lifetime of the neutral state against photoionization by single photons; and $\tau$, the lifetime of the ionized state before recapture of a free electron, leading to a return to the neutral state. We assume that the recaptured electron will appear in the conduction band of the SNP, with subsequent radiative transition to the valence band.
The solutions of Equations (6) through (11) are given by the following four expressions: $$\begin{aligned}
P_{0} = [1 + \tau_{1} (W_{1} + \frac{1}{\tau_{i}}) + \frac{W_{2} \tau_{1} (W_{1} + \frac{1}{\tau_{i}})}
{(\frac{1}{\tau_{2}} + \frac{1}{\tau_{A}})} + \tau (\frac{W_{2} \tau_{1}(W_{1}+ \frac{1}{\tau_{i}})}
{\tau_{A}(\frac{1}{\tau_{2}} + \frac{1}{\tau_{A}})} + \frac{1}{\tau_{i}})]^{-1}\end{aligned}$$ $$\begin{aligned}
P_{1} = \tau_{1}(W_{1} + \frac{1}{\tau_{i}}) \cdot P_{0}\end{aligned}$$ $$\begin{aligned}
P_{2} = \frac{\tau_{1} W_{2}(W_{1} + \frac{1}{\tau_{i}})}{(\frac{1}{\tau_{2}} + \frac{1}{\tau_{A}})} \cdot P_{0}\end{aligned}$$ $$\begin{aligned}
P_{+} = \tau[\frac{\tau_{1} W_{2}(W_{1} + \frac{1}{\tau_{i}})}{\tau_{A}(\frac{1}{\tau_{2}} +
\frac{1}{\tau_{A}})} + \frac{1}{\tau_{i}}] \cdot P_{0}\end{aligned}$$
The ionization rate per SNP for single-photon ionization can be found from $P_0/\tau_{i}$, while the 2-photon Auger ionization rate becomes $P_{2}/\tau_{A}$. These two rates equal each other, provided
$$\begin{aligned}
\tau_{1} = \frac{(\tau_{A}(\frac{1}{\tau_{2}} + \frac{1}{\tau_{A}}))}{\tau_{i}W_{2}(W_{1}+\frac{1}{\tau_{i}})}.\end{aligned}$$
Numerical Solutions
-------------------
For the numerical evaluations of these solutions, we need to find values for the rate coefficients and time constants involved in Equations (12) through (15). We adopt the direction-integrated intensity of the interstellar radiation field at the galactocentric distance of the Sun given by Mathis et al.(1983) and the optical absorption cross sections of SNPs by Kovalev et al. (2000a), extrapolated into the ultraviolet, to estimate the radiative excitation rate in the diffuse interstellar medium. We find $W_{1} \simeq 1.0 \cdot 10^{-5} \rm s^{-1}$ for nanoparticles with diameters near 3.5 nm under these conditions, and, as discussed above, we set $W_{1} = W_{2}$. The timescale $\tau$ for recombinations between ionized SNPs and free electrons is estimated from the work of Draine & Sutin (1987). With a characteristic diameter of 3.5 nm for a SNP with a single positive charge, an average electron density of $n_{e} = 0.03$ $\rm cm^{-3}$ from pulsar dispersion measures (Spitzer 1978), and an electron temperature $T_{e}$ = 100 K we find $\tau \simeq 1.0 \cdot 10^{6}$ s, which includes an enhancement of the particle cross section due to Coulomb focussing by a factor of about 200. The timescale for single-photon ionization, $\tau_{i}$, cannot be estimated directly in the absence of measured ionization cross sections. However, the observation of a large ERE efficiency in the diffuse ISM suggests a fairly high ratio of neutral to ionized SNPs in this environment, and we estimate that $P_{0} \sim 0.5$. Equation (12) then leads to the conclusion that $\tau_{i} \simeq 1.0 \cdot 10^{6}$ s for the diffuse ISM. Since single-photon ionization is the dominant photoionization process of SNPs in the diffuse ISM, we will use this result to estimate the cross section for this process later in Sect. 4.2. Moreover, as the radiation density increases, the single-photon ionization rate $\frac{1}{\tau_{i}}$ as well as the photoexcitation rates $W_{1}$ and $W_{2}$ scale in direct proportion of the radiation density. The radiative lifetime of the once-excited state, $\tau_{1}$, is strongly dependent upon size and temperature of the SNP and relatively long, consistent with the indirect-bandgap nature of silicon. From Takeoka et al. (2000) we estimate $\tau_{1}=2 \cdot 10^{-3}$ s for an appropriate average. The dynamics of the doubly-excited state is controlled by the small value of $\tau_{A} \sim 10^{-10}$ s (Efros & Rosen 1997). We will assume that radiative decay from the doubly-excited state is not entirely prohibited by setting $\tau_{2} = 10^{-6}$ s.
We computed the probabilities $P_{0}, P_{1}, P_{2}$ and $P_{+}$ for a range of radiation densities extending from the interstellar density to $1 \cdot 10^{6}$ times the interstellar value. We also varied the lifetime of the ionized state, $\tau$, from the interstellar value to $1 \cdot 10^{6}$ times shorter, in steps of factors of ten. $\tau$ scales in inverse proportion to the electron density when the single-charge state is the dominant state among ionized SNPs, but $\tau$ can be much longer than expected, if the bulk of SNPs are multiply ionized. At low electron energies, $\tau$ is relatively independent of the electron temperature (Draine & Sutin 1987), as the temperature dependences of electron velocity and recombination cross section cancel one another. Aside from a constant factor, which measures the unknown fraction of the SNP absorption to the total absorption by all interstellar grains, the probability $P_{0}$ measures the intrinsic ERE quantum efficiency. Similarly, the probability $P_{1}$ is directly proportional to the ERE intensity. Finally, the ratios $\frac{P_{0}}{\tau_{i}}$ and $\frac{P_{2}}{\tau_{A}}$ measure the rates of single-photon ionization and two-photon Auger ionization, respectively.
In Figure 5 we plot the probability $P_{1}$ as a function of the radiation field density superimposed upon our ERE intensity data, compiled in Section 2. The model curves are normalized to fit the ERE intensity in the diffuse ISM at $log(U)$ = 0. The model provides a consistent explanation for all observed ERE intensities and upper limits by suggesting that the more intense radiation fields in reflection nebulae and HII regions will lead to higher ERE intensities, as long as the resultant increased photoionization rate is at least partially balanced by increased electron capture rates. The model fit suggests that in reflection nebulae the rate of returning ionized SNPs to the neutral state is up to one order of magnitude higher than in the diffuse ISM, consistent with the increased densities and the absence of hydrogen ionization in these objects. In the Orion Nebula, electron capture rates higher by at least five orders of magnitude than in the ISM are expected, consistent with electron temperatures of $T_{e} \sim 10^{4}$ K and electron densities near $10^{4}$ $\rm cm^{-3}$ generally quoted for the Orion Nebula (Rubin et al. 1998; Baldwin et al. 1996). However, Figure 5 shows that the rate of returning ionized SNPs to the neutral state, $\frac{1}{\tau}$, increases by only three orders of magnitude between the ISM and the Orion Nebula. We attribute this smaller than expected increase to the dominance of multiple charge states among ionized SNPs in the Orion Nebula (Weingartner & Draine 2001; Bakes & Tielens 1994). Thus, most electron captures by ionized SNPs involve transitions between different charge states instead of transitions between the singly-charged state and the neutral state. It is of interest to note that only a fraction of about 1% neutral SNPs is needed in order to fit both the reflection nebulae and Orion Nebula intensity data. Despite this low fraction of SNPs capable of luminescing, the ERE intensities are nevertheless high, because (see Equation 3) the increase in the radiation field densities (factors of $10^{3}$ to $10^{5}$) more than balance the decrease in the efficiency (factor of $\sim 50$). In the Orion Nebula, a neutral SNP luminesces approximately once per second, while in the diffuse ISM only one luminescence event per neutral SNP occurs every 30 hours, on average.
In Figure 6 we plot the probability $P_{0}$ as a function of the radiation field density superimposed upon our ERE efficiency data produced in Section 2. Again, the model curves are normalized to fit the derived ERE efficiency for the diffuse ISM. The model matches the precipitous decline of the ERE quantum efficiency seen in reflection nebulae in the range $2 < log(U) < 4$ exceedingly well. With the illuminating stars in reflection nebulae having spectral types B 1 V or later, these nebulae, with H and He in the neutral state, form a sequence of low-electron-density environments. The Orion Nebula, by contrast, maintains a comparatively high ERE quantum efficiency, because its higher electron density effectively balances the ionizing effect of the more intense radiation upon the SNPs. The geometry of the Orion Nebula does not permit a reliable determination of whether the observed ERE in this object is produced in the ionized region or in the PDR on the face of the molecular cloud. However, the spatial distribution of the ERE detected in the HII region Sh 152 (Darbon et al. 2000) suggests a close association of the ERE with the ionized volume.
The comparison of the two SNP ionization processes is shown in Figure 7, where the predicted ionization rates per SNP are plotted as a function of the radiation field density. Two sets of curves are shown for values of the parameter $\tau$ differing by five orders of magnitude. Since the recombination of ionized SNPs with free electrons is independent of how a SNP was ionized, the ratio of the two ionization rates is dependent only upon the radiation density and not on $\tau$. We find that the single-photon ionization process clearly is dominant under all conditions of radiation density found in the ERE sources studied here. For the two-photon Auger process to become important under interstellar conditions, the lifetime of the excited state in SNPs, $\tau_{1}$, would need to be increased from the experimentally observed value by about 3 orders of magnitude, i.e $\tau_{1}$ $\simeq$ 2 s.
The Size-Dependence of the Ionization Equilibrium
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The ERE band exhibits a considerable width, which is interpreted by the the SNP model as reflecting the size distribution of the actively luminescing particles. The observations of the shift of the peak of the ERE intensity toward longer wavelengths with increasing radiation density (Figure 2), when combined with the relationship between particle size and luminescence wavelength imposed by quantum confinement (Ledoux et al. 1998, 2000), imply that this active size distribution is progressively eroded by radiation, starting with the smallest particles. While the single-photon ionization cross section has yet to be measured, we expect that it is large enough to make single-photon ionization of SNPs dominant over two-photon Auger ionizaton under astrophysical conditions (Figure 7). We expect the size dependence of single-photon ionization of SNPs to involve a scaling with the geometric cross section $\sim a^{2}$, despite the fact that for particles with radius $a \ll \lambda$, the photon absorption cross section $C_{abs} \sim a^{3}$ (Bohren & Huffman 1983, p. 140). The reason for this difference lies in the fact that electron escape lengths are small compared to the diameter range of the SNPs assumed responsible for the ERE (Weingartner & Draine 2001). Ionization experiments with single 193 nm (6.4 eV) photons on neutral silicon nanocrystals (Ehbrecht & Huisken 1999) have demonstrated the preferential ionization of the larger nanoparticles in a given, narrow, size distribution, covering the cluster size range from 1000 to 2000 atoms. According to Fuke et al. (1993), silicon clusters with more than 200 atoms have a first-ionization potential essentially equal to that of bulk silicon. Thus, the preferred ionization of larger clusters is not a result of a lower ionization potential. Therefore, if single-photon ionization is the only process considered as controlling the size distribution of actively luminescent SNPs, we conclude that the peak emission should shift progressively toward *shorter* wavelengths with increasing radiation density as a result of of the preferential ionization of larger particles, the opposite of what is being observed.
If the SNP model is to be maintained, other radiation-driven processes must become important in high-radiation-density environments, which lead to the efficient elimination of the smallest SNPs. We propose that photofragmentation of multiply-charged SNPs, combined with single-photon heating, could be such a process. Our determinations of the ERE efficiency (Figure 4) suggest that the fraction of neutral SNPs in RN is about 5%, and about 1% in the Orion Nebula. Calculations of the expected charge equilibrium of 3 nm diameter particles in these radiation environments predict distributions with peaks of multiple charge states of two to six positive charges per particle (Bakes & Tielens 1994; Weingartner & Draine 2001). Our own determinations of the lifetime of the ionized state, single or multiple, in RN and in the Orion Nebula (Sect. 3.4) also indicated a likely predominance of multiple positive charge states.
The balance between photoionization and electron capture by ionized nanoparticles in any given environment determines their charge distributions. For a particle of given radius, the resulting charge distribution is almost entirely determined by the parameter U$\frac{\sqrt{T}}{n_{e}}$ (Bakes & Tielens 1994; Weingartner & Draine 2001). Unfortunately, for most of our ERE sources, there are no independent estimates of the temperature T or the electron density $n_{e}$ of sufficient reliability to permit a valid examination of the variations of $\lambda_{p}(ERE)$ and $\eta$ with U$\frac{\sqrt{T}}{n_{e}}$. Of the two observed ERE parameters, $\eta$ represents additional complications, because its value is expected to depend also on the fractional abundance of SNPs relative to larger grains, in addition to the expected dependence on the fraction of SNPs remaining neutral. There is convincing evidence that the abundance of transiently heated particles, i.e. nanoparticles, varies substantially from cloud to cloud (Boulanger et al. 1990). This is in contrast to $\lambda_{p}(ERE)$, which only depends on the size distribution of the actively luminescent particles, not on their abundance. Therefore, in Figure 8 we employ generic environmental parameters for different ERE source environments to demonstrate the variation of $\lambda_{p}(ERE)$ with U$\frac{\sqrt{T}}{n_{e}}$ expected on the basis of our model. The control of the actively luminescent size distribution by photon interactions is different in two separate domains. In the first domain, defined physically by dense dark nebulae (DN) and the diffuse ISM (DISM), increasing photoionization shifts the charge distribution from a predominatly neutral state to a state roughly equally divided between neutral particles and singly-charged particles. In this regime, the ERE peak wavelength is expected to shift towards shorter wavelengths as shown in Figure 8. In the second domain, consisting of reflection nebulae (RN) and HII-regions (HII), particles with two or more positive charges appear. Photofragmentation of the smallest of such particles will rapidly shift the peak of the size distribution to larger sizes, resulting in larger values of $\lambda_{p}(ERE)$, for the following reasons.
The phase diagram of multiply-charged nanoparticles of a given positive charge state recognizes three possible size ranges (Li et al. 1998). Below a certain critical size, multiply-charged nanoparticles will disintegrate via Coulomb explosion even at 0 K and are thus unstable. Above this critical size, we encounter a range of diameters, where multiply-charged particles are metastable and where fragmentation occurs as a result of Coulomb repulsion and heating to temperatures 0 K $< T < T_{c}$, where $T_{c}$ is the critical temperature, corresponding to the size-dependent boiling temperature of the nanoparticles (Li et al. 1998, eqn. 11). Beyond this metastable size range, particles of a given charge state are stable and can only be vaporized by raising their temperature to $T > T_{c}$, the same as uncharged particles. Photofragmentation experiments of multiply-charged silicon nanocrystals in the 1 to 6 nm size range (Ehbrecht & Huisken 1999; Bescos et al. 2000) provide strong evidence that doubly and triply ionized silicon particles in this size range are in the metastable regime with respect to photofragmentation. Such particles decay by ejecting small, singly-charged ionic fragments, predominantly $\rm Si^{+}$ monomers and $\rm Si_{2}^{+}$ dimers, thus achieving a state of higher stability. However, this decay occurs only during the short interaction with a heating laser photon field and ceases immediately after the particles cool.
Under conditions present in astrophysical ERE sources, where photon fields are relatively diluted, the processes of excitation, ionization, and heating must be considered as sequential, time-separated events, involving single photons. In the diffuse ISM, the smallest stable SNP is a neutral particle which can survive the absorption of a 13.6 eV photon, emit a 2 eV ERE photon, and endure a temperature pulse associated with the addition of 11.6 eV of internal energy. This excess energy can also lead to an ionization, but each successive ionization increases the ionization potential of the particle to (Seidl et al. 1991)
$$\begin{aligned}
I(z,a) = W_{b} + (2z - 1) \frac{e^{2}}{2a} ,\end{aligned}$$
where $W_{b}$ is the work function of the bulk material and where the second right-hand term represents the additional energy required to remove a further electron from a (z-1)-fold positively charged, conducting sphere of radius $\rm a$. Photon absorptions by progressively multiply-charged SNPs, which are unable to lose energy via luminescence and which are progressively more difficult to ionize, lead to progressively increased single-photon heating. Recently, Li & Draine (2001a, 2001c) have calculated the expected temperature distributions of charged pure silicon nanoparticles and of SNPs with $SiO_{2}$ mantles. They predict maximum temperatures of $\sim 700$ K and of $\sim 380$ K, respectively, for particles of 2 nm diameter in a typical RN environment, such as NGC 2023, where multiple positive charge states would be expected for SNPs (Weingartner & Draine 2001). Particles of twice this diameter reach maximum temperatures of only about half these values under the same conditions.
In the light of these findings, we anticipate, therefore, that multiply charged, metastable SNPs, upon heating by energetic UV photons, achieve a state of increased stabilty by ejecting singly-charged ionic fragments, thus reducing their overall charge state and mass. This effect will be strongest for the smallest SNPs, which, for a given charge state, experience the highest Coulomb instability and which also gain the highest temperature increase upon absorption of a photon of given energy. The gain in increased stability resulting from the ejection of an ionic fragment is only temporary, however, because a renewed ionization to the previous charge state will leave the now less-massive particle in an even less-stable state. Consequently, in dense radiation fields, the small-size end of an SNP size distribution will be progressively destroyed by a combination of Coulomb explosion and thermal heating. The only particles permanently stable are the larger SNPs, which never reach charge/temperature combinations to place them into the metastable regime for photofragmentation.
DISCUSSION
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Limits on the Intrinsic Luminescence Efficiency of Interstellar SNPs
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The fit of our model calculations for the SNP quantum yield to the observed lower limits of the ERE luminescence efficiency was achieved by assuming that the intrinsic quantum yield of the SNP ensemble, under conditions present in the diffuse interstellar medium, is about 50%. This number is obtained by having an even balance between neutral SNPs, which have an intrinsic quantum yield of 100%, and ionized SNPs, which do not contribute to the ERE. Quantum yields between 50% and 100% have indeed been found for SNP laboratory samples (Wilson et al. 1993; Credo et al. 1998; Ledoux et al. 2001). It is straightforward to show that the interstellar ERE carrier must have similar efficiencies.
Gordon et al.(1998) estimated that the ERE represents about 3% of the energy of the total dust emission in the high-Galactic-latitude ISM. From a study of diffuse infrared Galactic emission with IRAS, Boulanger & Perault (1988) concluded that about 40% of the Galactic dust emission is attributable to dust grains small enough to undergo significant temperature fluctuations upon absorption of individual photons, which would include nano-sized ERE carriers. While cloud-to-cloud variations of this fraction were observed (Boulanger et al. 1990), this general result was confirmed by observations with DIRBE (Dwek et al. 1997; Bernard et al. 1994). A certain fraction, estimated as about 25% of the small-particle emission or about 10% of the total dust emission, is usually attributed to polycyclic aromatic hydrocarbons (PAHs) in order to account for the presence of the so-called unidentified infrared band (UIB) features in the Galactic spectrum (Desert et al. 1990; Dwek et al. 1997; Misselt et al. 2001). If SNPs are the ERE carrier, they would contribute to the remaining small-particle emission component, which amounts to about 30% of the total dust emission, after subtracting the PAH component. Other possible contributors to the small-particle emission are tiny carbonaceous grains and tiny amorphous silicate grains (Li & Draine 2001b).
Given the absorption spectrum of SNPs (Zubko et al. 1999) and the spectrum of the interstellar radiation field (Mathis et al. 1983), a typical photon exciting an SNP has a likely energy of $E_{abs}$ = 8 eV, of which about 2 eV would be emitted as ERE, if the SNP is capable of luminescing, and 6 eV would be thermalized by the particle and emitted in the infrared. Let us consider two limiting cases for the intrinsic efficiency of the ERE carrier. The highest value for the quantum yield is 100%, assuming that a single excitation can at most result in a single luminescence photon. Then, three times as much energy as is emitted in form of the ERE is contributed by thermal emission to the IR. With the estimate of Gordon et al.(1998), we find that about 9% of the total dust emission, or about 1/3 of the small particle emission not attributed to PAHs could be resulting from the presence of SNPs in the interstellar dust mixture. Conversely, a lower limit to the intrinsic efficiency of the interstellar ERE carrier, assuming they are nanoparticles, can be obtained by assuming that all the small-particle thermal emission not assigned to PAHs is produced by the ERE carrier. The associated efficiency can then be obtained from (Duley 1992):
$$\begin{aligned}
\frac{ERE}{Thermal} & \sim & \left(\frac{3\%}{30\%}\right) \sim \left(\frac{\eta \cdot E_{ERE}}{E_{abs}
- \eta \cdot E_{ERE}}\right)\end{aligned}$$
where $E_{ERE}$ is the bandgap energy of the ERE carrier. The resulting lower limit to the intrinsic efficiency of the ERE carrier is $\eta$ $\geq$ 36%. Since our SNP model equates the intrinsic efficiency of the SNP ensemble with the fraction of neutral SNPs, our assumption of 50% for this fraction is consistent with the observed characteristics of the actual ERE carrier.
Prediction of the SNP Cross Section for Single-Photon Ionization
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An intrinsic quantum yield of $\geq 36\%$ of the ERE carrier in the diffuse ISM implies that the value of $P_{0}$ (Eqn. 12) is $\geq$0.36 for the radiation field and the electron recombination rate applicable to that environment. This condition is met provided $\tau$ $\geq$ 1.8 $\tau_{i}$. Thus, we find:
$$\begin{aligned}
\frac{\tau}{\tau_{i}} & = &
\left(\frac{ \langle \sigma \rangle _{i} \cdot \int_{5.1 eV}^{13.6 eV} 4 \pi J_{\nu} \frac{d\nu}{h\nu}}
{ \pi a^{2} \cdot \tilde{\sigma} \cdot n_{e} \cdot v_{e} \cdot s}\right) \geq 1.8 ,\end{aligned}$$
where $\langle \sigma \rangle_{i}$ is the average cross section for single-photon ionization, averaged over the energy range from 5.1 eV to 13.6 eV of the interstellar radiation field. Eqn.(19) allows us to evaluate the value of $\langle \sigma \rangle_{i}$. With a geometric cross section of a typical SNP, $\pi a^{2} = 10^{-13}$ $\rm cm^{2}$, $\tilde{\sigma}$, the reduced cross section of Draine & Sutin (1987) estimated at 200, an electron density $n_{e}$ = 0.03 $\rm cm^{-3}$, an average electron speed $v_{e} = 3 \cdot 10^{6}$ $\rm cm$ $\rm s^{-1}$, a sticking coefficient *s* = 0.3, and a flux of ionizing photons of $2.9 \cdot 10^{8}$ $\rm cm^{-2} s^{-1}$, we find $ \langle \sigma \rangle _{i}$ $\leq$ 3.4 $\cdot 10^{-15}$ $\rm cm^{2}$. A published measurement of this quantity does not exist and our value is, therefore, a specific prediction made by the SNP model which can be used to either confirm or falsify our model. Our prediction applies to SNPs of average diameter 3.5 nm, expected to luminesce with peak emission around a wavelength of $\sim$ 700 nm (Ledoux et al. 2000). This ionization cross section is expected to vary with radius *a* as $\sim a^{2}$ (see Sect. 3.5).
The photoionization cross section measures the likelihood that the absorption of a single photon with energy above the ionization threshold of SNP’s ($\sim$ 5.1 eV) leads to the ejection of an electron. Experiments conducted with palladium nanoparticles (Schleicher et al. 1993), with sizes and work function comparable to those of SNPs (Fuke et al. 1993), present useful information with which to assess the likelihood that our cross section prediction is reasonable. The absolute photoelectron yield per photon of palladium nanoparticles for photons within 1 eV above the threshold is only of the order of $10^{-2}$, and it approaches unity only at a photon energy of $\sim$ 10 eV. Similar increases in the photoelectric yield per photon by two to three orders of magnitude over the photon energy range from 5 eV to 10 eV have been reported by Burtscher et al. (1984) for small silver and gold particles. Theoretical calculations of the photoelectron yield of small silicate grains by Ballester et al. (1995) predict a yield of only 0.1 at the Lyman limit, decreasing to 0.001 at a photon energy of 8 eV. SNPs more likely will behave like silicates or other insulators, where electron-phonon scattering interferes with the ejection of photoelectrons, rather than like small metal clusters, where electron-electron scattering takes the place of the process by which most of the energy of absorbed photons with energy $E > E_{i}$ is converted into heat (Gail & Sedlmayr 1980). This suggests, if the cited results are applicable to SNPs, that the photoionization cross section of SNPs is substantially smaller than the absorption cross section throughout most of the astrophysically relevant UV spectrum. Absorption cross sections measured for SNPs (Kovalev et al. 2000) cover only the photon energy range from 1.48 eV to 3.53 eV and extend from a few $10^{-19}$ $\rm cm^{2}$ to about $10^{-14}$ $\rm cm^{2}$, enough to estimate that the absorption cross sections of SNPs of about 3.5 nm diameter are not more than about $10^{-13}$ $\rm cm^{2}$ in the UV. According to our estimate, the single-photon photoionization cross section of SNPs, then, is not greater than about $3 \cdot 10^{-2}$ times the absorption cross section.
Another way to assess the likely validity of our result is to extrapolate the ionization cross section of SNPs from the ionization cross section of neutral silicon atoms. Near threshold, the latter cross section is about 5 $\cdot 10^{-17}$ $\rm cm^{2}$ (Verner et al. 1996). A typical SNP of 3.5 nm diameter has about 800 silicon atoms, with a combined ionization cross section of 4.0 $\cdot 10^{-14}$ $\rm cm^{2}$. Our estimate is about one order of magnitude smaller, corresponding to a photoelectron yield of about 0.1. This agrees surprisingly well with the yields found in experiments and in calculations, when averaged over the energy range of photons from 5.1 to 13.6 eV and a spectrum given by that of the interstellar radiation field.
The Special Role of Lynds 1780
------------------------------
Lynds 1780 (L 1780) is a high-latitude (b = $36.9^{\circ}$) dark nebula. Mattila (1979) noted the unusually red color of this object and determined its spectral energy distribution, but given that ERE as a general interstellar phenomenon was not to be recognized for several more years, he failed to find a likely source for the apparent red excess in the spectrum of L 1780. The identification of the red excess with ERE was proposed by Chlewicki & Laureijs (1987), who noted the similarities between the spectra of L 1780 and the Red Rectangle. L 1780 is an optically thick cloud ($A_{B} \sim$ 3 mag) with a density of about $10^{3}$ $\rm cm^{-3}$ (Mattila 1979), exposed to the local (D $\sim$ 100 pc) interstellar radiation field.
L 1780 occupies a special role among all ERE sources with likely interstellar dust mixtures: it has the highest efficiency determined so far ($\sim$ 13%; Fig. 4) for an object containing such a dust mixture and it exhibits a very broad emission band with a peak near 700 nm (Fig. 2). Until now, in view of the low density of the exciting radiation field this relatively long peak wavelength made L 1780 appear the odd outlier on graphs such as Figure 2. Our SNP model provides a natural explanation for the seemingly odd characteristics of the ERE in L 1780. In view of the much higher gas density of L 1780 compared to the diffuse interstellar medium in general, the electron density will be higher in L 1780 as well, while the UV-photon density is actually reduced due to the cloud’s intrinsic opacity, so that the overall ionization equilibrium tends more towards neutrality. Hence, $\frac{\tau}{\tau_{i}} \ll$ 1 in L 1780, while $\frac{\tau}{\tau_{i}} \sim$ 1 in the diffuse ISM and $\frac{\tau}{\tau_{i}} \gg$ 1 in reflection nebulae and HII regions. These are the conditions, respectively, under which our model predicts predominantly neutral SNPs, SNPs approximately equally balanced between neutral and ionized, and predominantly ionized SNPs. Thus, in L 1780 the SNPs are more predominantly neutral than is the case in the diffuse ISM. The larger SNPs, which are removed from luminescing in the diffuse ISM due to the size-dependent ionization, are now restored as ERE contributors. Consequently, both the overall quantum efficiency increases and the peak wavelength shifts towards larger values compared to the diffuse ISM. The ERE spectrum of L 1780 may, therefore, be regarded as most closely approximating that of a size distribution of luminescing SNPs, hardly modified by ionization.
Multiple Ionization and Photofragmentation
------------------------------------------
In Section 3.5 we have argued that Coulomb explosions of multiply-charged, metastable SNPs will gradually erode the small-size end of an existing SNP size distribution, once it is exposed to dense and hard radiation fields. An observable consequence of such conditions would be the increased gas-phase abundance of $\rm Si^{+}$ ions as well as SiO molecules in the photon-dominated regions between dense molecular clouds and nearby hot stars. An impressive example of such an enhanced gas-phase abundance of $\rm Si^{+}$ in the presence of a dense radiation field can be found in the ISO observations of the reflection nebula NGC 7023 by Fuente et al. (2000). In the inner cavity of this nebula, where ERE is almost totally absent (Witt & Boroson 1990), at least 20% to 30% of the cosmic abundance of silicon is found in the gas phase, while the inferred gas-phase abundance of silicon drops to about 5% in the high-density ERE filaments to the south and to the north-west of the central B 3 V star, HD 200775. This is consistent with an almost total photodisintegration of all SNPs in this volume, given that the SNP model requires about 20% of the total silicon abundance in the form of SNPs. We suggest that multiple ionization of SNPs and the resulting gradual photofragmentation may have destroyed the SNPs in the inner cavity of NGC 7023. We also suggest that this is the process by which the size distribution of SNPs in regions of high radiation density is changed permanently, so that only the larger SNPs capable of luminescing at longer wavelengths survive.
Infrared Band Emissions from SNPs
---------------------------------
A frequently raised question about interstellar SNPs concerns expectations that SNPs should give rise to infrared band emissions for which there is no clear present evidence. Li & Draine (2001a, 2001c) made a first attempt at calculating the expected emission spectrum of pure silicon particles and SNPs with oxide coatings. They predict the presence of a relatively sharp emission feature at 16.4 $\mu$m for pure silicon particles and a still stronger, broad emission peak at 20 $\mu$m due to $\rm SiO_{2}$ for oxide-coated SNPs. Several comments are in order regarding pure silicon particles: i) The case of pure silicon particles appears astrophysically less interesting; ii) such particles, with dangling bonds at the surface remaining unpassivated, would not luminesce in the ERE band; iii) they have never been proposed as the ERE carrier; iv) they would not be expected to be chemically stable under interstellar conditions, if they could form in the first place. In our view, the predicted 16.4 $\mu$m feature is, therefore, of questionable significance.
Adding an oxide coating to SNPs, according to Li & Draine (2001a), lowers the most likely temperature of the nanoparticles from 300 K to about 75 K, due to the oxide’s ability to radiate strongly in Si-O vibrational bands. This lower temperature, however, prohibits significant emission at the most commonly investigated band near 10 $\mu$m and leaves the radiation near 20 $\mu$m as the dominant mode. The detectability of the 20 $\mu$m band is a function of the SNP abundance as a fraction of the radiating dust mass, and thus is inversely proportional to the intrinsic ERE quantum efficiency of the oxide-covered SNPs. Experimentally determined quantum efficiencies for SNPs are found to be in the range from 50% to 100% (Wilson et al. 1993; Credo et al. 1999; Ledoux et al. 2001). Li & Draine (2001a), from their model calculations, conclude that that even for such high-efficiency SNPs, the predicted intensity of the 20 $\mu$m band emission from the diffuse ISM exceeds observed upper limits set by DIRBE observations by a factor of 7. Similarly, in the reflection nebula NGC 2023, they find the predicted 20 $\mu$m feature strength to exceed the observed intensity seen in an ISO spectrum by a significant factor. In order to decide, whether these constraints are in fact as severe as they appear, several objectives would need to be pursued in the future. These include sensitive spectroscopic observations of the 20 $\mu$m region in ERE sources, experimental determinations of the dielectric functions of oxygen-passivated SNPs for more solidly-based model calculations, as well as controlled oxidization experiments aimed at determining the minimum amount of $\rm SiO_{2}$ needed for the effective passivation of SNPs.
Detection of the predicted 20 $\mu$m emission band would be most likely in objects where the abundance of SNPs exceeds that in the interstellar medium by a wide margin. The Red Rectangle (Schmidt et al. 1980) has an unusually strong ERE band, which suggests that the dust locally produced in this proto-planetary nebula is exceptionally rich in the particles producing the ERE. The publically accessible ISO-SWS spectrum of the Red Rectangle $\it{(http://isowww.estec.esa.nl/galleries/cir/red\_rect.html)}$ does not exhibit any prominent feature at 20 $\mu$m. However, a detailed analysis of this spectrum has yet to be done. The Red Rectangle is a member of a class of similar objects, bi-polar proto-planetary nebulae with dust produced by mass-loosing AGB stars. Hrivnak et al. (2000) have used ISO to study the infrared spectra of nine such objects. Four of these exhibit a strong emission band at 21 $\mu$m, closely resembling the profile predicted by Li & Draine (2001a). In two additional objects, the identification of the 21 $\mu$m band is tentative; in three objects the band was not detectable. None of these nebulae has been studied for the presence of ERE, and none is expected, because the spectral types of the central stars are much later than that of the illuminating star in the Red Rectangle (B 9.5 III). The possibility that SNPs are formed in protoplanetary nebulae and are excited to emit ERE in the case where central stars with sufficient ultraviolet photons are present, deserves further investigation. There has also been an indication that the 21 $\mu$m band is present in the dust emission spectrum of the supernova remnant Cas A (Douvion et al. 2001), which, together with the strong presence of SiO emission from the remnant of SN 1987A (Roche et al. 1991), leads to the suggestion that supernova remnants could be locations for SNP formation. An alternate suggestion for the identification of the 21 $\mu$m feature in proto-planetary nebulae has been provided by von Helden et al. (2000), who proposed titanium carbide nanocrystals as the carrier of this feature. No estimates exist on whether the likely abundance of titanium carbide is sufficient to explain the intensity of the 21 $\mu$m band.
SUMMARY
=======
The existing body of observational data on the interstellar luminescence process known as ERE shows a wide range of variation in three observational characteristics: the band-integrated intensity, the wavelength of peak emission, and the efficiency with which absorbed UV/optical photons are converted into ERE photons. By systematically analysing all suitable ERE observations available so far, we have shown that the common environmental factor driving the variations in these three characteristics is the radiation field density in the different environments, where ERE has been detected. The ERE intensity increases with radiation density roughly linearly, as expected for photoluminescence; the peak of the ERE band shifts from about 610 nm in the diffuse ISM to beyond 800 nm in O-star dominated HII regions; and the photon conversion efficiency is found to be highest in environments with the lowest density of exciting photons, only to decline significantly in denser radiation fields. In the second part of this paper, we have introduced a model for the ERE carrier in which photoionization of nanoparticles, balanced by recombination with free electrons, is the controlling physical process which determines the total fraction of actively luminescing particles. The model relies on experimental findings, showing that ionization will quench photoluminescence in a luminescing semiconductor nanoparticle, with luminescence being restored upon recombination. This model successfully reproduces the observed variations of the ERE intensity and the ERE photon conversion efficiency for a large range of radiation field densities. We find that under the complete range of conditions where ERE is being observed in astronomical environments, the single-photon ionization process is always dominant over the two-photon Auger ionization process. In dense-radiation environments, photoionization will lead to relatively high positive charge states for SNPs, which will render the smallest SNPs unstable against photofragmentation upon single-photon heating. We suggest this process as the explanation for the observed shift of the wavelength of peak ERE intensity toward larger values in UV-radiation fields of increased density. Silicon nanoparticles have been proposed as carriers of the ERE in view of their ability to match the spectral characteristics of the ERE and the efficiency constraints better than any other current model candidate. In order to match the entire body of ERE data presented in this paper, we require that silicon nanoparticles with an average diameter of 3.5 nm have an ionization cross section of about $3.4 \cdot 10^{-15}$ $\rm cm^{2}$ when exposed to a radiation field with an energy distribution equal to that found in the diffuse ISM of the solar neighborhood. The experimental determination of this cross section would represent a critical test for the model proposed here. Finally, we note that sensitive spectroscopic searches for the 20 $\mu$m band predicted to be associated with oxygen-passivated SNPs in ERE-bright interstellar environments would provide severe constraints for the validity of the SNP model, especially if accompanied by experimental efforts to determine the dielectric functions for these particles in the laboratory.
Acknowledgement: We are extremely grateful to Bruce Draine, Friedrich Huisken, Gilles Ledoux, and Aigen Li, who reviewed an earlier version of this paper and who provided much critical input to the present version. We also acknowledge extensive stimulating discussions about ERE and nanoparticles with Louis Brus, Bob Deck, Walt Duley, Minoru Fujii, Dieter Gerlich, Karl Gordon, Thomas Henning, and Daniele Pierini. Finally, we acknowledge again Bruce Draine, who as referee provided further helpful criticism and several very constructive suggestions. This work was supported through Grant NAG5-9202 from the National Aeronautics and Space Administration to The University of Toledo.
Bakes, E.L.O., & Tielens, A.G.G.M. 1994, , 427, 822 Baldwin, J.A. et al. 1996, , 468, L115 Ballester, J.L., Shi, Y., & Dwek, E. 1995, J. Opt. Soc. Am. B, 12, 1211 Banin, U., Bruchez, M., Alivisatos, A.P., Ha, T., Weiss, S., & Chemia, D.S. 1999, J. Chem. Phys., 110, 1195 Bernard, J.P., Boulanger, F., Desert, F.X., Giard, M., Helou, G., & Puget, J.L. 1994, A&A, 291, L5 Bescos, B., Hoch, R., Schmidtke, H.-J., & Gerber, G. 2000, Appl. Phys. B, 71, 373 Bohren, C.F., & Huffman, D.R. 1983, Absorption and Scattering of Light by Small Particles, (New York: John Wiley & Sons, Inc.) Boulanger, F., & Perault, M. 1988, , 330, 964 Boulanger, F., Falgarone, E., Puget, J.L., & Helou, G. 1990, , 364, 136 Brongersma, M.L., Kik, P.G., Polman, A., Min, K.S., & Atwater, H.A. 2000, Appl. Phys. Lett., 76, 351 Burtscher, H., Schmidt-Ott, A., & Siegmann, H.C. 1984, Z. Phys. B - Cond. Matt., 56, 197 Castro-Carrizo, A., Lucas, R., Bujarrabal, V., Colomer, F., & Alcolea, J. 2001, A&A, 368, L34 Chepic, D.I., Efros, A.L., Ekimov, A.I., Ivanov, M.G., Kharchenko, I.A., Kudriavtsev, I.A., & Yazeva, T.V. 1990, J. Lumin. 47, 113 Chlewicki, G., & Laureijs, R.J. 1987, in Polycyclic Aromatic Hydrocarbons and Astrophysics, ed. A. Leger, L. d’Hendecourt, & N. Boccara, (Dordrecht: Reidel Publ. Co.), NATO ASI Series, Vol. 191, p. 335 - 337 Credo, G.M., Mason, M.D., & Buratto, S.K. 1999, Appl. Phys. Lett., 74 (14), 1998 Darbon, S., Perrin, J.-M., & Sivan, J.-P. 1999, A&A, 348, 990 Darbon, S., Savagno, A., Perrin, J.-M., Savine, C., Ducci, V., & Sivan, J.-P. 2000, A&A, 364, 723 Delerue, C., Allan, G., & Lannoo, M. 1993, Phys. Rev. B, 48, 11024 Delerue, C., Allan, G., & Lannoo, M. 1998, Semiconductors and Semimetals, 49, 253 Desert, F.-X., Boulanger, F., & Puget, J.L. 1990, A&A, 237, 215 Douvion, T., Lagage, P.O., & Pantin, E. 2001, , 369, 589 Draine, B.T., & Sutin, B. 1987, , 320, 803 Duley, W.W. 1985, MNRAS, 215, 259 Duley, W.W. 1992, MNRAS, 258, 773 Dwek, E. et al. 1997, , 475, 565 Ehbrecht, M., & Huisken, F. 1999, Phys. Rev. B, 59, 2975 Efros, A.L., & Rosen, M. 1997, , 78, 1110 Freedhoff, M.I., & Marchetti, A.P. 1997, in Handbook of Optical Properties. II. Optics of Small Particles, Interfaces, and Surfaces, ed.R.E. Hummel, & P. Wissmann, New York: CRC Press, 1-30 Fuente, A., Martin-Pintado, J., Rodriguez-Fernandez, N.J., Cernicharo, J., & Gerin, M. 2000, A&A, 354, 1053 Fujii, M., Mimura, A., Hayashi, S., & Yamamoto, K. 2000, J. Appl. Phys., 87, 1855 Fuke, K., Tsukamoto, K., Misaizu, F., & Sanekata, M. 1993, J. Chem. Phys., 99, 7807 Furton, D.G., & Witt, A.N. 1990, , 364, L45 Furton, D.G., & Witt, A.N. 1992, , 386, 587 Gail, H.-P., & Sedlmayr, E. 1980, A&A, 86, 380 Gail, H.-P., & Sedlmayr, E. 1999, A&A, 347, 594 Garray, G., Mardones, D., & Rodriguez, L.F. 2000, , 545, 861 Gordon, K.D., Witt, A.N., & Friedmann, B.C. 1998, , 498, 522 Gordon, K.D., et al. 2000, , 544, 859 Hirano, N., Mikami, H., Umemoto, T., Yamamoto, S., & Taniguchi, Y. 2001, , 547, 899 Hrivnak, B.J., Volk, K., & Kwok, S. 2000, , 535, 275 Huffman, D.R. 1977, Adv Phys., 29, 129 Kerr, T.H., Hurst, M.E., Miles, J.R., & Sarre, P.J. 1999, MNRAS, 303, 446 Kharchenko, V.A., & Rosen, M. 1996, J. Lumin., 70, 158 Koch, F., Kovalev, D., Averboukh, B., Pollisski, G., & Ben-Chorin, M. 1996, J. Lumin. 70, 320 Kovalev, D., Diener, J., Heckler, H., Polisski, G., Kuenzner, N., & Koch, F. 2000a, Phys. Rev. B, 61, 4485 Kovalev, D., Heckler, H., Ben-Chorin, M., Polisski, G., Schwartzkopff, M., & Koch, F. 2000b, J. Porous Mat., 7, 85 Kurucz, R.L., Peytremann, E., & Avrett, E.H. 1974, Blanketed Model Atmospheres of Early-Type Stars, (Washington, DC: Smithsonian Institution) Leach, S. 1986, J. Elect. Spectr. and Related Phenom., 41, 427 Ledoux, G., Guillois, O., Porterat, D., Renaud, C., Huisken, F., Kohn, B., & Paillard, V. 2000, Phys. Rev. B, 62 (23), 15942 Ledoux, G., Guillois, O., Huisken, F., Kohn, B., Porterat, D., & Reynaud, C. 2001, A&A, in press Li, A., & Draine, B.T. 2001a, , in press Li, A., & Draine, B.T. 2001b, , 554, 778 Li, A., & Draine, B.T. 2001c, http://xxx.lanl.gov/abs/astro-ph/0012509v2 Li, Y., Blaisten-Barojas, E., & Papaconstantopoulos, D.A. 1998, Phys. Rev. B, 57, 15519 Mason, M.D., Credo, G.M., Weston, K.D., & Buratto, S.K. 1998, , 80, 5405 Mathis, J.S., Mezger, P.G., & Panagia, N. 1983, A&A, 128, 212 Mattila, K. 1979, A&A, 78, 253 Meyer, B.K., Hofmann, D.M., Stadler, W., Petrova-Koch, V., Koch, F., Omling, P., & Emanuelsson, P. 1993, Appl. Phys. Lett., 63, 2120 Misselt, K.A., Gordon, K.D., Clayton, G.C., & Wolff, M.J. 2001, , 551, 277 Murakami, K., Suzuki, T., Makimura, T., & Tamura, M. 1999, Appl. Phys. A, 69, S13 Nirmal, M., & Brus, L. 1999, Acc. Chem. Res. 32, 407 Nirmal, M., Dabbousi, B.O., Bawendi, M.G., Macklin, J.J., Trautman, J.K., Harris, T.D., & Brus, L.E. 1996, Nature, 383, 802 Perrin, J.-M., & Sivan, J.-P. 1992, A&A, 255, 271 Rietmeijer, F.J.M., Nuth III, J.A., & Karner, J.M. 1999, , 527, 395 Rinnert, H., Vergnat, M., & Marchal, G. 1998, Appl. Phys. Lett., 72, 3157 Rinnert, H., Vergnat, M., Marchal, G., & Burneau, A. 1999, J. Lumin., 80, 445 Roche, P.F., Aitken, D.K., & Smith, C.H. 1991, MNRAS, 252, 39 Rouan, D., Lecoupanec, P., & Leger, A. 1995, Newsletter on Analysis of Astronomical Spectra, 22, 37 Rubin, R.H., Martin, P.G., Dufour, R.J., Ferland, G.J., Baldwin, J.A., Hester, J.J., & Walter, D.K. 1998, , 495, 891 Russell, R.W., Soifer, B.T., & Willner, S.P. 1978, , 220, 568 Sakata, A., Wada, S., Narisawa, T., Asano, Y., Iijima, Y., Onaka, T., & Tokunaga, A.T. 1992, , 393, L83 Schleicher, B., Burtscher, H., & Siegmann, H.C. 1993, Appl. Phys. Lett., 63, 1191 Schmidt, G.D., Cohen, M., & Margon, B. 1980, , L133 Schmidt, G.D., & Witt, A.N. 1991, , 383, 698 Seahra, S.S., & Duley, W.W. 1999, , 520, 719 Seidl, M., Meiwes-Broer, K.-H., & Brack, M. 1991, J. Chem. Phys., 95, 1295 Sitko, M.L., Savage, B.D., & Meade, M.R. 1981, , 246, 161 Sivan, J.-P., & Perrin, J.-M. 1993, , 404, 258 Smith, T.L. 2000, Ph.D. Thesis, The University of Toledo Snow, T.P., & Witt, A.N. 1996, , 468, L65 Spitzer, L. 1978, Physical Processes in the Interstellar Medium, (New York: John Wiley & Sons)
Szomoru, A., & Guhathakurta, P. 1998, , 494, L93 Takeoka, S., Fujii, M., & Hayashi, S. 2000, Phys. Rev. B, 62 (24), 16820 Theiss, W. 1997, Surf. Sci. Rep., 29, 91 Verner, D.A., Ferland, G.J., & Korista, K.T. 1996, , 465, 487 Vinciguerra, V., Franzo, G., Priolo, F., Iacona, F., & Spinella, C. 2000, J. Appl. Phys., 87, 8165 von Helden, G., Tielens, A.C.G.M., van Heijnsbergen, D., Duncan, M.A., Hony, S., Waters, L.B.F.M., & Meijer, G. 2000, Science, 288, 313 Wang, C., Shim, M., & Guyot-Sionnest, P. 2001, Science, 291, 2390 Weingartner, J.C., & Draine, B.T. 2001, ApJS, 134, 263 Waters, L.B.F.M. et al. 1998, Nature, 391, 868 Whittet, D.C.B. 1992, Dust in the Galactic Environment, (Bristol: IOP Publishing Ltd.) Wilson, W.L., Szajowski, P.F., & Brus, L.E. 1993, Science, 262, 1242 Witt, A.N., & Boroson, T.A. 1990, , 355, 182 Witt, A.N., & Gordon, K.D. 2000, , 528, 799 Witt, A.N., & Rogers, C. 1991, PASP, 103, 415 Witt, A.N., & Schild, R.E. 1985, , 294, 225 Witt, A.N., & Schild, R.E. 1986, ApJS, 62, 839 Witt, A.N., Gordon, K.D., & Furton, D.G. 1998, , 501, L111 Wolkin, M.V., Jorne, J., Fauchet, P.M., Allan, G., & Delerue, C. 1999, PRL, 82, 197 Zhang, Q., Ho, P.T.P., & Wright, M.C.H. 2000, , 119, 1345 Zubko, V., Smith, T.L., & Witt, A.N. 1999, , 511, L57
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abstract: '[ Forest-fire waiting times, defined as the time between successive events above a certain size in a given region, are calculated for Italy. The probability densities of the waiting times are found to verify a scaling law, despite that fact that the distribution of fire sizes is not a power law. The meaning of such behavior in terms of the possible self-similarity of the process in a nonstationary system is discussed. We find that the scaling law arises as a consequence of the stationarity of fire sizes and the existence of a non-trivial “instantaneous” scaling law, sustained by the correlations of the process. ]{}'
author:
- 'Álvaro Corral$^\dag$, Luciano Telesca$^*$, and Rosa Lasaponara$^*$'
bibliography:
- '../../biblio.bib'
title: ' Scaling and correlations in the dynamics of forest-fire occurrence '
---
In the last years, many natural hazards, like earthquakes, volcanic eruptions, landslides, rainfall, solar flares, etc., and other similar-in-spirit phenomena in condensed-matter physics have been shown to be characterized by a power-law distribution of event sizes, over many orders of magnitude in some cases [@Bak_book; @Turcotte_soc; @Malamud_hazards; @Sethna_nature]. This kind of distribution has profound implications for the nature of these phenomena, as it indicates that extreme events do not constitute a case separated from the smaller, ordinary ones; rather, the events are generated by a mechanism that operates in the same way for all the different scales involved, and a characteristic size of the events cannot be defined. In this way, a reasonable question such as “which is the typical size of the earthquakes in this region?” is impossible to answer.
Comparison with simple self-organized-critical (SOC) cellular-automaton models suggests that the events that define the dynamics in these phenomena consist of a small instability or excitation that propagates as a very rapid chain reaction or avalanche through a medium that is in a very particular state, similar to the critical points found at continuous phase transitions in condensed-matter physics [@Bak_book; @Turcotte_soc; @Sornette_critical_book; @Hergarten_book]. The dissipation produced by each avalanche would act as a feedback mechanism that balances a slow energy input and maintains the system close to the critical state.
Of special interest is the case of forest fires, for which cellular-automaton models yielded a power-law behavior for the distributions of burned areas (which are a measure of the size of the events), and showed the previous mechanisms at work [@Bak_forest_fire; @Drossel]; curiously, it was not until much later that Malamud [*et al.*]{} observed power law distributions for real forest fires, with exponents around 1.4 for the probability density (i.e., non-cumulative distribution) of fire sizes [@Malamud_science; @Turcotte_physA; @Malamud_fires_pnas].
Nevertheless, this issue is still open, as other studies with different data do not agree with a simple power-law behavior: Ricotta [*et al.*]{} [@Ricotta_99] postulated that fires of large sizes, due to negative economic and social effects, are reduced by the massive human intervention; therefore, less than expected large fires occur, leading to an increase (in absolute value) of the power-law exponent in that regime. Reed and McKelvey [@Reed_02], using the concept of extinguishments growth rate, presented a four-parameter “competing hazards” model providing the overall best fit. In a subsequent paper, Ricotta [*et al.*]{} [@Ricotta_01] have observed that a multiple power-law behavior, denoted by the presence of different power-law ranges delimited by cut-offs, is due to dynamical changes, linked to “more or less abrupt changes in the landscape-specific process-pattern interactions that control wildfire propagation, rather than statistical inaccuracies”. Therefore, the appearance of different size ranges with different power-law exponents can be accounted for different dynamics, involving topographic, climatic, vegetational, and human factors [@Telesca_05].
In addition to the size of the events, the dynamics of event occurrence is of fundamental interest. Notably, the temporal properties of some popular SOC cellular-automaton models were shown to be described by a trivial Poisson process, which prevented progress in this aspect until very recently, when it has been concluded that this picture is not appropriate [@Paczuski_btw]; in parallel, it has been found that real systems show a very rich behavior in time, with power-law distributions for the time between events and/or scaling laws for these distributions [@Boffetta; @Corral_prl.2004; @Baiesi_flares]. The existence of such scaling laws has implications no less deep than the fact of having a power-law distribution of event sizes, although they have been much less studied: (i) the scaling law reflects the fact that the occurrence of large events mimics the process of occurrence of smaller ones (and this behavior is not implicit in the distribution of event sizes), thus allowing to model the scarce big events on the basis of the abundant small ones; (ii) the scaling law is the signature of the invariance of the process under a renormalization-group transformation, which strengths the links between natural hazards and critical phenomena [@Corral_prl.2005].
We study in this paper the relation between the temporal properties of forest-fire occurrence and the size of the fires, using the AIB (Archivio Incendi Boschivi) fire catalog compiled by the Italian CFS (Corpo Forestale dello Stato) for all Italy [@Italy_fire_catalog], covering the years 1998–2002 (included) and containing 36821 fires. In order to characterize the overall behavior, we measure for the whole catalog the probability density of the burned areas $s$, defined as $$D(s) \equiv \frac{\mbox{Prob}[s \le \mbox{ area }
< s + ds]}{ds},
\label{distribution}$$ where $ds$ is the bin size (small enough to sample almost continuously $D(s)$ but large enough to guarantee statistical significance); the resulting shape for $D(s)$ is shown in Fig. \[Dm\]. Although a power law could be fit to the data, it is clearly seen that the curve is continuously bending downwards, which is the characteristic of a lognormal distribution, $$D(s)=
\frac C {\sqrt{2\pi} \, \sigma s} \, \exp\left(-\frac
{\left(\ln s - \mu\right)^2}{2\sigma^2}\right )
\propto
%{s^{-\left(1+\frac 1 {2\sigma^2} \ln \frac s {e^\mu}\right)}},
%$$
%$$
%= {s^{-\left(1+\ \ln (s /e^\mu)/(2\sigma^2)\right)}},
%=
\left( \frac {e^\mu} s \right)^{1+\ \frac{ \ln (s /e^\mu)}{2\sigma^2}},
\label{lognormal}$$ with $\mu$ and $\sigma$ the mean and standard deviation of $\ln s$, and $C$ a correction to normalization due to the fact that the fit is not valid for all $s$. In this way, for each $2\sigma^2$ that $\ln s$ is away from $\mu$ the exponent of the previous pseudo-power law increases in one unit (in other words, each decade $s$ is above $e^\mu$ increases the exponent in $\ln 10/ (2\sigma^2)$). When $s$ is measured in hectares (ha), the results of the best fit yield $\mu=-0.35$, $\sigma=9.5$, and $C=6.7$; this fit holds not only for the full data but it can be verified that also describes smaller parts of the country and shorter periods of time. In any case, we have no means to conclude if the deviation from a power-law behavior is due to human extinction efforts or to the territorial characteristics of a high-populated country.
From the distribution of sizes, knowing the total number of events, it is possible to calculate the mean waiting time (or recurrence time) for events above a certain size $s_c$ [@Malamud_fires_pnas]; however, looking at the individual values of the waiting times one sees that they are broadly distributed and therefore the mean values are not very informative about the dynamics of the process; so, in order to investigate the temporal properties of fire occurrence it is necessary to look at the whole waiting-time distribution. To be precise, the procedure is as follows: once a spatial area, a time period, and a minimum event size, $s_c$, are selected, the fire history is described as a simple point process, $\{t_0, t_1, t_2 \dots\}$, where $t_i$ denotes the time of occurrence of fire $i$. For this process, the set of waiting times, defined as the time intervals between consecutive events, is obtained straightforwardly as $
\tau_i \equiv t_i - t_{i-1}.
$ Important insight into the nature of the process may be obtained by considering $s_c$ not as a constant but as a variable parameter [@Bak.2002; @Corral_prl.2004], and then, the waiting-time probability density for the selected window, defined in the same way as in Eq. (\[distribution\]), will be also considered as a function $D(\tau;s_c)$ of the minimum size $s_c$.
For the whole country and the total temporal extension of the catalog we obtain the different set of curves displayed in Fig. \[Dt\](a). We might fit a (decreasing) power law for each distribution, but the exponent would decrease with the increase of the minimum size $s_c$. Instead, it is more convenient to rescale the distributions in order that all of them have the same mean and can be properly compared; this is accomplished by the scale transformation $\tau \rightarrow R(s_c) \tau$ and $D(\tau; s_c) \rightarrow D(\tau; s_c)/R(s_c)$, where $R(s_c)$ is the rate of fire occurrence, defined as the mean number of fires per unit time with $s\ge s_c$ (that is, the inverse of the mean of each distribution). The results of the rescaling, as shown in Fig. \[Dt\](b), lead to a collapse of the rescaled distributions into a single function $F$, signaling the fulfillment of a scaling law, $$D(\tau;s_c) = R(s_c) F(R(s_c) \tau),
\label{scaling}$$ in the same way as for several natural hazards [@Corral_prl.2004; @Baiesi_flares; @Bunde] and other avalanche-like processes [@Yamasaki; @Davidsen_fracture].
The rescaled plot unveils more clearly the behavior of the distributions: instead of different power laws, what we have is a unique shape, but at different scales. Again, the apparent continuous decrease of the exponent with the rescaled time, $\theta \equiv R(s_c) \tau$, suggest a lognormal shape for $F(\theta)$ as that of Eq. (\[lognormal\]), where now we will use tildes to denote the parameters. The best fit yields $\tilde \mu = -2.0$ and $\tilde \sigma= 2.0$, fixing $\tilde C\equiv 1$. Notice that now we have the constraint that the mean of the rescaled distribution, $\bar \theta$, has to be one; as $\bar \theta = e^{\mu+\sigma^2/2}$, this leads to $\mu=-\sigma^2/2$. It is remarkable that, unlike earthquakes, solar flares, or fractures [@Corral_prl.2004; @Baiesi_flares; @Davidsen_fracture; @Astrom], forest fires fulfill a scaling law for the waiting time distributions without displaying power-law distribution of event sizes. We could conclude that we have self-similarity in size-time without having scale invariance in size alone. This self-similarity means that for the linear scale transformation $\tau \rightarrow a \tau$ and $s_c \rightarrow b s_c$, the value of $b$ which guarantees scale invariance is given by $R(b s_c) a = R(s_c)$, which means that $b$ does not only depend on $a$, as in the case of a power-law distribution of sizes, but it also depends on $s_c$. This would be equivalent to define an artificial new size variable enforcing that it be power law distributed. However, although this picture describes a kind of self-similarity, it is not a sufficient condition. Indeed, the seasonality of fire occurrence prevents self-similarity in size-time: five years of fire occurrence cannot be equivalent to one year of smaller fires, as there is a clear annual modulation in fire occurrence; nevertheless, for a fixed time window still the small events are a model for the occurrence of the big ones.
Which is then the origin of the scaling law (\[scaling\])? It is not difficult to relate it with the stationarity of fire sizes and with the existence of a scaling law for the “instantaneous” waiting-time distributions. Indeed, $D(\tau; s_c)$ is a statistical mixture of those instantaneous waiting-time distributions $D_t(\tau; s_c)$, which, when the scale of variations of the rate is much larger than the corresponding mean waiting time, take into account that fire occurrence is not stationary but change with time $t$; if it is only the instantaneous rate $r(t;s_c)$ (defined as the number of fires per unit time in a small time interval around $t$) what determines fire occurrence, we can write $D_t(\tau; s_c) =D(\tau; s_c | r(t;s_c))$ and then, $$D(\tau; s_c) = \frac 1 {R(s_c)} \int_{r_{min}}^{r_{max}}
r D(\tau; s_c | r) \rho(r; s_c) \, dr,$$ where $\rho(r; s_c)$ is the density of rates, i.e., the fraction of the time the rate is in a particular small range of values, divided by that range [@Corral_Christensen]. Assuming the stationary nature of fire sizes (notice that this is not incompatible with the nonstationarity of time occurrence), this means that $r(t; s_c)=p r(t;s_0)$, where the fraction $p$ is the probability of having a fire larger than $s_c$ knowing that it has been larger than $s_0$, $p=\mbox{Prob}[s\ge s_c] / \mbox{Prob}[s\ge s_0]$; this implies that the density of rates fulfills a scaling law, $\rho(r; s_c)=p^{-1}\rho(p^{-1} r; s_0) \equiv p^{-1} g(p^{-1} r)$. Finally, with the hypothesis that $D(\tau; s_c | r)$ verifies as well a (instantaneous) scaling law, $D(\tau; s_c | r) = r f (r\tau)$, we get $$D(\tau; s_c) = \frac 1 {p R_0} \int_{p a}^{p b}
r f(r \tau) p^{-1} g(p^{-1} r)\, dr,$$ with $r_{min}= pa$, $r_{max}= pb$, and $R(s_c)=p R(s_0)\equiv p R_0$. A simple change of variables reveals that $D(\tau; s_c)$ is a function of the form $p \tilde F(p\tau) \equiv p\int_a^b x^2 f(p\tau x) g(x) dx$, which is equivalent to the scaling law (\[scaling\]). In other words, if fire occurrence under hypothetical stationary conditions verifies a scaling law for the waiting times (which in this case would be a reflection of the self-similarity of the stationary process, as explained above), non-stationary conditions keep that scaling valid (with a different scaling function) as long as fire size remains stationary and the rate does not become too small for this description to be invalid. \[On the other hand, for rates so small that the mean waiting time is much larger than the larger scale of variation of the rate itself (whose existence is not known), the structure of $r(t)$ would become irrelevant and the waiting-time distribution would tend to the exponential form characteristic of Poisson processes.\]
In order to support our argument for the existence of the scaling law (\[scaling\]) we show in Fig. \[rt\] the stationarity of fire sizes, by means of the evolution of $r(t;s_c)$ for different $s_c$, and how the different curves collapse when they are rescaled by their mean, $R(s_c)$; it is also easy to check that the distribution of rates verifies a scaling law. The last hypothesis, the scaling of $D(\tau; s_c | r)$ is more difficult to demonstrate due to the daily oscillations of $r(t;s_c)$, which makes that the rate can be considered approximately constant only for a few hours, corresponding to those of the daily maximum and minimum hazard (between 1 p.m. and 4 p.m. and between 1 a.m. and 9 a.m., respectively). This short range of variation leads to very low statistics; nevertheless, for the periods of the year of maximum fire occurrence (for about one month in the summer) the maximum and minimum daily rates are fairly constant for different days, which allows to improve the statistics. The results obtained in this way are shown in Fig. \[Dtinstant\], although they are not conclusive. Essentially, they are compatible with an instantaneous scaling law, with perhaps an exponential instantaneous distribution, $D(\tau; s_c | r) \simeq r e^{-r \tau}$, but the statistical errors are large; in any case, the hypothesis of the instantaneous scaling law cannot be rejected.
If we find an exponential form for the instantaneous distributions, does this mean that the dynamics can be described by a nonstationary Poisson process? This is the simplest model for nonstationary behavior, for which the events take place at a rate that does not depend on the occurrence of the other events, as in the simple (stationary) Poisson process, but with the difference that the rate changes with time (independently on the process, we can imagine the rate is related to the meteorological conditions, not affected by the presence of fire or not). This leads indeed to exponential instantaneous distributions (provided the rate is not too small), although the reciprocal is not true, in general. If, in addition, the size of the events constitutes an independent random process, this ensures the existence of a scaling law for the instantaneous distributions (as Poisson processes are invariant under random thinning plus rescaling, see [@Corral_prl.2005]). The nonstationary Poisson process has been recently used for earthquake occurrence, see Ref. [@Shcherbakov].
A test to verify if a process is of the nonstationary Poisson type was introduced by Bi [*et al.*]{} [@Bi]. One only needs to compute for each $i$ the statistics $h_i \equiv 2\tau_{{min }\, i}/(2\tau_{{min }\, i} + \tau_{{neig } \, i})$, where $\tau_{{min }\, i}$ is the minimum of $\tau_i$ and $\tau_{i+1}$, and $\tau_{{neig }\, i}$ is the length of the interval neighbor of the minimum one opposite to the one used in the comparison, i.e., $\tau_{i-1}$ or $\tau_{i+2}$ respectively. Under the hypothesis we want to test, both $\tau_{{neig }}$ and $2 \tau_{{min }}$ are independent and exponentially distributed with approximately the same rate, $r(t)$, and therefore it can be shown that $h$ is uniformly distributed between 0 and 1.
The application of the test to the fire data yields catastrophic results, see Fig. \[bi\]. The obtained probability density for $h$ is far from uniform, with very large peaks for precise $h$-values. This is due to the discretization of fire occurrences in the catalog, which are determined verbally and therefore rounded mainly in units of 10 or 15 min; this favors particular values of $\tau$ and there fore of $h$ (2/3, 4/5, 1/2, etc.). We can correct this effect by the addition of a uniform random value between -5 min and 5 min to each occurrence time $t_i$, then the peaks in the distribution of $h$ disappear and its shape gets closer to a uniform one; however, the difference is significant. We have verified that the difference is not due to the random addition we have performed: simulation of a non-stationary Poisson process where the occurrences are rounded in intervals of 10 min yields a perfect uniform distribution when this discretization is corrected by the uniform random addition just explained (Fig. \[bi\]). In consequence, this model does not seem suitable for fire occurrence, and although the instantaneous distributions are close to exponential (Fig. \[Dtinstant\]), this is not a sufficient condition to have a nonstationary Poisson process, as the absence of correlations is equally important for it.
If we reject the nonstationary Poisson process with independent sizes as a model of fire occurrence, the only way to get a scaling law for the instantaneous process is by means of orchestrated correlations between sizes and occurrence times [@Corral_prl.2005]. In order to establish the existence of such correlations we proceed to study conditional size distributions, defined as in Eq. (\[distribution\]) but with an additional condition for the computation of the probability. We consider $D(s \,|\, s_{pre} \ge s_c')$, which accounts for the size of the events for which the size of the immediate previous-in-time event, $s_{pre} $, is above a given threshold $s_c'$. The results in Fig. \[Dscond\_sold\] show that an increase of $s_c'$ triggers a greater proportion of large fires, i.e., large fires are followed by large fires. The dependence of a fire size on the previous size is small but significant, unlike to what happens for earthquakes, where correlation between their magnitudes has not been detected [@Corral_comment; @Corral_tectono] (nevertheless, for an alternative view see Ref. [@Lippiello]). On the other hand, the dependence of waiting times on the size of the event defining the starting of the waiting period can be measured by $D(\tau ; s_c \,|\, s_{pre} \ge s_c')$ showing how large fires cause a decrease in the number of long recurrence times, i.e., those fires tend to be closer in time to the next fires. The effect is again small, but clearly detectable, and in this case has a counterpart for earthquakes, where the Omori law for aftershocks implies the same behavior.
But the correlations between fires are not only with the previous event; its range can be quantified by means of the following auto-correlation function, $$c(j; s_c)=
\left \langle (\log s_i-\bar \ell)(\log s_{i+j}-\bar \ell)
\right \rangle \sigma_\ell^{-2},$$ where $\bar \ell$ is the arithmetic mean of the logarithm of the size (i.e., the logarithm of the geometric mean of the size), and $\sigma_\ell$ is the standard deviation of the logarithm; both $\ell$ and $\sigma_\ell$ depend on $s_c$. Notice that although the process is not stationary, the stationarity of the size gives sense to the autocorrelation function defined in this way. The results for this function are shown in Fig. \[corr\], and compared with the same correlation function calculated for a reshuffled version of the catalog, for which the size of the events are randomly permuted, breaking the correlations between them (which should yield an autocorrelation function fluctuation around zero). The conclusion is that positive correlations extend significantly beyond several hundreds of events (for events of size larger than 1 ha).
More clear is the behavior of the autocorrelation as a function of time; as the process is not stationary both functions are not equivalent. We define $$%%c(\Delta; s_c)= \langle (s(t)-\bar s)(s(t+\Delta)-\bar s)\rangle
%%\sigma_s^{-2},
\tilde c(\Delta; s_c)=
\left \langle (\log s(t)-\bar \ell)(\log s(t+\Delta)-\bar \ell)
\right \rangle \sigma_\ell^{-2},$$ where $s(t)$ denotes the size of the fire that happens at time $t$ (we slightly change notation, for convenience). The average is taken over all times $t$ and $t+\Delta$ for which there are fires, this yields the results of Fig. \[corr\]. The correlation is again positive, but larger in this case, suggesting that real time is a better variable to describe the evolution of correlations, which extend for about 10000 min, i.e., roughly 1 week. It is likely that these correlations are mediated through the meteorological conditions.
In summary, the dynamics of forest-fire occurrence shows a complex scale-invariant structure at any time, modulated by seasonal and daily variations and orchestrated by means of broad-range correlations.
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---
author:
- 'Matias Vera, Sol Alonso'
- Georgina Coldwell
date: 'Received xxx; accepted xxx'
title: Effect of bars on the galaxy properties
---
[ With the aim of assessing the effects of bars on disc galaxy properties, we present an analysis of different characteristics of spiral galaxies with strong, weak and without bars.]{} [ We identified barred galaxies from the Sloan Digital Sky Survey (SDSS). By visual inspection of SDSS images we classified the face-on spiral galaxies brighter than $g < 16.5$ mag into strong-bar, weak-bar and unbarred. With the goal of providing an appropiate quantification of the influence of bars on galaxy properties, we also constructed a suitable control sample of unbarred galaxies with similar redshift, magnitude, morphology, bulge sizes, and local density environment distributions to that of barred galaxies. ]{} [ We found 522 strong-barred and 770 weak-barred galaxies, which represent a bar fraction of 25.82$\%$, with respect to the full sample of spiral galaxies, in good agreement with several previous studies. We also found that strong-barred galaxies show lower efficient in star formation activity and older stellar populations (as derived with the $D_{n}(4000)$ spectral index), with respect to weak-barred and unbarred spirals from the control sample. In addition, there is a significant excess of strong barred galaxies with red colors. The color-color and color-magnitude diagrams show that unbarred and weak-barred galaxies are more extended towards the blue zone, while strong-barred disc objects are mostly grouped in the red region. Strong barred galaxies present an important excess of high metallicity values, compared to unbarred and weak-barred disc objects, which show similar $12+log\left(O/H\right)$ distributions. Regarding the mass-metallicity relation, we found that weak-barred and unbarred galaxies are fitted by similar curves, while strong-barred ones show a curve which falls abruptly, with more significance in the range of low stellar masses ($log(M_{*}/M_{\sun}) < 10.0$). These results would indicate that prominent bars produced an accelerating effect on the gas processing, reflected in the significant changes in the physical properties of the host galaxies. ]{}
Introduction
============
Galactic bars are structures observed in a significant fraction of spiral galaxies and are believed to have an important role in the dynamical evolution of their hosts. Several simulations show that bars can efficiently transport gas from the outer zones to the innermost central regions of the barred galaxies [@wei85; @deb98; @ath03]. By interaction with the edges of the bars, the gas clouds suffer shocks producing angular momentum losses and allowing a flow of material toward central kiloparcec scale [@SBF90]. Moreover, some works show that bars can be destroyed by a large central mass concentration (Roberts et al. 1979; Norman et al. 1996; Sellwood & Moore 1999; Athanassoula et al. 2005). This finding indicates that currently non-barred disc galaxies possibly had a bar in the past (Kormendy & Kennicutt 2004), and also, that bars may be recurrent in the galaxy life (Bournaud & Combes 2002; Berentzen et al. 2004; Gadotti & Souza 2006). So, in this context bars formed at different times, and with different conditions, might be present in the barred disc galaxies.
Due to the high efficiency of gas inflow, galactic bars can alter several properties of disc galaxies on relatively short timescales. In this sence, the presence of bars can affect the star formation activity, stellar population, colors, modify the galactic structure [@atha83; @buta96] and change the chemical composition [@comb93; @martin95], contributing to the evolution process of the host galaxies (Ellison et al. 2011, Zhou et al. 2015). In addition, the inflow processes have also been considered an efficient mechanism for trigger active galactic nuclei (AGN) [@comb93; @cor03; @alonso13; @alonso14], and to form bulges or pseudo-bulges (e.g., Combes & Sanders 1981; Kormendy & Kennicutt 2004; Debattista et al. 2005, 2006; Martinez-Valpuesta et al. 2006; Méndez-Abreu et al. 2008; Aguerri et al. 2009).
With respect to the relation between bars and host galaxy colors from statistical analysis, different studies show diverse results. Several observational works found that the bar fraction, $f_{bar}$, is higher in later-type spiral galaxies that are bluer and less concentrated systems (e.g. Barazza et al. 2008, Aguerri et al. 2009). However, other studies displayed an excess of barred galaxies with redder colors from different samples. [@master10a] found a high fraction of bars in passive red spiral galaxies for a sample obtained from the Galaxy Zoo catalog [@lintott11]. In addition, [@oh12] showed that a significant number of barred galaxies are redder than their counterparts of unbarred spiral galaxies. Recently, in our previous works (Alonso et al. 2013, 2014) we found an excess of red colors in spiral barred AGN with respect to unbarred active galaxies in a suitable control sample.
The role of the bars on star formation and metallicity have been the subject of several works, showing unclear conclusions. Many studies found that bars enhanced the star formation rates (SFR) in spiral galaxies compared with unbarred ones (e.g. Hawarden et al. 1986; Devereux 1987; Hummel 1990), while other works show that bars do not guarantee increase in star formation activity (Pompea & Rieke 1990; Martinet & Friedli 1997; Chapelon et al 1999). In the similar way, different authors found diverse results in the metalicity studies in barred galaxies with respect to their unbarred counterparts (e.g. Vila$-$Costas & Edmunds 1992; Oey & Kennicutt 1993; Martin & Roy 1994; Zaritsky et al 1994; Considere et al. 2000, Ellison et al. 2011). More recently, by using data from the CALIFA survey, Sánchez-Blázquez et al (2014) performed a comparative study of the stellar metallicity and age gradients in a sample of 62 spiral galaxies finding no differences with the presence or absence of bars.
The discrepancy in the results of the bar effects on SFR and metallicity may depend on the host galaxy morphology (Huang et al. 1996; Ho et al. 1997; James et al. 2009) and may be also due to the length$/$strength of the bar (Elmegreen & Elmegreen 1985, 1989; Erwin 2004; Menendez-Delmestre et al. 2007). Similarly, some studies (e.g. Athanassoula 1992; Friedli et al. 1994; Friedli & Benz 1995) from numerical simulations found such trends, showing that bar strength is related to the efficiency and quantity of gas inflow, and therefore with the star formation activity and metallicity gradients.
Furthermore, different authors have proposed diverse ways to build control samples from unbarred galaxies, used to obtain conclusions from comparative studies, and so the discrepancy in the results could be due to a biased selection of these samples. In this direction, [@perez09] found that a control sample for interacting galaxies should be selected matching, at least, redshift, morphology, stellar masses, and local density environment. This is also a suitable criteria for building control samples of barred galaxies (Alonso et al. 2013, 2014). Motivated by these finds, in this paper we conducted a detailed analysis of the effect of bars on host galaxy properties, with respect to the unbarred ones by studying different characteristics (e.g. color, stellar population, star formation activity, metallicity) with the aim of assessing whether bar structure in discs play a significant role in modifying galaxy properties, and how is this effect.
This paper is structured as follows. Section 2 presents the procedure used to construct the catalog of barred galaxies from Sloan Digital Sky Survey (SDSS), the classification of the bar structures and the control sample selection criteria. In section 3, we explore different properties of the barred spirals, in comparison with unbarred galaxies obtained from a suitable control sample. We analize in details the influence of bars on star formation activity, stellar population, color indexes and metallicity in host spiral galaxies, with respect to unbarred ones. Finally, section 4 summarizes our main results and conclusions. The adopted cosmology through this paper is $\Omega = 0.3$, $\Omega_{\lambda} = 0.7$, and $H_0 = 100~ \kms \rm Mpc$.
Catalog of barred galaxies
==========================
The analysis of this paper is based on the Sloan Digital Sky Server Data Release 7 (SDSS-DR7, Abazajian et al. 2009). It uses a 2.5m telescope to get photometric and spectroscopic data which cover near one-quarter of the celestial sphere and collect spectra of more than one million objects. DR7 includes 11663 square degrees of sky imaged in five wave-bands (u,g,r,i,z) containing photometric parameter of 357 million objects. The main galaxy sample, which contains about 900000 galaxies with measured spectra and photometry, is essentially a magnitude-limited spectroscopic sample $r_{lim}<17.77$ (Petrosian magnitude), and most of galaxies span a redshift range $0<z<0.25$ with a mean readshift of 0.1 (Strauss et al. 2002). For this work, several physical properties of galaxies have been derived and published for the SDSS-DR7 galaxies: gas-phase metallicities, stellar masses, current total and specific star-formation rates, concentration index, etc. (Brinchmann et al. 2004; Tremonti et al. 2004; Blanton et al. 2005). These data were obtained from the MPA/JHU[^1] and the NYU[^2] added-values catalogs.
With the aim of obtaining barred galaxies, we first cross-correlated the SDSS galaxies with the spiral objects obtained from the Galaxy Zoo catalog [@lintott11], which comprises a morphological classification of nearly 900000 galaxies drawn from the SDSS. In order to cover a wide coverage area, this survey is contributed by hundreds of thousands of volunteers, however due to the large number of classifiers it becomes complex to maintain an unified criteria. They define different categories (e.i. elliptical, spiral, merger, uncertain, etc) and give the fraction of votes in each category. In this study, we selected galaxies that were classified as spiral objects by the Galaxy Zoo team with a fraction of votes $>0.6$. Taking this into account, a low fraction of galaxies with non spiral morphological types could be included. In adittion, we exclude AGN objects (Coldwell et al. 2014), which could affect our interpretation of the results due to contributions from their emission line spectral features. Furthermore, as bars are objects which lie on the host disc plane (Sellwood & Wilkinson 1993) and visual inspection become less efficient while inclination increases, we applied another restriction on the ellipticity of the objects, selecting galaxies with axial ratio $b/a>0.4$. We also restricted the spiral edge-on galaxy sample in redshift ($z<0.06$) and imposed a magnitude cut such as the extinction corrected SDSS $g-$mag is brighter than 16.5. With these restrictions, our sample comprises 6771 galaxies, and therefore, we can make a plausible visual inspection of a good set of objects.
Classification
--------------
We proceeded to select barred galaxies by visual inspection. For this task, we used the $g$ + $r$ + $i$ combined color images, obtained from online SDSS-DR7 Image Tool[^3]. Then, by a detailed visual examination we classified the galaxies into four groups based on the presence of bars, taking into account their relative light contribution and length with respect to the structural properties of the host galaxies. We can summarize the classification as follows: 522 strong-barred (where the size of the bars have, at least, a 30% of their host galaxy sizes), 770 weak-barred (the size of the bars is smaller than 30% the size of their host galaxies), 688 ambiguous-barred (objects for which it is difficult to decide whether to have a bar or not), and 3711 non-barred galaxies. We also found some galaxies with elliptical and irregular appearance which were removed from our sample. The details of the classification are listed in Table 1, and Fig. 1 shows examples of each galaxy type studied in this work.
Therefore, the final catalogue of the barred galaxies has been constructed with 1292 spiral objects with strong and weak bars (we excluded ambiguous-barred galaxies that did not completely match the bar classification). This represents a fraction of 25.82$\%$ with respect to the sample of 5003 spiral galaxies with clear classification. In the same direction, several works carried out by means of visual inspection of different galaxy samples (e.g. the RC3 and UGC, [@deVau91; @nilson73; @marinova09; @alonso13]) finding a bar fraction of 25-30$\%$ in agreement with this work.
Galaxy type N$^0$ of objects Percentages
----------------------------------- ------------------ -------------
Non-barred galaxies 3711 54.80%
Strong-barred galaxies 522 7.71%
Weak-barred galaxies 770 11.37%
Ambiguous-barred galaxies 688 10.16%
Elliptical and irregular galaxies 1080 15.96%
Total sample 6771 100.0 %
: Galaxy classification, numbers and percentages of objects.
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Control sample
--------------
To provide a suitable quantification of the impact of bars on the host galaxy properties, we obtained a reliable control sample of unbarred disc objects following Alonso et al. (2013).
We constructed a control sample of galaxies using a Monte Carlo algorithm that selected objects classified as unbarred galaxies in the previous section with similar redshift and extinction and $K-$corrected [@blanton03] absolute $r-$band Petrosian magnitude distributions of the barred galaxy sample (see panels $a$ and $b$ in Fig. 2).
We have also considered unbarred galaxies in the control sample with similar concentration index ($C$) [^4] distribution to that of the barred catalog to obtain a similar bulge to disk ratio in both samples (panel $c$ in Fig.2). Furthermore, we restricted the control unbarred spirals to match the $fracdeV$ parameter defined as the fraction of the light fit by a de Vaucouleurs profile over an exponential profile, where a pure de Vaucouleurs elliptical should have $fracdeV = 1$, and a pure exponential disc spiral will have $fracdeV = 0$ (Masters et al. 2010a). Thus this index estimates the bulges surface brigthness distribution, so that it is a good indicator of its size in galaxies with disk morphology (Kuehn 2005, Bernardi et al. 2010, Skibba et al. 2012) (see panel $d$ in Fig. 2). Therefore, the possible differences in the results are associated with the presence of bars and not with the difference in the bulge prominences neither with the global galaxy morphology.
In addition, with the aim to obtain galaxies in the same density regions, we also selected objects without bars with similar distribution of the local density environment parameter ($\Sigma_5$) than that of barred galaxies. This parameter is calculated through the projected distance $d$ to the fifth nearest neighbor galaxy, $\Sigma_5 = 5/(\pi d^2)$, brighter than $M_r = -20.5$ and within a radial velocity difference of less than 1000 km $s^{-1}$ [@balo04]. Fig.2 (panel $e$) shows the distributions of the $log(\Sigma_5)$ for both samples.
With these restrictions we obtained a control sample of 2205 unbarred spiral objects with similar redshift, brightness, morphology, bulge prominence and local environment to that of barred galaxies. Then, any difference in the galaxy properties is associated only with the presence of the bar, and consequently, comparing the results, we will estimate the real difference between barred and unbarred galaxies, unveiling the effect of this structure on the disc galaxy features.
![Distributions of redshift, luminosities, concentration index, bulge size indicator, and local density parameter, $z$, $M_r$, $C$, $fracdeV$, and $log(\Sigma_5)$ ($a$, $b$, $c$, $d$, and $e$ panels), for barred galaxies (solid lines) and galaxies without bars in the control sample (dashed lines).[]{data-label="cont"}](histzMrFdeVBNewMatiasN.ps){width="90mm" height="120mm"}
Galaxy properties
=================
Different studies have shown that bars can induce several processes that modify many properties of the galaxies (Sellwood & Wilkinson 1993, Combes et al. 1993, Zaritzky et al. 1994, Lee et al. 2012, Oh et al. 2012, Alonso et al, 2013, 2014). However, there are still many questions about how the properties are modified by the presence of a bar structure in the disc of the spiral galaxies.
In this section we explore the effect of bars, with different structural strength, on the stellar population, star formation activity, colors and metallicity of the host galaxies, in comparison with the unbarred objects in a suitable control sample, obtained from the previous section. This analysis may help to deepen our understanding of this issue which has been explored by different authors under diverse approaches.
Star formation and stellar population
-------------------------------------
With the aim of assesing the effect of bars on the star formation and stellar age population, in the following analysis we use the specific star formation rate parameter, $log\left(SFR/M_{*}\right)$, as a good indicator of the star formation activity in non-AGN galaxies. It is estimated as a function of the $H\alpha$ line luminosity, and normalized using stellar masses (Brinchmann et al. 2004). We also employ the spectral index $D_{n}(4000)$ [@kauff03], that estimates the age of stellar populations. It is calculated from the spectral discontinuity occurring at 4000 $\AA $, produced by an accumulation of a large number of spectral lines in a narrow region of the spectrum, specially important in old stars. In this analysis, we use the $D_n(4000)$ definition obtained by [@balo99], as the ratio of the average flux densities in the narrow continuum bands (3850-3950 $\r{A}$ and 4000-4100 $\r{A}$). The spectroscopic data in SDSS are obtained within the aperture of a spectroscopic fiber (3 arcsec in diameter). This corresponds to a typical physical size of the fiber of $\approx$ 2kpc, at the mean redshift of our sample (z $\approx$ 0.03). Nevertheless, the SDSS spectroscopic parameters (e.g., SFR, metallicity) outside of the fiber are estimated following different methodologies, using the galaxy photometry (see for details, Kauffmann et al. 2003, Tremonti at al. 2004, Brinchmann et al. 2004).
In Fig. 3 (upper panel), we show the star formation activity distributions for each classified galaxy type. It can be clearly appreciated that strong-barred galaxies show a significant excess toward lower $log\left(SFR/M_{*}\right)$ values, with respect to weak-barred and unbarred objects in the control sample. In addition, a remarkable bimodality can be observed in galaxies with strong bar prominences. This behaviour clearly shows that there is an excess of strong barred galaxies with low star formation activity, indicating that the amount of gas may be not sufficient, after consumed by star formation in the previous process, during prior stages of the galaxy life. The value located near $log\left(SFR/M_{*}\right)\approx-11.3$ divides both distributions. Furthermore, weak-barred and unbarred galaxies show similar distribution of specific star formation rate.
Moreover, while several barred galaxies have more concentrated CO in the central region, some early type disks have a lack of CO in this region (e.g. Sheth et al. 2005; Sakamoto et al. 1999). In this sense, Sheth et al. (2005), using CO observations of the six barred spirals, finding that their sample of barred galaxies have very little molecular gas in the central regions of the galaxies. It could indicate that the gas was consumed in star formation processes at an earlier stage of the galaxy evolution. More recently, James & Percival (2016) studied the central regions of four barred galaxies, showing that the star formation activity is observed inhibited within each of these galaxies, suggesting that star formation appears to have been suppressed by the bar.
Lower panel in Fig. 3 shows the distributions of the spectral index $D_{n}(4000)$ for spiral galaxies in the different samples. As we can see, strong-barred galaxies show an important excess toward higher $D_{n}(4000)$ values in comparison with weak-barred and unbarred spiral objects, indicating that strong bars tend to exist in host galaxies with older stellar population. This finding could suggest that: i) strong bars preferentially formed in galaxies with old stellar population or ii) strong bars formed long time ago and thus the stellar population of galaxies became old. In this direction, Sheth et al. (2008) show that bars were formed first in massive and luminous galaxies, and later less massive and bluer systems acquired the majority of their bars. Therefore, the strong-bar excess toward higher $D_{n}(4000)$ values could represent bars formed in the first stage, which have grown together with their hosts. In addition, different authors (e.g., Weinzirl et al. 2009, Laurikainen et al. 2007, 2009) show that, in general, strong bars are more frequently found in early type disk galaxies which are massive than late type objects. Similar to that observed in the star formation distributions, strong-barred galaxies show a bimodality in the stellar population around $D_{n}(4000)\approx1.8$. Table 2 quantifies the percentages of galaxies in our different samples with efficient star formation activity and young stellar population. In the similar way, Sanchez-Blazquez et al. (2011) found old stellar population in four barred galaxies. This is in agreement with the findings of James & Percival (2016) from the spectroscopic analysis, in four different barred disc objects. Besides, old stellar populations have also been found in galaxies with bars by several studies (Perez, Sanchez-Blazquez & Zurita 2007; Perez, Sanchez-Blazquez & Zurita 2009; de Lorenzo-Caceres et al. 2012; de Lorenzo-Caceres, Falcon-Barroso, & Vazdekis 2013), in agreement with our result.
In addition, we divide strong barred galaxies into two subsamples: a group of galaxies that belong to the minor peak of the distributions of both $log\left(SFR/M_{*}\right)$ and $D_{n}(4000)$ (Group 1, G1) and a group of strong barred galaxies belonging to the major peak distributions in Fig. 3 (Group 2, G2). The limits to separate both groups are: $log\left(SFR/M_{*}\right) = -11.3$ and $D_{n}(4000) = 1.8$. In this way, we can reveal whether the differences in galaxy properties are mainly driven from having different star formation and stellar population or hosting a bar.
Type $log(SFR/M_{*})>-11.3$ $D_{n}(4000)<1.8$
--------------- ------------------------ -------------------
Strong-barred 79% 77%
Weak-barred 97% 97%
Unbarred 93% 91%
: Percentages of spiral galaxies with strong, weak and without bars with $log(SFR/M_{*})>-11.3$ and $D_{n}(4000)<1.8$.
We checked disc galaxies in the smaller peak in the previous bimodal distributions with older stellar population and low efficient in star formation activity. We noticed that there are 97 strong-barred lenticular galaxies. In Fig. 4 some examples can be seen. Therefore, an important fraction (about the 20%) of strong-barred galaxies are SB0 morphological types. It could be indicating that usually, when a lenticular galaxy contains a bar, it is an strong structure. These results are consistent with those obtained by Aguerri et al. (2009), who found that bar length (normalised by the galaxy size) in lenticular galaxies tend to be longer than ones in late-types objects. Furthermore, our finding agrees with Laurikainen et al. (2009), who found that prominent bars (which measures are calculated using the maximum m$=$2 Fourier density amplitude) are more common in lenticular galaxies than weak bars (considering that lenticular galaxies in their work are sub-divided in S0 and S0/a types, and that a medium amplitude is defined between strong and weak structures).
  
In order to understand the behaviour of star formation and stellar populations of spirals with strong, weak and without bars, with respect to the stellar masses and the morphology of the host galaxy, we have analysed $log\left(SFR/M_{*}\right)$ and $D_{n}(4000)$, as a function of $log(M_{*})$ and concentrarion index, $C$. Fig. 5 shows the mean $log\left(SFR/M_{*}\right)$ and $D_{n}(4000)$ as a function of the stellar mass. Errors were estimated by applying the bootstrap resampling technique in all figures (Barrow et al. 1984). For the Fig. 5 and 6 we considered barred galaxies that belong to the major peaks in Fig. 3, with the aim to exclude the fraction of the lenticular galaxies. As can be seen, star formation activity decreses towards higher stellar masses and, in the same direction, young stellar population diminish with $log (M_{*})$. Clearly host galaxies with strong bars show a systematically lower efficient of star formation activity and older stellar population in all stellar mass bins, with respect to the other samples studied in this paper. Furthermore, disc objects in the control sample show efficient activity in star formation and younger stellar population. In addition, the trends for weak barred galaxies are observed between strong barred objects and control sample.
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In addition, in Fig. 6 we can see clearly that strong barred galaxies become less efficient star formers while increases the $C$ index. On the other hand, for unbarred and weak-barred galaxies $log\left(SFR/M_{*}\right)$ remains almost constant for the all $C$ values. Moreover, strong-barred objects show older stellar population towards galaxies with earlier morphology. Unbarred spirals show younger stellar age population and higher SFR values, for different morphological types, while weak barred galaxies present intermediate tendencies. This fact could be indicating that bars tend to modify quickly their host galaxy properties (i.e., bars could accelerate the gas processing), when they have became prominent enough.
In the same direction, Masters et al. (2012) found a significantly lower bar fraction on the gas-rich galaxies with respect to gas-poor disc objects. The authors suggest that in gas-rich discs the bars funnelling the gas into the central region of the galaxy. Then, this material can turned into molecular gas and eventually trigger star formation acticity (e.g. Ho et al. 1997; Sheth et al. 2005; Ellison et al. 2011; Lee et al. 2012). So, this process could accelerate the gas consumption, ceasing the formation of the new stars by removing gas from the outer regions of the disk and become red host galaxies. These mechanisms may indicate different evolutionary stages of the bars in spiral galaxies, which depend on the strength of the bar structure. In this context, Jogee et al. (2005), based on the properties of circumnuclear gas and star formation, proposed a possible scenario of bar-driven dynamical evolution of the galaxies. In the first phases, large amounts of gas are transported by the bars towards the galactic central regions, along with an efficient star formation activity. Then, in the poststarburst phase, the gas has been consumed by circumnuclear starburst, showing low SFR (Sheth et al. 2005). In this context, we argue that during these different stages, the length/strength of the bars and host galaxy properties are modified.
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Colors
------
With the aim to explore the colors of the barred galaxies with different structural strength, in Fig. 7 we illustrate the $\left(u-r\right)$ and $\left(g-r\right)$ color distributions for different galaxy types classified previously. Strong-barred objects show a clear excess of redder colors, while unbarred and weak-barred galaxies have similar ($u-r$) and ($g-r$) distributions. In particular, strong bars in the Group 1 show a significant fraction of host galaxies with extremely red colors ($u - r > 2.0$ and $g - r > 0.7$). This finding could be reflected low efficiency in star formation activity, old stellar population and earlier morphological galaxy types. In the same direction, Masters et al. (2011) found that red spiral galaxies have a higher fraction of bars than that in the blue ones, in a sample obtained from Galaxy Zoo catalogue. Similarly, Oh et al. (2012) and Alonso et al. (2013, 2014) observed the same behaviour for AGN barred hosts. Therefore, it seems that bars play an important role in the modification of the host galaxy colors, but only when this structure has become prominent enough.
In addition, in Fig. 8 we present color-magnitude diagram for the different studied galaxy types. It can be seen that strong-barred galaxies are principally concentrated in the top area (red color region), while unbarred and weak-barred objects are more uniformly distributed. It is clear that galaxies with strong bars in the Group 1 are located in the redder region of the color-magnitude diagram with respect to galaxies from the other samples. We have also plotted the color fit, developed by Masters et al. (2010b), which separate blue and red populations ($(g-r) = 0.67-0.02\left[M_{r}+22\right]$). As we can see, galaxies with strong bars are mostly located above the line, while unbarred and weak-barred objects lies mostly under the line (blue color region). This findings could indicate that, at the same magnitude, strong barred galaxies are usually redder objects with respect to the other samples studied in this work. Table 3 quantifies the percentage of different galaxy types located in the red color region.
Galaxies $(g-r)\geq0.67-0.02\,(M_{r}+22)$
------------------ ----------------------------------
Unbarred 34.82%
Weak-barred 33.50%
Strong-barred G1 100.00%
Strong-barred G2 53.50%
: Percentages of galaxies located above the line fitted by Masters et al. (2010b).
Moreover, Fig. 9 illustrates color-color diagram for unbarred, weak- and strong-barred galaxies in both groups. We can note that the three galaxy types lie on a same straight line, although strong-barred ones are mostly grouped in the red region of the diagram (mainly those belonging to G1), while the other types show more dispersion and are more extended towards the blue region. This configuration could be indicating an evolutive relation between the different classified galaxy types. In the first time unbarred disc galaxies could start forming a bar in its central region, from the instability in the disc. This bar would become increasingly prominent while it consumes gas from the disc. In the beginning, the bar would not be able to modify significantly the host galaxy characteristics. Then, when it reaches a strong prominence, it could start to affect the host galaxy, producing an ascent in the color-color diagram, inducing many important changes in the host galaxy properties.
Fig. 10 shows the mean $(g-r)$ and $(u-r)$ colors as a function of the concentration index, $C$. It is clear that red objects increase towards the more concentrated galaxies, for the different samples. This result is consistent with expectations, since galaxies with higher values of the concentration index are related to the bulge type morphology, and lower concentration objects to spiral galaxies. On the other hand, galaxies by evolving passively can become red without increasing concentration parameter, and also some galaxies can increase $C$ without becoming red (e.g. by merging events). It can be also seen that strong-barred galaxies are redder, for the whole range of the $C$ parameter, compared to their counterparts of weak-barred objects and unbarred hosts in the control sample. We notice that this tendency is clearly more significant in galaxies with strong bars that belong to Group 1. In this context, the results could indicate that galaxies with strong bars could become redder than their counterparts of unbarred and weak-barred disc objects. This fact also supports the idea that intense bars accelerate the gas processing and it is reflected in a reddened population.
Metallicity
-----------
The chemical features of the galaxies can store fossil records of their history of formation since they are the result of diverse physical mechanisms acting at different stages of evolution (Freeman & Bland-Hawthorn 2002). In this sense, metallicity is one of the fundamental physical properties of galaxies, which principally reflects the amount of gas reprocessed by the stars. In addition, it depends strongly on the evolutive state of a galaxy so it is a good indicator of its age. In this analysis, as metallicity parameter, we used $12+log\left(O/H\right)$ which represents the ratio between oxygen and hydrogen abundances (Tremonti et al. 2004). We found that $\approx$ 80$\%$ of the objects in our samples have $12+log\left(O/H\right)$ measurement, and there is a null fraction of strong barred galaxies in G1 with this parameter. Therefore, in this section, the sample of the strong-barred galaxies belong to the Group 2.
The influence of the bars in the metallicity can be seen in Fig. 11, in the histograms of the $12+log\left(O/H\right)$ for disc galaxies with strong, weak and without bars. We also define the low and high metallicity galaxies by selecting two ranges of $12+log\left(O/H\right)$ values to have equal number of objects in the control sample. This threshold is $12+log\left(O/H\right) = 9.05$. From this figure, it can be appreciated that strong-barred galaxies present an important excess towards high metallicity values, while unbarred and weak-barred objects show similar distributions. Table 4 quantifies the excess of disc objects with high metallicity for the different samples. This result supports the previous ones, meaning that strong-barred galaxies show low star formation activity, with older/redder stellar populations and higher gas metallicity than weak-barred and unbarred spiral galaxies. From the chemodynamical simulation studies, Martel et al. (2013) found that the chemical evolution observed within the central region of the disc galaxies depends critically of the prominence of the bar, which evolves strongly with time.
Galaxy Type $12+log\left(O/H\right)>9.05$
--------------- -------------------------------
Strong-barred 64.1%
Weak-barred 51.4%
Unbarred 50.0%
: Fraction of galaxies with metallicites higher than 9.05.
We also study the mass-metallicity relation (MZR; Lequeux et al.1979) of barred galaxies as a tool to study the effects of bars on the galaxy metallicity. In the local Universe, Tremonti et al. (2004) have confirmed the dependence of metallicity on stellar mass with high statistical signal. Erb et al. (2006) has extended the study to high redshift finding a similar correlation, although displaced to lower metallicity (Maiolino et al. 2007). Fig. 12 shows the mass-metallicity relation for each galaxy type. We also compare these with the results obtained of Tremonti et al. (2004), who studied mass-metallicity relation for a sample of 53000 star-forming galaxies from SDSS. It can be seen that our galaxy samples are more metallic, according to the results of Ellison et al. (2008, 2011). These authors found that mass-metallicity relation is modulated by the star formation rate. They suggest that the metal enhancement without an accompanying increase in the star formation activity may be due to a short lived phase of bar-triggered star formation in the past. However, the most interesting point is that strong-barred objects show a relation which falls abruptly, with respect to ones to the other samples. This tendency is more significant in low stellar mass galaxies. Nevertheless, for strong barred galaxies with $log(M_{*}/M_{\sun}) > 10.0$ there is not a clear fall in the metallicity, and also similar trends are observed for different samples. This fact could be indicating that prominent bars produce an accelerating effect on the gas processing and hence on the host galaxy evolution towards earlier morphological types.
In addition, it can be seen that weak-barred and unbarred galaxies do not show significant differences in the metallicity. In a same direction, this behavior is reflected in the other galaxy properties: colors, star formation activity and stellar population. These findings could be indicating that weak bars do not produce noticeable changes in the galaxy properties and the effects on the physical characteristics begin to be felt when bar became prominent enough.
Summary and conclusions
=======================
We have performed a statistical study of physical properties of barred galaxies in contrast with unbarred ones. Our analysis is based on a sample derived from SDSS release. We complemented these data with a naked-eye classification of a sample of face-on spiral galaxies brigther than $g = 16.5$ mag, based on the presence of the bar, and taking into account the strength of the bar with respect to the structural properties of the host galaxies. With the purpose of providing an appropriate quantification of the effects of bars on host galaxies, we also constructed a suitable control sample of unbarred galaxies with the same redshift, $r-band$ magnitude, concentration index, bulge size parameter, and local environment distributions, following Alonso et al. (2013).
We can summarize the principal results of our analysis in the following conclusions:
1. We found 522 strong-barred, 770 weak-barred, and 3711 non-barred galaxies, which represents a bar fraction of 25.82$\%$, with respect to the full sample of spiral galaxies. This fraction agrees with previous studies found by other authors by visual inspection of different galaxy samples from optical images (Nilson 1973, de Vaucouleurs et al. 1991, Marinova et al. 2009, Masters et al. 2010b, Alonso et al. 2013).
2. We observed that strong-barred galaxies show lower star formation activity and older stellar populations, with respect to weak-barred and unbarred disc objects. We also found a significant fraction ($\approx20\%$) of strong-barred galaxies with older stellar population and low efficient star formation rate that have lenticular morphology (SB0 type). This result shows that, when S0 galaxies contain bar, it is usually a strong structure, in agreement with Aguerri et al. (2009).
3. We also studied the star formation activity and the age of stellar populations of galaxies as a function of $log(M_{*})$ and concentration index, $C$, in barred galaxies with weak/strong bars, and in the control sample. We found that strong-barred galaxies show a systematically lesser efficient star formation activity and older stellar population for different stellar mass bins, and towards earlier morphology, with respect to the other samples of galaxies with weak and without bars.
4. We examined the color distributions of different samples studied in this work, and we found that there is a significant excess of strong barred host galaxies with red colors. We also found that galaxies with strong bars are redder, for the whole concentration index range, with respect to their counterparts of weak-barred and unbarred disc objects. In particular, for strong barred galaxies that belong to the minor peaks of the star formation and stellar population distributions (see Fig. 3) these tendencies are clearly more significant, showing a high fraction of host galaxies with extremely red colors. These findings suggest that bar perturbations have a considerable effect in modifying galaxy colors in the host galaxies, producing an acceleration of the gas processing, when bar became prominent enough.
5. The color-magnitude and color-color diagrams show that strong-barred galaxies are mostly grouped in the red region, while unbarred and weak-barred objects are more extended to the blue region. The positions in the color diagrams, could indicate the existence of an evolutive relation between the different considered galaxy type. In this scenario, an unbarred galaxy would begin to form a bar as a consequence of a gravitational disturbance in the disk. Then, matter would fall down to the center of the galaxy, making place to a weak bar which would become gradually more prominent while the inflow accumulates material in the center. At first, the weak bar would not be able to alter significantly the host characteristics, but then, when this structure is strong enough, it could affect significantly the galaxy properties.
6. We also explore the metallicity, which principally reflects the amount of gas reprocessed by the stars. It shows that galaxies with strong bars present an important excess towards high metallicity values, while unbarred and weak-barred disc objects have similar distributions. The mass-metallicity relation reflects that although unbarred and weak-barred galaxies are fitted by similar curves, strong-barred ones show a curve which falls abruptly. It is more important in low stellar mass galaxies ($log(M_{*}/M_{\sun}) < 10.0$). This behaviour could be suggesting that prominent bars produce an accelerating effect on the gas processing, producing significant changes in the physical properties, also reflected in the evolutionary stages of the host galaxies.
This work was partially supported by the Consejo Nacional de Investigaciones Científicas y Técnicas and the Secretaría de Ciencia y Técnica de la Universidad Nacional de San Juan.
Funding for the SDSS has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The participating institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
Abazajian K. N., Adelman-McCarthy J. K., Agüeros M. A., Allam S. S., Allende Prieto C., An D., Anderson K. S. J., Anderson S. F., , 2009, [[*ApJS*]{}]{}, 182, 543.
Aguerri, J. A. L., & Gonzalez-Garcia, A. C. 2009, A&A, 494, 891.
Alonso S., Coldwell G. & Lambas D.G., 2013, A&A, 549, 141.
Alonso S., Coldwell G. & Lambas D.G., 2014, A&A, 572, 86.
Athanassoula, E. 1983, IAU Symp., 100, 243.
Athanassoula, E. 1992, MNRAS, 259, 345.
Athanassoula E., 2003, MNRAS, 341, 1179.
Athanassoula E., Lambert J. & Dehnen W., 2005, [[*MNRAS*]{}]{}, 363, 496.
Balogh M., Morris, S. L., Yee, H. K. C., Carlberg, R. G., Ellingson, E., 1999, [[*ApJ*]{}]{}, 527, 54.
Balogh M., Baldry I. K., Nichol R., Miller C., Bower R., Glazebrook K., 2004, [[*ApJ*]{}]{}Letters, 615, 101.
Barazza F. D., Jogee S., Marinova I., 2008, ApJ, 675, 1194.
Barrow J.D., Bhavsar S.P. & Sonoda B.H. 1984, [[*MNRAS*]{}]{}, 210, 19p.
Berentzen I., Athanassoula E., Heller C. H. & Fricke K. J., 2004, [[*MNRAS*]{}]{}, 347, 220.
Bernardi M., Shankar F., Hyde J. B., Mei S., Marulli F., Sheth R. K., 2010, [[*MNRAS*]{}]{}, 404, 2087.
Blanton M. R., Brinkmann J., Csabai I., Doi M., Eisenstein D., Fukugita M., Gunn J. E., Hogg D. W., Schlegel D. J., 2003, [[*AJ*]{}]{}., 125, 2348.
Bournaud F. & Combes F., 2002, [[*A*]{}&[*A*]{}]{}, 392, 83.
Brinchmann J., Charlot S, White S.D.M., Tremonti C., Kauffmann G., Heckman T. & Brinkmann J., 2004, [[*MNRAS*]{}]{}, 351, 1151.
Buta, R., & Combes, F. 1996, Fund. Cosm. Phys., 17, 95.
Chapelon, S., Contini, T., Davoust, E., 1999, A&A, 345, 81.
Coldwell G., Gurovich S., Díaz Tello J., Soechting I.K. & Lambas D.G., 2014, [[*MNRAS*]{}]{}, 437, 1199.
Combes, F., & Sanders, R. H. 1981, A&A, 96, 164.
Combes F., Elmegreen B. G., 1993, A&A, 271, 391.
Considere, S., Coziol, R., Contini, T., Davoust, E., 2000, A&A, 356, 89.
Corsini E. M., Debattista V. P., Aguerri J. A. L., 2003, ApJ, 599, 29.
Debattista V. P., Sellwood J. A., 1998, ApJ, 493, 5.
Debattista V. P., Carollo C. M., Mayer L., Moore B., 2005, ApJ, 628, 678.
Debattista V. P., Mayer L., Carollo C. M., Moore B., Wadsley J., Quinn T., 2006, ApJ, 645, 209.
de Lorenzo-Caceres A., Vazdekis A., Aguerri J. A. L., Corsini E. M., Debattista V. P., 2012, MNRAS, 420, 1092.
de Lorenzo-Caceres A., Falcon-Barroso J., Vazdekis A., 2013, MNRAS, 431, 2397.
de Vaucouleurs G., de Vaucouleurs A., Corwin H. G., Jr., Buta R. J., Paturel G., Fouque P., 1991, Third e Reference Catalogue of Bright Galaxies, Springer, New York, NY (USA).
Devereux N.A., [[*ApJ*]{}]{}, 1987, 323, 91.
Ellison S. L., Patton D. R., Simard L., McConnachie A. W., 2008, ApJ, 672, 107.
Ellison S.L., Nair P., Patton D.R., Scudder J.M., Mendel J.T. & Simard L., 2011, [[*MNRAS*]{}]{}, 416, 2182.
Elmegreen, B. G., & Elmegreen, D. M., 1985, ApJ, 288, 438.
Elmegreen, B. G., & Elmegreen, D. M., 1989, ApJ, 342, 677.
Erb D. K., Shapley A. E., Pettini M., Steidel C. C., Reddy N. A., Adelberger K. L., 2006, ApJ, 644, 813.
Erwin P., 2004, [[*A*]{}&[*A*]{}]{}, 415, 941.
Freeman K. & Bland-Hawthorn J., 2002, ARA&A, 40, 487.
Friedli, D., Benz, W., & Kennicutt, R. 1994, ApJ, 430, 105.
Friedli D. & Benz, W., 1995, A&A, 301, 649.
Gadotti, D. A., & de Souza, R. E. 2006, ApJS, 163, 270.
Hawarden T. G., Mountain C. M., Leggett S. K., Puxley, P. J., 1986, MNRAS, 221, 41.
Hawarden, T. G., Huang, J. H., Gu, Q. S., 1996, ASPC, 91, 54.
Huang, J. H., Gu, Q. S., Su, H. J., Hawarden, T. G., Liao, X. H., Wu, G. X., 1996, A&A, 313, 13.
Hummel, E., van der Hulst, J. M., Kennicutt, R. C., Keel, W. C., 1990, A&A, 236, 333.
Ho L.C., Filippenko A.V & Sargent W.L.W., 1997, [[*ApJ*]{}]{}, 487, 591.
Kauffmann G., Heckman T. M., Tremonti C. , 2003, [[*MNRAS*]{}]{}, 346, 1055.
James, P. A., Bretherton, C. F., Knapen, J. H., 2009, A&A, 501, 207.
James P. A. & Percival S. M., 2016, MNRAS, 457, 917.
Jogee S., Scoville N., & Kenney J. D. P. 2005, ApJ, 630, 837.
Kormendy, J., & Kennicutt, R. C. 2004, ARA&A, 42, 603.
Kuehn F., Ryden B. S., 2005, [[*ApJ*]{}]{}, 634, 1032.
Laurikainen E., Salo H., Buta R., Knapen J. H., 2007, MNRAS, 381, 401.
Laurikainen E., Salo H., Buta R., Knapen J. H., [[*ApJ*]{}]{}, 692, 34.
Lee G., Woo J., Lee M.G., Hwang H.S., Lee J.C., Sohn J. & Lee J.H.,2012, [[*ApJ*]{}]{}, 750, 141.
Lequeux J., Peimbert M., Rayo J. F., Serrano A., Torres-Peimbert S., 1979, A&A, 80, 155.
Lintott 2011, [[*MNRAS*]{}]{}, 410, 166.
Maiolino R., , 2007, NCimB, 122, 935.
Marinova, I., , 2009, [[*ApJ*]{}]{}, 698, 1639.
Martin P., 1995, [[*AJ*]{}]{}, 109, 2428.
Martin P., & Roy J. R. 1994, [[*ApJ*]{}]{}, 424, 599.
Martinez$-$Valpuesta, I., Shlosman, I., & Heller, C. 2006, ApJ, 637, 214.
Martinet, L., & Friedli, D., 1997, A&A, 323, 363.
Masters K.L., 2010, [[*MNRAS*]{}]{}, 404, 792.
Masters K.L., 2010, [[*MNRAS*]{}]{}, 405, 783.
Masters K.L., 2011, [[*MNRAS*]{}]{}, 411, 2026.
Masters K., Nichol R., Haynes M., Keel W., 2012, [[*MNRAS*]{}]{}, 424, 2180.
Martel H., Kawata D., Ellison S., 2013, [[*MNRAS*]{}]{}, 431, 2560.
Menéndez-Delmestre K., Sheth K., Schinnerer E., Jarrett T.H & Scoville N.Z., [[*ApJ*]{}]{}, 2007, 657, 790.
Méndez-Abreu J., Corsini E. M., Debattista V. P., De Rijcke S., Aguerri J. A. L., Pizzella A, 2008, ApJ, 679, 73.
Nilson P. 1973, Acta Universitatis Upsaliensis. Nova Acta Regiae Societatis Scientiarum Upsaliensis - Upp- sala Astronomiska Observatoriums Annaler, Uppsala: Astronomiska Observatorium, 1973.
Norman, C. A., Sellwood, J. A., & Hasan, H. 1996, ApJ, 462, 114.
Oey, M. S., Kennicutt, R. C., 1993, ApJ, 411, 137O.
Oh S., Oh K. & Yi S. K., 2012, [[*ApJS*]{}]{}, 198, 40.
Perez I., Sanchez-Blazquez P., Zurita A., 2007, A&A, 465.
Perez I., Sanchez-Blazquez P. Zurita, A., 2009, A&A, 495, 775.
Perez J., Tissera P. & Blaizot J., 2009, [[*MNRAS*]{}]{}, 397, 748.
Pompea, S. M., & Rieke, G. H., 1990, ApJ, 356, 416.
Roberts, W. W., Jr., Huntley, J. M., & van Albada, G. D. 1979, ApJ, 233, 67.
Sakamoto K., Okumura S. K., Ishizuki S., Scoville N. Z., 1999, ApJ, 525, 691.
Sánchez-Blázquez P., Rosales-Ortega F.F., Méndez-Abreu J.M., 2014, A&A, 570, 6.
Sellwood J. A., & Wilkinson A., 1993, Rep. Prog. Phys., 56, 173.
Sellwood, J. A., & Moore, E. M. 1999, ApJ, 510, 125
Sheth K., Vogel S. N., Regan M. W., Thornley M. D., Teuben, P. J. 2005, ApJ, 632, 217.
Sheth K., Elmegreen D.M., Elmegreen B.G., Capak P. Abraham R.G.,Athanassoula E., 2008, ApJ, 675, 1141.
Shlosman I., Begelman M.C.& Frank J., 1990, [[*Nature*]{}]{}, 345, 679.
Skibba R. A. , 2012, [[*MNRAS*]{}]{}, 423, 1485.
Strauss 2002, [[*AJ*]{}]{}, 124, 1810.
Tremonti C., Heckman T. M., Kauffmann, G. , 2004, [[*ApJ*]{}]{}, 613, 898.
Vila$-$Costas, M. B., & Edmunds, M. G., 1992, MNRAS, 259, 121.
Weinberg M. D., 1985, MNRAS, 213, 451.
Weinzirl T., Jogee S., Khochfar S., Burkert A., Kormendy J., 2009, ASP, 419,419.
Zaritsky D., Kennicutt R. C., & Huchra J. P. 1994, [[*ApJ*]{}]{}, 420, 87.
Zhou Z., Cao C., Wu H., 2015 [[*AJ*]{}]{}, 149, 1.
[^1]: http://www.mpa-garching.mpg.de/SDSS/DR7/
[^2]: http://sdss.physics.nyu.edu/vagc/
[^3]: http://skyserver.sdss.org/dr7/en/tools/chart/list.asp
[^4]: $C=r90/r50$ is the ratio of Petrosian $90\%-50\%$ r-band light radii.
|
---
abstract: |
We present spectral analysis of early observations of the Type IIn supernova 1998S using the general non-local thermodynamic equilibrium atmosphere code [PHOENIX]{}. We model both the underlying supernova spectrum and the overlying circumstellar interaction region and produce spectra in good agreement with observations. The early spectra are well fit by lines produced primarily in the circumstellar region itself, and later spectra are due primarily to the supernova ejecta. Intermediate spectra are affected by both regions. A mass-loss rate of order $\dot M \sim
0.0001-0.001$[${\textrm{M}_\odot}$]{} yr$^{-1}$ is inferred for a wind speed of $100-1000$ [km s$^{-1}$]{}. We discuss how future self-consistent models will better clarify the underlying progenitor structure.
author:
- '[Eric J. Lentz]{}, [E. Baron]{}, [Peter Lundqvist]{}, [David Branch]{}, [Peter H. Hauschildt]{}, [Claes Fransson]{}, [Peter Garnavich]{}, [Nate Bastian]{}, [Alexei V. Filippenko]{}, [R. P. Kirshner]{}, [P. M. Challis]{}, [S. Jha]{}, [Bruno Leibundgut]{}, [R. McCray]{}, [E. Michael ]{}, [Nino Panagia]{}, [M. M. Phillips]{}, [C. S. J. Pun]{}, [Brian Schmidt]{}, [George Sonneborn]{}, [N. B. Suntzeff]{}, [L. Wang]{}, and [J. C. Wheeler]{}'
title: 'Analysis of the Type IIn Supernova 1998S: Effects of Circumstellar Interaction on Observed Spectra'
---
Introduction
============
SN 1998S was discovered on Mar. 3 UT by Zhou Wan [@IAUC6829] as part of the Beijing Astronomical Observatory (BAO) Supernova Survey [@IAUC6612]. The discovery was confirmed by the Katzman Automatic Imaging Telescope (KAIT) during the Lick Observatory Supernova Search [@IAUC6627; @flipper00]. SN 1998S is located in NGC 3877, a spiral galaxy classified as SA, with a heliocentric velocity of 902 [km s$^{-1}$]{} [@UGC73] and a Galactic extinction of $A_B=0.01$ mag [@burheilred].
Filippenko & Moran [@IAUC6829] obtained a high-resolution spectrum of SN 1998S on Mar. 4 with the Keck-1 telescope and classified SN 1998S as a Type II supernova (SN II) on the basis of broad H$\alpha$ emission superposed on a featureless continuum. Further spectra were obtained at the Fred L. Whipple Observatory (FLWO) [@Garn98S00] and a campaign to monitor SN 1998S in the UV from the *Hubble Space Telescope* (*HST*) was mounted by the Supernova INtensive Study (SINS) team. Three epochs have been observed with *HST* — March 16, March 30, and May 13. SN 1998S is a Type IIn supernova [SN IIn; @schlegel2n90], a classification which shows wide variations in the spectra [@filarev97], but includes narrow lines on top of an underlying broad-line supernova spectrum. This has been taken as strong evidence that the supernova ejecta were interacting with a slow-moving circumstellar wind [@leon98S00], probably in a fashion similar to (but possibly more extreme than) that of SN 1979C and SN 1980K [@lentz98Saas99; @bao98s00].
In order to get an initial understanding and to confirm the basic picture of SNe IIn as strong circumstellar interacters, a set of parameterized models of SN 1998S was examined with the fully relativistic, NLTE, multi-purpose, expanding atmosphere code, [PHOENIX]{}, [cf. @hbjcam99 and references therein]. [PHOENIX]{} solves the spherically symmetric radiation transport along with the NLTE rate equations and the condition of radiative equilibrium (including deviations due to time dependence of the deposition of non-thermal gamma rays).
A modification of [PHOENIX]{} is underway in order to treat self-consistently both the underlying supernova spectrum and the overlying circumstellar interaction region; here we treat the two regions separately. Figure \[fig:diagram\] shows a schematic representation of SN 1998S. Some of the important coupling between the two regions is not included in the calculations and therefore not all of the features observed can be expected to be reproduced by the synthetic spectra. However, our models serve to confirm the basic picture of a SN IIn as a Type II supernova that interacts strongly with a near-constant velocity wind. We are able to identify important physical effects that need to be included in future simulations.
While our models are spherically symmetric, @leon98S00 have shown that the spectra of SN 1998S are significantly polarized, which could be due to asymmetry in the outermost SN ejecta, the circumstellar medium (CSM), or both. @gerardetal00 suggest that dust and CO are likely to have formed in the SN ejecta while @Fassia98S00 argue that the early dust is likely to come from the CSM.
Models
======
In this paper we focus on three epochs in particular: an early epoch on Mar 16, $\sim 20$ days after explosion, using combined *HST* and ground-based spectra, where effects from the circumstellar region dominate; Mar 30, $\sim 34$ days after explosion, again where there are combined *HST* and ground-based data, and where effects from both photospheric SN ejecta and the circumstellar region are important, and a later ground-based spectrum from April 17, $\sim 50$ days after explosion, where the densest circumstellar gas has been largely, but not completely, overrun by the supernova ejecta. In the earlier spectrum most of the observed lines are formed in the low-velocity circumstellar material, whereas in the later spectrum the lines show the characteristic width of a Type II supernova. A detailed analysis of the light curve and other observed spectra will be presented elsewhere [@Garn98S00]; see also @leon98S00 and @Fassia98S00 [@Fassia98S01].
March 16
--------
We have modeled the circumstellar region as a constant-velocity wind with a density profile $\rho \propto r^{-2}$. While the underlying radiation below the circumstellar region is in fact due to the supernova itself and should show broad P-Cygni profiles as well as a UV deficit due to line blanketing in the differentially expanding supernova atmosphere, we ignore these complications for the present discussion and assume that the underlying radiation is given by a Planck function, with $T_{\textrm{Planck}} = 13250$ K. In future work we will treat the effects of the circumstellar interaction region on the supernova itself, and couple the proper supernova boundary condition into the circumstellar region. Nevertheless, our present decoupled prescription allows us to model the important physics, and to estimate velocities, density profiles, and the radial extent of the circumstellar interaction region and the supernova. The region modeled in these calculations coincides with the region labeled “High Velocity CS Wind” in Figure \[fig:diagram\]. High-resolution spectra [@Fassia98S01] have shown that there may be several velocity components present in the circumstellar medium with velocities as low as 80 [km s$^{-1}$]{}. We focus here on only the higher velocity (but possibly still unresolved) components of the CS spectrum.
Figure \[csoverview\] presents an overview of our best model fit compared with the observed *HST* UV + FLWO optical spectrum taken on Mar 16, 1998. The observed spectrum has been dereddened using the reddening law of @card89 and a color excess $E_{B-V}
=0.15$ mag [@Garn98S00]. The assumed extinction is also in agreement with the results of @Fassia98S00 who find $E(B-V)=0.18 \pm 0.10$ mag. The overall agreement in the line positions and shape of the spectrum is excellent, particularly the pseudo-continuum near 2000 . The model consists of a constant-velocity circumstellar wind with $v_{wind} = 1000$ [km s$^{-1}$]{}, an inner density of $\rho_0 = 2.0 \times 10^{-15}$ [g cm$^{-3}$]{}, an inner radius $R_{inner} = 1.0 \times 10^{15}$ cm, and an outer radius $R_{outer} =
1.5 \times 10^{15}$ cm. The total continuum optical depth at $5000$ (roughly the electron scattering optical depth) is $\tstd
= 0.2$, (where $\tstd$ is the total continuum optical depth at 5000 ), and the mass of the wind is $6\times
10^{-3}$ [${\textrm{M}_\odot}$]{}. Assuming the *ejected* wind velocity was 100 [km s$^{-1}$]{} this corresponds to a mass-loss rate of 0.0012 [${\textrm{M}_\odot}$]{} yr$^{-1}$, which should be accurate to an order of magnitude. We believe that the high velocity seen here is due to radiative acceleration of a wind that was ejected at a lower velocity. Since the mass-loss rate depends inversely on the wind velocity at ejection and wind velocities typical of red-giants are $v_{wind} \approx 10$ [km s$^{-1}$]{}, we think that assuming an ejection velocity of 100 [km s$^{-1}$]{} allows us to estimate the mass loss rate to an order of magnitude.
In Figures \[lineidfig\] we expand the wavelength scale and identify the features in the observed spectrum. Several of them are clearly pairs of interstellar absorption features where one member is due to absorption in our Galaxy and the other is due to absorption in the parent galaxy (Mg II h+k shows this effect clearly). Table \[lineidtab\] lists the line identifications. We note that the “interstellar” absorption lines in the parent galaxy may also have a circumstellar contribution. Close examination of Figure 3 shows that the observed lines are significantly wider than those in the synthetic spectrum. On the other hand, @Fassia98S01 observed IR features with velocities as low as 90 [km s$^{-1}$]{}, and @bowen98S00 observed UV P-Cygni features with velocities of $\sim 100$ [km s$^{-1}$]{}. Convolving the synthetic spectrum with a Gaussian of width 400 [km s$^{-1}$]{} improves the fit, but since we have assumed a velocity higher than that of the lowest velocity observed [@Fassia98S01], it is difficult to separate out the instrumental resolution ($\sim 300-400$ [km s$^{-1}$]{}) from the velocity of the circumstellar medium. It could be that the velocity structure of the circumstellar region is quite complicated with a higher velocity component radiatively accelerated by the supernova, as was suggested for SN 1993J [@flc96], and a lower velocity component wind further away from the progenitor star.
Our model spectrum clearly does an extremely good job in reproducing the overall shape and position of the observed features; nevertheless, the line features are somewhat weaker in general than those observed. This could be due to the effects of the radiation from the circumstellar interaction. The effects of this radiation are not included in these simple preliminary calculations. The effects of the “top-lighting” or “shine-back” are not limited to radiative transfer effects alone, but will also affect the ionization state of the matter, particularly if there is significant X-ray emission from a reverse shock. In future work we will include the effects of external irradiation from the circumstellar region and replace the simple inner Planck function boundary condition that we have used here with a model supernova spectrum. Such a spectrum would be hotter, but diluted and contain both the UV deficit of a normal Type II supernova as well as broad P-Cygni features for which there is evidence in the observed spectrum.
March 30
--------
Figure \[mar30data\] displays the combined *HST* spectra with an optical spectrum obtained at the FLWO. It is interesting to note that the narrow features present on Mar. 16 seem to have disappeared, and the broad lines are all quite weak. A simple analytical explanation of this is presented in @toplight00, which shows that with the additional emission (“toplighting” or “shine-back”) from the circumstellar shell, one expects the supernova features to appear muted. Figure \[nate\] displays a [`PHOENIX`]{} spectrum, along with the results obtained when it has been muted according to the prescription in @toplight00. The regular [`PHOENIX`]{}spectrum is based upon the simplest assumptions: homogeneous solar abundances, a model temperature $\Tmod = 6000$ K [the model temperature is simply a way of parameterizing the total bolometric luminosity in the observers frame, see @hbjcam99], a velocity of 5000 [km s$^{-1}$]{} at $\tstd=1$, and a density structure $\rho \propto
r^{-8}$. Using Eqn. 23 of @toplight00, we have calculated the muting, using $E=0.9$, where $E$ is the ratio of the CS intensity to that of the SN intensity given in Eqn. 22 of @toplight00, we have assumed a ratio of $R_{CS}/R_{Ph} =1.5$, where the ratio is the radius of the circumstellar shell to the radius of the “SN photosphere”. While the fit is not terribly good, the trend is evident. Naturally, a fully consistent model would be better, but it would require significant computational resources to resolve both the ejecta and circumstellar region. @fransson79c84 calculated lineshapes expected from the CS wind and the cool, dense shocked material and compared them with those observed in SN 1979C.
@leon98S00 suggest that SN 1998S underwent a significant mass-loss episode that ended about 60 years prior to explosion and that there was a second, weaker mass-loss episode 7 years prior to explosion. Thus, we may be seeing the over-running of the closest CS shell and still observing effects of the more distant CS shells.
April 17
--------
During the early evolution the nearest circumstellar material is overrun by the supernova ejecta so the effects of the CSM on the optical and UV spectra become smaller. Inspection of the observed optical spectra [@leon98S00; @Garn98S00] shows an increasing contrast in the broad features typical of Type II SNe during the time from the initial *HST* observation, March 16, to the FLWO spectrum of April 17. @Blaylock98S00 show that the strengthening of these features during this transition is well reproduced by including the effects of radiation from the circumstellar interaction region along with the scattering of light from the supernova photosphere in the circumstellar region.
Figure \[apr17opt\] displays our best model fit to the observed optical spectrum taken at the FLWO [@Garn98S00]. We again use simple assumptions: homogeneous solar abundances, a model temperature $\Tmod = 5700$ K, a velocity of 5000 [km s$^{-1}$]{} at $\tstd=1$, and a density structure $\rho \propto
r^{-8}$. The highest velocity in the model is only 6,000 [km s$^{-1}$]{}, which gives an indication that the ejecta are entrained by the circumstellar material, but this is not well constrained by our models. Again, overall the fit is very good. The Na D line in the observed spectrum is too weak in our synthetic spectrum, which may indicate the need to self-consistently include the effect of the circumstellar region or may be due to enhanced sodium. The extended absorption wing of H$\alpha$ is due in our model to blending of weak Fe II lines, although some of the absorption may be due to Si II. In any case it is not evidence for high-velocity hydrogen.
Conclusions
===========
We have shown that a simple model of an ordinary Type II supernova atmosphere interacting strongly with a radiatively accelerated wind reasonably well reproduces the observed line-widths and many of the observed features in both the UV and the optical spectra. This model is robust in that it works well at both very early times and more than a month after the explosion. This confirms the general picture of SNe IIn as being the core collapse of massive stars that have experienced a significant mass-loss epoch and thus are surrounded by a circumstellar medium with which the supernova ejecta interact. As expected from our models [@lentz98Saas99], SN 1998S has been detected about 600 days after explosion at 6 cm [@IAUC7322]. Although SN 1998S is about 5 times less luminous than SN 1988Z, further monitoring of the radio light curve will be very interesting and will help determine the mass-loss rate. From the light curve @Fassia98S00 find that the mass of the ejected envelope was quite low and the wind was weaker than that of SN 1988Z. SN 1998S may well be more closely related to SN 1979C and SN 1980K. In future work we will develop a more self-consistent model of the supernova circumstellar interaction, and will be able to constrain mass-loss rates and total mass loss which are of great interest for the theory of stellar evolution.
We thank the anonymous referee for suggestions which considerably improved the presentation of this paper. PHH was supported in part by the Pôle Scientifique de Modélisation Numérique at ENS-Lyon. NB was supported in part by an NSF REU supplement to the Univ. of Oklahoma. This work was supported in part by NSF grants AST-9731450, AST-9417102, AST-9987438, and AST-9417213; by NASA grant NAG5-3505 and an IBM SUR grant to the University of Oklahoma; by NSF grant AST-9720704, NASA ATP grant NAG 5-8425, and LTSA grant NAG 5-3619 to the University of Georgia; and by NASA GO–2563.001 to the SINS group from the Space Telescope Science Institute, which is operated by AURA, Inc. under NASA contract NAS 5–26555. Some of the calculations presented in this paper were performed at the San Diego Supercomputer Center (SDSC), supported by the NSF, and at the National Energy Research Supercomputer Center (NERSC), supported by the U.S. DOE; we thank both these institutions for a generous allocation of computer time. This research has 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.
[23]{} natexlab\#1[\#1]{}
Blaylock, M. [et al.]{} 2000, ApJ, in preparation
Bowen, D., Roth, K., Meyer, D., & Blades, J. 2000, ApJ, 536, 225
Branch, D., Jeffery, D., Blaylock, M., & Hatano, K. 2000, PASP, 112, 217
Burstein, D. & Heiles, C. 1982, AJ, 87, 1165
Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245
Fassia, A. [et al.]{} 2000, MNRAS, in press, astro-ph/0006080
—. 2001, MNRAS, in preparation
Filippenko, A. V. 1997, Ann. Rev. Astr. Ap., 35, 309
Filippenko, A. V. [et al.]{} 2000, ApJ, in preparation
, C. 1984, A&A, 132, 115
Fransson, C., Lundqvist, P., & Chevalier, R. 1996, ApJ, 461, 993
Garnavich, P. [et al.]{} 2000, in preparation
Gerardy, C., Fesen, R., H[ö]{}flich, P., & Wheeler, J. C. 2000, AJ, 119, 2968
Hauschildt, P. H. & Baron, E. 1999, J. Comp. Applied Math., 109, 41
Lentz, E., Baron, E., Branch, D., Hauschildt, P. H., Fransson, C., Lundqvist, P., Garnavich, P., & Kirshner, R. P. 1999, BAAS, 30, 1324
Leonard, D., Filippenko, A. V., Barth, A., & Matheson, T. 2000, ApJ, 536, 239
, W. D., [Li]{}, C., [Filippenko]{}, A. V., & [Moran]{}, E. C. 1998, IAU Circ., 6829
Liu, Q., Hu, J., Hang, H., Qiu, Y., Zhu, Z., & Qiao, Q. 2000, A&A, 144, 219
Nilson, P. 1973, V, Vol. 1, Uppsala General Catalogue of Galaxies (Uppsala, Sweden: Nova Regiae Societatis Upsaliensis)
Qiao, Q., Li, W., Qiu, Y., Zhang, Y., & Hu, J. 1997, IAU Circ., 6612
Schlegel, E. M. 1990, MNRAS, 224, 269
Treffers, R. R., Peng, C. Y., & Filippenko, A. V. 1997, IAU Circ., 6627
, S. D., [Lacey]{}, C. K., [Sramek]{}, R. A., & [Weiler]{}, K. W. 1999, IAU Circ., 7322
[llll]{} 1168& N I 1168?& 1550& C IV 1550\
1668& S II 1668& 1561& C I 1561?\
1176& C III 1176& 1601& Fe III 1601,1607\
1192& Si II 1192, S III 1198& 1625& Fe II 1625\
1216& Ly alpha & 1657& C I 1657\
1234& S II 1324& 1666& S I 1666?, O III 1665, Al II 1671\
1243& N I 1243? & 1698& Si I 1698?\
1227& C III 1247& 1719& N IV 1719\
1249.5& Si II 1250, 1263& 1750& N III 1750\
1252& S II 1256& 1805& S II 1805\
1299& Si III 1299& 1815& Si II 1815\
1304& O I 1304& 1854& Al III 1854,1862\
1335& C II 1335& 1892& Si III 1982?\
1338& O IV 1338& 1930& C I 1930?\
1342& Si III 1342& 2287& Co II 2287\
1346& N II 1346?& 2297& C III 2297\
1364.3& Si III 1364& 2344& Fe II 2344\
1371& O V 1371?& 2374& Fe II 2374\
1394& S IV 1394& 2383& Fe II 2383\
1403& S IV 1403& 2396& Fe II 2396\
1428& C III 1428?& 2406& Fe II 2406\
1493& N I 1493?& 2586& Fe II 2586,2600\
1527& Si II 1527,1533?& 2798& Mg II h+k 2796,2804\
& & 2853& Mg I 2853\
|
---
abstract: |
We review some relevant results in the context of higher spin black holes in three-dimensional spacetimes, focusing on their asymptotic behaviour and thermodynamic properties. For simplicity, we mainly discuss the case of gravity nonminimally coupled to spin-$3$ fields, being nonperturbatively described by a Chern-Simons theory of two independent $sl\left(
3,\mathbb{R}\right) $ gauge fields. Since the analysis is particularly transparent in the Hamiltonian formalism, we provide a concise discussion of their basic aspects in this context; and as a warming up exercise, we briefly analyze the asymptotic behaviour of pure gravity, as well as the BTZ black hole and its thermodynamics, exclusively in terms of gauge fields. The discussion is then extended to the case of black holes endowed with higher spin fields, briefly signaling the agreements and discrepancies found through different approaches. We conclude explaining how the puzzles become resolved once the fall off of the fields is precisely specified and extended to include chemical potentials, in a way that it is compatible with the asymptotic symmetries. Hence, the global charges become completely identified in an unambiguous way, so that different sets of asymptotic conditions turn out to contain inequivalent classes of black hole solutions being characterized by a different set of global charges.
author:
- 'Alfredo Pérez$^{1}$, David Tempo$^{1}$, Ricardo Troncoso$^{1,2}$'
title: Brief review on higher spin black holes
---
Introduction
============
Fundamental particles of spin greater than two are hitherto unknown, which from a purely theoretical point of view, appears to agree with the widespread belief that massless fields of spin $s>2$ are doomed to suffer from inconsistencies. Indeed, the lore is reflected through a well-known claim in the context of supergravity (see e.g., [@PTT-SUGRA]), which asserts that the maximum number of local supersymmetries is bounded by eight; otherwise, since the supersymmetry generators act as raising or lowering operators for spin, a supermultiplet would contain fields of spin greater than two. In turn, through the Kaluza-Klein mechanism, this also sets an upper bound on the spacetime dimension to be at most eleven. The supposed inconsistency of higher spin fields relies on solid no-go theorems (see [@PTT-BBS] for a good review about this subject). In particular, it is worth mentioning the result of Aragone and Deser [@PTT-Aragone-Deser], which states that the higher spin gauge symmetries of the free theory around flat spacetime, cannot be preserved once the field is minimally coupled to gravity.
A consistent way to circumvent the incompatibility of higher spin gauge symmetries with interactions was pioneered by Vasiliev [@PTT-VV1], [@PTT-VV2], who was able to formulate the field equations for a whole tower of nonminimally coupled fields of spin $s=0$, $1$, $2$, ..., $\infty$, in presence of a cosmological constant (For recent reviews see e.g., [@PTT-ReviewHS1-1], [@PTT-ReviewHS1-2]). It is worth pointing out that, since the hypotheses of the Coleman-Mandula theorem are not fulfilled by Vasiliev theory, spacetime and gauge symmetries become inherently mixed in an unaccustomed form [@PTT-Vasiliev-Paros]. It then goes without saying that the very existence of Vasiliev theory, naturally suggests a possible reformulation of supergravity theories from scratch, which would may in turn elucidate new alternative approaches to strings and M-theory. Indeed, in eleven dimensions and in presence of a negative cosmological constant, a supergravity theory that shares some of these features, as the mixing of spacetime and gauge symmetries, is known to exist [@PTT-TZ].
In order to gain some insights about this counterintuitive subject, one may instead follow the less ambitious approach of finding a simpler set up that still captures some of the relevant features that characterize the dynamics of higher spin fields. In this sense, the three-dimensional case turns out to be particularly appealing, since the dynamics is described through a standard field theory with a Chern-Simons action [@PTT-3D-2], [@PTT-3D-3], [@PTT-3D-4]. The generic theory can be further simplified, since it admits a consistent truncation to the case of a finite number of nonpropagating fields with spin $s=2$, $3$, ..., $N$. Hence the simplest case with the desired properties corresponds to $N=3$, so that the theory describes gravity with negative cosmological constant, nonminimally coupled to an interacting spin-three field. The remarkable simplification of the theory then allows the possibility of finding different classes of exact black hole solutions endowed with a nontrivial spin-three field, as the ones in [@PTT-GK], [@PTT-CM], and [@PTT-HPTT], respectively. However, despite the simplicity of these solutions, the subject has not been free of controversy, mainly due to the puzzling discrepancies that have been found in the characterization of their global charges and their entropy.
The purpose of this brief review, is overviewing some of the relevant results about this ongoing subject, as well as explaining how the apparent tension between different approaches is fully resolved once the chemical potentials are suitably identified along the lines of [@PTT-HPTT], [@PTT-BHPTT], so that the asymptotic symmetries, and hence the global charges, are completely characterized in an unambiguous way.
It is worth highlighting that the action principle in terms of the metric and the spin-3 field is currently known as a weak field expansion of the spin-3 field up to quadratic order [@PTT-CFPT2]. Thus, in order to deal with the full nonperturbative treatment of the higher spin black hole solutions, it turns out to be useful to describe them only in terms of gauge fields and the topology of the manifold, without making any reference neither to the metric nor to the spin-3 field.
Since the analysis becomes particularly transparent in the Hamiltonian formalism, in the next section we concisely discuss some of their basic aspects in the context of Chern-Simons theories in three dimensions. As a useful warming up exercise, in section \[PTT-Pure grav\], the asymptotic behaviour of pure gravity with negative cosmological constant [@PTT-Brown-Henneaux], as well as the BTZ black hole [@PTT-BTZ], [@PTT-BHTZ] and its thermodynamics, are briefly analyzed exclusively in terms of gauge fields. Section \[PTT-HSG3D\] is devoted to the case of gravity coupled to spin-3 fields, including the asymptotic behaviour described in [@PTT-Henneaux-Rey], [@PTT-CFPT1], the higher spin black hole solution of [@PTT-GK], [@PTT-AGKP], and its thermodynamics [@PTT-PTT1], [@PTT-PTT2], briefly signaling the agreements and discrepancies found through different approaches. We conclude with section \[PTT-Puzzles\], where it is explained how these puzzling differences become fully resolved once the fall off of the fields is precisely specified, so that different sets of asymptotic conditions turn out to contain inequivalent classes of black hole solutions [@PTT-HPTT], [@PTT-BHPTT] being characterized by a different set of global charges.
Basic aspects and Hamiltonian formulation of Chern-Simons theories in three dimensions
======================================================================================
In three-dimensional spacetimes, gauge theories described by a Chern-Simons action are much simpler than their corresponding Yang-Mills analogues, in the sense that less structure is required in order to formulate them. Indeed, the manifold $M$, locally described by a set of coordinates $x^{\mu}$, is only endowed with a gauge field $A=A_{\mu}^{I}T_{I}dx^{\mu}$, where $T_{I}$ stand for the generators of a Lie algebra $\mathfrak{g}$, which is assumed to admit an invariant nondegenerate bilinear form $g_{IJ}=\left\langle T_{I},T_{J}\right\rangle $. These ingredients are enough to construct the action, given by$$I_{CS}\left[ A\right] =\frac{k}{4\pi}\int_{M}\left\langle AdA+\frac{2}{3}A^{3}\right\rangle \ , \label{PTT-ICS}$$ where $k$ is a constant, and wedge product between forms has been assumed. Consequently, the action does not require the existence of a spacetime metric, but it is sensitive to the topology of $M$. The field equations imply the vanishing of curvature, i.e., $F=dA+A^{2}=0$, so that the connection becomes locally flat on shell, and then the theory is devoid of local propagating degrees of freedom. Note that the action (\[PTT-ICS\]) is already in Hamiltonian form. Indeed, if the topology of $M$ is of the form $M=\Sigma
\times\mathbb{R}
$, where $\Sigma$ stands for the spacelike section, the connection splits as $A=A_{i}dx^{i}+A_{t}dt$, and hence the action (\[PTT-ICS\]) reduces to$$I_{H}=-\frac{k}{4\pi}\int_{M}dtd^{2}x\varepsilon^{ij}\left\langle A_{i}\dot
{A}_{j}-A_{t}F_{ij}\right\rangle \ , \label{PTT-ICSH}$$ up to a boundary term. It is then apparent that $A_{i}$ correspond to the dynamical fields, whose Poisson brackets are given by $\left\{ A_{i}^{I}\left( x\right) ,A_{j}^{J}\left( x^{\prime}\right) \right\}
=\frac{2\pi}{k}g^{IJ}\varepsilon_{ij}\delta\left( x-x^{\prime}\right) $, while $A_{t}$ become Lagrange multipliers associated to the constraints $G=\frac{k}{4\pi}\varepsilon^{ij}F_{ij}$. Then, the smeared generator of the gauge transformations reads$$G\left( \Lambda\right) =\int_{\Sigma}d^{2}x\left\langle \Lambda
G\right\rangle \ ,$$ so that $\delta A_{i}=\left\{ A_{i},G\left( \Lambda\right) \right\}
=\partial_{i}\Lambda+\left[ A_{i},\Lambda\right] $ (see, e.g., [@PTT-Balachandran], [@PTT-Banados-Q], [@PTT-Carlip-Q]). However, when $\Sigma$ has a boundary, according to the Regge-Teitelboim approach [@PTT-Regge-Teitelboim], the generator of the gauge transformations has to be improved by a boundary term $Q\left( \Lambda\right) $, i.e., $$\tilde{G}\left( \Lambda\right) =G\left( \Lambda\right) +Q\left(
\Lambda\right) \ , \label{PTT-improved-G}$$ being such that its functional variation is well-defined everywhere. This implies that the variation of the the conserved charge associated to an asymptotic gauge symmetry, generated by a Lie algebra valued parameter $\Lambda$, is determined by the dynamical fields at a fixed time slice at the boundary, which reads$$\delta Q\left( \Lambda\right) =-\frac{k}{2\pi}\int_{\partial\Sigma
}\left\langle \Lambda\delta A_{\theta}\right\rangle d\theta\ ,
\label{PTT-deltaqeta}$$ where $\partial\Sigma$ stands for the boundary of the spacelike section $\Sigma$.
The transformation law of the Lagrange multipliers, $\delta A_{t}=\partial
_{t}\Lambda+\left[ A_{t},\Lambda\right] $, is then recovered requiring the improved action to be invariant under gauge transformations. Note that on-shell, by virtue of the identity $\mathcal{L}_{\xi}A_{\mu}=\nabla_{\mu
}\left( \xi^{\nu}A_{\nu}\right) +\xi^{\nu}F_{\nu\mu}$, diffeomorphisms $\delta_{\xi}A_{\mu}=-\mathcal{L}_{\xi}A_{\mu}$ are equivalent to gauge transformations with parameter $\Lambda=-\xi^{\mu}A_{\mu}$, and hence, the variation of the generator of an asymptotic symmetry spanned by an asymptotic killing vector $\xi^{\mu}$, reads$$\delta Q\left( \xi\right) =\frac{k}{2\pi}\int_{\partial\Sigma}\xi^{\mu
}\left\langle A_{\mu}\delta A_{\theta}\right\rangle d\theta\ .
\label{PTT-deltaqchi}$$ This means that the variation of the total energy of the system, which takes into account the contribution of all the constraints, is given by$$\delta E=\delta Q\left( \partial_{t}\right) =\frac{k}{2\pi}\int
_{\partial\Sigma}\left\langle A_{t}\delta A_{\theta}\right\rangle d\theta\ .
\label{PTT-deltaE}$$
It should be stressed that the whole canonical structure only makes sense provided the variation of the canonical generators can be integrated. This can be generically done once a precise set of asymptotic conditions is specified, which in turn determines the asymptotic symmetries. This will be explicitly discussed in the next section for the case of pure gravity with negative cosmological constant, as well as in section \[PTT-HSG3D\], and further elaborated in section \[PTT-Puzzles\] in the case of gravity coupled to a spin-3 field.
General Relativity with negative cosmological constant in three dimensions {#PTT-Pure grav}
==========================================================================
As it was shown in [@PTT-AT], [@PTT-W] General Relativity in vacuum can be described in terms of a Chern-Simons action. In the case of negative cosmological constant the corresponding Lie algebra is of the form $\mathfrak{g}=\mathfrak{g}_{+}+\mathfrak{g}_{-}$, where $\mathfrak{g}_{\pm}$ stand for two independent copies of $sl\left( 2,\mathbb{R}\right) $, which will be assumed to be described by the same set of matrices $L_{i}$, with $i=-1,0,1$, given by $$L_{-1}=\begin{pmatrix}
0 & 0\\
1 & 0
\end{pmatrix}
\quad;\quad L_{0}=\begin{pmatrix}
-\frac{1}{2} & 0\\
0 & \frac{1}{2}\end{pmatrix}
\quad;\quad L_{1}=\begin{pmatrix}
0 & -1\\
0 & 0
\end{pmatrix}
\;, \label{PTT-sl(2,R)-MR}$$ so that the $sl\left( 2,\mathbb{R}\right) $ algebra reads$$\left[ L_{i},L_{j}\right] =\left( i-j\right) L_{i+j}\;.$$ The connection then splits in two independent $sl\left( 2,\mathbb{R}\right)
$-valued gauge fields, according to $A=A^{+}+A^{-}$, while the invariant nondegenerate bilinear form is chosen such that the action (\[PTT-ICS\]) reduces to$$I=I_{CS}\left[ A^{+}\right] -I_{CS}\left[ A^{-}\right] \ ,$$ so that the bracket now corresponds to just the trace, i.e., in the representation of (\[PTT-sl(2,R)-MR\]), $\left\langle \cdots\right\rangle
=\mathrm{tr}\left( \cdots\right) $, and the level is fixed by the AdS radius and the Newton constant as $k=\frac{l}{4G}$. The link between the gauge fields and spacetime geometry is made through$$A^{\pm}=\omega\pm\frac{e}{l}\ , \label{PTT-Amn-GR}$$ where $\omega$ and $e$ correspond to the spin connection and the dreibein, respectively. The field equations, $F^{\pm}=0$, then imply that the spacetime curvature is constant and the torsion vanishes, while the metric is recovered from$$g_{\mu\nu}=2\mathrm{tr}\left( e_{\mu}e_{\nu}\right) \ , \label{PTT-gmunu}$$ which is manifestly invariant under the diagonal subgroup of $SL\left(
2,\mathbb{R}\right) \times SL\left( 2,\mathbb{R}\right) $, that corresponds to the local Lorentz transformations. Note that diffeomorphisms can always be expressed in terms of the remaining gauge symmetries.
Brown-Henneaux boundary conditions {#PTT-Brown-Henneaux-Section}
----------------------------------
As explained in [@PTT-CHvD], the asymptotic behaviour of gravity with negative cosmological constant, as originally described by Brown and Henneaux [@PTT-Brown-Henneaux], can be readily formulated in terms of the gauge fields $A^{\pm}$. The gauge can be chosen such that the radial dependence is entirely captured by the group elements $$g_{\pm}=e^{\pm\rho L_{0}}\ , \label{PTT-gmn}$$ so that the asymptotic form of the connections is given by$$A^{\pm}=g_{\pm}^{-1}a^{\pm}g_{\pm}+g_{\pm}^{-1}dg_{\pm}\ ,
\label{PTT-Acontuti}$$ where $a^{\pm}=a_{\theta}^{\pm}d\theta+a_{t}^{\pm}dt$, read$$a^{\pm}=\pm\left( L_{\pm1}-\frac{2\pi}{k}\mathcal{L}_{\pm}L_{\mp1}\right)
dx^{\pm}\ , \label{PTT-BH-BCs}$$ with $x^{\pm}=\frac{t}{l}\pm\theta$, and the functions $\mathcal{L}_{\pm}$ depend only on time and the angular coordinate.
The asymptotic form of the dynamical fields $a_{\theta}^{\pm}$ is preserved under gauge transformations, $\delta a_{\theta}^{\pm}=\partial_{\theta}\Lambda^{\pm}+\left[ a_{\theta}^{\pm},\Lambda^{\pm}\right] $, generated by$$\Lambda^{\pm}\left( \varepsilon_{\pm}\right) =\varepsilon_{\pm}L_{\pm1}\mp\varepsilon_{\pm}^{\prime}L_{0}+\frac{1}{2}\left( \varepsilon_{\pm
}^{\prime\prime}-\frac{4\pi}{k}\varepsilon_{\pm}\mathcal{L}_{\pm}\right)
L_{\mp1}\ , \label{PTT-Lambda-GR}$$ where $\varepsilon_{\pm}$ are arbitrary functions of $t$, $\theta$, provided the functions $\mathcal{L}_{\pm}$ transform as$$\delta\mathcal{L}_{\pm}=\varepsilon_{\pm}\mathcal{L}_{\pm}^{\prime
}+2\mathcal{L}_{\pm}\varepsilon_{\pm}^{\prime}-\frac{k}{4\pi}\varepsilon_{\pm
}^{\prime\prime\prime}\ . \label{PTT-deltaL-GR}$$ Hereafter, prime denotes the derivative with respect to $\theta$. Furthermore, requiring the components of the gauge fields along time, $a_{t}^{\pm}$, to be mapped into themselves under the same gauge transformations, together with the transformation laws in (\[PTT-deltaL-GR\]), implies that the functions $\mathcal{L}_{\pm}$ and the parameters $\varepsilon_{\pm}$ are chiral, i.e.,$$\partial_{\mp}\mathcal{L}_{\pm}=0\ \ ,\ \ \partial_{\mp}\varepsilon_{\pm}=0\ .
\label{PTT-FE+CC-GR}$$ Note that the first condition in (\[PTT-FE+CC-GR\]) means that the field equations have to be fulfilled in the asymptotic region.
Consequently, according to (\[PTT-deltaqeta\]), the variation of the canonical generators associated to the asymptotic gauge symmetries generated by $\Lambda=\Lambda^{+}+\Lambda^{-}$, in this case reduces to$$\delta Q\left( \Lambda\right) =\delta Q_{+}\left( \Lambda^{+}\right)
-\delta Q_{-}\left( \Lambda^{-}\right) \ ,$$ with$$\delta Q_{\pm}\left( \Lambda^{\pm}\right) =-\frac{k}{2\pi}\int\left\langle
\Lambda^{\pm}\delta a_{\theta}^{\pm}\right\rangle d\theta=-\int\varepsilon
_{\pm}\delta\mathcal{L}_{\pm}d\theta\ ,$$ which can be readily integrated as$$Q_{\pm}\left( \Lambda^{\pm}\right) =-\int\varepsilon_{\pm}\mathcal{L}_{\pm
}d\theta\ . \label{PTT-Q-Brown-Henneaux}$$ Therefore, since the canonical generators fulfill $\delta_{\Lambda_{1}}Q\left[ \Lambda_{2}\right] =\left\{ Q\left[ \Lambda_{2}\right] ,Q\left[
\Lambda_{1}\right] \right\} $, their algebra can be directly obtained by virtue of (\[PTT-deltaL-GR\]), which reduces to two copies of the Virasoro algebra with the same central extension $c=\frac{3l}{2G}$ [@PTT-Brown-Henneaux]. Expanding in Fourier modes, according to $\mathcal{L}=\frac{1}{2\pi}\sum_{m}\mathcal{L}_{m}e^{im\theta}$, the algebra explicitly reads$$i\left\{ \mathcal{L}_{m},\mathcal{L}_{n}\right\} =\left( m-n\right)
\mathcal{L}_{m+n}+\frac{k}{2}m^{3}\delta_{m+n,0}\ ,$$ for both copies.
BTZ black hole and its thermodynamics
-------------------------------------
The asymptotic conditions described above, manifestly contain the BTZ black hole solution [@PTT-BTZ], [@PTT-BHTZ], being described by$$a^{\pm}=\pm\left( L_{\pm1}-\frac{2\pi}{k}\mathcal{L}_{\pm}L_{\mp1}\right)
dx^{\pm}\ , \label{PTT-amn-BTZ}$$ when $\mathcal{L}_{\pm}$ are nonnegative constants. Indeed, by virtue of eqs. (\[PTT-Amn-GR\]) and (\[PTT-gmunu\]), the spacetime metric is recovered in normal coordinates:$$\begin{aligned}
ds^{2} & =l^{2}\left[ d\rho^{2}+\frac{2\pi}{k}\left( \mathcal{L}_{+}\left( dx^{+}\right) ^{2}+\mathcal{L}_{-}\left( dx^{-}\right)
^{2}\right) \right. \nonumber\\
& \left. -\left( e^{2\rho}+\frac{4\pi^{2}}{k^{2}}\mathcal{L}_{+}\mathcal{L}_{-}e^{-2\rho}\right) dx^{+}dx^{-}\right] \ .
\label{PTT-BTZ-metricNC}$$ As shown in [@PTT-Carlip-Teiltelboim] (see also [@PTT-MS]), the topology of the Euclidean black hole corresponds to $\mathbb{R}
^{2}\times S^{1}$, where $\mathbb{R}
^{2}$ stands for the one of the $\rho-\tau$ plane, and $\tau=-it$ is the Euclidean time, fulfilling $0\leq\tau<\beta$, where $\beta=T^{-1}$ is the inverse of the Hawking temperature. Since $\mathbb{R}
^{2}$ can be mapped into a disk through a conformal compactification, the black hole topology is then equivalent to the one of a solid torus.
As explained in the introduction, and for later purposes, afterwards we will perform the remaining analysis exclusively in terms of the gauge fields (\[PTT-amn-BTZ\]) and the topology of the manifold, without making any reference to the spacetime metric.
The simplest gauge covariant object that is sensitive to the global properties of the manifold turns out to be the holonomy of the gauge field around a closed cycle $\mathcal{C}$, defined as$$\mathcal{H}_{\mathcal{C}}=P\exp\left( \int_{\mathcal{C}}A_{\mu}dx^{\mu
}\right) \ ,$$ which is an element of the gauge group. Hence, since in this case the gauge group corresponds to $SL\left( 2,\mathbb{R}
\right) \times SL\left( 2,\mathbb{R}
\right) $, the holonomy around $\mathcal{C}$ is of the form $\mathcal{H}_{\mathcal{C}}=\mathcal{H}_{\mathcal{C}}^{+}\otimes\mathcal{H}_{\mathcal{C}}^{-}$, with$$\mathcal{H}_{\mathcal{C}}^{\pm}=P\exp\left( \int_{\mathcal{C}}A_{\mu}^{\pm
}dx^{\mu}\right) \ .$$ As the topology of the manifold is the one of a solid torus, there are two inequivalent classes of cycles: (I) the ones that wind around the handle, and (II) those that do not. This means that the former ones are noncontractible, while the latter can be continuously shrunk to a point. Then, the holonomies along contractible cycles are trivial, i.e.,$$\mathcal{H}_{\mathcal{C}_{II}}^{\pm}=-\mathrm{1}\ , \label{PTT-Hii-BTZ}$$ where the negative sign is due to the fact that, according to (\[PTT-sl(2,R)-MR\]), we are dealing with the fundamental (spinorial) representation of $SL\left( 2,\mathbb{R}
\right) $; while the holonomies along noncontractible cycles $\mathcal{H}_{\mathcal{C}_{I}}^{\pm}$ are necessarily nontrivial. Indeed, it is easy to verify that this is the case for the gauge fields that describe the BTZ black hole (\[PTT-amn-BTZ\]). For simplicity, we explicitly carry out the computation in the static case, i.e., for $\mathcal{L}:=\mathcal{L}_{\pm}$, since the inclusion of rotation is straightforward.
A simple noncontractible cycle in this case is parametrized by $\rho=\rho_{0}$, and $\tau=\tau_{0}$, with $\rho_{0}$, $\tau_{0}$ constants, so that the corresponding holonomies around it read$$\mathcal{H}_{\theta}^{\pm}=e^{2\pi a_{\theta}^{\pm}}\ . \label{PTT-Hphi-BTZ}$$ These holonomies are then fully characterized, up to conjugacy by elements of $SL\left( 2,\mathbb{R}
\right) $, by the eigenvalues of $2\pi a_{\theta}^{\pm}$, given by$$\lambda_{\pm}^{2}=2\pi^{2}\mathrm{tr}\left[ \left( a_{\theta}^{\pm}\right)
^{2}\right] =\frac{8\pi^{3}}{k}\mathcal{L}\ ,$$ and hence, since $\mathcal{L}$ is nonnegative, they are manifestly nontrivial.
Analogously, a simple contractible cycle is parametrized by $\rho=\rho_{0}$, and $\theta=\theta_{0}$. Since the holonomies around this cycle are trivial, the conditions in (\[PTT-Hii-BTZ\]) reduce to$$\mathcal{H}_{\tau}^{\pm}=e^{\beta a_{\tau}^{\pm}}=e^{i\beta a_{t}^{\pm}}=-\mathrm{1}\ , \label{PTT-Htau-BTZ}$$ and since the cycle winds once, the eigenvalues of $i\beta a_{t}$ are given by $\pm i\pi$, which equivalently implies that$$\beta^{2}\mathrm{tr}\left[ \left( a_{t}^{\pm}\right) ^{2}\right] =2\pi
^{2}\ . \label{PTT-BetaEq-BTZ}$$ Therefore, the triviality of the holonomies around this cycle amounts to fix the Euclidean time period as$$\beta=l\sqrt{\frac{\pi k}{2\mathcal{L}}}\ , \label{PTT-beta-BTZ}$$ in full agreement with the Hawking temperature.
Note that the variation of the total energy (\[PTT-deltaE\]) in this case reads$$\delta E=\frac{k}{2\pi}\int_{\partial\Sigma}\left( \left\langle a_{t}^{+}\delta a_{\theta}^{+}\right\rangle -\left\langle a_{t}^{-}\delta
a_{\theta}^{-}\right\rangle \right) d\theta=\frac{4\pi}{l}\delta
\mathcal{L\ },$$ from which, by virtue of (\[PTT-beta-BTZ\]) and the first law, implies that$$\delta S=\beta\delta E=\delta\left( 4\pi\sqrt{2\pi k\mathcal{L}}\right) \ ,$$ which means that the entropy can be expressed in terms of the global charges (\[PTT-Q-Brown-Henneaux\]), as$$S=4\pi\sqrt{2\pi k\mathcal{L}}\ . \label{PTT-S-BTZ}$$
The black hole entropy found in this way agrees with the standard result obtained in the metric formalism. Indeed, according to (\[PTT-BTZ-metricNC\]), in the static case the event horizon is located at $e^{2\rho_{+}}=\frac{2\pi}{k}\mathcal{L}$, so that its area is given by $A=4\pi l\sqrt
{\frac{2\pi}{k}\mathcal{L}}$, and hence (\[PTT-S-BTZ\]) is equivalent to the Bekenstein-Hawking formula $S=\frac{A}{4G}$.
Higher spin gravity in 3D {#PTT-HSG3D}
=========================
As explained in the introduction, gravity with negative cosmological constant, nonminimally coupled to an interacting spin-$3$ field can be described in terms of a Chern-Simons theory [@PTT-3D-2], [@PTT-3D-3], [@PTT-3D-4]. The action is then of the form (\[PTT-ICS\]), and as in the case of pure gravity, the corresponding Lie algebra is of the form $\mathfrak{g}=\mathfrak{g}_{+}+\mathfrak{g}_{-}$, but where now $\mathfrak{g}_{\pm}$ are enlarged to two independent copies of $sl\left( 3,\mathbb{R}\right) $. Both copies of the algebra will be assumed to be spanned by the same set of matrices $L_{i}$, $W_{m}$, with $i=-1,0,1$, and $m=-2,-1,0,1,2$, given by (see e.g., [@PTT-CFPT1])$$L_{-1}=\begin{pmatrix}
0 & -2 & 0\\
0 & 0 & -2\\
0 & 0 & 0
\end{pmatrix}
\quad;\quad L_{0}=\begin{pmatrix}
1 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & -1
\end{pmatrix}
\quad;\quad L_{1}=\begin{pmatrix}
0 & 0 & 0\\
1 & 0 & 0\\
0 & 1 & 0
\end{pmatrix}
\;,$$$$W_{-2}=\begin{pmatrix}
0 & 0 & 8\\
0 & 0 & 0\\
0 & 0 & 0
\end{pmatrix}
\quad;\quad W_{-1}=\begin{pmatrix}
0 & -2 & 0\\
0 & 0 & 2\\
0 & 0 & 0
\end{pmatrix}
\quad;\quad W_{0}=\frac{2}{3}\begin{pmatrix}
1 & 0 & 0\\
0 & -2 & 0\\
0 & 0 & 1
\end{pmatrix}
\;, \label{PTT-MR}$$$$W_{1}=\begin{pmatrix}
0 & 0 & 0\\
1 & 0 & 0\\
0 & -1 & 0
\end{pmatrix}
\quad;\quad W_{2}=\begin{pmatrix}
0 & 0 & 0\\
0 & 0 & 0\\
2 & 0 & 0
\end{pmatrix}
\;,$$ whose commutation relations read$$\begin{aligned}
\left[ L_{i},L_{j}\right] & =\left( i-j\right) L_{i+j}\;,\nonumber\\
\left[ L_{i},W_{m}\right] & =\left( 2i-m\right) W_{i+m}\;,\label{PTT-sl(3,R)-algebra}\\
\left[ W_{m},W_{n}\right] & =-\frac{1}{3}\left( m-n\right) \left(
2m^{2}+2n^{2}-mn-8\right) L_{m+n}\;,\nonumber\end{aligned}$$ so that the subset of generators $L_{i}$ span the algebra $sl\left( 2,\mathbb{R}
\right) $ in the so-called principal embedding.
The invariant nondegenerate bilinear form can also be chosen so that the action (\[PTT-ICS\]) reads$$I=I_{CS}\left[ A^{+}\right] -I_{CS}\left[ A^{-}\right] \ ,$$ where $A^{\pm}$ stand for the gauge fields that correspond to both copies of $sl\left( 3,\mathbb{R}\right) $, and now the bracket is given by a quarter of the trace in the representation of (\[PTT-MR\]), i.e., $\left\langle
\cdots\right\rangle =\frac{1}{4}\mathrm{tr}\left( \cdots\right) $. As in the case of pure gravity, the level is also chosen as $k=\frac{l}{4G}$.
It is useful to introduce a generalization of the dreibein and the spin connection, which relate with the gauge fields according to$$A^{\pm}=\omega\pm\frac{e}{l}\ ,$$ so that the spacetime metric and the spin-$3$ field can be recovered as$$g_{\mu\nu}=\frac{1}{2}\mathrm{tr}\left( e_{\mu}e_{\nu}\right) \ ;\ \varphi
_{\mu\nu\rho}=\frac{1}{3!}\mathrm{tr}\left( e_{(\mu}e_{\nu}e_{\rho)}\right)
\ , \label{PTT-metric+spin3}$$ being manifestly invariant under the diagonal subgroup of $SL\left(
3,\mathbb{R}\right) \times SL\left( 3,\mathbb{R}\right) $, which corresponds to an extension of the local Lorentz group. The remaining gauge symmetries are then not only related to diffeomorphisms, but also with the higher spin gauge transformations. It is worth pointing out that, since the metric transforms in a nontrivial way under the action of the higher spin gauge symmetries, some standard geometric and physical notions turn out to be ambiguous, since they are no longer invariant. This last observation can be regarded as an additional motivation to explore the physical properties of the theory directly in terms of its original variables, given by the gauge fields $A^{\pm}$.
Asymptotic conditions with $W_{3}$ symmetries
---------------------------------------------
A consistent set of asymptotic conditions for the theory described above was found in [@PTT-Henneaux-Rey], [@PTT-CFPT1]. Using the gauge choice as in [@PTT-CHvD], the radial dependence can be completely absorbed by $SL\left( 3,\mathbb{R}
\right) $ group elements of the form (\[PTT-gmn\]), so that the asymptotic behaviour of the gauge fields can be written as in eq. (\[PTT-Acontuti\]), where $a^{\pm}$ are now given by$$a^{\pm}=\pm\left( L_{\pm1}-\frac{2\pi}{k}\mathcal{L}_{\pm}L_{\mp1}-\frac{\pi
}{2k}\mathcal{W}_{\pm}W_{\mp2}\right) dx^{\pm}\ , \label{PTT-amn-Standard W3}$$ and $\mathcal{L}_{\pm}$, $\mathcal{W}_{\pm}$ stand for arbitrary functions of $t$, $\theta$. The asymptotic symmetries can then be readily found following the same steps as in the case of pure gravity, previously discussed in section \[PTT-Brown-Henneaux-Section\].
The asymptotic form of the fields $a_{\theta}^{\pm}$ is maintained under gauge transformations generated by$$\begin{aligned}
\Lambda^{\pm}\left( \varepsilon_{\pm},\chi_{\pm}\right) & =\varepsilon
_{\pm}L_{\pm1}+\chi_{\pm}W_{\pm2}\mp\varepsilon_{\pm}^{\prime}L_{0}\mp
\chi_{\pm}^{\prime}W_{\pm1}+\frac{1}{2}\left( \varepsilon_{\pm}^{\prime
\prime}-\frac{4\pi}{k}\varepsilon_{\pm}\mathcal{L}_{\pm}+\frac{8\pi}{k}\mathcal{W}_{\pm}\chi_{\pm}\right) L_{\mp1}\nonumber\\
& -\left( \frac{\pi}{2k}\mathcal{W}_{\pm}\varepsilon_{\pm}+\frac{7\pi}{6k}\mathcal{L}_{\pm}^{\prime}\chi_{\pm}^{\prime}+\frac{\pi}{3k}\chi_{\pm
}\mathcal{L}_{\pm}^{\prime\prime}+\frac{4\pi}{3k}\mathcal{L}_{\pm}\chi_{\pm
}^{\prime\prime}\right. \left. -\frac{4\pi^{2}}{k^{2}}\mathcal{L}_{\pm}^{2}\chi_{\pm}-\frac{1}{24}\chi_{\pm}^{\prime\prime\prime\prime}\right)
W_{\mp2}\nonumber\\
& +\frac{1}{2}\left( \chi_{\pm}^{\prime\prime}-\frac{8\pi}{k}\mathcal{L}_{\pm}\chi_{\pm}\right) W_{0}\mp\frac{1}{6}\left( \chi_{\pm}^{\prime
\prime\prime}-\frac{8\pi}{k}\chi_{\pm}\mathcal{L}_{\pm}^{\prime}-\frac{20\pi
}{k}\mathcal{L}_{\pm}\chi_{\pm}^{\prime}\right) W_{\mp1}\ ,
\label{PTT-Lambda-W3}$$ which depend on two arbitrary parameters per copy, $\varepsilon_{\pm}$, $\chi_{\pm}$, being functions of $t$ and $\theta$, provided the transformation law of the fields $\mathcal{L}_{\pm}$, $\mathcal{W}_{\pm}$ reads$$\begin{aligned}
\delta\mathcal{L}_{\pm} & =\varepsilon_{\pm}\mathcal{L}_{\pm}^{\prime
}+2\mathcal{L}_{\pm}\varepsilon_{\pm}^{\prime}-\frac{k}{4\pi}\varepsilon_{\pm
}^{\prime\prime\prime}-2\chi_{\pm}\mathcal{W}_{\pm}^{\prime}-3\mathcal{W}_{\pm}\chi_{\pm}^{\prime}\ ,\label{PTT-deltaL-W3}\\
\delta\mathcal{W}_{\pm} & =\varepsilon_{\pm}\mathcal{W}_{\pm}^{\prime
}+3\mathcal{W}_{\pm}\varepsilon_{\pm}^{\prime}-\frac{64\pi}{3k}\mathcal{L}_{\pm}^{2}\chi_{\pm}^{\prime}+3\chi_{\pm}^{\prime}\mathcal{L}_{\pm}^{\prime\prime}+5\mathcal{L}_{\pm}^{\prime}\chi_{\pm}^{\prime\prime}+\frac
{2}{3}\chi_{\pm}\mathcal{L}_{\pm}^{\prime\prime\prime}-\frac{k}{12\pi}\chi_{\pm}^{\prime\prime\prime\prime\prime}\nonumber\\
& -\frac{64\pi}{3k}\left( \chi_{\pm}\mathcal{L}_{\pm}^{\prime}-\frac
{5k}{32\pi}\chi_{\pm}^{\prime\prime\prime}\right) \mathcal{L}_{\pm}\ .
\label{PTT-deltaW-W3}$$ Then, the time component of the gauge fields $a_{t}^{\pm}$, is preserved under the gauge transformations generated by (\[PTT-Lambda-W3\]), with the transformation rules in (\[PTT-deltaL-W3\]), (\[PTT-deltaW-W3\]), provided the fields and the parameters are chiral:$$\begin{aligned}
\partial_{\mp}\mathcal{L}_{\pm} & =\partial_{\mp}\mathcal{W}_{\pm
}=0\ ,\label{PTT-FE-S-W3}\\
\partial_{\mp}\varepsilon_{\pm} & =\partial_{\mp}\chi_{\pm}=0\ .
\label{PTT-CC-S-W3}$$ As in the case of pure gravity, the chirality of the fields in eq. (\[PTT-FE-S-W3\]) reflects the fact that the field equations in the asymptotic region are satisfied.
The variation of the canonical generators that correspond to the asymptotic symmetries spanned by (\[PTT-Lambda-W3\]) now reads$$\delta Q_{\pm}\left( \Lambda^{\pm}\right) =-\frac{k}{2\pi}\int\left\langle
\Lambda^{\pm}\delta a_{\theta}^{\pm}\right\rangle d\theta=-\int\left(
\varepsilon_{\pm}\delta\mathcal{L}_{\pm}-\chi_{\pm}\delta\mathcal{W}_{\pm
}\right) d\theta\ , \label{PTT-deltaQ-W3}$$ and then integrates as$$Q_{\pm}\left( \Lambda^{\pm}\right) =-\int\left( \varepsilon_{\pm
}\mathcal{L}_{\pm}-\chi_{\pm}\mathcal{W}_{\pm}\right) d\theta\ .
\label{PTT-Q-W3}$$ This means that generic gauge fields that fulfill the asymptotic conditions described here, do not only carry spin-$2$ charges associated to $\mathcal{L}_{\pm}$, whose zero modes are related to the energy and the angular momentum, but they also possess spin-$3$ charges corresponding to $\mathcal{W}_{\pm}$.
The algebra of the canonical generators can be straightforwardly recovered from the transformation law of the fields in (\[PTT-deltaL-W3\]), (\[PTT-deltaW-W3\]) and it is found to be given by two copies of the $W_{3}$ algebra with the same central extension as in pure gravity, i.e., $c=\frac
{3l}{2G}$. Once the fields are expanded in modes, the Poisson bracket algebra is such that both copies fulfill $$\begin{aligned}
i\left\{ \mathcal{L}_{m},\mathcal{L}_{n}\right\} & =\left( m-n\right)
\mathcal{L}_{m+n}+\frac{k}{2}m^{3}\delta_{m+n,0}\ ,\nonumber\\
i\left\{ \mathcal{L}_{m},\mathcal{W}_{n}\right\} & =\left( 2m-n\right)
\mathcal{W}_{m+n}\ ,\label{PTT-W3-Algebra}\\
i\left\{ \mathcal{W}_{m},\mathcal{W}_{n}\right\} & =\frac{1}{3}\left(
m-n\right) \left( 2m^{2}-mn+2n^{2}\right) \mathcal{L}_{m+n}+\frac{16}{3k}\left( m-n\right) \Lambda_{m+n}+\frac{k}{6}m^{5}\delta_{m+n,0}\ ,\nonumber\end{aligned}$$ where $$\Lambda_{n}=\sum_{m}\mathcal{L}_{n-m}\mathcal{L}_{m}\ ,$$ so that the algebra is manifestly nonlinear.
It has also been shown that once the asymptotic conditions (\[PTT-amn-Standard W3\]) are expressed in a suitable decoupling gauge choice, they admit a consistent vanishing cosmological constant limit, so that the asymptotic symmetries are spanned by a higher spin extension of the BMS$_{3}$ algebra with an appropriate central extension [@PTT-GMPT] (see also [@PTT-Grumi-Flat]). Related results along these lines, including Hamiltonian reduction [@PTT-HP-Flat], unitarity [@PTT-GRR-Flat], and the analysis of cosmologies endowed with higher spin fields have been discussed in [@PTT-Chethan], [@PTT-PZ], [@PTT-Milne], [@PTT-Grassmann-Flat]).
Higher spin black hole proposal and its thermodynamics
------------------------------------------------------
It is simple to verify that, for the case of constant functions $\mathcal{L}_{\pm}$ and $\mathcal{W}_{\pm}$, the asymptotic conditions described in the previous subsection do not accommodate black holes carrying nontrivial spin-$3$ charges. This is because once the holonomies along a thermal cycle are required to be trivial, the spin-$3$ charges $\mathcal{W}_{\pm}$ are forced to vanish. Thus, with the aim of finding black holes solutions which could in principle be endowed with spin-$3$ charges, a different set of asymptotic conditions was proposed in [@PTT-GK] (see section \[PTT-Puzzles\]) and further analyzed in [@PTT-CS], [@PTT-CJS]. Indeed, this set includes interesting new black holes solutions, which in the static case are described by three constants, and the gauge fields are of the form (\[PTT-Acontuti\]), with$$\begin{aligned}
a^{\pm} & =\pm\left( L_{\pm1}-\frac{2\pi}{k}\mathcal{\tilde{L}}L_{\mp1}\mp\frac{\pi}{2k}\mathcal{\tilde{W}}W_{\mp2}\right) dx^{\pm}\nonumber\\
& +\tilde{\mu}\left( W_{\pm2}-\frac{4\pi}{k}\mathcal{\tilde{L}}W_{0}+\frac{4\pi^{2}}{k^{2}}\mathcal{\tilde{L}}^{2}W_{\mp2}\pm\frac{4\pi}{k}\mathcal{\tilde{W}}L_{\mp1}\right) dx^{\mp}\ . \label{PTT-amn-GK}$$ The precise form of the $SL\left( 3,\mathbb{R}
\right) $ group elements $g_{\pm}=g_{\pm}\left( \rho\right) $, which was further specified in [@PTT-AGKP], would be needed in order to reconstruct the metric and the spin-$3$ field according to eq. (\[PTT-metric+spin3\]). In the case of $sl\left( 3,\mathbb{R}
\right) $ gauge fields, the conditions that guarantee the triviality of the their holonomies around the thermal circle, since the representation in (\[PTT-MR\]) is vectorial, now read$$\mathcal{H}_{\tau}^{\pm}=e^{i\beta a_{t}^{\pm}}=\mathrm{1}\ ,
\label{PTT-Hat-GK}$$ which turn out to be equivalent to$$\mathrm{tr}\left[ \left( a_{t}^{\pm}\right) ^{3}\right] =0\ \ ;\ \ \beta
^{2}\mathrm{tr}\left[ \left( a_{t}^{\pm}\right) ^{2}\right] =8\pi^{2}\ .
\label{PTT-HOL-GK-Tr}$$ For the gauge fields (\[PTT-amn-GK\]), conditions (\[PTT-HOL-GK-Tr\]) reduce to$$\begin{aligned}
64\pi\mathcal{\tilde{L}}^{2}\tilde{\mu}\left( 32\pi\mathcal{\tilde{L}}\tilde{\mu}^{2}-9k\right) +27k\mathcal{\tilde{W}}\left( 32\pi\mathcal{\tilde
{L}}\tilde{\mu}^{2}+k\right) -864\pi k\mathcal{\tilde{W}}^{2}\tilde{\mu}^{3}
& =0\ ,\label{PTT-Hat-1}\\
\frac{l^{2}\pi k}{2}\left( \mathcal{\tilde{L}}-3\tilde{\mu}\mathcal{\tilde
{W}}+\frac{32\pi}{3k}\tilde{\mu}^{2}\mathcal{\tilde{L}}^{2}\right) ^{-1} &
=\beta^{2}\ , \label{PTT-Hat-2}$$ respectively, which for the branch that is connected to the BTZ black hole, being such that $\tilde{\mu}\rightarrow0$ when $\mathcal{\tilde{W}}\rightarrow0$, can be solved for $\beta$ and $\tilde{\mu}$ in terms of $\mathcal{\tilde{L}}$ and $\mathcal{\tilde{W}}$, according to$$\begin{aligned}
\beta & =\frac{l}{2}\sqrt{\frac{\pi k}{2\mathcal{\tilde{L}}}}\ \frac
{2C-3}{C-3}\left( 1-\frac{3}{4C}\right) ^{-1/2}\ ,\label{PTT-betaGK}\\
\tilde{\mu} & =\frac{3}{4}\sqrt{\frac{kC}{2\pi\mathcal{\tilde{L}}}}\ \frac{1}{2C-3}\ , \label{PTT-muGK}$$ where the constant $C$ is defined through$$\frac{C-1}{C^{3/2}}=\sqrt{\frac{k}{32\pi\mathcal{\tilde{L}}^{3}}}\mathcal{\tilde{W}\ }.$$
A proposal to deal with the global charges and the thermodynamics of this black hole solution, being based on a holographic approach, was put forward in [@PTT-GK], [@PTT-AGKP]. The bulk field equations were identified with the Ward identities for the stress tensor and the spin-$3$ current of an underlying dual CFT in two dimensions, so that the integration constant $\mathcal{\tilde{L}}$ was interpreted as the stress tensor, while $\mathcal{\tilde{W}}$ and $\tilde{\mu}$ were associated to the spin-$3$ current and its source, respectively. According to this prescription, the first law of thermodynamics implies that the variation of the entropy should be given by$$\delta\tilde{S}=\frac{4\pi}{l}\beta\left( \delta\mathcal{\tilde{L}}-\tilde{\mu}\delta\mathcal{\tilde{W}}\right) \ ,$$ which by virtue of (\[PTT-betaGK\]), (\[PTT-muGK\]) integrates as$$\tilde{S}=4\pi\sqrt{2\pi k\mathcal{\tilde{L}}}\sqrt{1-\frac{3}{4C}}\ ,
\label{PTT-S-GK}$$ so that the trivial holonomy conditions around the thermal circle agree with the integrability conditions of thermodynamics.
It is worth mentioning that the black hole entropy formula (\[PTT-S-GK\]) remarkably agrees with the result found in [@PTT-GHJ], which was obtained from a completely different approach. Indeed, the computation of the free energy was carried out directly in the dual CFT with extended conformal symmetry in two dimensions, exploiting the properties of the partition function under the S-modular transformation, making then no reference to the holonomies in the bulk.
These approaches have been reviewed in [@PTT-Rew-GG], [@PTT-Rew-AGKP], [@PTT-Rew-J], and further results about black hole thermodynamics along these lines have been found in [@PTT-F-GK1], [@PTT-F-GK2], [@PTT-F-GK3z], [@PTT-KU], [@PTT-dBJ], [@PTT-dBJ2], [@PTT-F-GK3], [@PTT-F-GK4], [@PTT-Last-1], [@PTT-Last].
However, it should be stressed that identifying the integration constants $\mathcal{\tilde{L}}$ and $\mathcal{\tilde{W}}$ with global charges, appears to be very counterintuitive from the point of view of the canonical formalism. This is because, in spite of the fact that the components of the gauge fields along $dx^{\pm}$ for the black hole solution (\[PTT-amn-GK\]) agree with the ones of the asymptotic fall-off in (\[PTT-amn-Standard W3\]), once a nonvanishing constant $\tilde{\mu}$ is included, the additional terms along $dx^{\mp}$ amount to a severe modification of the asymptotic form of the dynamical fields $a_{\theta}^{\pm}$, so that the expression for the global charges in eq. (\[PTT-Q-W3\]) no longer applies for this class of black hole solutions. Hence, as shown in [@PTT-PTT1], in full analogy with what occurs in the case of three-dimensional General Relativity coupled to scalar fields with slow fall-off at infinity [@PTT-HMTZ-2+1], [@PTT-HMTZ-Log], the effect of modifying the asymptotic behaviour is such that the total energy acquires additional nonlinear contributions in the deviation of the fields with respect to the reference background. Indeed, the variation of the total energy can be obtained directly from (\[PTT-deltaE\]), which for the case of the black hole solution (\[PTT-amn-GK\]), reads$$\begin{aligned}
\delta E & =\frac{k}{2\pi}\int\left( \left\langle a_{t}^{+}\delta
a_{\theta}^{+}\right\rangle -\left\langle a_{t}^{-}\delta a_{\theta}^{-}\right\rangle \right) d\theta\ ,\nonumber\\
& =\frac{4\pi}{l}\left[ \delta\mathcal{\tilde{L}}-\frac{32\pi}{3k}\delta(\mathcal{\tilde{L}}^{2}\mu^{2})+\tilde{\mu}\delta\mathcal{\tilde{W}}+3\mathcal{\tilde{W}}\delta\tilde{\mu}\right] \ . \label{PTT-deltaE1-GK}$$
Note that (\[PTT-deltaE1-GK\]) is not an exact differential. This is natural because the variation of the total energy not only includes the variation of the mass, but also the contribution coming from all the constraints. Therefore, in order to suitably disentangle the mass (internal energy) from the work terms, one should provide a consistent set of asymptotic conditions that allows the precise identification of the global charges as well as the chemical potentials. This is discussed in section \[PTT-Puzzles\]. Nonetheless, the expression (\[PTT-deltaE1-GK\]) provides a nice shortcut to compute the black hole entropy, circumventing the explicit computation of higher spin charges and their chemical potentials [@PTT-PTT1], [@PTT-PTT2]. This is because, by virtue of the first law, the inverse temperature $\beta$ acts as an integrating factor, being such that the product $\beta\delta E$ becomes an exact differential that corresponds to the variation of the entropy, i.e.,$$\delta S=\beta\delta E=\delta\left[ 4\pi\sqrt{2\pi k\mathcal{\tilde{L}}}\left( 1-\frac{3}{2C}\right) ^{-1}\sqrt{1-\frac{3}{4C}}\right] \ ,$$ so that the black hole entropy is given by$$S=4\pi\sqrt{2\pi k\mathcal{\tilde{L}}}\left( 1-\frac{3}{2C}\right)
^{-1}\sqrt{1-\frac{3}{4C}}\ . \label{PTT-Sc-GK}$$
As explained in [@PTT-PTT2], the entropy (\[PTT-Sc-GK\]) can be recovered from a suitable generalization of the Bekenstein-Hawking formula, given by $$S=\frac{A}{4G}\cos\left[ \frac{1}{3}\arcsin\left( 3^{3/2}\frac{\varphi_{+}}{A^{3}}\right) \right] \ , \label{PTT-EntropyHS}$$ which depends on the reparametrization invariant integrals of the pullback of the metric and the spin-$3$ field at the spacelike section of the horizon, i.e., on the horizon area $A$ and its spin-$3$ analogue:$$\varphi_{+}^{1/3}:=\int_{\partial\Sigma_{+}}\left( \varphi_{\mu\nu\rho}\frac{dx^{\mu}}{d\sigma}\frac{dx^{\nu}}{d\sigma}\frac{dx^{\rho}}{d\sigma
}\right) ^{1/3}d\sigma\ .$$
It is worth highlighting that, for the static case, and in the weak spin-$3$ field limit, our expression for the entropy (\[PTT-EntropyHS\]) reduces to$$S=\frac{A}{4G}\left. \left( 1-\frac{3}{2}\left( g^{\theta\theta}\right)
^{3}\varphi_{\theta\theta\theta}^{2}+\mathcal{O}\left( \varphi^{4}\right)
\right) \right\vert _{\rho_{+}}\ , \label{PTT-Entropy-pert}$$ in full agreement with the result found in [@PTT-CFPT2], which was obtained from a completely different approach. Indeed, in [@PTT-CFPT2] the action was written in terms of the metric and the perturbative expansion of the spin-$3$ field up to quadratic order, so that the correction to the area law in (\[PTT-Entropy-pert\]) was found by means of Wald’s formula [@PTT-Wald-Entropy].
Further results about black hole thermodynamics and along these lines have been found in [@PTT-dBJ], [@PTT-dBJ2], [@PTT-ACI], [@PTT-V], [@PTT-LLW], [@PTT-F-PTT1], and the variation of the total energy (\[PTT-deltaE1-GK\]) has also been recovered through different methods in [@PTT-CS], [@PTT-CJS].
Since the entropy is expected to be an intrinsic property of the black hole, the fact that the nonperturbative expression for the entropy $S$ in eq. (\[PTT-Sc-GK\]) differs from $\tilde{S}$ in (\[PTT-S-GK\]) by a factor that characterizes the presence of the spin-three field, i.e.,$\ S=\tilde
{S}\left( 1-\frac{3}{2C}\right) ^{-1}$, is certainly disturbing. Indeed, curiously, a variety of different approaches either lead to $\tilde{S}$ or $S$, in refs. [@PTT-GK], [@PTT-GHJ], [@PTT-KU], [@PTT-Last-1], [@PTT-Last], and [@PTT-PTT2], [@PTT-ACI], [@PTT-V], [@PTT-LLW], respectively, or even to both results [@PTT-dBJ], [@PTT-dBJ2] for the black hole entropy.
As explained in [@PTT-PTT1], [@PTT-PTT2], the discrepancy of these results stems from the mismatch in the definition of global charges aforementioned, which turns out to be inherited by the entropy once computed through the first law, even in the weak spin-$3$ field limit.
Nonetheless, some puzzles still remain to be clarified, as it is the question about how the entropy (\[PTT-Sc-GK\]) fulfills the first law of thermodynamics in the grand canonical ensemble, which is related to whether the black hole solution (\[PTT-amn-GK\]) actually carries or not a global a spin-$3$ charge. This is discussed in the next section \[PTT-Puzzles\].
Solving the puzzles: asymptotic conditions revisited and different classes of black holes {#PTT-Puzzles}
=========================================================================================
As explained in [@PTT-HPTT], [@PTT-BHPTT], the puzzles mentioned above become resolved once the asymptotic conditions are extended so as to admit a generic choice of chemical potentials associated to the higher spin charges, so that the original asymptotic $W_{3}$ symmetries are manifestly preserved by construction. In this way, any possible ambiguity is removed. This can be seen as follows. At a slice of fixed time, according to (\[PTT-amn-Standard W3\]), the asymptotic behaviour of the dynamical fields is of the form$$a_{\theta}^{\pm}=\left( L_{\pm1}-\frac{2\pi}{k}\mathcal{L}_{\pm}L_{\mp
1}-\frac{\pi}{2k}\mathcal{W}_{\pm}W_{\mp2}\right) d\theta\ ,
\label{PTT-a-theta-W3}$$ which is maintained under the gauge transformations $\Lambda^{\pm}$, defined through (\[PTT-Lambda-W3\]), with (\[PTT-deltaL-W3\]) and (\[PTT-deltaW-W3\]). In order to determine the asymptotic form of the gauge fields along time evolution, note that the field equations $F_{ti}=0$ read$$\dot{A}_{i}=\partial_{i}A_{t}+\left[ A_{i},A_{t}\right] \ ,$$ which implies that the time evolution of the dynamical fields corresponds to a gauge transformation parametrized by $A_{t}$. Hence, in order to preserve the asymptotic symmetries along the evolution in time, the Lagrange multipliers must be of the allowed form (\[PTT-Lambda-W3\]), i.e., $a_{t}^{\pm}=\Lambda^{\pm}$. Thus, following [@PTT-HPTT], the chemical potentials are included in the time component of the gauge fields only, so that the asymptotic form of the gauge fields is given by $$a^{\pm}=\pm\left( L_{\pm1}-\frac{2\pi}{k}\mathcal{L}_{\pm}L_{\mp1}-\frac{\pi
}{2k}\mathcal{W}_{\pm}W_{\mp2}\right) dx^{\pm}\pm\frac{1}{l}\Lambda^{\pm}(\nu_{\pm},\mu_{\pm})dt\ , \label{PTT-amn-pot-w3}$$ where $\nu_{\pm}$, $\mu_{\pm}$ stand for arbitrary fixed functions of $t$, $\theta$ without variation ($\delta\nu_{\pm}=\delta\mu_{\pm}=0$), that correspond to the chemical potentials. Note that, since the asymptotic form of the dynamical fields (\[PTT-a-theta-W3\]) is unchanged as compared with (\[PTT-amn-Standard W3\]), the expression for the global charges remains the same, i.e., at a fixed $t$ slice, the global charges are again given by (\[PTT-Q-W3\]), so that the asymptotic symmetries are still generated by two copies of the $W_{3}$ algebra.
Consistency then requires that the asymptotic form of $a_{t}^{\pm}$, should also be preserved under the asymptotic symmetries, which implies that the field equations have to be fulfilled in the asymptotic region, and the parameters of the asymptotic symmetries satisfy deformed chirality conditions, which read$$\begin{aligned}
l\mathcal{\dot{L}}_{\pm} & =\pm\left( 1+\nu_{\pm}\right) \mathcal{L}_{\pm
}^{\prime}\mp2\mu_{\pm}\mathcal{W}_{\pm}^{\prime}\ ,\nonumber\\
l\mathcal{\dot{W}}_{\pm} & =\pm\left( 1+\nu_{\pm}\right) \mathcal{W}_{\pm
}^{\prime}\pm\frac{2}{3}\mu_{\pm}\left( \mathcal{L}_{\pm}^{\prime\prime
\prime}-\frac{16\pi}{k}\left( \mathcal{L}_{\pm}^{2}\right) ^{\prime}\right)
\ , \label{PTT-WLpunto}$$ and$$\begin{aligned}
l\dot{\chi}_{\pm} & =\pm\left( 1+\nu_{\pm}\right) \chi_{\pm}^{\prime}\pm2\mu_{\pm}\varepsilon_{\pm}^{\prime}\ ,\nonumber\\
l\dot{\varepsilon}_{\pm} & =\pm\left( 1+\nu_{\pm}\right) \varepsilon_{\pm
}^{\prime}\mp\frac{2}{3}\mu_{\pm}\left( \chi_{\pm}^{\prime\prime\prime}-\frac{32\pi}{k}\chi_{\pm}^{\prime}\mathcal{L}_{\pm}\right) \ ,
\label{PTT-EpsilonXipunto}$$ respectively, where for simplicity, in eqs. (\[PTT-WLpunto\]), (\[PTT-EpsilonXipunto\]), the chemical potentials associated to the spin-$2$ and spin-$3$ charges, given by $\nu_{\pm}$ and $\mu_{\pm}$, were assumed to be constants.
Therefore, by construction, the functions $\mathcal{L}_{\pm}$, $\mathcal{W}_{\pm}$ are really what they mean, since their Poisson brackets fulfill the $W_{3}$ algebra with the same central extension. Note that this is so regardless the choice of chemical potentials, because the canonical generators do no depend on the Lagrange multipliers.
The asymptotic conditions given by (\[PTT-amn-pot-w3\]) then provide the required extension of the ones in [@PTT-Henneaux-Rey], [@PTT-CFPT1], since the latter are recovered when the chemical potentials are switched off, i.e., for $\nu_{\pm}=0$, $\mu_{\pm}=0$. In this case, eqs. (\[PTT-WLpunto\]) and (\[PTT-EpsilonXipunto\]) reduce to (\[PTT-FE-S-W3\]) and (\[PTT-CC-S-W3\]), respectively, expressing the fact that the fields and the parameters become chiral.
From a different perspective, the case of $\nu_{\pm}=-1$, $\mu_{\pm}=1$ has also been discussed in [@PTT-G-Lif].
It is worth emphasizing that since the Lagrange multipliers appear in the improved action through the improved generators (\[PTT-improved-G\]), the interpretation of $\nu_{\pm}$, $\mu_{\pm}$ as chemical potentials, is also guaranteed by construction. Note that this corresponds to the standard procedure one follows in the case of Reissner-Nordström black holes, where the chemical potential associated to the electric charge corresponds to the time component of the electromagnetic field, being the Lagrange multiplier of the $U(1)$ constraint.
The extended asymptotic conditions (\[PTT-amn-pot-w3\]), in the case of constant functions $\mathcal{L}_{\pm}$, $\mathcal{W}_{\pm}$ and chemical potentials $\nu_{\pm}$, $\mu_{\pm}$, then accommodate a new class of black hole solutions, endowed not only with with mass and angular momentum, but also with nontrivial well-defined spin-$3$ charges [@PTT-HPTT]. Their asymptotic and thermodynamical properties are further discussed in [@PTT-BHPTT], where it is explicitly shown that for this solution, there is no tension between the different approaches mentioned above.
Note that in the standard approach for black hole thermodynamics, the temperature and the chemical potential for the angular momentum do not explicitly appear in the fields. Instead, they enter through the identifications involving the Euclidean time and the angle, so that the range of the coordinates is not fixed and depends on the solution. The presence of nonvanishing chemical potentials $\nu_{\pm}$ associated to the spin-$2$ charges, then allows performing the description keeping the range of the coordinates fixed once and for all, i.e., $0\leq\theta<2\pi$ and $0\leq
\tau<2\pi l$, which amounts to introduce a non trivial lapse and shift in the metric formalism. Both approaches are indeed equivalent, but in the case of higher spin black holes, since the chemical potentials that correspond to the spin-$3$ charges cannot be absorbed into the modular parameter of the torus, it becomes conceptually safer to follow the latter approach, since all the chemical potentials become introduced and treated unambiguously in the same footing.
Otherwise, for instance, if the chemical potentials were not introduced along the thermal circles, but instead along additional non-vanishing components of the gauge fields along the conjugate null directions, as in the case of [@PTT-GK], the asymptotic form of the gauge fields would be given by$$a^{\pm}=\pm\left( L_{\pm1}-\frac{2\pi}{k}\mathcal{\tilde{L}}_{\pm}L_{\mp
1}-\frac{\pi}{2k}\mathcal{\tilde{W}}_{\pm}W_{\mp2}\right) dx^{\pm}\pm
\Lambda^{\pm}\left( \tilde{\nu}_{\pm},\tilde{\mu}_{\pm}\right) dx^{\mp},$$ which severely modifies the components of the dynamical fields $a_{\theta
}^{\pm}$, in a way that is incompatible with the asymptotic $W_{3}$ symmetry. This is because at a fixed $t$ slice, the terms proportional to $\tilde{\mu
}_{\pm}$ contribute to $a_{\theta}^{\pm}$ with additional terms of the form$$\begin{aligned}
a_{\theta}^{\pm} & =\left( L_{\pm1}-\frac{2\pi}{k}\mathcal{\tilde{L}}_{\pm
}L_{\mp1}-\frac{\pi}{2k}\mathcal{\tilde{W}}_{\pm}W_{\mp2}\right) +(\tilde
{\nu}_{\pm}L_{\pm1}+\tilde{\mu}_{\pm}W_{\pm2})\nonumber\\
& +\left[ \frac{1}{2}\left( -\frac{4\pi}{k}\tilde{\nu}_{\pm}\mathcal{\tilde
{L}}_{\pm}+\frac{8\pi}{k}\mathcal{\tilde{W}}_{\pm}\tilde{\mu}_{\pm}\right)
L_{\mp1}-\left( \frac{\pi}{2k}\mathcal{\tilde{W}}_{\pm}\tilde{\nu}_{\pm
}-\frac{4\pi^{2}}{k^{2}}\mathcal{\tilde{L}}_{\pm}^{2}\tilde{\mu}_{\pm}\right)
W_{\mp2}\right] \nonumber\\
& -\frac{4\pi}{k}\mathcal{\tilde{L}}_{\pm}\tilde{\mu}_{\pm}W_{0}\ ,\end{aligned}$$ that are not of highest (or lowest) weight, and hence incompatible with the asymptotic conditions (\[PTT-amn-pot-w3\]) that implement the Hamiltonian reduction of the current algebra associated to $sl(3,\mathbb{R}
)$ to the $W_{3}$ algebra. Indeed, in this case, the asymptotic symmetries that preserve the asymptotic form of $a_{\theta}$ are shown to be spanned by two copies of the Bershardsky-Polyakov algebra $W_{3}^{2}$ [@PTT-Polyakov], [@PTT-Bershadsky], corresponding to the other non trivial (so-called diagonal) embedding of $sl\left( 2,\mathbb{R}
\right) $ into $sl\left( 3,\mathbb{R}
\right) $ [@PTT-BHPTT]. Therefore, in spite of dealing with the same action, the effect of this drastic modification of the boundary conditions amounts to deal with a completely different theory, being characterized by a different field content, and hence with an inequivalent spectrum, so that their corresponding black hole solutions, as the one in (\[PTT-amn-GK\]), are characterized by another set of global charges of lower spin.
It is worth pointing out that our procedure to incorporate chemical potentials can be straightforwardly extended to the case of $\mathfrak{g}_{\pm}=sl\left(
N,\mathbb{R}
\right) $, regardless the way in which $sl\left( 2,\mathbb{R}
\right) $ is embedded, as well as to the case of infinite-dimensional higher spin algebras.
Some closing remarks are in order. It should be mentioned that the case of three-dimensional gravity nonminimally coupled with spin-$3$ fields, also appears to be consistently formulated in the second-order formalism by introducing a suitable set of auxiliary fields [@PTT-FN]. Besides, in the case of spin-$3$ and higher, consistent sets of asymptotic conditions have also been proposed in [@PTT-Henneaux-Rey], [@PTT-CFP], [@PTT-GH], while exact solutions and their properties have been explored in [@PTT-PK], [@PTT-TAN], [@PTT-GGR], [@PTT-BCT], [@PTT-FHK]. In the context of higher spin supergravity in three dimensions, the asymptotic structure was analyzed in [@PTT-Super-HS], and exact solutions have also been found in [@PTT-Super-1], [@PTT-Super-SD], [@PTT-Super-T]. Moreover, along the lines of holography and the corresponding dual CFT theory with extended conformal symmetry at the boundary [@PTT-GGS], [@PTT-H1], [@PTT-H2], further interesting results can also be found in [@PTT-CGGR], [@PTT-H4], [@PTT-H5], [@PTT-H6], [@PTT-CPR], [@PTT-H7], [@PTT-GJP], [@PTT-CF].
We thank G. Barnich, X. Bekaert, E. Bergshoeff, C. Bunster, A. Campoleoni, R. Canto, D. Grumiller, M. Henneaux, J. Jottar, C. Martínez, J. Matulich, J. Ovalle, R. Rahman, S-J Rey, J. Rosseel, C. Troessaert and M. Vasiliev for stimulating discussions. R.T. also thanks L. Papantonopoulos and the organizers of the Seventh Aegean Summer School, Beyond Einstein’s theory of gravity, for the opportunity to give this lecture in a wonderful atmosphere. This work is partially funded by the Fondecyt grants N${^{\circ}}$ 1130658, 1121031, 11130260, 11130262. The Centro de Estudios Científicos (CECs) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of Conicyt.
[99]{}
P. Van Nieuwenhuizen, Supergravity, Phys. Rept. **68**, 189 (1981).
X. Bekaert, N. Boulanger and P. Sundell, “How higher-spin gravity surpasses the spin two barrier: no-go theorems versus yes-go examples,” Rev. Mod. Phys. **84**, 987 (2012) \[arXiv:1007.0435 \[hep-th\]\].
C. Aragone and S. Deser, “Consistency Problems of Hypergravity,” Phys. Lett. B **86**, 161 (1979).
M. A. Vasiliev, Consistent equation for interacting gauge fields of all spins in (3+1)-dimensions, Phys. Lett. B **243**, 378 (1990).
M. A. Vasiliev, Nonlinear equations for symmetric massless higher spin fields in (A)dS(d), Phys. Lett. B **567**, 139 (2003) \[hep-th/0304049\].
X. Bekaert, S. Cnockaert, C. Iazeolla and M. A. Vasiliev, Nonlinear higher spin theories in various dimensions, hep-th/0503128.
A. Sagnotti, Notes on Strings and Higher Spins, J. Phys. A **46**, 214006 (2013) \[arXiv:1112.4285 \[hep-th\]\].
M. A. Vasiliev, “Higher-Spin Theory and Space-Time Metamorphoses,” arXiv:1404.1948 \[hep-th\].
R. Troncoso and J. Zanelli, “New gauge supergravity in seven-dimensions and eleven-dimensions,” Phys. Rev. D **58**, 101703 (1998) \[hep-th/9710180\].
M. P. Blencowe, A Consistent Interacting Massless Higher Spin Field Theory In D = (2+1), Class. Quant. Grav. **6**, 443 (1989).
E. Bergshoeff, M. P. Blencowe and K. S. Stelle, Area Preserving Diffeomorphisms And Higher Spin Algebra, Commun. Math. Phys. **128**, 213 (1990).
M. A. Vasiliev, Higher spin gauge theories in four-dimensions, three-dimensions, and two-dimensions, Int. J. Mod. Phys. D **5**, 763 (1996) \[hep-th/9611024\].
M. Gutperle and P. Kraus, “Higher Spin Black Holes,” JHEP **1105**, 022 (2011) \[arXiv:1103.4304 \[hep-th\]\].
A. Castro, E. Hijano, A. Lepage-Jutier and A. Maloney, “Black Holes and Singularity Resolution in Higher Spin Gravity,” JHEP **1201**, 031 (2012) \[arXiv:1110.4117 \[hep-th\]\].
M. Henneaux, A. Pérez, D. Tempo and R. Troncoso, Chemical potentials in three-dimensional higher spin anti-de Sitter gravity, JHEP **1312**, 048 (2013) \[arXiv:1309.4362 \[hep-th\]\].
C. Bunster, M. Henneaux, A. Pérez, D. Tempo and R. Troncoso, Generalized Black Holes in Three-dimensional Spacetime, JHEP **1405**, 031 (2014) \[arXiv:1404.3305 \[hep-th\]\].
A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, “Towards metric-like higher-spin gauge theories in three dimensions,” J. Phys. A **46**, 214017 (2013) \[arXiv:1208.1851 \[hep-th\]\].
J. D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. **104**, 207 (1986).
M. Bañados, C. Teitelboim and J. Zanelli, The Black hole in three-dimensional space-time, Phys. Rev. Lett. **69**, 1849 (1992) \[hep-th/9204099\].
M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2+1) black hole, Phys. Rev. D **48**, 1506 (1993) \[gr-qc/9302012\].
M. Henneaux and S. -J. Rey, Nonlinear $W_{infinity}$ as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity, JHEP **1012**, 007 (2010) \[arXiv:1008.4579 \[hep-th\]\].
A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields, JHEP **1011**, 007 (2010) \[arXiv:1008.4744 \[hep-th\]\].
M. Ammon, M. Gutperle, P. Kraus and E. Perlmutter, “Spacetime Geometry in Higher Spin Gravity,” JHEP **1110**, 053 (2011) \[arXiv:1106.4788 \[hep-th\]\].
A. Pérez, D. Tempo and R. Troncoso, Higher spin gravity in 3D: Black holes, global charges and thermodynamics, Phys. Lett. B **726**, 444 (2013) \[arXiv:1207.2844 \[hep-th\]\].
A. Pérez, D. Tempo and R. Troncoso, Higher spin black hole entropy in three dimensions, JHEP **1304**, 143 (2013) \[arXiv:1301.0847 \[hep-th\]\].
A. P. Balachandran, G. Bimonte, K. S. Gupta and A. Stern, Conformal edge currents in Chern-Simons theories, Int. J. Mod. Phys. A **7**, 4655 (1992) \[hep-th/9110072\].
M. Bañados, Global charges in Chern-Simons field theory and the (2+1) black hole, Phys. Rev. D **52**, 5816 (1996) \[hep-th/9405171\].
S. Carlip, Conformal field theory, (2+1)-dimensional gravity, and the BTZ black hole, Class. Quant. Grav. **22**, R85 (2005) \[gr-qc/0503022\].
T. Regge and C. Teitelboim, Role Of Surface Integrals In The Hamiltonian Formulation Of General Relativity, Annals Phys. **88**, 286 (1974).
A. Achucarro and P. K. Townsend, A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories, Phys. Lett. B **180**, 89 (1986).
E. Witten, (2+1)-Dimensional Gravity as an Exactly Soluble System, Nucl. Phys. B **311**, 46 (1988).
O. Coussaert, M. Henneaux and P. van Driel, “The Asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant,” Class. Quant. Grav. **12**, 2961 (1995) \[gr-qc/9506019\].
S. Carlip and C. Teitelboim, “Aspects of black hole quantum mechanics and thermodynamics in (2+1)-dimensions,” Phys. Rev. D **51**, 622 (1995) \[gr-qc/9405070\].
J. M. Maldacena and A. Strominger, “AdS(3) black holes and a stringy exclusion principle,” JHEP **9812**, 005 (1998) \[hep-th/9804085\].
H. A. González, J. Matulich, M. Pino and R. Troncoso, Asymptotically flat spacetimes in three-dimensional higher spin gravity, JHEP **1309**, 016 (2013) \[arXiv:1307.5651 \[hep-th\]\].
H. Afshar, A. Bagchi, R. Fareghbal, D. Grumiller and J. Rosseel, Higher spin theory in 3-dimensional flat space, Phys. Rev. Lett. **111**, 121603 (2013) \[arXiv:1307.4768 \[hep-th\]\].
H. A. González and M. Pino, Boundary dynamics of asymptotically flat 3D gravity coupled to higher spin fields, arXiv:1403.4898 \[hep-th\].
D. Grumiller, M. Riegler and J. Rosseel, “Unitarity in three-dimensional flat space higher spin theories,” arXiv:1403.5297 \[hep-th\].
C. Krishnan and S. Roy, “Higher Spin Resolution of a Toy Big Bang,” Phys. Rev. D **88**, 044049 (2013) \[arXiv:1305.1277 \[hep-th\]\].
B. Burrington, L. A. Pando Zayas and N. Rombes, On Resolutions of Cosmological Singularities in Higher-Spin Gravity, arXiv:1309.1087 \[hep-th\].
C. Krishnan and S. Roy, “Desingularization of the Milne Universe,” arXiv:1311.7315 \[hep-th\].
C. Krishnan, A. Raju and S. Roy, A Grassmann Path From Ad$S_{3}$ to Flat Space, arXiv:1312.2941 \[hep-th\].
G. Compère and W. Song, “$\mathcal{W}$ symmetry and integrability of higher spin black holes,” JHEP **1309**, 144 (2013) \[arXiv:1306.0014 \[hep-th\]\].
G. Compère, J. I. Jottar and W. Song, “Observables and Microscopic Entropy of Higher Spin Black Holes,” JHEP **1311**, 054 (2013) \[arXiv:1308.2175 \[hep-th\]\].
M. R. Gaberdiel, T. Hartman and K. Jin, Higher Spin Black Holes from CFT, JHEP **1204**, 103 (2012) \[arXiv:1203.0015 \[hep-th\]\].
M. R. Gaberdiel and R. Gopakumar, “Minimal Model Holography,” J. Phys. A **46**, 214002 (2013) \[arXiv:1207.6697 \[hep-th\]\].
M. Ammon, M. Gutperle, P. Kraus and E. Perlmutter, “Black holes in three dimensional higher spin gravity: A review,” J. Phys. A **46**, 214001 (2013) \[arXiv:1208.5182 \[hep-th\]\].
K. Jin, “Higher Spin Gravity and Exact Holography,” PoS Corfu **2012**, 086 (2013) \[arXiv:1304.0258 \[hep-th\]\].
B. Chen, J. Long and Y. -n. Wang, “Black holes in Truncated Higher Spin AdS$_{3}$ Gravity,” JHEP **1212**, 052 (2012) \[arXiv:1209.6185 \[hep-th\]\].
J. R. David, M. Ferlaino and S. P. Kumar, Thermodynamics of higher spin black holes in 3D, JHEP **1211**, 135 (2012) \[arXiv:1210.0284 \[hep-th\]\].
B. Chen, J. Long and Y. -N. Wang, Phase Structure of Higher Spin Black Hole,JHEP **1303**, 017 (2013) \[arXiv:1212.6593\].
P. Kraus and T. Ugajin, “An Entropy Formula for Higher Spin Black Holes via Conical Singularities,” JHEP **1305**, 160 (2013) \[arXiv:1302.1583 \[hep-th\]\].
J. de Boer and J. I. Jottar, “Thermodynamics of higher spin black holes in $AdS_{3}$,” JHEP **1401**, 023 (2014) \[arXiv:1302.0816 \[hep-th\]\].
J. de Boer and J. I. Jottar, Entanglement Entropy and Higher Spin Holography in AdS$_{3}$, arXiv:1306.4347 \[hep-th\].
M. Beccaria and G. Macorini, “On the partition functions of higher spin black holes,” JHEP **1312**, 027 (2013) \[arXiv:1310.4410 \[hep-th\]\].
M. Beccaria and G. Macorini, Analysis of higher spin black holes with spin-4 chemical potential, arXiv:1312.5599 \[hep-th\].
B. Chen, J. Long and J. -j. Zhang, “Holographic Rényi entropy for CFT with $W$ symmetry,” arXiv:1312.5510 \[hep-th\].
S. Datta, J. R. David, M. Ferlaino and S. P. Kumar, “Higher spin entanglement entropy from CFT,” arXiv:1402.0007 \[hep-th\].
M. Henneaux, C. Martínez, R. Troncoso and J. Zanelli, Black holes and asymptotics of 2+1 gravity coupled to a scalar field, Phys. Rev. D **65**, 104007 (2002) [\[]{}arXiv:hep-th/0201170[\]]{}.
M. Henneaux, C. Martínez, R. Troncoso and J. Zanelli, Asymptotically anti-de Sitter spacetimes and scalar fields with a logarithmic branch, Phys. Rev. D **70**, 044034 (2004) \[hep-th/0404236\].
R. M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D **48**, 3427 (1993) \[gr-qc/9307038\].
M. Ammon, A. Castro and N. Iqbal, “Wilson Lines and Entanglement Entropy in Higher Spin Gravity,” JHEP **1310**, 110 (2013) \[arXiv:1306.4338 \[hep-th\]\].
L. McGough and H. Verlinde, “Bekenstein-Hawking Entropy as Topological Entanglement Entropy,” JHEP **1311**, 208 (2013) \[arXiv:1308.2342 \[hep-th\]\].
W. Li, F. -L. Lin and C. -W. Wang, “Modular Properties of 3D Higher Spin Theory,” JHEP **1312**, 094 (2013) \[arXiv:1308.2959 \[hep-th\]\].
A. Chowdhury and A. Saha, “Phase Structure of Higher Spin Black Holes,” arXiv:1312.7017 \[hep-th\].
M. Gutperle, E. Hijano and J. Samani, “Lifshitz black holes in higher spin gravity,” arXiv:1310.0837 \[hep-th\].
A. M. Polyakov, “Gauge Transformations and Diffeomorphisms,” Int. J. Mod. Phys. A **5**, 833 (1990).
M. Bershadsky, Conformal field theories via Hamiltonian reduction, Commun. Math. Phys. **139**, 71 (1991).
I. Fujisawa and R. Nakayama, Second-Order Formalism for 3D Spin-3 Gravity, Class. Quant. Grav. **30**, 035003 (2013) \[arXiv:1209.0894 \[hep-th\]\].
A. Campoleoni, S. Fredenhagen and S. Pfenninger, Asymptotic W-symmetries in three-dimensional higher-spin gauge theories, JHEP **1109**, 113 (2011) \[arXiv:1107.0290 \[hep-th\]\].
M. R. Gaberdiel and T. Hartman, “Symmetries of Holographic Minimal Models,” JHEP **1105**, 031 (2011) \[arXiv:1101.2910 \[hep-th\]\].
P. Kraus and E. Perlmutter, Partition functions of higher spin black holes and their CFT duals, JHEP **1111**, 061 (2011) \[arXiv:1108.2567 \[hep-th\]\].
H. -S. Tan, Aspects of Three-dimensional Spin-4 Gravity, JHEP **1202**, 035 (2012) \[arXiv:1111.2834 \[hep-th\]\].
M. Gary, D. Grumiller and R. Rashkov, Towards non-AdS holography in 3-dimensional higher spin gravity, JHEP **1203**, 022 (2012) \[arXiv:1201.0013 \[hep-th\]\].
M. Bañados, R. Canto and S. Theisen, The Action for higher spin black holes in three dimensions, JHEP **1207**, 147 (2012) \[arXiv:1204.5105 \[hep-th\]\].
M. Ferlaino, T. Hollowood and S. P. Kumar, Asymptotic symmetries and thermodynamics of higher spin black holes in AdS3, Phys. Rev. D **88**, 066010 (2013) \[arXiv:1305.2011 \[hep-th\]\].
M. Henneaux, G. Lucena Gómez, J. Park and S. -J. Rey, Super- W(infinity) Asymptotic Symmetry of Higher-Spin $AdS_{3}$ Supergravity, JHEP **1206**, 037 (2012) \[arXiv:1203.5152 \[hep-th\]\].
B. Chen, J. Long and Y. -N. Wang, Conical Defects, Black Holes and Higher Spin (Super-)Symmetry, JHEP **1306**, 025 (2013) \[arXiv:1303.0109 \[hep-th\]\].
S. Datta and J. R. David, “Black holes in higher spin supergravity,” JHEP **1307**, 110 (2013) \[arXiv:1303.1946 \[hep-th\]\].
H. S. Tan, Exploring Three-dimensional Higher-Spin Supergravity based on sl(N $\vert$N - 1) Chern-Simons theories, JHEP **1211**, 063 (2012) \[arXiv:1208.2277 \[hep-th\]\].
M. R. Gaberdiel, R. Gopakumar and A. Saha, Quantum $W$-symmetry in $AdS_{3}$, JHEP **1102**, 004 (2011) \[arXiv:1009.6087 \[hep-th\]\].
M. R. Gaberdiel and R. Gopakumar, An $AdS_{3}$ Dual for Minimal Model CFTs, Phys. Rev. D **83**, 066007 (2011) \[arXiv:1011.2986 \[hep-th\]\].
M. R. Gaberdiel, R. Gopakumar, T. Hartman and S. Raju, Partition Functions of Holographic Minimal Models, JHEP **1108**, 077 (2011) \[arXiv:1106.1897 \[hep-th\]\].
A. Castro, R. Gopakumar, M. Gutperle and J. Raeymaekers, “Conical Defects in Higher Spin Theories,” JHEP **1202**, 096 (2012) \[arXiv:1111.3381 \[hep-th\]\].
M. Ammon, P. Kraus and E. Perlmutter, Scalar fields and three-point functions in D=3 higher spin gravity, JHEP **1207**, 113 (2012) \[arXiv:1111.3926 \[hep-th\]\].
M. R. Gaberdiel and P. Suchanek, Limits of Minimal Models and Continuous Orbifolds, JHEP **1203**, 104 (2012) \[arXiv:1112.1708 \[hep-th\]\].
P. Kraus and E. Perlmutter, “Probing higher spin black holes,” JHEP **1302**, 096 (2013) \[arXiv:1209.4937 \[hep-th\]\].
S. Banerjee, A. Castro, S. Hellerman, E. Hijano, A. Lepage-Jutier, A. Maloney and S. Shenker, Smoothed Transitions in Higher Spin AdS Gravity, Class. Quant. Grav. **30**, 104001 (2013) \[arXiv:1209.5396 \[hep-th\]\].
A. Campoleoni, T. Prochazka and J. Raeymaekers, “A note on conical solutions in 3D Vasiliev theory,” JHEP **1305**, 052 (2013) \[arXiv:1303.0880 \[hep-th\]\].
M. R. Gaberdiel, K. Jin and E. Perlmutter, Probing higher spin black holes from CFT, JHEP **1310**, 045 (2013) \[arXiv:1307.2221 \[hep-th\]\].
A. Campoleoni and S. Fredenhagen, “On the higher-spin charges of conical defects,” Phys. Lett. B **726**, 387 (2013) \[arXiv:1307.3745\].
|
---
abstract: 'We propose an experiment, which would allow to pinpoint the role of spin-orbit coupling in the metal-nonmetal transition observed in a number of two-dimensional systems at low densities. Namely, we demonstrate that in a parallel magnetic field the interplay between the spin-orbit coupling and the Zeeman splitting leads to a characteristic anisotropy of resistivity [*with respect to the direction of the in-plane magnetic field*]{}. Though our analytic calculation is done in the deeply insulating regime, the anisotropy is expected to persist far beyond that regime.'
address: 'Department of Physics, University of Utah, Salt Lake City, Utah 84112'
author:
- 'Guang-Hong Chen, M. E. Raikh, and Yong-Shi Wu'
title: 'Does spin-orbit coupling play a role in metal-nonmetal transition in two-dimensional systems?'
---
In a recent paper[@papada99] an interesting experimental observation was reported. It was demonstrated that the period of beats of the Shubnikov-de Haas oscillations in a two-dimensional hole system is strongly correlated with the zero-magnetic-field temperature dependence of the resistivity. The beats of the Shubnikov-de Haas oscillations have their origin in the splitting of the spin subbands in a zero magnetic field,[@luo; @Das]. The authors of Ref. were able to tune the zero-field splitting by changing the gate voltage. They observed that, while in the absence of the subband splitting, the zero-magnetic-field resistivity was temperature-independent below $T=0.7$K , a pronounced rise (by $5$ percent) in resistivity with temperature emerged in the interval $0.2K < T < 0.7K$ at the maximal subband splitting, indicating a metallic-like behavior. This close correlation suggests that it is a mechanism causing the spin subband splitting that plays an important role in the crossover from the metallic-like to the insulating-like temperature dependence of resistivity with decreasing carrier density (the metal-nonmetal transition). This transition by now has been experimentally observed in a number of different two-dimensional electron[@kra; @krav; @pop; @han] and hole[@lam; @col; @pap; @hane; @sim] systems. By challenging the commonly accepted concepts, it has attracted a lot of theoretical interest and attempts to identify the underlying mechanism. Possible relevance of zero-field splitting to the transition was first conjectured in Ref. . The evidence presented in Ref. about the importance of the subband splitting for metallic-like behavior of resistivity is further supported by the very recent data reported in Ref. .
Another important feature of the metal-nonmetal transition, which might also provide a clue for the understanding of its origin, is that the metallic phase is destroyed by a relatively weak [*parallel*]{} magnetic field[@sim; @simonian97; @prinz; @kravchenko98; @simonian98; @mertes99; @okamoto98]. At the same time, no quenching of the metallic phase in a parallel magnetic field was observed in SiGe hole gas[@senz99], in which the strain, caused by the lattice mismatch, splits the light and heavy holes.
As far as the theory is concerned, the role of the parallel magnetic field was previously accounted for exclusively through the Zeeman energy, which either alters the exchange interactions (and, thus, electron-ion binding energy[@klap; @alt]) or suppresses the liquid phase,[@he] or affects the transmittancy of the point contact between the phase-coherent regions[@meir].
It is appealing to combine the observations [@papada99; @Yaish99] of the subband splitting in zero field and the results [@sim; @simonian97; @prinz; @kravchenko98; @simonian98; @mertes99; @okamoto98] in a parallel magnetic field within a single picture. The spin-orbit (SO) coupling appears to be a promising candidate for such a unifying mechanism. Indeed, on one hand, it is known to lead to spin subband splitting. On the other hand, a parallel magnetic field, though not affecting the orbital in-plane motion, destroys the SO coupling and, thus, suppresses the intersubband transitions. Possible importance of these transitions was emphasized in Ref. . Their suppression with increasing magnetic field is caused by the fact that the corresponding subband wave functions become orthogonal for [*all*]{} wave vectors.
At the present moment there is no consensus in the literature about the role of the SO coupling. Several authors [@skvortsov97; @dmitriev; @lyanda'] have explored the role of the SO coupling as a possible source of the metallic-like behavior, by considering noninteracting two-dimensional system and including the SO terms into the calculation of the weak-localization corrections. At the same time, the majority of theoretical works [@klap; @alt; @he; @meir; @dobro; @belitz; @chak1; @phillips; @cast; @si; @das; @neil; @chak2], stimulated by the experimental observation of the transition, disregarded the SO coupling.
To pinpoint the role of the SO coupling in the metal-nonmetal transition, it seems important to find a [*qualitative*]{} effect which exists only in the presence of the SO coupling. Such an effect is proposed in the present paper. We show that an interplay between the SO coupling and the Zeeman splitting gives rise to a characteristic [*anisotropy*]{} of resistivity with respect to the [*direction*]{} of the parallel magnetic field. Obviously, the Zeeman splitting alone cannot induce any anisotropy. To demonstrate the effect, we consider the deeply insulating regime, where the physical picture of transport is transparent.
We choose the simplest form for the spin-orbit Hamiltonian [@rashba; @rashba'] $$\label{hso}
\hat{H}_{SO}=\alpha{\bf k}\cdot({\bf\bbox{ \sigma}}\times {\bf \hat{z}}).$$ Here $\alpha$ is the SO coupling constant, ${\bf k}$ is the wave vector, ${\bf \hat{z}}$ is the unit vector normal to the 2D plane, $\bbox{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ are the Pauli matrices. In the presence of the parallel megnetic field, the single particle Hamiltonian can be written as $$\label{ham}
\hat{H}=\frac{\hbar^2k^2}{2m}+\alpha{\bf k}\cdot({\bf\bbox{ \sigma}}\times
{\bf \hat{z}})+g\mu_B{\bf\bbox{ \sigma}}\cdot{\bf
B}=\left(\begin{array}{cc}\frac{\hbar^2k^2}{2m} & \frac{\Delta_Z}{2}
e^{-i\phi_{\bf B}}-i\alpha k e^{-i\phi_{{\bf k}}}\\ \frac{\Delta_Z}{2}
e^{i\phi_{\bf B}}+i\alpha k e^{i\phi_{\bf k}} &
\frac{\hbar^2k^2}{2m}\end{array}\right),$$ where $m$ is the effective mass, $g$ and $\mu_B$ are the g-factor and the Bohr magneton respectively; $\Delta_Z=2g\mu_B B$ is the Zeeman splitting; $\phi_{\bf B}$ and $\phi_{\bf k}$ are, correspondingly, the azimuthal angles of magnetic field ${\bf B}$ (Fig. 1) and the wave vector ${\bf k}$. The energy spectrum of the Hamiltonian Eq. (\[ham\]) is given by $$\label{spectrum}
E_{\pm}({\bf
k})=\frac{\hbar^2k^2}{2m}\pm\frac{1}{2}\sqrt{\Delta_Z^2+4\alpha^2k^2+4\alpha
k \Delta_Z\sin(\phi_{{\bf B}}-\phi_{{\bf k}})}.$$ Note that the spectrum is anisotropic only if [*both*]{} $\Delta_Z$ and $\alpha$ are nonzero.
The standard procedure for the calculation of the hopping conductance is the following[@book]. Denote with $P_{12}$ the hopping probability between the localized states $1$ and $2$. The logarithm of $P_{12}$ represents the sum of two terms $$\label{log}
\ln P_{12}=-\frac{\varepsilon_{12}}{T}-\ln|G({\bf R})|^2,$$ where the first term originates from the activation; $\varepsilon_{12}$ is the activation energy[@book] and $T$ is the temperature. The second term in Eq. (\[log\]) describes the overlap of the wave functions of the localized states centered at points ${\bf R}_1$ and ${\bf R}_2$, so that ${\bf R}={\bf R}_1-{\bf R}_2$. In Eq. (\[log\]) we use the fact that within the prefactor the overlap integral coincides with the Green function $G({\bf R})$. For the matrix Hamiltonian Eq. (\[ham\]), the Green function is also a matrix $$\label{green}
\hat{G}({\bf R})=\int\frac{d^2{\bf k}}{(2\pi)^2}\frac{e^{i{\bf k}\cdot{\bf
R}}}{E-\hat{H}({\bf k})}.$$ By projecting onto the eigen-space of Hamiltonian Eq. (\[ham\]), the above expression can be presented as $$\label{proj}
\hat{G}({\bf R})=\int\frac{dk k d\phi_{\bf
k}}{(2\pi)^2}e^{ikR\cos(\phi_{\bf k}-\phi_{\bf
R})}\biggl[\frac{\hat{P}_{+}({\bf k})}{E-E_{+}({\bf
k})}+\frac{\hat{P}_{-}({\bf k})}{E-E_{-}({\bf k})}\biggr],$$ where the projection operators $P_{\pm}({\bf k})$ are defined as $$\label{p+-}
\hat{P}_{+}({\bf k})=\frac{1}{2}\left(\begin{array}{cc}1&
O({\bf k})\\ O^{*}({\bf k}) & 1\end{array}\right),
\hspace{1.2cm}\hat{P}_{-}({\bf k})=1-\hat{P}_{+}({\bf k}),$$ where $O^{*}({\bf k})$ is the complex conjugate of $O({\bf k})$, which is defined as $$\label{off}
O({\bf k})=\frac{\frac{\Delta_Z}{2}\exp(-i\phi_{\bf B})-i\alpha k\exp(-i\phi_{{\bf
k}})}{E_{+}({\bf k})-E_{-}({\bf k})}.$$
When the distance $R$ is much larger than the localization radius, $a_0$, the integral over $\phi_{\bf k}$ is determined by a narrow interval $|\phi_{\bf k}-\phi_{\bf R}|\sim (kR)^{-1/2}\ll 1$. This allows to replace $\phi_{\bf k}$ by $\phi_{\bf B}$ in the square brackets and perform the angular integration. Then we obtain $$\label{green2}
\hat{G}({\bf R})=\sqrt{\frac{2\pi}{iR}}\int_0^{\infty}\frac{dk
\sqrt{k}}{(2\pi)^2}e^{ikR}\biggl[ \frac{\hat{P}_{+}(k,\phi_{\bf
R})}{E-E_{+}(k,\phi_{\bf R})}+\frac{\hat{P}_{-}(k,\phi_{\bf
R})}{E-E_{-}(k,\phi_{\bf R})}\biggr].$$ The next step of the integration is also standard. Namely, for large $R$, the $\hat{G}({\bf R})$ is determined by the poles of the integrand. However, in the case under the consideration, the equation $E_{\pm}(k)=E$ leads to a fourth-order algebraic equation. To simplify the calculations we will restrict ourselves to the strongly localized regime $|E|\gg
m\alpha^2/\hbar^2$. In this case the poles can be found by the successive approximations. In the zero-order approximation, we get the standard result $k=ik_0$, where $k_0$ is defined as $$\label{k0}
k_0=a_0^{-1}=\frac{\sqrt{2m|E|}}{\hbar}.$$ In the first order approximation, we have $k=ik_0+k_1$ where $k_1$ is given by $$\label{k1}
k_1=\pm
i\frac{m\alpha}{\hbar^2}\sqrt{\Delta_1^2-1+2i\Delta_1\sin(\phi_{\bf
B}-\phi_{\bf R})},$$ where the dimensionless Zeeman splitting $\Delta_1$ is defined as $$\label{delta}
\Delta_1=\Delta_Z/2\alpha k_0.$$ Within this approximation, the long-distance asymptotics of the Green function is $$\label{asym}
\hat{G}(R)\propto e^{-R/a(\phi_{\bf B},\phi_{\bf R})},$$ where the decay length is given by $$\label{decay}
a(\phi_{\bf B}, \phi_{\bf R})^{-1}
=k_0\biggl(1-\frac{m\alpha}
{\hbar^2} Re\sqrt{\Delta_1^2-1+2i\Delta_1\sin(\phi_{\bf B}
-\phi_{\bf R})}\biggr).$$ In the last equation it is assumed that the real part, $Re(...)$, has a positive sign. Our main observation is that the decay length and, concommitantly, the probability of hopping are [*anisotropic*]{}, when the parallel megnetic field and the SO coupling are present [*simultaneously*]{}. By evaluating the real part in Eq. (\[decay\]) we obtain $$\label{1/a}
a(\phi_{\bf B}, \phi_{\bf R})^{-1}
=k_0\biggl(1-\frac{m\alpha}{\sqrt{2}\hbar^2k_0}
\sqrt{\Delta_1^2-1+\sqrt{1+\Delta_1^4-2\Delta_1^2
\cos2(\phi_{\bf B}-\phi_{\bf R})}}\biggr).$$ To characterize the anisotropy quantitatively, we introduce the perpendicular decay length $a_{\perp}=
a(\phi_{\bf B}-\phi_{\bf R}=\pm\frac{\pi}{2})$ and the parallel decay length $a_{\parallel}=a(\phi_{\bf B}=\phi_{\bf R})$. Then a quantitative measure of the anisotropy can be defined as $$\label{anisotropy}
\frac{a_{\perp}-a_{\parallel}}{a_0}
=\frac{m\alpha}{\hbar^2k_0}f(\Delta_1),$$ where the function $f(x)$ is given by $$\label{universal}
f(x)=x-(x^2-1)^{1/2}\theta(x-1),$$ where $\theta(x)$ is the step-function. In the strongly localized regime ($\alpha k_0\ll |E|$) the anisotropy is weak. The magnetic field dependence of the anisotropy is shown in Fig. 2. It can be seen that the maximal anisotropy corresponds to $\Delta_1=1$ and it vanishes both in strong and weak magnetic fields. The theory of hopping transport in the systems with anisotropic localization radius is presented in Ref. . The principal outcome of this theory is that the anisotropy of the localization radius (and, consequently, the exponential anisotropy of the hopping probability (\[log\])) [*does not*]{} lead to the [*exponential*]{} anisotropy of the hopping resistance. In fact, the exponent of the resistance is the same as for the isotropic hopping with localization radius $\sqrt{a_{\parallel}a_{\perp}}$. However, the anisotropy in the Green function manifests itself in the prefactor of the hopping resistance[@book] $$\label{hopres}
\frac{\rho_{\parallel}-\rho{\perp}}{\rho_{\parallel}+\rho{\perp}}=C\frac{a_{\parallel}-a_{\perp}}{a_{\parallel}+a_{\perp}}\approx C\frac{m\alpha}{2\hbar^2k_0}f(\Delta_1),$$ where $C \sim 1$ is the numerical factor, determined by the perturbation theory in the method of invariants for random bond percolation problem[@book]. The exact value of the constant $C$ depends on the regime of hopping (nearest-neighbor or variable-range hopping).
The microscopic origin of the SO Hamiltonaian Eq. (\[hso\]) is the asymmery of the confinement potential. In III-V semiconductor quantum wells there exists another mechanism of the SO coupling, which originates from the absence of the inversion symmetry in the bulk (the Dresselhaus mechanism[@dress]). Within this mechanism $\hat{H}_{SO}=\beta(\sigma_xk_x-\sigma_yk_y)$ (for \[001\] growth direction). Then the calculation similar to the above leads to the following result for the anisotropic decay length $$\label{anidress}
a(\phi_{\bf B}, \phi_{\bf R})^{-1}
=k_0\biggl(1-\frac{m\beta} {\sqrt{2}\hbar^2k_0}
\sqrt{\Delta_2^2-1+\sqrt{1+\Delta_2^4
+2\Delta_2^2\cos(\phi_{\bf B}+\phi_{\bf R})}}\biggr),$$ where $\Delta_2$ is related to the Zeeman splitting as $$\label{dell}
\Delta_2=\Delta_Z/2\beta k_0.$$ By using Eq. (\[anidress\]) we get for anisotropy $$\label{dress}
\frac{a_{\perp}-a_{\parallel}}{a_0}=
-\frac{m\beta}{\hbar^2k_0}f(\Delta_2),$$ where the function $f$ is determined by Eq. (\[universal\]).
In conclusion, we have demonstrated that, due to the SO coupling, the rotation of an in-plane magnetic field with respect to the direction of current should lead to a characteristic angular variation of resistivity with a period $\pi$. The anisotropy is maximal for intermediate magnetic fields and vanishes in the weak and the strong-field limits. In the strongly localized regime, considered in the present paper, the magnitude of anisotropy is small. However, as seen from Eqs. (\[anisotropy\]) and (\[hopres\]), the magnitude of anisotropy should increase as the Fermi level moves up with increasing carrier concentration (since $k_0$ decreases). So the resistivity is expected to remain anisotropic, perhaps with a modified angular dependence, far beyond the deeply insulating regime. For high enough concentrations the SO coupling (subband splitting) is negligible, so that the anisotropy should be also weak. If the intersubband scattering governs the metal-nonmetal transition, then the resistivity anisotropy should reach maximum around the critical density.
Finally, let us discuss two possible complications for the experimental observation of the anisotropy in resistivity. Both of them stem from the fact that a realistic two-dimensional system has a finite thickness. Firstly, with finite thickness, even a small deviation of the magnetic field direction from the in-plane position would cause a certain anisotropy even without SO coupling. However, in this case, the anisotropy would only increase with increasing magnetic field, while the SO-induced anisotropy should vanish in the strong-field limit. The second effect of the finite thickness is that it causes the anisotropy of the Dresselhaus term with respect to the crystalline axes. As it is shown in Ref. , the interplay of anisotropic Dresselhaus and isotropic Bychkov-Rashba terms results in the [*crystalline*]{} anisotropy of the resistivity in the weak-localization regime. This effect should be distinguished from the anisotropy [*with respect to the direction of current*]{} predicted in the present paper.
[*Acknowledgements*]{}. The authors are grateful to R. R. Du for helpful discussions. M.E.R. acknowledges the support of the NSF grant INT-9815194, and Y.S.W. the NSF grant PHY-9601277.
S. J. Papadakis, E. P. De Poortere, H. C. Manohoran, M. Shayegan, and R. Winkler, Science [**283**]{}, 2056 (1999).
J. Luo, H. Munekata, F. F. Fang, and P. J. Stiles, Phys. Rev. B [**38**]{}, 10142 (1988).
B. Das, D. C. Miller, S. Datta, R. Reifenberger, W. P. Hong, P. K. Bhattacharya, J. Singh, and M. Jaffe, Phys. Rev. B [**39**]{}, 1411 (1989).
S. V. Kravchenko, W. Mason, J. E. Furneaux, and V. M. Pudalov, Phys. Rev. Lett. [**75**]{}, 910 (1995).
S.V. Kravchenko, D. Simonian, M. P. Sarachik, W. Mason and J. E. Furneaux, Phys. Rev. Lett. [**77**]{}, 4938 (1996).
D. Popović, A. B. Fowler, and S. Washburn, Phys. Rev. Lett. [**79**]{}, 1543 (1997).
Y. Hanein, D. Shahar, J. Yoon, C. C. Li, D. C. Tsui, and H. Shtrikman, Phys. Rev. B [**58**]{}, R13338 (1998);
J. Lam, M. D’Iorio, D. Brown, and H. Lafontaine, Phys. Rev. B [**56**]{}, R12741 (1997).
P. T. Coleridge, R. L. Williams, Y. Feng, and P. Zawadzki, Phys. Rev. B [**56**]{}, R12764 (1997).
S. J. Papadakis and M. Shayegan, Phys. Rev. B [**57**]{}, R15068 (1998).
Y. Hanein, U. Meirav, D. Shahar, C. C. Li, D. C. Tsui, and H. Shtrikman, Phys. Rev. Lett. [**80**]{}, 1288 (1998);
M. Y. Simmons, A. R. Hamilton, M. Pepper, E. H. Linfield, P. D. Rose, D. A. Ritchie, A. K. Savchenko, and T. G. Griffiths, Phys. Rev. Lett. [**80**]{}, 1292 (1998).
V. M. Pudalov, Pis’ma Zh. Eksp. Teor. Fiz. [**66**]{}, 168 (1997) \[JETP Lett. [**66**]{}, 175 (1997)\].
Y. Yaish, O. Prus, E. Buchstab, G. Ben Yosef, U. Sivan, and A. Stern, preprint cond -mat/9904324.
D. Simonian, S. V. Kravchenko, M. P. Sarachik, and V. M. Pudalov, Phys. Rev. Lett. [**79**]{}, 2304 (1997).
V. M. Pudalov, G. Brunthaler, A. Prinz, and G. Bauer, Pis’ma Zh. Eksp. Teor. Fiz. [**65**]{}, 887 (1997) \[JETP Lett. [**65**]{}, 932 (1997)\].
S. V. Kravchenko, D. Simonian, M. P. Sarachik, A. D. Kent, and V. M. Pudalov, Phys. Rev. B [**58**]{}, 3553 (1998).
D. Simonian, S. V. Kravchenko, M. P. Sarachik, and V. M. Pudalov, Phys. Rev. B [**57**]{}, R9420 (1998).
K. M. Mertes, D. Simonian, M. P. Sarachik, S. V. Kravchenko, and T. M. Klapwijk, preprint cond-mat/9903179.
T. Okamoto, K. Hosoya, S. Kawaji, and A. Yagi, preprint cond-mat/9904058.
V. Senz, U. Dötsch, Gennser, T. Ihn, T. Heinzel, K. Ensslin, R. Hartmann, and D. Grützmacher, preprint cond-mat/9903367.
T. M. Klapwijk and S. Das Sarma, preprint cond-mat/9810349.
B. L. Altshuler and D. L. Maslov, Phys. Rev. Lett. [**82**]{}, 145 (1999).
S. He and X. C. Xie, Phys. Rev. Lett. [**80**]{}, 3324 (1998).
Y. Meir, preprint cond-mat/9904361.
M. A. Skvortsov, JETP Lett. [**67**]{}, 133 (1998).
I. V. Gornyi, A. P. Dmitriev, and V. Yu. Kachorovskii, JETP Lett. [**68**]{}, 338 (1998).
Y. Lyanda-Geller, Phys. Rev. Lett. [**80**]{}, 4273 (1998).
V. Dobrosavljevic, E. Abrahams, E. Miranda, and S. Chakravarty, Phys. Rev. Lett. [**79**]{}, 455 (1997).
D. Belitz and T. R. Kirkpatrick, Phys. Rev. B, [**58**]{}, 8214 (1998).
S. Chakravarty, L. Yin, and E. Abrahams, Phys. Rev. B [**58**]{}, R559 (1998).
P. Phillips, Y. Wan, I. Martin, S. Knysh, and D. Dalidovich, Nature (London) [**395**]{}, 253 (1998).
C. Castellani, C. Di Castro, and P. A. Lee, Phys. Rev. B, [**57**]{}, R9381 (1998).
Q. Si and C. M. Varma, Phys. Rev. Lett. [**81**]{}, 4951 (1998).
S. Das Sarma and E. H. Hwang, preprint cond-mat/9812216.
J. S. Thakur and D. Neilson, Phys. Rev. B [**58**]{}, 13717 (1998); J. S. Thakur and D. Neilson, Phys. Rev. B [**59**]{}, R5280 (1999); J. S. Thakur, L. Liu, and D. Neilson, Phys. Rev. B [**59**]{}, 7255 (1999).
S. Chakravarty, S. Kivelson, C. Nayak, and K. Völker, preprint cond-mat/9805383.
G. Benenti, X. Waintal, J.-L. Pichard, preprint cond-mat/9812339; cond-mat/9904096.
Yu. A. Bychkov and E. I. Rashba, Pis’ma Zh. Eksp. Teor. Fiz. [**39**]{}, 64 (1984) \[JETP Lett. [**39**]{}, 78 (1984)\]. Yu. A. Bychkov and E. I. Rashba, J. Phys. C [**17**]{}, 6039 (1984).
B. I. Shklovskii and A. L. Efros, [*Electronic Properties of Doped Semiconductors*]{}, (Springer, 1984).
G. Dresselhaus, Phys. Rev. [**100**]{}, 580 (1955).
A. G. Mal’shukov, V. A. Froltsov, and K. A. Chao, Phys. Rev. B [**59**]{}, 5702 (1999).
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---
abstract: 'The diagonal hydrodynamic reductions of a hierarchy of integrable hydrodynamic chains are explicitly characterized. Their compatibility with previously introduced reductions of differential type is analyzed and their associated class of hodograph solutions is discussed.'
author:
- |
L. Martínez Alonso$^{1 }$ and A. B. Shabat $^{2}$\
*$^1$Departamento de Física Teórica II, Universidad Complutense*\
*E28040 Madrid, Spain*\
*$^2$Landau Institute for Theoretical Physics*\
*RAS, Moscow 117 334, Russia*
title: Hydrodynamic reductions and solutions of a universal hierarchy
---
*Key words:* Hydrodynamic Systems, Differential Reductions, Hodograph Solutions.
*MSC:* 35L40,58B20.
Introduction
============
In a series of papers [@1]-[@4] we have considered an infinite hierarchy of integrable systems [@5] which admits many interesting $(1+1)$-dimensional reductions like the Burgers, KdV and NLS hierarchies as well as different classes of *energy-dependent* hierarchies [@6]-[@9] including the Camassa-Holm model also [@10] . It motivated the term *universal hierarchy* we proposed in [@1].
This universal hierarchy can be defined in terms of a generating function $G=G(\lambda,{{\boldsymbol{x}}})$ depending on an spectral parameter $\lambda$ and an infinite set of variables ${{\boldsymbol{x}}}:=(\ldots,x_{-1},x_0,x_1,\ldots)$, which admits expansions $$\begin{aligned}
\label{1}
G&=1+\frac{g_1({{\boldsymbol{x}}})}{\lambda}+\frac{g_2({{\boldsymbol{x}}})}{\lambda^2}+\cdots,\quad
\lambda\rightarrow\infty,\\
\nonumber \\\label{1p}
G&=b_0({{\boldsymbol{x}}})+b_1({{\boldsymbol{x}}})\lambda+b_2({{\boldsymbol{x}}})\lambda^2+\cdots,\quad
\lambda\rightarrow 0.\end{aligned}$$ The hierarchy is provided by the system of flows $$\label{2}
\partial_n G=\langle A_n,G\rangle, \quad n\in \mathbb{Z},$$ where $$\langle U,V\rangle:=U(\partial_xV)-(\partial_xU)V,$$ and $$\begin{aligned}
\label{3}
A_n:&=\lambda^n+g_1({{\boldsymbol{x}}})\lambda^{n-1}+\cdots+g_{n-1}({{\boldsymbol{x}}})\lambda+g_n({{\boldsymbol{x}}}),\quad n\geq 0\\
\nonumber \\
A_{-n}:&=\frac{b_0({{\boldsymbol{x}}})}{\lambda^n}+\frac{b_1({{\boldsymbol{x}}})}{\lambda^{n-1}}+\cdots
+\frac{b_{n-1}({{\boldsymbol{x}}})}{\lambda},\quad n>0.\end{aligned}$$ In terms of the coefficients $\{g_n({{\boldsymbol{x}}})\}_{n\geq
1}\bigcup\{b_n({{\boldsymbol{x}}})\}_{n\geq 0}$ the system becomes a hierarchy of hydrodynamic chains.
An alternative and useful formulation of is obtained by introducing the generating function $$\label{4}
H:=\frac{1}{G}.$$ Thus Eq. is equivalent to $$\label{7}
\partial_n H=\partial_x\Big(A_nH\Big),\quad n\in \mathbb{Z},$$ which means, in particular, that the coefficients $\{h_n({{\boldsymbol{x}}})\}_{n\geq 1}$ supply an infinite set of conservation laws for .
The hierarchy forms a compatible system. Indeed, as a consequence of one derives the consistency conditions [@1]-[@4] $$\label{8}
\partial_n A_m-\partial_m A_n=\langle A_n,A_m\rangle,\quad n,m \in \mathbb{Z},$$
It is important to notice that implies that the pencil of differential forms $$\label{9}
\omega(\lambda):=\sum_{n \in \mathbb{Z}}(H\,A_n){\operatorname{d}}x_n,$$ is closed with respect to the variables ${{\boldsymbol{x}}}$ since it satisfies $$\begin{aligned}
\partial_m(H\,A_n)-&\partial_n(H\, A_m)=\partial_mH\,A_n-
\partial_nH\,A_m+H(\partial_mA_n-\partial_nA_m)\\
=&\partial_x(HA_m)A_n-\partial_x(HA_n)A_m+H\langle A_n,A_m\rangle=0.\end{aligned}$$ The potential function $Q=Q(\lambda,{{\boldsymbol{x}}})$ corresponding to $\omega$ $$\label{10a}
{\operatorname{d}}Q=\omega,$$ leads to another useful formulation of our hierarchy. Indeed, according to $$\label{11a}
\partial_n Q=A_n\,\partial_x Q,\quad n\in \mathbb{Z}.$$ Moreover, is completely determined by $Q$ as $$\label{12a}
G=\frac{1}{\partial_x Q}.$$
In Section 2 of this paper the formulation is applied to prove that includes multidimensional models such as $$\begin{aligned}
\label{10}
&u_{tx}=u_xu_{yz}-u_{yx}u_z,\\
\label{10p}
&u_{yy}=u_yu_{xz}-u_{xy}u_z,\\
\label{10pp}
&u_{zx}=u_xu_{yy}-u_{xy}u_y,\\
\label{10ppp}
&\Big(\frac{u_t}{u_x}\Big)_t=\Big(\frac{u_y}{u_x}\Big)_z,\\
\label{10pppp}
&\Big(\frac{u_t}{u_x}\Big)_t=\Big(\frac{u_y}{u_x}\Big)_x.\end{aligned}$$
Section 3 deals with the main aim of the present paper: to characterize the *hydrodynamic reductions* of . These reductions are given by the solutions of of the form $G=G(\lambda,{{\boldsymbol{R}}})$, where ${{\boldsymbol{R}}}=(R^1,\ldots,R^{N})$ denotes a finite set of functions (*Riemann invariants*) satisfying a system of hydrodynamic equations of diagonal form $$\label{13}
\partial_n R^i=\Lambda ^i_n({{\boldsymbol{R}}}) \partial_x R^i,\quad n\in \mathbb{Z}.$$ This type of reductions appeared in the context of the dispersionless KP hierarchy [@11]-[@14] and has been also used in [@15] to characterize the integrability of $(2+1)$-dimensional quasilinear systems. In the present paper we study these reductions for the whole set of flows of the hierarchy and we obtain their explicit form . Our results are summarized in Theorem 1. It should be noticed that a similar analysis for the first member ($t_1$-flow) of has been recently performed in [@16]. The compatibility between hydrodynamic and differential reductions considered in part 2 of Section 3. The paper finishes with a discussion of the solutions supplied by the generalized hodograph method [@17].
The following notation conventions are henceforth used. Firstly, $G_{\infty}$ and $G_0$ stand for the expansions of $G$ as $\lambda\rightarrow\infty$ and $\lambda\rightarrow 0$ , respectively. Furthermore, let $\mathbb{V}$ be the space of formal Laurent series $$V=\sum_{n=-\infty}^{\infty} a_n\lambda^n.$$ We will denote by $\mathbb{V}_{r,s}$ $(r\leq s)$ the subspaces of elements $$V=\sum_{n=r}^{s} a_n\lambda^n.$$ and by $P_{r,s}:\mathbb{V}\mapsto\mathbb{V}_{r,s}$ the corresponding projectors. Given $V\in\mathbb{V}$ we will also denote $$(V)_{r,s}:=P_{r,s} (V).$$ In particular, notice that we can write $$A_n=\Big(\lambda^n G_{\infty}\Big)_{0,+\infty},\quad
A_{-n}=\Big(\lambda^{-n} G_0\Big)_{-\infty,-1},\quad n\geq 1.$$
Integrable models arising in the hierarchy
==========================================
Multidimensional models
-----------------------
If we use the potential function $Q(\lambda,{{\boldsymbol{x}}})$ for the differential form $$\label{a1}
{\operatorname{d}}Q=\sum_{n \in \mathbb{Z}}(H\,A_n){\operatorname{d}}x_n,$$ then from one readily deduces that $Q$ admits expansions of the form $$\begin{aligned}
\label{a2}
Q&=\sum_{n\geq 0}\lambda^n
x_n+\frac{q_1({{\boldsymbol{x}}})}{\lambda}+\frac{q_2({{\boldsymbol{x}}})}{\lambda^2}+\cdots,\quad
\lambda\rightarrow\infty,\\
\nonumber \\
Q&=\sum_{n\geq
1}\frac{x_n}{\lambda^n}+p_0({{\boldsymbol{x}}})+p_1({{\boldsymbol{x}}})\lambda+p_2({{\boldsymbol{x}}})\lambda^2+\cdots,\quad
\lambda\rightarrow 0.\end{aligned}$$
By substituting into and by identifying coefficients of equal powers of $\lambda$ one obtains formulas for the differentials of the functions $q_n({{\boldsymbol{x}}})$ and $p_n({{\boldsymbol{x}}})$ in terms of the coefficients of the expansions and . For example, the simplest ones are $$\begin{aligned}
\label{a3}
{\operatorname{d}}q_1&=b_0{\operatorname{d}}x_{-1}+\sum_{n\geq 1}(b_n\,{\operatorname{d}}x_{-n-1}-g_n\,{\operatorname{d}}x_{n-1}),\\ \nonumber \\ \label{a3p}{\operatorname{d}}p_0&=\frac{1}{b_0}\Big({\operatorname{d}}x_0+\sum_{n\geq 1} (g_n\, {\operatorname{d}}x_n-b_n\, {\operatorname{d}}x_{-n})\Big).\end{aligned}$$ They imply $$\begin{aligned}
\label{a4}
{\operatorname{d}}q_1=\frac{1}{\partial_xp_0}\Big({\operatorname{d}}x_{-1}-\sum_{n\neq -1}
\partial_{n+1}p_0
\,{\operatorname{d}}x_n \Big)\\\nonumber\\
\label{a4p} {\operatorname{d}}p_0=\frac{1}{\partial_{-1}q_1}\Big({\operatorname{d}}x_0-\sum_{n\neq
0}\partial_{n-1}q_1\,{\operatorname{d}}x_n \Big).\end{aligned}$$ Permutability of crossing derivatives of $q_1$ and $p_0$ in these identities lead at once to multidimensional nonlinear equations for the functions $g_n$ and $b_n$. For example, starting from $$\partial_m\partial_{-1}q_1=\partial_{-1}\partial_mq_1,
\quad m\neq -1,$$ and using , the following nonlinear equation results $$\label{a5}
\partial_n\partial_0p_0= \partial_x
p_0\,(\partial_{-1}\partial_{n+1}p_0)-(\partial_{-1}\partial_xp_0)\,D_{n+1}p_0,\quad n\neq -1.$$ In the same way, from the crossing relation $$\partial_m\partial_nq_1=\partial_n\partial_mq_1,\quad m,n\neq -1,$$ and we get the following nonlinear equation $$\label{a6}
\partial_m\Big(\frac{\partial_{n+1}p_0}{\partial_xp_0}\Big)
=\partial_n\Big(\frac{\partial_{m+1}p_0}{\partial_xp_0}\Big),
\quad m,n\neq -1.$$ The same type of equations can be derived for $q_1$. The different choices available for $n,m$ in the equations and give rise to the models -.
2-dimensional integrable models
-------------------------------
In [@1]-[@3] we developed a theory of differential reductions of our hierarchy based on imposing differential constraints on $G\approx G_{\infty}$ of the form $$\label{28}
\Big(\mathcal{F}(\lambda,G,G_x,G_{xx},\ldots)\Big)_{-\infty,-1}=0,\quad
x:=x_0.$$ In particular the following three classes of reductions associated to arbitrary polynomials $a=a(\lambda)$ in $\lambda$ were characterized:
[**Zero-order reductions**]{}
$$\label{r0}
a(\lambda) G=U(\lambda,{{\boldsymbol{x}}}),\quad U:=\Big(a(\lambda)
G\Big)_{0,+\infty},$$
[**First-order reductions**]{}
$$\label{r1}
G_x+a(\lambda)=U(\lambda,{{\boldsymbol{x}}})G,\quad
U:=\Big(\frac{a}{G}\Big)_{0,+\infty},$$
[**Second-order reductions**]{}
$$\label{r2}
\frac{1}{2}GG_{xx}-\frac{1}{4}G_x^2+a(\lambda)=U(\lambda,{{\boldsymbol{x}}})G^2,\quad
U:=\Big(\frac{a}{G^2}\Big)_{0,+\infty}.$$
The first-order reduction for a linear function $a(\lambda)$ determines the Burgers hierarchy. On the other hand, under the differential constraints the hierarchy describes the KdV hierarchy and its generalizations associated to energy-dependent Schrödinger spectral problems. In particular, the linear and quadratic choices for $a(\lambda)$ lead to the KdV (Korteweg-deVries) and NLS (Nonlinear-Schrödinger) hierarchies, respectively. Indeed, if we define the functions $\psi(\lambda,{{\boldsymbol{x}}})$ by $$\label{s1}
\psi(\lambda,{{\boldsymbol{x}}}):=\exp(D_x^{-1}\phi),\quad
\phi:=-\frac{1}{2}\frac{H_x}{H}\pm \sqrt{a(\lambda)}\,H,$$ then from it is straightforward to deduce that $$\begin{aligned}
\partial_n\psi&=-\frac{1}{2}(\partial_n\log H)\psi\pm \sqrt{a}A_nH\psi\\&=
A_n(-\frac{1}{2}D_x\log H\pm \sqrt{a}H)\psi-\frac{1}{2} A_{n,x}\psi\\
&=A_n\psi_x-\frac{1}{2}A_{n,x}\psi,\\
\psi_{xx}&=(\phi_x+\phi^2)\psi=(\{D_x,H\}+aH^2)\psi= U\psi.\end{aligned}$$ In other words, the functions $\psi$ are wave functions for the integrable hierarchies associated to energy-dependent Schrödinger problems. The evolution law of the potential function $U$ under the flows can be determined from the equation $$\label{s2}
\partial_nU=-\frac{1}{2}A_{n,xxx}+2UA_{n,x}+U_xA_n,$$ which arises as an straightforward consequence of and .
Additional reduced hierarchies including nonlinear integrable models such as the Camassa-Holm equation can be also deduced (see [@4]).
Hydrodynamic reductions and solutions
=====================================
Hydrodynamic reductions
-----------------------
Let us consider now the hydrodynamic reductions of . We look for classes a solutions $$\label{3.0}
G=G(\lambda,{{\boldsymbol{R}}}),$$ of where ${{\boldsymbol{R}}}=(R^1,\ldots,R^{N})$ satisfies a infinite system of hydrodynamic equations of diagonal form . Our aim is to characterize both the form of $G=G(\lambda,{{\boldsymbol{R}}})$ and the *characteristic speeds* $\Lambda^i_n$ defining the system . By substituting into then by using the identification of coefficients of the derivatives $\partial_x R^i,\; i=1,\ldots,N$ implies $$\label{3.1}
(D_iG)\Lambda^i_n=A_n(D_iG)-(D_iA_n)G,\quad 1\leq i\leq N,\;\; n\in\mathbb{Z}$$ where $$\quad D_i:=
\frac{\partial}{\partial R^i}.$$ In addition to these equations we impose the requirement of the commutativity of the flows , which is equivalent to the following restrictions on the characteristic speeds $$\label{3.2}
\frac{D_j \Lambda_n^i}{\Lambda_n^j-\Lambda_n^i}=
\frac{D_j \Lambda_m^i}{\Lambda_m^j-\Lambda_m^i},\quad
i\neq j,\quad m\neq n.$$
We start our analysis by considering the *positive flows $n\geq 1$*. From we get the following system for $G\approx G_{\infty}$ $$\label{3.3}
(D_i G)\Lambda^i_n=(\lambda^n G)_{-\infty,0}(D_iG)-
(D_i(\lambda^n G)_{-\infty,0})G, \quad n\geq 1.$$ By substituting in these equations the expansion $G\approx G_{\infty}$ and identifying the coefficients in $\frac{1}{\lambda}$ and $\frac{1}{\lambda^2}$ we get $$\begin{aligned}
\label{3.4a}
D_ig_{n+1}&=\Lambda^i_n D_i g_1,\\
\label{3.4b}
D_ig_{n+2}&=\Lambda^i_n D_i g_2+g_{n+1}(D_i g_1)-(D_i g_{n+1})g_1.\end{aligned}$$ As an immediate consequence it follows $$\label{3.5}
\Lambda^i_{n+1}=g_{n+1}+\Lambda^i_n(\Lambda_1^i-g_1),$$ which implies $$\label{3.6}
\Lambda^i_n=A_n\Big(\lambda=\Lambda^i_1-g_1\Big),\quad n\geq 1.$$
We now look for a system characterizing $\Lambda^i_1$ and $g_1$. To this end we differentiate and find $$\begin{aligned}
&D_jD_ig_{n+1}=D_jD_ig_1\Lambda^i_n+D_i g_1D_j\Lambda^i_n\\
&=D_iD_jg_1\Lambda^j_n+D_j g_1D_i\Lambda^i_n,\end{aligned}$$ so that $$D_{ij}g_1=\frac{D_j\Lambda^i_n}{\Lambda^j_n-\Lambda^i_n}D_ig_1
+\frac{D_i\Lambda^j_n}{\Lambda^i_n-\Lambda^j_n}D_jg_1,$$ and from we may write $$\label{3.7}
D_{ij}g_1=\frac{D_j\Lambda^i_1}{\Lambda^j_1-\Lambda^i_1}D_ig_1
+\frac{D_i\Lambda^j_1}{\Lambda^i_1-\Lambda^j_1}D_jg_1,\quad i\neq j.$$ On the other hand according to $$\Lambda^i_2=g_2+\Lambda^i_1(\Lambda^i_1-g_1),$$ so that from and with the help of we find $$\frac{D_j\Lambda^i_1}{\Lambda^j_1-\Lambda^i_1}=\frac{D_j\Lambda^i_2}{\Lambda^j_2-\Lambda^i_2}
=\frac{D_j\Lambda^i_1(2\Lambda^i_1-g_1)+(\Lambda^j_1-\Lambda^i_1)D_jg_1}
{(\Lambda^j_1-\Lambda^i_1)(\Lambda^j_1+\Lambda^i_1-g_1)},\quad i\neq j,$$ which reduces to $$D_j\Lambda^i_1=D_j g_1,\quad i\neq j.$$ In this way $$\label{3.8}
\Lambda^i_1=g_1+f^i(R^i),$$ where the functions $f^i$ are arbitrary. By using this result in it follows $$D_{ij}g_1=0,\quad i\neq j,$$ so that $$\label{3.9}
g_1=\sum_{k=1}^N h^k(R^k),$$ where the functions $h^k$ are arbitrary.
By using these results we may determine $G(\lambda,{{\boldsymbol{R}}})$ since from the equation with $n=1$ $$D_i\ln G=\frac{D_i A_1}{A_1-\Lambda^i_1}=\frac{\dot{h}^i(R^i)}{
\lambda-f^i(R^i)},$$ where $\dot{h}^i:=D_ih^i$. Thus we get $$\label{3.10}
G(\lambda,{{\boldsymbol{R}}})=\exp\Big(\sum_{i=1}^N \int^{R^i}\frac{\dot{h}^i(R^i)}{
\lambda-f^i(R^i)}{\operatorname{d}}R^i\Big),$$ where the undefinite integrations are determined up to a function of $\lambda$ decaying at $\lambda\rightarrow\infty$. The expression coincides with the generating function of the conservation laws densities found in [@16].
Let us now determine the characteristic speeds $\Lambda^i_{n}$ for the *negative flows $n\leq -1$*. The equations imply the following system for $G\approx G_{0}$ $$\label{3.11}
(D_i G)\Lambda^i_{-n}=(\lambda^{-n} G)_{0,\infty}(D_iG)-
(D_i(\lambda^{-n} G)_{0,\infty})G, \quad n\geq 1.$$ Then by inserting the expansion $G\approx G_{0}$ of and identifying the coefficients in $\lambda^0$ and $\lambda$ we get $$\begin{aligned}
\label{3.12a}
&D_ib_n=(b_n+\Lambda^i_{-n}) D_i\ln b_0,\\
\label{3.12b}
&b_0D_ib_{n+1}=(b_n+\Lambda^i_n) D_i b_1+b_{n+1}(D_i b_0)-
(D_i b_n)b_1.\end{aligned}$$ It implies the following recurrence relation for the characteristic speeds $$\label{3.13}
\Lambda^i_{-n-1}=\frac{\Lambda^i_{-1}}{b_0}(b_n+\Lambda^i_{-n}),$$ which implies $$\label{3.14}
\Lambda^i_{-n}=A_{-n}\Big(\lambda=\frac{b_0}{\Lambda^i_{-1}}\Big),\quad n\geq 1.$$ The only unknown now is $\Lambda^i_{-1}$ since according to $$b_0=\exp\Big(-\sum_{i=1}^N \int^{R^i}\frac{\dot{h}^i(R^i)}{
f^i(R^i)}{\operatorname{d}}R^i\Big).$$ To find $\Lambda^i_{-1}$ we use for $n=1$ $$\Lambda^i_{-2}=\frac{\Lambda^i_{-1}}{b_0}(b_1+\Lambda^i_{-1}),$$ and the commutativity condition for the $n=-1$ and $n=-2$ flows $$\frac{D_j\Lambda^i_{-1}}{\Lambda^j_{-1}-\Lambda^i_{-1}}
=\frac{D_j\Lambda^i_{-2}}{\Lambda^j_{-2}-\Lambda^i_{-2}}.$$ Thus by eliminating $\lambda^i_{-2}$ one finds at once $$D_j\ln \Lambda^i_{-1}=D_j\ln b_0,\quad i\neq j.$$ Therefore, $$\label{3.15}
\Lambda^i_{-1}=b_0 \,g^i(R^i),$$ with $g^i$ being arbitrary functions.
At this point we have determined all the unknowns of our problem. However, in our calculation we only used a subset of the equations required, so we must prove that our solution satisfies the full system of equations -.
Let us begin with for $n\geq 1$ $$D_i\ln G=\frac{D_i A_n}{A_n-\Lambda^i_n},\quad n\geq 1,$$ which according to , - reads $$\label{3.16}
D_iA_n(\lambda)=\frac{\dot{h}^i}{
\lambda-f^i}\Big(A_n(\lambda)-A_n(\lambda=f^i)\Big).$$ To prove this identity we express $A_n(\lambda)$ as $$\begin{aligned}
A_n(\lambda)&=(\lambda^n G)_{0,\infty}=\frac{1}{2\pi i}\int_{\gamma_{\infty}}
\frac{(\widetilde{\lambda}^n G(\widetilde{\lambda}))_{0,\infty}}
{\widetilde{\lambda}-\lambda}{\operatorname{d}}\widetilde{\lambda}=
\frac{1}{2\pi i}\int_{\gamma_{\infty}}
\frac{\widetilde{\lambda}^n G(\widetilde{\lambda})}
{\widetilde{\lambda}-\lambda}{\operatorname{d}}\widetilde{\lambda}\\\\
&=\frac{1}{2\pi i}\int_{\gamma_{\infty}}
\frac{\widetilde{\lambda}^n}
{\widetilde{\lambda}-\lambda}
\exp\Big(\sum_{i=1}^N \int^{R^i}\frac{\dot{h}^i}{
\widetilde{\lambda}-f^i}{\operatorname{d}}R^i\Big)
{\operatorname{d}}\widetilde{\lambda},\end{aligned}$$ where $\gamma_{\infty}$ is a positively oriented closed loop around $\infty$ in the complex plane of $\widetilde{\lambda}$ ($\lambda$ and $f^i$ are assumed to lie inside the loop). Now by differentiating this expression with respect to $R^i$ one finds $$\begin{aligned}
D_iA_n(\lambda)&=
\frac{1}{2\pi i}\int_{\gamma_{\infty}}
\frac{\dot{h}^i\widetilde{\lambda}^n G(\widetilde{\lambda})}
{(\widetilde{\lambda}-\lambda)(\widetilde{\lambda}-f^i)}
{\operatorname{d}}\widetilde{\lambda}=\frac{1}{2\pi i}\int_{\gamma_{\infty}}
\frac{\dot{h}^i A_n(\widetilde{\lambda})}
{(\widetilde{\lambda}-\lambda)(\widetilde{\lambda}-f^i)}
{\operatorname{d}}\widetilde{\lambda}\\\\
&=\frac{\dot{h}^i}{\lambda-f^i}\Big( A_n(\lambda)-A_n(\lambda=f^i)\Big),\end{aligned}$$ which proves . This identity leads also at once to since it implies $$\frac{D_j A_n(\lambda=f^i)}{A_n(\lambda=f^j)-A_n(\lambda=f^i)}=
\frac{\dot{h}^j}{f^j-f^i},\quad i\neq j,$$ which means that $$\frac{D_j \Lambda^i_n}{\Lambda^j_n-\Lambda^i_n}=
\frac{D_j \Lambda^i_1}{\Lambda^j_1-\Lambda^i_1},\quad n>1.$$
Let us consider now for $n\leq -1$ $$D_i\ln G=\frac{D_i A_{-n}}{A_{-n}-\Lambda^i_{-n}},\quad n\geq 1.$$ From , and it takes the form $$\label{3.17}
D_iA_{-n}(\lambda)=\frac{\dot{h}^i}{
\lambda-f^i}\Big(A_{-n}(\lambda)-A_{-n}(\lambda=\frac{1}{g^i})\Big).$$ In order to proof we express $A_{-n}(\lambda)$ in the form $$\begin{aligned}
A_{-n}(\lambda)&=(\lambda^{-n} G)_{-\infty,-1}=
\frac{1}{2\pi i}\int_{\gamma_0}
\frac{(\widetilde{\lambda}^{-n} G(\widetilde{\lambda}))_{-\infty,-1}}
{\widetilde{\lambda}-\lambda}{\operatorname{d}}\widetilde{\lambda}=
\frac{1}{2\pi i}\int_{\gamma_0}
\frac{\widetilde{\lambda}^{-n} G(\widetilde{\lambda})}
{\widetilde{\lambda}-\lambda}{\operatorname{d}}\widetilde{\lambda}\\\\
&=\frac{1}{2\pi i}\int_{\gamma_0}
\frac{\widetilde{\lambda}^{-n}}
{\widetilde{\lambda}-\lambda}
\exp\Big(\sum_{i=1}^N \int^{R^i}\frac{\dot{h}^i}{
\widetilde{\lambda}-f^i}{\operatorname{d}}R^i\Big)
{\operatorname{d}}\widetilde{\lambda},\end{aligned}$$ where $\gamma_0$ is a closed small loop with negative orientation around $\widetilde{\lambda}=0$ ($\lambda$ and $f^i$ are assumed to lie outside the loop ). By differentiating with respect to $R^i$ it yields $$\begin{aligned}
D_iA_{-n}(\lambda)&=
\frac{1}{2\pi i}\int_{\gamma_0}
\frac{\dot{h}^i\widetilde{\lambda}^{-n} G(\widetilde{\lambda})}
{(\widetilde{\lambda}-\lambda)(\widetilde{\lambda}-f^i)}
{\operatorname{d}}\widetilde{\lambda}\\\\
&=\frac{1}{2\pi i}\int_{\gamma_0}
\frac{\dot{h}^i A_{-n}(\widetilde{\lambda})}
{(\widetilde{\lambda}-\lambda)(\widetilde{\lambda}-f^i)}
{\operatorname{d}}\widetilde{\lambda}
=\frac{\dot{h}^i}{\lambda-f^i}\Big( A_{-n}(\lambda)-A_{-n}(\lambda=f^i)\Big),\end{aligned}$$ Hence holds if we set $$g^i(R^i)=\frac{1}{f^i(R^i)}.$$ The remaining commutativity conditions follow at once.
Therefore we may summarize our analysis in the next theorem
The hydrodynamic reductions of the hierarchy are determined by $$\begin{aligned}
\label{3.18}
&G(\lambda,{{\boldsymbol{R}}})=\exp\Big(\sum_{i=1}^N \int^{R^i}\frac{\dot{h}^i(R^i)}{
\lambda-f^i(R^i)}{\operatorname{d}}R^i\Big),
\\\nonumber\\
\label{3.19}
&\partial_n R^i=\Lambda^i_n({{\boldsymbol{R}}})\, \partial_x R^i,\quad \Lambda^i_n({{\boldsymbol{R}}}):=A_n(\lambda=f^i(R^i)),\end{aligned}$$ where the functions $h^i$ and $f^i$ are arbitrary.
Since the systems are invariant under local transformations of the form $R^i\rightarrow\widetilde{R^i}(R^i)$, without loss of generality we will henceforth set $$\label{gauge}
h^i(R^i)=R^i,$$ so that the form of the reduced generating function is $$\label{3.20a}
G(\lambda,{{\boldsymbol{R}}})=\exp\Big(\sum_{i=1}^N \int^{R^i}\frac{{\operatorname{d}}R^i}{
\lambda-f^i(R^i)}\Big),$$
In this way some of the simplest hydrodynamic reductions are given by $$\begin{aligned}
\label{3.20}
&f^i(R^i)=c_i,\quad G=\exp\Big(\sum_{i=1}^N \frac{R^i}{\lambda-c_i}\Big),
\\
\label{3.21}
&f^i(R^i)=-\frac{R^i}{\epsilon_i},\quad G=\prod_{i=1}^N \Big(
\frac{\epsilon_i\lambda+R^i}{\epsilon_i\lambda+\lambda_i}\Big)^{\epsilon_i}.\end{aligned}$$
We also notice that the characteristic velocities of the hydrodynamic systems can be written in terms of the Schur polynomials $$\exp(\sum_{n\geq 1}k^n x_n)=\sum_{n\geq 0}k^n S_n(x_1,\ldots,x_n),$$ as $$\begin{aligned}
&\Lambda^i_n({{\boldsymbol{R}}})=\sum_{j=0}^n S_j(I_1,\ldots,I_j)\Big(f^i(R^i)\Big)^{n-j},
\quad n\geq 0,\\
&\Lambda^i_{-n}({{\boldsymbol{R}}})=\exp(-I_0)\sum_{j=0}^{n-1} S_j(I_{-1},\ldots,I_{-j})\Big
(f^i(R^i)\Big)^
{-n+j},\quad n>0,\end{aligned}$$ where $$I_n:=sgn(n)\sum_{i=1}^N \int^{R^i}(f^i(R^i))^{n-1}{\operatorname{d}}R^i.$$
Compatibility with differential reductions
------------------------------------------
A natural question is to find the hydrodynamic reductions compatible with the differential reductions and . Let us prove the following result:
The only hydrodynamic reductions compatible with either first or second order differential constraints are characterized by $$f^i(R^i)=-R^i+c_i,\quad i=1,\ldots,N,$$ which correspond to generating functions of the form $$G(\lambda,{{\boldsymbol{R}}})=\alpha(\lambda)\prod_{i=1}^N (\lambda+R^i-c_i).$$
If we substitute into the differential constraint for second-order differential reductions we get $$\frac{a(\lambda)}{G^2}=U-\frac{1}{2}\sum_i\Big( \dot{f}^i\frac{(\partial_x R^i)^2}
{(\lambda-f^i)^2}+\frac{\partial_{xx} R^i}{\lambda-f^i}\Big)
-\frac{1}{4}\Big(\sum_i\frac{\partial_x R^i}{\lambda-f^i}\Big)^2.$$ This means that $G$ has a simple zero at each $\lambda=f^i(R^i)$ so that from we have $$\exp\Big(\int^{R^i}\frac{{\operatorname{d}}R^i}{\lambda-f^i(R^i)}\Big)=(\lambda-f^i(R^i))H(\lambda,R^i),$$ where $H$ is different from zero at $\lambda=f^i(R^i)$. If we now differentiate with respect to $R^i$ we deduce that $$\frac{1}{\lambda-f^i}=-\frac{\dot{f}^i}{\lambda-f^i}+\mathcal{O}(1),\quad \lambda\rightarrow\infty.$$ Therefore the statement of the theorem for second-order differential constraints follows. The corresponding proof for first-order constraints is similar.
Hodograph solutions
-------------------
The general solution of the infinite system is provided by the implicit *generalized hodograph* formula [@17] $$\label{3.22}
x+\sum_{n\in \mathbb{Z}-\{0\}}\Lambda^i_n({{\boldsymbol{R}}})\, x_n=\Gamma^i({{\boldsymbol{R}}}),
\quad i=1,\ldots,N,$$ where the functions $\Gamma^i$ are the general solution of the linear system $$\label{3.23}
\frac{D_j \Gamma^i}{\Gamma^j-\Gamma^i}=
\frac{1}{f^j-f^i},\quad i\neq j.$$ By introducing the potential function $\Phi({{\boldsymbol{R}}})$ $$\Gamma^i=D_i\Phi,\quad i=1,\ldots,N,$$ the system reduces to the Laplace type form $$\label{3.24}
(f^i-f^j)D_iD_j\Phi=D_i\Phi-D_j\Phi,\quad i\neq j,$$ the general solution of which depends on $N$ arbitrary functions of one variable.
In particular it is immediate to deduce [@16] that the generating function provides a one-parameter family of solutions of . Thus we can produce important solutions of from linear superpositions of $G(\lambda,{{\boldsymbol{R}}})$. For example $$\begin{aligned}
\Lambda^i_n&=D_i\Phi,\;\; \Phi:=\frac{1}{2\pi i}\int_{\gamma_{\infty}}
\lambda^n G(\lambda,{{\boldsymbol{R}}}){\operatorname{d}}\lambda,\quad n\geq 0,\\
\Lambda^i_{-n}&=D_i\Phi,\;\; \Phi:=\frac{1}{2\pi i}\int_{\gamma_0}
\lambda^{-n} G(\lambda,{{\boldsymbol{R}}}){\operatorname{d}}\lambda,\quad n> 0.\end{aligned}$$ Furthermore, in several important cases the general solution of can be written in terms of $G(\lambda,{{\boldsymbol{R}}})$.
[**Examples** ]{}
If we set $$f^i(R^i)=\frac{R^i}{n},\quad n=1,2,\ldots,$$ then the generating function (non normalized as $\lambda\rightarrow\infty$) is $$G(\lambda,{{\boldsymbol{R}}})=\prod_{i=1}^N\Big(\lambda-\frac{R^i}{n}\Big)^{-n}.$$ Hence, by taking for each $i=1,\ldots,N$ a closed loop $\gamma_i$ in the complex $\lambda$-plane with positive orientation around $\lambda_i=\frac{R^i}{n}$, the general solution of can be expressed as $$\Phi:=\frac{1}{2\pi i}\sum_{i=1}^N\int_{\gamma_i}
\phi_i(\lambda) G(\lambda,{{\boldsymbol{R}}}){\operatorname{d}}\lambda,$$ where the functions $\phi_i(\lambda)$ are arbitrary. For example, if $n=1$ it takes the form $$\Phi=\sum_{i=1}^N\phi_i(R^i)\prod_{k\neq i}\frac{1}{R^i-R^k}.$$
Similarly we may deal with the case $$f^i(R^i)=-\frac{R^i}{n},\quad n=1,2,\ldots,$$ which leads to $$G(\lambda,{{\boldsymbol{R}}})=\prod_{i=1}^N\Big(\lambda+\frac{R^i}{n}\Big)^{n}.$$ Now to generate the general solution of we take for each $i=1,\ldots,N$ a path $\gamma_i(R^i)$ in the complex $\lambda$-plane ending at $\lambda_i=-R^i/n$ so that we can write $$\Phi:=\frac{1}{2\pi i}\sum_{i=1}^N\int_{\gamma_i(R^i)}
\phi_i(\lambda) G(\lambda,{{\boldsymbol{R}}}){\operatorname{d}}\lambda,$$ where the functions $\phi_i(\lambda)$ are arbitrary.
Let us consider in detail the case $$f^i(R^i)=-R^i,\quad i=1,2.$$ One finds $$\Phi=\Big(\theta_1(-R^1)-\theta_2(-R^2)\Big)(R^1-R^2)+
2\int^{-R^1}\theta_1(\lambda){\operatorname{d}}\lambda+2\int^{-R^2}\theta_2(\lambda){\operatorname{d}}\lambda,$$ with $\theta_i(\lambda)$ being arbitrary functions. A normalized generating function is given by $$\label{3.26a}
G(\lambda,{{\boldsymbol{R}}})=\frac{(\lambda+R^1)(\lambda+R^2)}{(\lambda+\lambda_0)^2},
\quad \lambda_0\neq 0.$$
We may characterize the general solution of the hydrodynamic flows corresponding to $n=-1$ and $n=-2$ by means of the hodograph formula $$\label{3.26}
x+A_{-1}(\lambda=-R^i)\,y+A_{-2}(\lambda=-R^i)\,z=D_i\Phi,\quad i=1,2,$$ where $y:=x_{-1},\; z:=x_{-2}$. From one calculates $$b_0=c^2R^1R^2,\;\;\; b_1=c^2(R^1+R^2)
-2c^3R^1R^2,\quad
c:=\frac{1}{\lambda_0},$$ and gets that the system reads $$\begin{aligned}
\nonumber
(x-c^2z)-c^2(y-2cz)R^2=\dot{\theta}_1(-R^1)(R^2-R^1)-\theta_1(-R^1)
-\theta_2(-R^2),\\
\label{3.27}
\\\nonumber
(x-c^2z)-c^2(y-2cz)R^1=\dot{\theta}_2(-R^2)(R^1-R^2)-\theta_1(-R^1)
-\theta_2(-R^2).\end{aligned}$$ In particular, it implies $${{\boldsymbol{R}}}={{\boldsymbol{R}}}(x-c^2z,y-2cz),$$ so that ${{\boldsymbol{R}}}$ is constant on the straight lines $$\vec{x}:=(x,y,z)=(x_0,y_0,0)+(c^2+2c,1)s.$$ We notice that starting from these solutions we may generate solutions $$u(x,y,z):=\int^y b_0({{\boldsymbol{R}}}){\operatorname{d}}y+\int^z b_1({{\boldsymbol{R}}}) {\operatorname{d}}z,$$ of the nonlinear equation $$u_{yy}=u_uu_{zx}-u_{yx}u_z.$$
[**Acknowledgements**]{}
The authors thank Professor Eugeni Ferapontov for useful discussions. A.B. Shabat was supported by a grant of the Ministerio de Cultura y Deporte of Spain. L. Martinez Alonso was supported by the DGCYT project BFM2002-01607.
[99]{}
L. Martinez Alonso and A. B. Shabat, Phys.Lett. [**A 300**]{}, 58 (2002).
L. Martinez Alonso and A. B. Shabat, J. Non. Math. Phys. [**10**]{} No 2, 1 (2003).
L. Martinez Alonso and A. B. Shabat, *“ On the prolongation of a hierarchy of hydrodynamic chains”* to appear in *New Trends in Integrability and Partial Solvability* proceedings NATO ARW.978791, NATO Science Series , Kluwer Academic Publishers.
A. Shabat, Theor. and Math. Phys. [**136**]{} No 2, 1066 (2003).
V. G. Mikhalev, Funct. Anal. Appl. [**26**]{} No 2, 140 (1992). M. Jaulent and I. Miodek, Lett. Math. Phys. [**1**]{}, 243 (1976); Lett. Nuovo Cimento [**20**]{}, 655 (1977).
L. Martínez Alonso, J. Math. Phys. [**21**]{}, 2342 (1980).
M. Antonowicz and A. P. Fordy, Physica [**28 D**]{}, 345 (1987).
A. N. W. Hone, Phys. Lett. [**A 249**]{}, 46 (1998).
R. Camassa and D. Holm, Phys. Rev. Letters [**71**]{}, 1661 (1993).
J. Gibbons and Y. Kodama. Phys. Lett. [**A 135**]{}, 167 (1989).
J. Gibbons and S. P. Tsarev. Phys. Lett. [**A 211**]{} , 19 (1996). J. Gibbons and S. P. Tsarev. Phys. Lett. [**A 258**]{} , 263 (1999). M. Mañas, L. Martinez Alonso and E. Medina. J. Phys. A: Math. Gen. [**35**]{}, 401 (2002) E. V. Ferapontov and K. R. Khusnutdinova. *On integrability of $(2+1)$-dimensional quasilinear systems* nlin.SI/0305044, (2003). M. V. Pavlov. *Integrable hydrodynamic chains* nlin.SI/0301010, (2003). S. P. Tsarev. Izvestija AN USSR Math. [**54**]{}, N5, 1048 (1990).
|
---
abstract: |
The General Intensional Programming System () has been built around the Lucid family of intensional programming languages that rely on the higher-order intensional logic (HOIL) to provide context-oriented multidimensional reasoning of intensional expressions. HOIL combines functional programming with various intensional logics to allow explicit context expressions to be evaluated as first-class values that can be passed as parameters to functions and return as results with an appropriate set of operators defined on contexts. ’s frameworks are implemented in as a collection of replaceable components for the compilers of various Lucid dialects and the demand-driven eductive evaluation engine that can run distributively. provides support for hybrid programming models that couple intensional and imperative languages for a variety of needs. Explicit context expressions limit the scope of evaluation of math expressions (effectively a Lucid program is a mathematics or physics expression constrained by the context) in tensor physics, regular math in multiple dimensions, etc., and for cyberforensic reasoning as one of the use-cases of interest. Thus, is a support testbed for [[HOIL]{}]{}-based languages some of which enable such reasoning, as in formal cyberforensic case analysis with event reconstruction. In this paper we discuss the architecture, its evaluation engine and example use-cases.\
\
[**Keywords:**]{} Intensional Programming, Higher-Order Intensional Logic (HOIL), Run-Time System, General Intensional Programming System (GIPSY), Multi-Tier Architecture, Peer-to-Peer Architecture
author:
- |
Serguei A. Mokhov\
[Concordia University, Montreal, Canada]{}\
[[mokhov@cse.concordia.ca](mokhov@cse.concordia.ca)]{}\
- |
Joey Paquet\
[Concordia University, Montreal, Canada]{}\
[[paquet@cse.concordia.ca](paquet@cse.concordia.ca)]{}
bibliography:
- 'gipsy-hoil-arXiv.bib'
title: 'Using the General Intensional Programming System (GIPSY) for Evaluation of Higher-Order Intensional Logic (HOIL) Expressions'
---
Introduction
============
The project is an ongoing effort aiming at providing a flexible platform for the investigation on the intensional programming model as realized by the latest versions of the programming language [@lucid76; @lucid77; @lucid85; @lucid95; @nonprocedural-iterative-lucid-77], a multidimensional context-aware language whose semantics is based on possible worlds semantics [@kripke59; @kripke69]. provides an integrated framework for compiling programs written in theoretically all variants of , and even any language of intensional nature that can be translated into some kind of “generic Lucid” (e.g. [@paquetThesis; @gipsy-simple-context-calculus-08] or [[TransLucid]{}]{} [@eager-translucid-secasa08; @multithreaded-translucid-secasa08]).
Historically, the concept of was conceived as a very modular collection of frameworks geared towards sustainable support for the intensional programming languages and embracing continuous iterative revision and development overcoming defects of an earlier system [@glu1; @glu2; @agi95glu] that did not survive for very long due to its inflexibility to extend to the newer dialects, and its unmaintainability defects [@paquetThesis; @gipsy-arch-2000; @gipsy2005]. The ’s design centered around the compiler framework (), the eduction execution engine (), a.k.a the run-time execution environment, a sort of virtual machine for execution of the intensional logic expressions, and the programming environment (). The former of the three is responsible to support multiple compilers in a similar compiler framework that all produce a consistent, well agreed on binary format, essentially a compiled GIPSY program, as a binary output. The second performs lazy demand-driven, potentially parallel/distributed evaluation of the compiled Lucid programs, or as we call them, programs.
Eductive Model of Computation
-----------------------------
The first operational model for computing Lucid programs was designed independently by Cargill at the University of Waterloo and May at the University of Warwick, based directly on the formal semantics of , itself based on Kripke models and possible-worlds semantics [@kripke59; @kripke69]. This technique was later extended by Ostrum for the implementation of the Luthid interpreter [@luthid]. Luthid being tangential to standard , its implementation model was later used as a basis to the design of the pLucid interpreter by Faustini and Wadge [@eductive-interpreter]. This program evaluation model is now called [*eduction*]{} and opens doors for distributed execution of such programs [@swobodaphd04; @intensionalisation-tools; @distributed-context-computing; @marf-gipsy-distributed-ispdc08].
The concept of eduction can be described as “tagged-token demand-driven dataflow” computing (whereupon influenced a popular media platform and language called [[PureData]{}]{} [@puredata]). The central concept to this model of execution is the notion of generation, propagation, and consumption of [*demands*]{} and their resulting [*values*]{}. Lucid programs are declarative programs where every identifier is defined as a [[HOIL]{}]{} expression using other identifiers and an underlying algebra. An initial demand for the value of a certain identifier is generated, and the eduction engine, using the defining expression of this identifier, generates demands for the constituting identifiers of this expression, on which operators are applied in their embedding expressions. These demands in turn generate other demands, until some demands eventually evaluate to some values, which are then propagated back in the chain of demands, operators are applied to compute expression values, until eventually the value of the initial demand is computed and returned.
Lucid identifiers and expressions inherently vary in a [*multidimensional context space*]{}, i.e. any identifier or expression can be evaluated in a multidimensional context, thus leading to have identifiers and expressions representing a set of values, one value for each possible context in which the identifier or expression can be evaluated. This is brining the notion of [*intensionality*]{}, where identifiers are defined by intensional expressions i.e. expressions whose evaluation varies in a multidimensional context space, which can then be constrained by a particular multidimensional context specification. Note that Lucid variables and expressions represent “dimensionally abstract” concepts, i.e. they do not explicitly mention their dimensionality. For example, Newton’s Law of Universal Gravitation can be written literally in as:
[F = (G \* m1 \* m2) / r \* r;]{} and can then be evaluated in different dimensional manifolds (i.e. $n$-dimensional spaces), keeping the same definition, but being evaluated in contexts varying in their dimensionality. For example, [F]{} can be evaluated in a one-dimensional space, yielding a single scalar, or in a three-dimensional manifold, yielding a three-dimensional vector. Note that a [time]{} dimension could also be added where, for example, the masses ([m1]{} and [m2]{}) and/or the distance between them ([r)]{} might be defined as to vary in time. In such a case, the expression would then inherently be varying in the time dimension, due to some of its constituents varying in this dimension.
Intensional Logic and Programming
---------------------------------
Intensional programming, in the sense of the latest evolutions of , is a programming language paradigm based on the notion of declarative programming where the declarations are evaluated in an inherent multidimensional context space. The context space being in most cases infinite, intensional programs are evaluated using a lazy demand-driven model of execution that we introduced earlier – [*eduction*]{} [@eductive-interpreter], where the program identifiers are evaluated in a restricted context space, in fact, a [*point*]{} in space, where each demand is generated, propagated, computed and stored as an [*identifier-context*]{} pair [@bolu04].
Many problem domains are intensional in nature, e.g. computation of differential and tensor equations [@paquetThesis], temporal computation and temporal databases [@paquet-intensional-databases-95; @active-functional-idatabase], multidimensional signal processing [@agi95glu], context-driven computing [@crr05], constraint programming [@wanphd06], negotiation protocols [@wanphd06], automated reasoning in cyberforensics [@flucid-imf08; @flucid-isabelle-techrep-tphols08; @marf-into-flucid-cisse08], multimedia and pattern recognition [@marfl-context-secasa08] among others. The current mainstream programming languages are not well adapted for the natural expression of the intensional aspects of such problems, requiring the expression of the intensional nature of the problem statement into a procedural (and therefore sequential) approach in order to provide a computational solution.
Intensional programming can be used to solve widely diversified problems, which can be expressed using diversified languages of intensional nature. There also has been a wide array of flavors of Lucid languages developed over the years. Yet, very few of these languages have made it to the implementation level. The project aims at the creation of a programming environment encompassing compiler generation for all flavors of , a generic run-time system enabling the execution of programs written in all flavors of . Our goal is to provide a flexible platform for the investigation on programming languages of intensional nature, in order to prove the applicability of intensional programming to solve important problems.
Intensional programming is based on intensional (or multidimensional) logics, which, in turn, are based on natural language understanding (aspects, such as, time, belief, situation, and direction are considered). Intensional programming brings in [*dimensions*]{} and [*context*]{} to programs (e.g. space and time in physics or chemistry). Intensional logic adds dimensions to logical expressions; thus, a non-intensional logic can be seen as a constant or a snapshot in all possible dimensions. [*Intensions are dimensions*]{} at which a certain statement is true or false (or has some other than a Boolean value). [*Intensional operators*]{} are operators that allow us to navigate within these dimensions [@paquetThesis].
### Temporal Intensional Logic Example
Temporal intensional logic is an extension of temporal logic that allows to specify the time in the future or in the past [@paquetThesis].
\(1) $E_1$ := it is raining [**here**]{} [**today**]{}
Context: {`place:`[**here**]{}, `time:`[**today**]{}}
\(2) $E_2$ := it was raining [**here**]{} [*before*]{}([**today**]{}) = [*yesterday*]{}
\(3) $E_3$ := it is going to rain [*at*]{} (altitude [**here**]{} + 500 m) [*after*]{}([**today**]{}) = [*tomorrow*]{}
Let’s take $E_1$ from (1) above. The context is a collection of the dimensions and with the corresponding tag values of [**here**]{} and [**today**]{}. Then let us fix [**here**]{} to [**Montreal**]{} and assume it is a [*constant*]{}. In the month of May, 2009, with granularity of day, for every day, we can evaluate $E_1$ to either [*true*]{} or [*false*]{}, as shown in .
------------------------------------------------------------------------
Tags days in May: 1 2 3 4 5 6 7 8 9 ...
Values (raining?): F F T T T F F F T ...
------------------------------------------------------------------------
If one starts varying the [**here**]{} dimension (which could even be broken down to $X$, $Y$, $Z$), one gets a two-dimensional (or 4D respectively) evaluation of $E_1$, as shown in .
------------------------------------------------------------------------
Place/Time 1 2 3 4 5 6 7 8 9 ...
Montreal T F T F T F T F T ...
Honolulu F F T T T F F F T ...
New York F F F F T T T F F ...
Tampa F T T T T T F F F ...
------------------------------------------------------------------------
Even with these toy examples we can immediately illustrate the hierarchical notion of the dimensions in the context: so far the place and time we treated as atomic values fixed at days and cities. In some cases, we need finer subdivisions of the context evaluation, where time can become fixed at hour, minute, second and finer values, and so is the place broken down into boroughs, regions, streets, etc. and finally the $X,Y,Z$ coordinates in the Euclidean space with the values of millimeters or finer. This notion becomes more apparent and important e.g. in , a forensic case specification language for automated reasoning in cybercrime and other investigations.
[[HOIL]{}]{}
------------
To summarize, expressions written in virtually all dialects are correspond to higher-order intensional logic ([[HOIL]{}]{}) expressions with some dialect-specific instantiations. They all can alter the context of their evaluation given a set of operators and in some cases types of contexts, their range, and so on. [[HOIL]{}]{} combines functional programming and intensional logics, e.g. temporal intensional logic mentioned earlier. The contextual expression can be passed as parameters and returned as results of a function and constitute the multi-dimensional constraint on the Lucid expression being evaluated. The corresponding context calculus [@wanphd06; @gipsy-simple-context-calculus-08; @tongxinmcthesis08] defines a comprehensive set of context operators, most of which are set operators and the baseline operators are and that allow to switch the current context or query it, respectively. Other operators allow defined a context space and a point in that context corresponding to the current context. The context can be arbitrary large in its rank. The identified variables of the dimension type within the context can take on any data type, e.g. an integer, or a string, during lazy binding of the resulting context to a dimension identifier.
’s Architecture
===============
evolved from a modular collection of frameworks for local execution into a multi-tier architecture [@gipsy-multi-tier-secasa09]. With the bright but short-lived story of in mind, efforts were made to design a new system with similar capacities, but with more flexibility in mind. The new system would have to be able to cope with the fast evolution and diversity of the Lucid family of languages, thus necessitating a flexible compiler architecture, and a language-independent run-time system for the execution of Lucid programs. The architecture of the GIPSY compiler, the General Intensional Programming Compiler (GIPC) is framework-based, allowing the modular development of compiler components (e.g. parser, semantic analyzer and translator). It is based on the notion of the Generic Intensional Programming Language (), a core language into which all other flavors of the Lucid language can be translated to. The notion of a generic language also solved the problem of language-independence of the run-time system by allowing a common representation for all compiled programs, the Generic Eduction Engine Resources (), which is a dictionary of run-time resources compiled from a GIPL program, that had been previously generated from the original program using semantic translation rules defining how the original Lucid program can be translated into the . For a more complete description of the GIPSY compiler framework, see [@aihuawu02; @chunleiren02; @wuf04; @mokhovmcthesis05]. The architecture necessitates the presence of the intensional-imperative type system and support links to imperative languages being presented elsewhere [@gipsy-type-system-c3s2e09].
General Intensional Program Compiler ()
---------------------------------------
The , conceptually represented at the high level in . The type abstractions and implementations are located in the package and serve as a glue between the compiler (known as the – a General Intensional Program Compiler) and the run-time system (known as the – a General Eduction Engine) to do the static and dynamic semantic analyses and evaluation respectively. The is a very modular system allowing most components to be replaceable as long as they comply with some general architectural interface or API. One of such API interfaces is the (conceptually represented as – GEE Resource – a dictionary of run-time resources) that contains among other things the type annotations that can be statically inferred during compilation. At run-time, the engine does it’s own type checking and evaluation when traversing the AST stored in the GEER and evaluating expressions represented in the tree. Since both the and the use the same type system to do their analysis, they consistently apply the semantics and rules of the type system with the only difference that the , in addition to the type checks, does the actual evaluation. The is primarily implemented in .
![’s -to- Flow Overview in Relation to the GIPSY Type System.[]{data-label="fig:gipsy"}](GIPSY-GEER-flow){width=".5\textwidth"}
![GIPC Framework[]{data-label="fig:gipc-preprocessor"}](GIPC){width=".5\textwidth"}
The [@mokhovmcthesis05; @mokhovgicf2005] is something that is invoked first by the GIPC (see ) on incoming GIPSY program’s source code stream. The ’s role is to do preliminary program analysis, processing, and splitting the source GIPSY program into “chunks”, each written in a different language and identified by a [*language tag*]{}. In a very general view, a GIPSY program is a hybrid program consisting of different languages in one or more source file; then, there has to be an interface between all these code segments. Thus, the after some initial parsing (using its own preprocessor syntax) and producing the initial parse tree, constructs a preliminary dictionary of symbols used throughout the program. This is the basis for type matching and semantic analysis applied later on. This is also where the first step of type assignment occurs, especially on the boundary between typed and typeless parts of the program, e.g. and a specific Lucid dialect. The then splits the code segments of the GIPSY program into chunks preparing them to be fed to the respective concrete compilers for those chunks. The chunks are represented through the class that the collects.
General Eduction Engine ()
--------------------------
The design architecture adopted is a distributed multi-tier architecture, where each tier can have any number of instances. The architecture bears resemblance with a peer-to-peer architecture, e.g.:
- Demands are propagated without knowing where they will be processed or stored.
- Any tier or node can fail without the system to be fatally affected.
- Nodes and tiers can seamlessly be added or removed on the fly as computation is happening.
- Nodes and tiers can be affected at run-time to the execution of any GIPSY program, i.e. a specific node or tier could be computing demands for different programs.
### Generic Eduction Engine Resources
One of the central concepts of our solution is [*language independence*]{} of the run-time system. In order to achieve that, we rely on an intermediate representation that is generated by the compiler: the Generic Eduction Engine Resources (GEER). The General Intensional Programming Compiler (GIPC) compiles a program into an instance of the GEER, a dictionary of identifiers compiled from the program by the compiler [@wuf04; @mokhovmcthesis05]. The compiler framework provides with the potential to allow the easy addition of any flavor of the Lucid language to be added through automated compiler generation taking semantic translation rules in input [@aihuawu02]. As the name suggests, the GEER structure is generic, in the sense that the data structure and semantics of the GEER are independent of the language in which its corresponding source code was written. This is necessitated by the fact that the engine was designed to be “source language independent”, an important feature made possible by the presence of the Generic Intensional Programming Language (GIPL) as a generic language in the Lucid family of languages. Thus, the compiler first translates the source program (written in any flavor of Lucid) into “generic Lucid”, then generate the GEER run-time resources for this program, which is then made available at run-time to the various tiers upon demand. The GEER contains, for all Lucid identifiers in a given program, typing information, rank i.e. dimensionality information, as well as an abstract syntax tree representation of the declarative definition of the identifier. It is this latter tree that is later on traversed by the demand generator in order to proceed with demand generation. In the case of hybrid Lucid programs, the GEER also contains a dictionary of procedures called by the Lucid program, known as Procedure Classes, as they in fact are [*wrapper classes*]{} wrapping procedures inside a Java class in cases where the functions being called are not written in Java [@mokhovmcthesis05; @mokhovgicf2005].
### GIPSY Tier
The architecture adopted for this new evolution of the GIPSY is a multi-tier architecture where the execution of GIPSY programs is divided in three different tasks assigned to separate tiers. Each GIPSY tier is a separate process that communicates with other tiers using [*demands*]{}, i.e. the GIPSY Multi-Tier Architecture operational mode is fully [*demand-driven*]{}. The demands are generated by the tiers and migrated to other tiers using the Demand Store Tier. In this paper, we refer to a “tier” as an abstract and generic entity that represents a computational unit independent of other tiers and that collaborates with other tiers to achieve program execution as a group.
### GIPSY Node
Abstractly, a GIPSY node is a computer that has registered for the hosting of one or more GIPSY tier. GIPSY Nodes are registered through a GIPSY Manager instance. Technically, a GIPSY Node is a controller that wraps GIPSY Tier instances, and that is remotely reporting and being controlled by a GIPSY Manager. Operationally, a GIPSY Node hosts one tier Controller for each kind of tier (see ). The Tier Controller acts as a factory that will, upon necessity, create [*instances*]{} of this tier, which provide the concrete operational features of the tier in question. This model permits scalability of computation by allowing the creation of new tiers instances as existing tier instances get overloaded or lost.
![Design of the GIPSY Node[]{data-label="fig:GIPSYnodeDesign"}](GIPSYnode.jpg){width=".5\textwidth"}
### GIPSY Instance
A GIPSY Instance is a set of interconnected GIPSY Tiers deployed on GIPSY Nodes executing GIPSY programs by sharing their respective GEER instances. A GIPSY Instance can be executing across different GIPSY Nodes, and the same GIPSY Node may host GIPSY Tiers that a part of separate GIPSY Instances
### Demand Generator Tier
The Demand Generator Tier (DGT) generates demands according to the program declarations and definitions stored in one of the instances of GEER that it hosts. The demands generated by the Demand Generator Tier instance can be further processed by other Demand Generator Tiers instances (in the case of intensional demands) or Demand Worker Tier instances (in the case of procedural demands), the demands being migrated across tier instances through a Demand Store Tier instance. Each DGT instance hosts a set of GEER instances that corresponds to the Lucid programs it can process demands for. A demand-driven mechanism allows the Demand Generator Tier to issue system demands requesting for additional GEER instances to be added to its GEER Pool, thus enabling DST instances to process demands for additional programs as they are executed on the GIPSY instances they belong to.
### Demand Store Tier
The Demand Store Tier (DST) acts as a tier middleware in order to migrate demands between tiers. In addition to the migration of the demands and values across different tiers, the Demand Store Tier provide persistent storage of demands and their resulting values, thus achieving better processing performances by not having to re-compute the value of every demand every time it is re-generated after having been processed. From this latter perspective, it is equivalent to the historical notion of [*warehouse*]{} in the eduction model of computation. A centralized communication point or warehouse is likely to become an execution bottleneck. In order to avoid that, the Demand Store Tier uses a peer-to-peer architecture and mechanism to connect all Demand Store Tier instances in a given GIPSY instance. This allows any demand or its resulting value to be stored on any DST instance, but yet allows abstract querying for a specific demand value on any of the DST instances. If the demanded value is not found on the DST instance receiving the demand, it will contact its DST instance peers using a peer-to-peer mechanism. This mechanism allows to see the Demand Store abstractly as a single store that is, behind the scenes, a distributed one.
### Demand Worker Tier
The Demand Worker Tier (DWT) processes [*procedural demands*]{} i.e. demands for the execution of functions or methods defined in a procedural language, which are only present in the case where hybrid intensional programs are being executed. The DGT and DWT duo is an evolution of the generator-worker architecture adopted in GLU [@glu1; @glu2]. It is through the operation of the DWT that increased granularity of computation is achieved. Similarly to the DGT, each DWT instance hosts a set of compiled procedures (Procedure Classes) that corresponds to the procedural demands it can process. A demand-driven mechanism allows the Demand Worker Tier to issue system demands requesting for additional Procedure Classes to be added to its Procedure Class Pool, thus achieving increasing capacities over time, on demand.
### GIPSY Instance Manager
A GIPSY Instance Manager (GIM) is a component that enables the registration of GIPSY Nodes and Tiers, and to allocate them to the GIPSY Instances that it manages. The GIPSY Instance Manager interacts with the allocated tiers in order to determine if new tiers and/or nodes are necessary to be created, and issue demands to GIPSY Nodes to spawn new tier instances if need there be. In order to ease the node registration, the GIPSY Instance Manager tier can be implemented using a web interface, so that users can register nodes using a standard web browser, rather than requiring a client. GIPSY Instance Managers are peer-to-peer components, i.e. users can register a node through any GIPSY Instance Manager, which will then inform all the others of the presence of the new node, which will then be available for hosting new GIPSY Tiers at the request of any of the GIPSY Instance Managers currently running. The GIM uses [*system demands*]{} to communicate with Nodes and Tiers.
Context-Oriented Reasoning {#sect:context-reasoning}
==========================
As mentioned earlier, the reasoning aspect of is a particularity of a dialect rather than the architecture, and in this paper we look at it from the reasoning angle. The architecture is general enough to go beyond reasoning – in the essence it is an evaluation of intensional logic expressions. Now if those expressions form a language dialect, that helps us with reasoning, such as to reason about cybercrime incidents and claims. Some other dialects have less relevance in that regard, but still can be included into the system – such as for tensor fields evaluation of particles in plasma [@paquetThesis], reactive programming, multi-core processing [@eager-translucid-secasa08; @multithreaded-translucid-secasa08], software versioning, and others.
[@kaiyulucx; @wanphd06] is a fundamental extension of and the family as a whole that promotes the contexts as first-class values thereby creating a “true” generic language. Wan in [@kaiyulucx; @wanphd06] defined a new collection of set operators (e.g. union, intersection, box, etc.) on the multidimensional contexts, which will help with the multiple explanations of the evidential statements in forensic evaluation where the context sets are often defined as cross products (boxes), intersections, and unions. Its further specification, refinement, and implementation details are presented in [@tongxinmcthesis08; @gipsy-simple-context-calculus-08].
Operational Semantics for Reasoning about Lucid Expressions {#sect:semantics}
-----------------------------------------------------------
\[appdx:semantics\]
Here for convenience we provide the semantic rules of [@paquetThesis] (see ), [@wanphd06] (see ).
$$\begin{aligned}
{\mathbf{E_{cid}}} &:& \frac
{\johndef(\myid)=(\texttt{const},c)} {\context{\myid}{c}}\\\nonumber\\
{\mathbf{E_{opid}}} &:& \frac
{\johndef(\myid)=(\texttt{op},f)}
{\context{\myid}{\myid}}\\\nonumber\\
{\mathbf{E_{did}}} &:& \frac
{\johndef(\myid)=(\texttt{dim})}
{\context{\myid}{\myid}}\\\nonumber\\
{\mathbf{E_{fid}}} &:& \frac
{\johndef(\myid)=(\texttt{func},\myid_i,E)}
{\context{\myid}{\myid}}\\\nonumber\\
{\mathbf{E_{vid}}} &:& \frac
{\johndef(\myid)=(\texttt{var},E)\qquad
\context{E}{v}}
{\context{\myid}{v}}\\\nonumber\\
{\mathbf{E_{c_T}}} &:& \frac
{\context{E}{\textit{true}}\qquad
\context{E'}{v'}
}
{\context{\myifthenelse}{v'}}\\\nonumber\\
{\mathbf{E_{c_F}}} &:& \frac
{\context{E}{\textit{false}}\qquad
\context{E''}{v''}
}
{\context{\myifthenelse}{v''}}\\\nonumber\\
{\mathbf{E_{tag}}} &:& \frac
{\context{E}{\myid}\qquad
\johndef(\myid)=(\texttt{dim})
}
{\context{\#E}{\mathcal{P}(\myid)}}\\\nonumber\\
{\mathbf{E_{at}}} &:& \frac
{\context{E'}{\myid}\qquad
\johndef(\myid)=(\texttt{dim})\qquad
\context{E''}{v''}\qquad
\mathcal{D},\mathcal{P}\mydagger[\myid\mapsto v''] \vdash E : v
}
{\context{E\;@E'\;E''}{v}}\\\nonumber\\
{\mathbf{E_{w}}} &:& \frac
{\qcontext{Q}{\mathcal{D}',\mathcal{P}'}\qquad
\mathcal{D}',\mathcal{P}' \vdash E : v
}
{\context{E\;\mathtt{where}\;Q}{v}}\\\nonumber\\
{\mathbf{Q_{dim}}} &:& \frac
{}
{\qcontext{\texttt{dimension}\;\myid}
{\mathcal{D}\mydagger[\myid\mapsto(\texttt{dim})],
\mathcal{P}\mydagger[\myid\mapsto 0]}
}\\\nonumber\\
{\mathbf{Q_{id}}} &:& \frac
{}
{\qcontext{\myid=E}
{\mathcal{D}\mydagger[\myid\mapsto(\texttt{var},E)],
\mathcal{P}}
}\\\nonumber\\
{\mathbf{QQ}} &:& \frac
{\qcontext{Q}{\mathcal{D}',\mathcal{P}'}\qquad
\mathcal{D}',\mathcal{P}' \vdash Q' : \mathcal{D}'',\mathcal{P}''
}
{\qcontext{Q\;Q'}{\mathcal{D}'',\mathcal{P}''}}\\\nonumber\\
{\mathbf{E_{op}}} &:& \frac
{\context{E}{\myid}\qquad
\johndef(\myid)=(\texttt{op},f)\qquad
\context{E_i}{v_i}
}
{\context{E(E_1,\ldots,E_n)}{f(v_1,\ldots,v_n)}}\\\nonumber\\
{\mathbf{E_{fct}}} &:& \frac
{\context{E}{\myid}\qquad
\johndef(\myid)=(\texttt{func},\myid_i,E')\qquad
\context{E'[\myid_i\leftarrow E_i]}{v}
}
{\context{E(E_1,\ldots,E_n)}{v}}
$$
$$\begin{aligned}
{\mathbf{E_{\#}}} &:& \frac
{}
{\context{\#}{\mathcal{P}}}\\\nonumber\\
{\mathbf{E_{.}}} &:& \frac
{\context{E_{2}}{\myid_{2}}\qquad
\johndef(\myid_{2})=(\texttt{dim})
}
{\context{E_{1}.E_{2}}{\mathit{tag }(E_{1} \downarrow \{\myid_{2}\})}
}\\\nonumber\\
{\mathbf{E_{tuple}}} &:& \frac
{\context{E}{\myid}\qquad
\mathcal{D}\mydagger[\myid \mapsto (\texttt{dim})]\qquad
\mathcal{P}\mydagger[\myid \mapsto 0]\qquad
\context{E_{i}}{v_{i}}
}
{\context{\langle E_{1},E_{2},\ldots,E_{n}\rangle E}{v_{1} \;\; \mathit{fby}.\myid \;\; v_{2} \;\; \mathit{ fby}.\myid \;\; \ldots \;\; v_{n} \;\; \mathit{fby}.\myid \;\; \texttt{eod}}
}\\\nonumber\\
{\mathbf{E_{select}}} &:& \frac
{E=[\texttt{d}:\texttt{v'}]\qquad
E' = \langle \texttt{E}_{1},\ldots,\texttt{E}_{n} \rangle\texttt{d}
\mathcal{P'}=\mathcal{P}\mydagger[d \mapsto v']\qquad
\mathcal{D},\mathcal{P}' \vdash E' : v
}
{\context{\mathit{select}(E,E')}{v}
}\\\nonumber\\
{\mathbf{E_{at(c)}}} &:& \frac
{\context{\mathcal{C}}{\mathcal{P}}\qquad
\context{E}{v}
}
{\context{E\;@C}{v}}\\\nonumber\\
{\mathbf{E_{at(s)}}} &:& \frac
{\context{\mathcal{C}}{\{\mathcal{P}_{1},\ldots,\mathcal{P}_{2}\}}\qquad
\mathcal{D},\mathcal{P}_{i:1 \ldots m} \vdash E : v_{i}
}
{\context{E\;@C}{\{v_{1},\ldots,v_{m}\}}}\\\nonumber\\
{\mathbf{C_{context}}} &\!\!\!\!\!\!\!\!\!\!:\!\!\!\!\!\!\!\!\!\!& \frac
{
\begin{array}{l}
\context{E_{d_{j}}}{\myid_{j}}\qquad
\johndef(\myid_{j})=(\texttt{dim})\\
\context{E_{i_{j}}}{v_{j}}\qquad
\mathcal{P}' = \mathcal{P}_{0}\mydagger[\myid_{1}\mapsto v_{1}]\mydagger \ldots \mydagger [\myid_{n}\mapsto v_{n}]
\end{array}
}
{
\context{[E_{d_{1}}:E_{i_{1}},E_{d_{2}}:E_{i_{2}},\ldots,E_{d_{n}}:E_{i_{n}}]}{\mathcal{P}'}
}\\\nonumber\\
{\mathbf{C_{box}}} &\!\!\!\!\!\!\!\!\!\!:\!\!\!\!\!\!\!\!\!\!& \frac
{
\begin{array}{l}
\context{E_{d_{i}}}{\myid_{i}}\qquad\qquad\qquad
\johndef(\myid_{i})=(\texttt{dim})\\
\{E_1,\ldots,E_n\} = \mathit{dim}(\mathcal{P}_1)=\ldots=\mathit{dim}(\mathcal{P}_m)\\
E' = \texttt{f}_{p}(\texttt{tag}(\mathcal{P}_1),\ldots,\texttt{tag}(\mathcal{P}_m))\qquad
\context{E'}{\mathit{true}}
\end{array}
}
{
\context{\mathit{Box}[E_1,\ldots,E_n|E']}{\{\mathcal{P}_1,\ldots,\mathcal{P}_m\}}
}\\\nonumber\\
{\mathbf{C_{set}}} &:& \frac
{\context{E_{w:1 \ldots m}}{\mathcal{P}_m}
}
{\context{\{E_1,\ldots,E_m\}}{\{\mathcal{P}_1,\ldots,\mathcal{P}_w\}}
}\\\nonumber\\
{\mathbf{C_{op}}} &:& \frac
{
\context{E}{\myid}\qquad
\johndef(\myid)=(\texttt{cop}, f)\qquad
\context{C_i}{v_i}
}
{\context{E(C_1,\ldots,C_n)}{f(v_1,\ldots,v_n)}
}\\\nonumber\\
{\mathbf{C_{sop}}} &:& \frac
{
\context{E}{\myid}\qquad
\johndef(\myid)=(\texttt{sop}, f)\qquad
\context{C_i}{\{v_{i_{1}},\ldots,v_{i_{k}}\}}
}
{\context{E(C_1,\ldots,C_n)}{f(\{v_{1_{1}},\ldots,v_{1_{s}}\},\ldots,\{v_{n_{1}},\ldots,v_{n_{m}}\})}
}$$
Higher Order Context
--------------------
HOCs represent essentially nested contexts, e.g. as conceptually shown in modeling evidential statement for forensic specification evaluation. Such a context representation can be modeled as a tree in an OO ontology or a context-set as in . The early notion and specification of nested context first appeared Swoboda’s works [@swobodaphd04; @intensionalisation-tools; @distributed-context-computing], but there the evaluation has taken place only at the leaf context nodes. Another, more recent work on the configuration specification as a context in the intensional manner was the MARFL language [@marfl-context-secasa08; @marf-into-flucid-cisse08], allowing evaluation at arbitrary nesting of the configuration context with some defaults in the intermediate and leaf context nodes.
{width="\textwidth"}
Reasoning About Cyberforensic Cases and
----------------------------------------
A dialect, [@flucid-imf08; @flucid-isabelle-techrep-tphols08] develops a specification of a cyber incident for analysis of claims of witnesses against encoded evidential statements to see if they agree or not and if they do provide potential backtraces of event reconstruction. The hierarchical context space similar to the one defined in the previous section is also used in to denote a context of a cybercrime case as a hierarchy consisting of the evidential statement $es$, that consists of observation sequences (“stories” told by evidence and witnesses) $os$, which in turn consist of observations $o$, that denote a certain observer property $P$ and its duration $[\min, \min+\max]$, as shown in . draws from an earlier formal approach using finite state automata by Gladyshev [@blackmail-case; @printer-case] that is not very usable and all of the benefits of Lucid and intensional evaluation with functions and operators that navigate withing the higher-order context space of evidence and witness stories to evaluate claims. The Demptser-Shafer theory [@prob-argumentation-systems; @shafer-evidence-theory] is used to assigned weights, such as credibility and admissibility to witnesses evidence as a part of reasoning parameters.
Conclusion
==========
We presented a modular intensional programming research platform, , for reasoning tasks of HOIL expressions. The concept of context as a first-class value is central in the programming paradigms is build to explore, such as a family of the programming languages. At the time of this writing has support for compilation of , , , , and and the execution of if the former two with the other being completed. The DMS for distributed transport of the demands has implementations in Jini, plain RMI, and JMS.
Future Work
-----------
The future and ongoing work within the context of is a complete formalization of its hybrid intensional-imperative type system [@gipsy-type-system-c3s2e09], the revision of the syntax and semantics of the language, and the multi-tier overhaul of the evaluation engine () including support for OO intensional dialects.
Acknowledgments
---------------
This work was funded by NSERC and the Faculty of Computer Science and Engineering, Concordia University. Thanks to many of the GIPSY project team members for their valuable contributions, suggestions, and reviews, including Peter Grogono, Emil Vassev, Xin Tong, and Amir Pourteymour.
\[sect:bib\]
|
---
abstract: 'We present a class of models with radiative neutrino mass and stable dark-matter candidates. Neutrino mass is generated by a one-loop diagram with the same topography as Ma’s 2006 proposal (which used an inert scalar-doublet and singlet fermion). We generalize this approach and determine all variants with new fields no larger than the adjoint representation. When the neutrino mass diagram contains a Majorana mass insertion there are two possibilities, both of which are known. If the mass insertion is of the Dirac type there are seven additional models, two of which are excluded by direct-detection experiments. The other five models are also constrained, such that only scalar dark-matter is viable. There are cases with an inert singlet, an inert doublet, and an inert triplet, providing a natural setting for inert $N$-tuplet models of dark matter, with the additional feature of achieving radiative neutrino mass. We show that some of the models admit a simple explanation for the (requisite) discrete symmetry, and briefly discuss cases with representations larger than the adjoint, which can admit a connection to the astrophysical gamma-ray signal.'
---
A Class of Inert N-tuplet Models with Radiative Neutrino Mass and Dark Matter\
[Sandy S. C. Law$^{*}$[^1] and Kristian L. McDonald$^{\dagger}$[^2]]{}\
\* Department of Physics, National Cheng-Kung University,\
Tainan 701, Taiwan\
$\dagger$ ARC Centre of Excellence for Particle Physics at the Terascale,\
School of Physics, The University of Sydney, NSW 2006, Australia\
Introduction\[sec:introduction\]
================================
The experimental evidence for neutrino mass, acquired in recent decades, provides concrete evidence for physics beyond the Standard Model (SM) (see e.g. [@GonzalezGarcia:2012sz]). Although the requisite new degrees of freedom cannot yet be determined it is clear that additional particles are likely to exist, in order to generate the masses. Similarly, there is by now a large amount of evidence for an additional galactic constituent, an unknown substance referred to as dark matter (see e.g. [@Peter:2012rz]). This may or may not require new degree’s of freedom, but the hypothesis that the dark matter is comprised of a stable (or long lived) new particle species provides a simple explanation for this observed feature of the Universe. Given that these two indicators for beyond-SM physics can be explained by extending the particle spectrum of the SM, it is natural to ask if the requisite new particles can be related. Could the mechanism of neutrino mass be related to the existence of a stable dark-matter candidate?
A particularly simple model realizing this idea was proposed by Ma in 2006 [@Ma:2006km]. This model extends the SM to include an additional SM-like scalar doublet and gauge-singlet fermions, all of which are odd under a discrete $Z_2$ symmetry. The extended field content allows for radiative neutrino mass, generated at the one-loop level, while the lightest beyond-SM field is absolutely stable. One thus arrives at a simple synergetic model of radiative neutrino mass and dark matter.
In this work we generalize Ma’s approach. We present a class of related models, all of which generate neutrino mass via a loop diagram with the same topography as Ma’s, whilst simultaneously admitting stable dark-matter candidates. The loop-diagram employed by Ma contains a mass insertion on the internal fermion line (see Figure \[fig:yukawa\_loop\]), and our generalizations fall naturally into two categories; those which break lepton-number symmetry via a Majorana mass insertion, and those with a Dirac mass insertion, such that lepton-number symmetry is broken at a vertex. Although the basic mechanism is very similar in both cases, this difference modifies one’s expectations for the beyond-SM field content and the associated phenomenology.
It turns out that, in both cases, this approach is very general and many realizations are possible. However, restricting attention to models in which the beyond-SM multiplets are no larger than the adjoint representation significantly reduces the possibilities. As we shall see, there are only two such (minimal) models with a Majorana mass-insertion, both of which are known [@Ma:2006km; @Ma:2008cu]. We find seven additional models with a mass insertion of the Dirac type, all of which achieve radiative neutrino mass and dark-matter candidates. We detail these models, finding a subset that are compatible with direct-detection experiments. There are cases with an inert singlet, an inert doublet, and an inert triplet; the models therefore provide a natural setting for inert $N$-tuplet theories of dark matter, such that radiative neutrino mass is also achieved.
Interestingly, three of the new models admit a simple extension that can explain the origin of the (formerly imposed) discrete symmetry. By upgrading the discrete symmetry to a gauged $U(1)$ symmetry, and extending the field content by a single SM-singlet scalar, the discrete symmetry can arise as an accidental symmetry of the low-energy Lagrangian, after $U(1)$ symmetry breaking takes place. This provides a simple explanation for the discrete symmetry.
Though we focus on models with representations no larger than the adjoint, we also briefly discuss cases where the beyond-SM fields can be quadruplet and/or quintuplet representations of $SU(2)_L$. We present the candidate models in these cases, and mention some key issues, based on the lessons learned from our preceding studies. Despite the use of larger multiplets, these models can still be of interest; in addition to allowing for radiative neutrino mass, the exotics with larger electric-charges in these multiplets can enhance the $2\gamma$ and/or $\gamma+Z$ signal from dark-matter annihilation when they appear inside loops [@Kopp:2013mi]. This can provide a simple connection between the mechanism of neutrino mass and the astrophysical gamma-ray signal [@Weniger:2012tx].
Before proceeding we note that the connection between radiative neutrino mass and dark matter has been explored in a number of different models, including Ref. [@Krauss:2002px], which precedes Ma’s work; for other examples see [@Boehm:2006mi]. Previous works on inert-singlet models [@McDonald:1993ex], inert-doublet models [@Barbieri:2006dq], and inert-triplet models [@Cirelli:2005uq] are also well known. Additional relevant works dealing with inert-multiplet dark matter and/or radiative neutrino mass are cited in the text. Note also that Refs. [@Ma:1998dn; @Bonnet:2012kz] have detailed the one-loop realizations of the $d=5$ operator for neutrino mass. Inert scalar dark matter can also help cure the little hierarchy problem found in low-scale seesaws [@Fabbrichesi:2013qca]. The present work follows on from the generalized tree-level seesaws presented in Ref. [@McDonald:2013kca].
The plan of this paper is as follows. In Section \[sec:rad\_nu\_plus\_dm\] we discuss Ma’s model and present the generalizations that similarly achieve radiative neutrino mass and dark-matter candidates. Section \[sec:fermion\_dm\] considers the case of fermionic dark-matter, while Section \[sec:scalar\_dm\] discusses scalar dark-matter. One of the generalized models is presented in more detail in Section \[sec:real\_and\_complex\_scalar\]. In Section \[sec:gauge\_symmetry\] we show that some of the models allow a simple extension, such that the discrete symmetry appears as an accidental symmetry in the low-energy theory. Models with exotics forming larger $SU(2)$ representations are discussed in Section \[sec:non-minimal\_models\] (and explicitly displayed in the Appendix). We conclude in Section \[sec:conc\].
Radiative Neutrino Mass and Dark Matter\[sec:rad\_nu\_plus\_dm\]
================================================================
We are interested in the class of models that generate neutrino mass radiatively by the diagram in Figure \[fig:yukawa\_loop\]. Here $\mathcal{F}$ is a beyond-SM fermion and $S_{1,2}$ are new scalars (which can be identical in some cases). A basic feature of this diagram is that the three vertices can all involve two beyond-SM fields. Consequently one can always consider a discrete $Z_2$ symmetry whose action on the beyond-SM multiplets is {, S\_1, S\_2}&& - {, S\_1, S\_2}, while all SM fields transform trivially. The lightest field within the multiplets $\mathcal{F}$ and $S_{1,2}$ will thus be stable, and provided this field is electrically neutral and colorless, one arrives at a dark-matter candidate. These comments are generic for all models of this type; the connection between loop masses and dark matter is simple to realize in this class of models.
Figure \[fig:yukawa\_loop\] produces Majorana neutrino masses, so the loop diagram must contain a source of lepton number violation. Choosing the convention for lepton-number symmetry such that the new fermion $\mathcal{F}$ has the same value as the SM leptons, there are two ways to explicitly break lepton-number symmetry; it can be broken at either the mass insertion or at one of the vertices. The simplest models, in terms of the requisite number of beyond-SM multiplets, are those with a lepton number violating (Majorana) mass insertion. In this case one has $\mathcal{F}_L\equiv \mathcal{F}_R^c$ and minimal cases occur for $S_1=S_2\equiv S$. Thus, only two beyond-SM multiplets are required. The general loop-diagram for this subset of models is given in Figure \[fig:mass\_loop\]. We consider this case first.
Models with a Majorana Mass Insertion\[sec:maj\_mass\]
------------------------------------------------------
We seek models that achieve neutrino mass via Figure \[fig:mass\_loop\] and give rise to dark-matter candidates. Clearly the fermion must form a real representation of the SM gauge symmetry, $\mathcal{F}_R\sim(1,R_{\mathcal{F}},0)$, in order to allow a bare Majorana mass. As we are considering dark-matter candidates we do not consider colored fields. Note also that $R_{\mathcal{F}}$ must be odd-valued to ensure there is no fractionally charged particles (the lightest of which would be stable and thus cosmologically excluded). Odd-valued $R_{\mathcal{F}}$ also ensures that $\mathcal{F}_R$ contains an electrically neutral component, so no additional constraint is imposed by this demand.
With this information one can obtain the viable combinations of $\mathcal{F}_R$ and $S$ that generate Figure \[fig:mass\_loop\]. The basic Lagrangian terms are && i|[\_R]{}\^D\_\_R - \_R +|D\^S|\^2 -M\_S\^2 |S|\^2 +|[L]{} \_R+\_(S\^H)\^2 +,\[eq:ma\_lagrangian\] where $L$ ($H$) is the SM lepton (scalar) doublet and $\tilde{S}$ denotes the charge-conjugate of $S$. It turns out that the possible combinations for $\mathcal{F}$ and $S$ are not restricted by quantum numbers; one can consider increasingly large multiplets, presumably up to some unitarity limits [@Hally:2012pu], and realize a model with Figure \[fig:mass\_loop\] and a dark-matter candidate. However, if we restrict our attention to models with $R_{\mathcal{F}},R_S\le 3$, such that no new multiplet is larger than the adjoint representation, there are only two possibilities. The first case is Ma’s original proposal, which employs an additional (inert) scalar doublet $S\sim (1,2,1)$, and a gauge-singlet fermion $\mathcal{F}_R\sim(1,1,0)$ [@Ma:2006km]. This model is the prototype for the class we are considering. The second model also employs the scalar doublet $S\sim (1,2,1)$, but instead utilizes the triplet fermion $\mathcal{F}_R\sim(1,3,0)$ [@Ma:2008cu], familiar from the Type-III seesaw [@Foot:1988aq]. Thus, both of the models are known in the literature, and there are no additional possibilities unless one considers larger multiplets.
In each of these models the dark-matter candidate can be a neutral component of $S$ or a Majorana fermion. There is, however, an important difference between the singlet case, $\mathcal{F}\sim(1,1,0)$, and the other model; the singlet does not participate in weak interactions and is therefore brought into thermal contact with the SM sector via the Yukawa coupling. For fermionic dark-matter, this can produce conflict between the need to keep the Yukawa coupling large to ensure thermal dark-matter, and the need to suppress the Yukawa coupling to limit the size of flavor changing effects. This issue does not arise in the triplet fermion model, as the fermions can maintain equilibrium with the SM sector via weak interactions in these cases, even if the Yukawa couplings are suppressed. For an analysis of Ma’s model, incorporating recent LHC data on the Higgs, see e.g. Ref. [@Ho:2013hia]. Also note that loop effects can induce observable interactions between dark matter and experimental detectors in Ma’s model [@Schmidt:2012yg].
Models with a Dirac Mass Insertion\[sec:dirac\_mass\]
-----------------------------------------------------
Having exhausted the minimal models with a lepton-number violating mass insertion, we now consider models with a Dirac mass insertion; i.e. the beyond-SM fermion has nonzero hypercharge. In this case the general mass-diagram has the form shown in Figure \[fig:yukawa\_loop\]. The fields $\mathcal{F}_R$ and $\mathcal{F}_L$ are no longer related by charge conjugation, so the mass insertion is of the Dirac type and $\mathcal{F}$ is a vector-like fermion. In addition one requires $S_1\ne S_2$. We again consider a $Z_2$ symmetry under which the SM fields transform trivially but the new fields are odd. The Lagrangian contains the following pertinent terms: && i|\^D\_ - M\_ + \_[i=1,2]{}{|D\^S\_i|\^2 - M\_i\^2 |S\_i|\^2 }\
& & + \_1|[L]{} \_RS\_1 + \_2 |[L]{}\^c\_L\_2 + \_S\_1\_2 H\^2 +,\[eq:L\_ykawa\_lagrange\] where, in our convention, lepton number symmetry is broken by a Yukawa coupling. With the $Z_2$ symmetry present, there are no terms in the scalar potential that are linear in *just one* of the new scalars $S_{1,2}$. Therefore the beyond-SM scalars do not acquire an induced VEV and parameter space exists for which $\langle S_{1,2}\rangle=0$, so the $Z_2$ symmetry remains exact.
In selecting viable multiplets one must ensure that no new multiplet contains a fractionally charged field, to avoid a (cosmologically excluded) stable charged field. To ensure that the lightest $Z_2$-odd field is a neutral dark-matter candidate one must demand that the new multiplets contain at least one neutral field. Note that the neutral field does not have to appear as an explicit propagating degree of freedom inside the loop diagram; it is sufficient merely that the loop-diagram exists and that the particle content includes a neutral field that can play the role of dark matter.
With these conditions in mind we search for viable combinations of the beyond-SM multiplets that realize Figure \[fig:yukawa\_loop\]. We find that the size of the beyond-SM multiplets is not restricted by our demands; one can consider increasingly large multiplets and realize the loop diagram. However, if attention is restricted to models in which none of the beyond-SM multiplets are larger than the adjoint representation, only seven distinct models are found. These are listed in Table \[L\_yukawa\]. Of the seven models, one employs the exotic lepton triplet $\mathcal{F}\sim(1,3,-2)$, studied in Refs. [@Chua:2010me; @DelNobile:2009st], three contain an exotic vector-like (and SM-like) lepton doublet $\mathcal{F}\sim(1,2,-1)$ [@DelNobile:2009st; @Joglekar:2012vc; @Arina:2012aj], and two contain a charged lepton doublet $\mathcal{F}\sim(1,2,-3)$ [@DelNobile:2009st; @Law:2011qe]. There is also a model with a SM-like charged singlet fermion, $\mathcal{F}\sim(1,1,-2)$, which already appeared in Ref. [@Aoki:2011yk].
Neutrino masses take a standard calculable form in these models. For example, in models $(A)$ and $(B)$ only singly-charged exotics propagate in the loop, and the SM neutrino mass matrix is given by[^3] (\_)\_&&\
& & Here $M_{\mathcal{F},a}$ is the mass for the charged component of the exotic fermion $\mathcal{F}_a$, and summation is implied for the repeated index $a$ (which labels the exotic-fermion generations).[^4] The masses $M_{>,<}$ refer to the charged scalar mass-eigenstates, which are linear combinations of the charged scalars $S_1^+$ and $S_2^+$. The mixing results from the $\lambda_{{\text{\tiny SH}}}$-term in Eq. , which takes the explicit form $\lambda_{{\text{\tiny SH}}} \tilde{H}^\dagger S_1 S_2^\dagger H\subset\mathcal{L}$, for models $(A)$ and $(B)$. If all the exotics are at the TeV scale one requires dimensionless couplings of $\mathcal{O}(10^{-3})$ to obtain $m_\nu\sim0.1$ eV. The scenario with all exotics at the TeV scale is most interesting from a phenomenological perspective. However, strictly speaking one only requires the lightest exotic to have a mass of $\lesssim\mathcal{O}(\mathrm{TeV})$ in order to realize a dark-matter candidate. The other exotics can be much heavier, allowing larger dimensionless couplings.
Before moving on to discuss dark matter in detail, we note that, of the models in Table \[L\_yukawa\], only models $(A)$, $(B)$ and $(D)$ are expected to produce (dominant) radiative neutrino masses in the absence of the $Z_2$ symmetry, as we briefly discuss in Appendix \[app:mass\_no\_Z2\].
----------- -------------------- ------------------- --------------- -------------------------------------
Model $\mathcal{F}$ $S_1$ $S_2$ Dark Matter Status
$(A)$ $\ \ (1,1,-2)\ \ $ $\ \ (1,2,1)\ \ $ $(1,2,3)$ Inert Doublet (Ref. [@Aoki:2011yk])
$(B)$ $\ \ (1,3,-2)\ \ $ $\ \ (1,2,1)\ \ $ $(1,2,3)$ Inert Doublet
$(C)$ $(1,2,-1)$ $(1,1,0)$ $(1,3,2)$ Inert Singlet or Triplet
$(D)$ $(1,2,-1)$ $(1,3,0)$ $(1,1,2)$ Inert Real Triplet
$(E)$ $(1,2,-1)$ $(1,3,0)$ $(1,3,2)$ Inert Triplet
$(F)$ $(1,2,-3)$ $(1,3,2)$ $(1,1,4)$ Excluded (Direct Detection)
$(G)$ $(1,2,-3)$ $(1,3,2)$ $(1,3,4)$ Excluded (Direct Detection)
----------- -------------------- ------------------- --------------- -------------------------------------
: \[L\_yukawa\] Minimal models with a Dirac mass insertion. These models allow radiative neutrino mass and contain at least one stable electrically-neutral beyond-SM field.
Inert Fermionic Dark Matter\[sec:fermion\_dm\]
==============================================
We now turn our attention to the dark-matter candidates in these models. It is *a priori* possible that both fermionic and scalar dark-matter candidates are possible, as in Ma’s original proposal [@Ma:2006km]. In this section we consider fermionic dark-matter. Note that not all the models contain neutral beyond-SM fermions; specifically, model $(A)$ has $\mathcal{F}\sim(1,1,-2)$, and models $(F)$ and $(G)$ use $\mathcal{F}\sim(1,2,-3)$. In these cases all beyond-SM fermions are charged and only scalar dark-matter is possible. One can already exclude the parameter space with light fermions in these models, namely $M_{\mathcal{F}}<M_{1,2}$, due to the appearance of a stable charged-fermion.
On the other hand, models $(B)$ through $(E)$ all contain neutral fermions and can, in principle, admit fermionic dark-matter. However, all of the fermion multiplets in these models have nonzero hypercharge, which can lead to strong constraints from direct-detection experiments. More precisely, if the dark-matter abundance is generated by a standard thermal WIMP one can exclude Dirac-fermion dark-matter with nonzero hypercharge, due to the strong constraints from e.g. XENON100 [@Aprile:2011hi]. Thus, it is important to determine whether the neutral fermion is Dirac or Majorana.
At tree-level the fermion $\mathcal{F}$ remains a Dirac particle. However, its coupling to the SM neutrinos, which obtain Majorana masses via Figure \[fig:yukawa\_loop\], leads to a small radiative Majorana-mass. For the case of $\mathcal{F}\sim(1,3,-2)$, the typical diagram is shown in Figure \[fig:F\_maj\_mass\]. Similar diagrams can occur for models $(C)$ through $(E)$, though the scalar $S_1$ is real in these cases. The loop-induced Majorana mass will, in general, split the Dirac fermion $\mathcal{F}$ into a pair of Majorana fermions. However, one can already see that the mass-splitting will be very small. The sub-loop in Figure \[fig:F\_maj\_mass\] is the same loop-diagram that generates SM neutrino masses in Figure \[fig:yukawa\_loop\]. Thus, in the limit that SM neutrino masses vanish, $m_\nu\rightarrow0$, the Majorana mass for $\mathcal{F}$ will also vanish. We therefore expect $\Delta M_{\mathcal{F}}\propto m_\nu$, where $\Delta M_{\mathcal{F}}$ is the Majorana mass for $\mathcal{F}$. This is born out by explicit calculations. For example, with $M_{\mathcal{F}}\ll M_S$, where $M_S$ denotes an approximate common mass for the beyond-SM scalars, one obtains M\_&\~& m\_, where the Lagrangian contains the term $\lambda_{11{\text{\tiny H}}}^*(H^\dagger S_1)^2\subset\mathcal{L}$ to generate the uppermost vertex in Figure \[fig:F\_maj\_mass\]. The beyond-SM neutral fermions therefore form pseudo-Dirac particles with a tiny splitting.
Direct detection experiments give strong constraints on spin-independent elastic-scattering events that can occur when fermionic dark-matter couples to the $Z$ boson [@Aprile:2011hi]. These constraints can be avoided if the Dirac fermion has a mass-split, as the resulting pair of Majorana fermions has non-diagonal couplings to the $Z$ boson (to leading order). Provided the mass split exceeds the average kinetic-energy of the local dark-matter particles, $Z$-boson exchange with SM detectors is highly suppressed, as the heavier fermion is kinematically inaccessible. However, for the models with neutral fermions in Table \[L\_yukawa\] the mass split satisfies & <& 10\^[-12]{}M\_=(), which is (much) too small to evade direct-detection bounds, given typical DM speeds of $v_{{\text{\tiny DM}}}\sim10^{-3}$. We conclude that none of the models in Table \[L\_yukawa\] are viable when the lightest beyond-SM field is a fermion, due to either a cosmologically excluded stable charged-particle or a dark-matter candidate that contradicts direct-detection constraints. The entire region of parameter space in which a fermion is the lightest beyond-SM state is thus excluded for these models.
Inert Scalar Dark Matter\[sec:scalar\_dm\]
==========================================
With the above information we can restrict our attention to the limit $M_{\mathcal{F}}\gg M_{1,2}$ for the models in Table \[L\_yukawa\], for which the stable particle is a scalar. In this limit the models are effectively inert $N$-tuplet models with the additional feature of realizing radiative neutrino mass. In this section, we consider the viability of the scalar dark-matter candidates in the different models.
We first consider models $(A)$ and $(B)$, whose common features allow them to be discussed together. Both these models have a single (candidate) dark-matter multiplet, which is an inert SM-like scalar doublet, $S_1\sim(1,2,1)$; i.e. an inert doublet [@Barbieri:2006dq]. Also, in both models the second scalar is a charged doublet, $S_2=(S_2^{++},S_2^+)^T$, whose components must be heavier than the dark matter. Inert-doublet dark matter is well-studied in the literature, and it is known that a viable dark-matter abundance can be realized [@Barbieri:2006dq]. The inert-doublet leads to three new scalars, which we denote as $H'^{\pm}$, $H'^0$ and $A^0$, and either of the last two can be the dark matter. As per usual for an inert-doublet model, the neutral components of $S_1$ cannot mix with the SM Higgs in models $(A)$ and $(B)$, due to the discrete symmetry. However, the charged scalar $S_1^+$ will mix with $S_2^+$, as mentioned already. If this mixing is large, the phenomenology of the lightest charged-scalar will differ from that of $H'^+$ in a standard inert-doublet model. For small mixing the lightest charged scalar will correspond mostly to $H'^{+}$ and the phenomenology of $S_1$ will be well approximated by a standard inert-doublet. Note that one cannot take the limit $\lambda_{{\text{\tiny SH}}}\rightarrow0$ without turning off the radiative neutrino mass in Figure \[fig:yukawa\_loop\]. The demand that radiative neutrino mass is realized therefore requires nonzero mixing between $S_1^+$ and $S_2^+$. However, given that $S_2$ must be heavier than the dark-matter, one generally expects the mixing to be of order $\langle H^0\rangle^2 /M_2^2$ which is $\lesssim 10^{-1}$ for $M_2\gtrsim$ TeV. Thus, $S_1$ can be well-approximated by a standard inert-doublet.
The inert-doublet model contains five main regions of parameter space in which the observed relic-abundance is obtained [@Dolle:2009fn]. Four of these have a light particle-spectrum that can be probed at the LHC. The discovery of an SM-like scalar with mass of roughly $125$ GeV at the LHC allows one to update the viable parameter space and phenomenology of the inert-doublet models. Recent analysis, incorporating the LHC data, shows that the low-mass regions for the dark-matter candidate can already be in tension with constraints from XENON100 [@Aprile:2011hi] and WMAP [@Komatsu:2010fb], while the heavier region with $M_{{\text{\tiny DM}}}\gtrsim500$ GeV is essentially unaffected [@Goudelis:2013uca]. Specifically, the surviving region for lighter dark-matter lies close to the Higgs-resonance/$WW$-production threshold [@Goudelis:2013uca].
The region of parameter space with $M_{{\text{\tiny DM}}}=\mathcal{O}(10)$ GeV is particularly interesting for the present models as, in this case, the additional beyond-SM multiplets can be light enough to appear at the LHC. This was discussed already in Ref. [@Aoki:2011yk] for model $(A)$, where it was shown that the charged scalar-doublet $S_2\sim(1,2,3)$ can produce observable signals when the inert-doublet dark matter is light. Although the region of parameter space with heavier dark-matter ($M_{{\text{\tiny DM}}}\gtrsim500$ GeV) will not be accessible at the LHC, it is expected that XENON-1T will probe this parameter space, potentially giving observable direct-detection signals [@Klasen:2013btp]. One deduces that viable dark matter is possible in both models $(A)$ and $(B)$, and that the combined (projected) LHC and XENON-1T data sets are expected to probe the viable parameter space in these models.
We note that, in general, models with hypercharge-less dark matter are not as strongly constrained by direct-detection experiments. Such candidates do not couple directly to the $Z$ boson so interactions with detectors do not arise at tree-level. Provided the mass-splitting between the charged and neutral components of the dark-matter multiplet exceeds the average kinetic-energy of the dark matter in the local halo, interactions with the $W$ boson are also highly suppressed (or absent). Even if the neutral and charged components of a dark-matter multiplet are degenerate at tree-level, an $\mathcal{O}(100)$ MeV split is induced radiatively, which is sufficient to ensure scattering via $W$ boson exchange is suppressed/absent.
These comments apply to model $(D)$, in which the sole dark-matter candidate is the neutral component of the inert real-triplet, $S_1\sim(1,3,0)$. The neutral component of this multiplet does not interact with the $Z$ boson, and the charged component can be sufficiently split (by radiative effects) to ensure the neutral state is the lightest field. This alleviates potential tension with direct-detection experiments. The possibility of inert real-triplet dark matter is well known in the literature [@Cirelli:2005uq; @FileviezPerez:2008bj; @Hambye:2009pw; @Araki:2011hm]. The neutral component of $S_1$ is a viable cold dark-matter candidate that saturates the observed relic abundance of $\Omega_{{\text{\tiny CDM}}}\hat{h}^2\simeq0.11$ [@Komatsu:2010fb] for $M_{{\text{\tiny DM}}}\approx 2.5$ TeV [@Cirelli:2005uq]. If the real-triplet is lighter it can only comprise part of the dark-matter abundance and additional candidates are needed. In model $(D)$ the dark-matter abundance must be comprised solely of $S_1$, so that $M_{{\text{\tiny DM}}}\approx 2.5$ TeV is a necessary requirement for this model. Unfortunately this makes it difficult to directly produce the exotic states at the LHC; both $\mathcal{F}\sim(1,2,-1)$ and $S_2\sim(1,1,2)$ must be heavier than $2.5$ TeV to ensure the dark matter is the lightest exotic, pushing them beyond projected experimental reach. Model $(D)$ is, however, a viable model of dark matter and radiative neutrino mass for $M_{{\text{\tiny DM}}}\approx 2.5$ TeV.
Next we turn our attention to models $(F)$ and $(G)$, which both employ $\mathcal{F}\sim(1,2,-3)$ and $S_1\sim(1,3,2)$. In both models $S_2$ is comprised purely of electrically-charged fields, so the neutral component of $S_1$ is the sole dark-matter candidate. This complex neutral-field cannot mix with the SM scalar, due to the $Z_2$ symmetry, so its particle and antiparticle sates remain degenerate. The dark-matter abundance is therefore comprised of both states, posing a serious difficulty for these models. Due to the nonzero hypercharge for $S_1$, the neutral field can scatter off SM-detectors via tree-level $Z$-boson exchange. This process is strongly constrained by direct-detection data sets. Previous works show that one requires a mass of $\sim 2.6$ TeV to obtain the correct abundance, however, the spin-independent cross-section exceeds $10^{-37}$cm$^2$ in the regions of parameter space compatible with the LEP experiments [@Hambye:2009pw; @Araki:2011hm]. Such a large cross section is incompatible with the constraints from, e.g., XENON100 [@Aprile:2011hi]. Thus, although one can successfully generate the requisite dark-matter abundance, direct-detection constraints prove fatal for models $(F)$ and $(G)$ and both models can be excluded.
It remains for us to consider models $(C)$ and $(E)$. These models admit two distinct scalar dark-matter candidates and thus allow more possibilities, as we shall see in following section.
Models with a Real Scalar and a Complex Triplet\[sec:real\_and\_complex\_scalar\]
=================================================================================
Models $(C)$ and $(E)$ both employ the SM-like fermion $\mathcal{F}\sim(1,2,-1)$ and the complex scalar triplet $S_2\sim(1,3,2)$. Furthermore, in both cases $S_1$ is a real scalar. These models differ from the other cases as both scalars now possess a neutral component, giving two dark-matter candidates. We saw that models $(F)$ and $(G)$ could be excluded precisely because the dark-matter abundance was comprised of the neutral component of the complex scalar triplet. This difficulty is avoided in models $(C)$ and $(E)$, however, due to allowed mass-mixing between the neutral components of $S_1$ and $S_2$. In this section we discuss model $(C)$ in some detail, to elucidate the possibilities. The analysis of model $(E)$ is rather similar, due to the related field content, and we limit ourselves to some brief comments on this model at the end of the section.
Model $(C)$ contains the beyond-SM scalars $S_1\sim(1,1,0)\equiv S$ and $S_2\sim(1,3,2)\equiv \Delta$. The full scalar-potential can be written as V(H,S,)&=&|H|\^2 +S\^2 +\_\^2 (\^) +\_1 |H|\^4 +\_2\[(\^) \]\^2\
& &+\_2’(\^\^) +S\^4+\_4 |H|\^2(\^) + \_4’ H\^\^H\
& &+\_5|H|\^2S\^2 +\_6 S\^2(\^)+\_ S{ \^\^H +H\^},\[model\_c\_potential\] where the overall phase of $\Delta$ has been used to choose $\lambda_{{\text{\tiny SH}}}$ real without loss of generality.[^5] The discrete symmetry $\{S,\,\Delta\}\rightarrow -\{S,\,\Delta\}$ ensures there is no mass-mixing between the SM scalar and the beyond-SM fields. This symmetry also forbids terms linear in a single beyond-SM scalar, like $H\Delta^\dagger H$ or $SH^2$, which would otherwise induce a non-zero VEV for $\Delta$ and $S$ after electroweak symmetry breaking. Consequently, parameter space exists in which neither $S$ nor $\Delta$ acquire a VEV. The scalar $S$ and the neutral components of $\Delta$ will mass-mix, however, due to the $\lambda_{{\text{\tiny SH}}}$ term in the potential.
Expanding the neutral SM-scalar around its VEV, and expanding the neutral component of $\Delta$ as H\^0= (v+h\^0+i\^0) \^0=(\_R+i\_I), respectively, the mass-mixing Lagrangian for the neutral scalars is && - \^T\^2 . Here the basis vector is $\mathcal{S}=(S, \Delta_R,\Delta_I)^T$, and the squared-mass matrix has the form \^2 = (
[ccc]{} \_S\^2 +\_5 v\^2& v\^2&0\
v\^2& \_\^2 +v\^2&0\
0&0& \_\^2 +v\^2
). Thus, the CP-odd scalar $\Delta_I$ is a mass eigenstate with mass $\tilde{M}_{\Delta}^2+\lambda_4 v^2/2$, while the CP-even scalars $S$ and $\Delta_R$ mass-mix to produce two physical scalars that are linear combinations of these fields.
The dark-matter candidate will be one of the neutral-scalar mass eigenstates. To determine which one, we must find the masses for the mixed CP-even states. Let us define $M_S^2=\tilde{M}_S^2 +\lambda_5 v^2$ and $M^2_\Delta= \tilde{M}_\Delta^2 +\lambda_4v^2/2$, which are the CP-even mass eigenstates in the limit $\lambda_{{\text{\tiny SH}}}\rightarrow 0$. In this limit $\Delta_R$ and $\Delta_I$ are degenerate and form a single complex-scalar with mass $M^2_\Delta$. For nonzero $\lambda_{{\text{\tiny SH}}}$, the CP-even mass eigenvalues can be written as M\_ = {M\_S\^2+M\^2\_\^[1/2]{}}, where the eigenstates are related to the original fields as (
[c]{} S\_+\
S\_-
)= (
[cc]{} &\
-&
) (
[c]{} S\
\_R
). Here, the mixing angle is 2&=&.
In the limit where the singlet-scalar is heaviest, $M_S^2\gg M^2_\Delta$, the mass eigenvalues are approximately M\_+\^2M\_S\^2+M\_-\^2M\^2\_- M\_S\^2M\^2\_. Noting that $M_-^2<M^2_\Delta$, reveals that $S_-$ is the lightest exotic state and is thus the DM candidate. Simple expressions are obtained for the mass eigenvectors in this limit: .
[c]{} S\_+S+\_R\
\
S\_-\_R-S
. M\_S\^2M\^2\_, so that the lightest scalar $S_-$ is comprised mostly of $\Delta_R$.
Thus, for $M_S^2\gg M^2_\Delta$ the dark matter is comprised of $S_-$, which mostly consists of the CP-even part ($\Delta_R$) of the neutral field in the scalar triplet $S_2$. The mass-splitting between $S_-$ and the CP-odd state $\Delta_I$ is $|\Delta M^2|= \lambda_{{\text{\tiny SH}}}^2v^4/8M_S^2$. Provided this splitting exceeds the dark-matter kinetic-energy, $\sqrt{|\Delta M^2|}> \mathrm{KE}_{{\text{\tiny DM}}}$, the state $\Delta_I$ will not be kinematically accessible via tree-level processes in direct-detection experiments. This significantly weakens the bounds on dark matter arising from a complex triplet. This also gives an upper bound on the mass parameter $M_S^2$, beyond which the splitting between the dark matter and $\Delta_I$ is so small that tree-level scattering via $Z$ exchange is expected in present-day experiments. One finds M\_S &<& () ()\^2()10\^2 . Thus, the heavier state $S_+$ cannot be made arbitrarily heavy if the dark matter is to avoid exclusion via, e.g., the XENON100 data.
With $M_{{\text{\tiny DM}}}\equiv M_-=\mathcal{O}(\mathrm{TeV})$, the mass-split between $S_-$ and $\Delta_I$ is smaller than $M_{\Delta}$ (the mass of $\Delta_I$). Once the temperature drops below $M_\Delta$, the heavier state $\Delta_I$ will decay, with the decay products necessarily containing $S_-$, due to the conserved discrete-symmetry. The expression $|\Delta M^2|= \lambda_{{\text{\tiny SH}}}^2v^4/8M_S^2$ shows that the mass-splitting between $S_-$ and $\Delta_I$ is bounded as $\sqrt{|\Delta M^2|}\lesssim |\lambda_{{\text{\tiny SH}}}|\times 4$ GeV, given that $M_{\Delta}\gtrsim3$ TeV is needed to achieve the correct relic abundance and we are working with $M_S>M_\Delta$. In this mass range $\Delta_I$ can decay as $\Delta_I\rightarrow S_-+Z^*\rightarrow S_-+\bar{f}f$, where $f$ is a SM fermion with mass $m_f<|\lambda_{{\text{\tiny SH}}}|\times2$ GeV. Therefore, even if charged SM fermions are not kinematically available, final-states containing neutrinos will be accessible unless $\lambda_{{\text{\tiny SH}}}$ is exceptionally small. After $\Delta_I$ has decayed away, the primordial plasma is comprised of $S_-$ and the SM fields. $S_-$ can maintain equilibrium with the SM sector via gauge interactions and via the $\lambda_4$ and $\lambda_4'$ (hereafter $\lambda_4$) quartic terms in Eq. .[^6] When the quartic interactions are dominant, the model is similar to an inert real-triplet model; the dark-matter abundance will be obtained for $M_{{\text{\tiny DM}}}\simeq 2.5$ TeV, in line with the analysis of Ref. [@Cirelli:2005uq]. As one makes $\lambda_4$ smaller, the tree-level mass splitting between the charged and neutral components of $\Delta$ diminishes and coannihilation channels like $\Delta^-\Delta^{++}\rightarrow W^+\gamma$ become available. At this point, making $\lambda_4$ smaller does not modify the requisite dark-matter mass as gauge interactions dominate. The analysis of Ref. [@Araki:2011hm] for an inert complex-triplet finds that $M_{{\text{\tiny DM}}}\gtrsim 2.8$ TeV is required for the entire region of parameter space.[^7] We thus expect that $M_{{\text{\tiny DM}}}\gtrsim 2.5$ TeV will be required even when the gauge interactions dominate the quartic interactions during freeze-out for the present scenario. With this value fixed, model $(C)$ becomes a viable model of neutrino mass and dark matter. It will be difficult, however, to produce the exotics in this model, given that the lightest exotic mass is $\gtrsim 2.5$ TeV.[^8]
We now briefly discuss the alternative limit with $M_S^2\ll M^2_\Delta$. In this case the CP-even mass eigenvalues are M\_+\^2M\_\^2+M\_-\^2M\^2\_S - M\_\^2M\^2\_S. We see that the dark-matter candidate remains as the lightest CP-even eigenstate $S_-$, with mass $M_-$. The mass eigenstates are now .
[c]{} S\_+\_R+S\
\
S\_-S-\_R
. M\_\^2M\^2\_S, so the dark matter is comprised mostly of the singlet-scalar $S$. Singlet-scalar dark matter is well known [@McDonald:1993ex] and detailed analysis show that $M_{{\text{\tiny DM}}}\gtrsim 80$ GeV is compatible with direct-detection constraints and WMAP data for a Higgs mass of $m_h\simeq125$ GeV [@Djouadi:2011aa]. The viable region of parameter space can be probed by XENON1T, excepting a small resonant window with $M_{{\text{\tiny DM}}}\simeq62$ GeV, where the dark-matter-Higgs coupling can be very small. Lighter dark-matter with $M_{{\text{\tiny DM}}}\lesssim60$ GeV is ruled out by LHC bounds on invisible Higgs decays [@Djouadi:2011aa].
We see that model $(C)$ has viable parameter space in which it behaves like an inert-triplet model or an inert singlet model. This analysis is sufficient to demonstrate that model $(E)$ also has viable regions of parameter space. In model $(E)$ one has $S_1\sim(1,3,0)$, while the second scalar remains as $S_2\sim(1,3,2)$. The scalar potential for this model contains a term similar to the $\lambda_{{\text{\tiny SH}}}$ term in Eq. , which mixes the neutral component of $S_1$ with the CP-even neutral component of $S_2$. If $S_2$ is heaviest the model behaves like an inert real-triplet model, while if $S_1$ is heaviest the lightest scalar is mostly comprised of $\Delta_R$ (the neutral CP-even part of $S_2$). Direct-detection constraints can be evaded due to the mass mixing, and the model is again an effective model of inert-triplet dark matter. In both cases we expect that a viable dark-matter abundance and viable neutrino masses can be obtained, though the dark matter will be heavy, with $M_{{\text{\tiny DM}}}\gtrsim2.5$ TeV (neglecting resonant regions).
In terms of the observational prospects for the beyond-SM multiplets at the LHC, the limit $M_\Delta^2\gg M^2_S$ in model $(C)$ appears to be the most optimistic scenario for the models in Table \[L\_yukawa\] (excepting the resonant regions, which also allow lighter fields). In this limit the dark matter can be relatively light, $M_{{\text{\tiny DM}}}\simeq100$ GeV, and thus the exotic states $\Delta$ and $\mathcal{F}$ can both be of order a few hundred GeV. In principle it could be possible to observe all three beyond-SM multiplets in this limit. For the other viable models in Table \[L\_yukawa\] the dark matter has to be relatively heavy: $M_{{\text{\tiny DM}}}\gtrsim 500$ GeV for the inert doublet models, and $M_{{\text{\tiny DM}}}\gtrsim 2.5$ TeV for the inert triplet cases. This pushes the additional beyond-SM fields beyond the reach of the LHC.
Note that any mass degeneracy between charged and neutral members of an inert multiplet at tree-level is lifted by radiative effects, making the charged components heavier than the neutral components. The heavier members of a given multiplet can decay to lighter members of the same multiplet via the weak interactions, e.g. $\mathcal{F}^+\rightarrow W^++\mathcal{F}^0$, where the $W$ can be virtual. A heavier multiplet can also decay to a lighter multiplet via the Yukawa coupling; e.g. $\mathcal{F}^-\rightarrow S^0 + \ell^-$ if $M_{\mathcal{F}}\gg M_S$. Because of the discrete symmetry the new fields can only be pair produced in colliders, and conservation of the $Z_2$ charge means final states resulting from exotic decay chains necessarily include stable electrically-neutral fields that will escape the detector.
On the Origin of the Discrete Symmetry\[sec:gauge\_symmetry\]
=============================================================
Following Ma’s original proposal, we employed a discrete $Z_2$ symmetry to ensure stability of the lightest beyond-SM field appearing in the neutrino-mass diagram. One can argue that the use of a discrete symmetry is not completely satisfying, either because it seems ad hoc, or because of the view that quantum gravity effects are not expected to conserve global symmetries. This motivates one to consider whether a simple explanation for the discrete symmetry can be found. The simplest possibility is to replace the discrete symmetry with a gauged $U^\prime(1)$ symmetry, which would not be broken by quantum gravity effects. With enough additional ingredients one can presumably achieve this goal for all the models we have discussed. However, we would like to know which models allow for a minimal extension, such that $Z_2\rightarrow U^\prime(1)$, and a single SM-singlet scalar $\eta$ is added to the particle spectrum to break the $U^\prime(1)$ symmetry.
Writing the full gauge group as $\mathcal{G}_{{\text{\tiny SM}}}\times U(1)^\prime$, where $\mathcal{G}_{{\text{\tiny SM}}}$ is the SM gauge group, we have the following transformation properties for the beyond-SM fields[^9] \~(1\_,Q\_), S\_[1,2]{}\~( Q\^[,]{}\_[1,2]{},Q),\~(Q\^\_,-Q), where the “SM" superscript denotes the charges under $\mathcal{G}_{{\text{\tiny SM}}}$, given in Table \[L\_yukawa\]. Inspection of Eq. shows that all Lagrangian terms needed to generate neutrino mass are allowed by the $U^\prime(1)$ symmetry. However, in the case of models $(A)$ and $(B)$, which are inert-doublet models, the enhanced symmetry prevents the additional term $(S_1^\dagger H)^2$. This term is not needed for neutrino mass but is required to split the neutral components of $S_1\sim(1,2,1)$ in order to avoid direct-detection constraints [@Barbieri:2006dq]. Thus, models $(A)$ and $(B)$ are not compatible with this minimal symmetry extension.
On the other hand, we find that models $(C)$, $(D)$ and $(E)$, which have one scalar forming a real representation of the SM gauge symmetry, remain as viable models of dark matter provided $Q_\eta=-2Q$. This relationship is needed to lift a mass-degeneracy of neutral beyond-SM fields. For example, consider model $(C)$, which now has the following terms in the scalar potential V(H,S,,)&&\_ { S\^\^H +S\^\*H\^} + {S\^2+ S\^[\*2]{}\^},\[model\_c\_potential’\] in addition to the terms in Eq. . All other terms containing $\eta$ depend only on the modulus $|\eta|^2$. We have used the relative phase of $S$ and $\Delta$ to choose $\lambda_{{\text{\tiny SH}}}$ real and the phase of $S$ to choose $\mu_{\eta }$ real. Note that the symmetry breaking $U^\prime(1)\rightarrow Z_2$ is achieved by nonzero $\langle \eta\rangle$, motivating the discrete symmetry as an accidental subgroup of the gauged $U^\prime(1)$ symmetry.
In the basis $\mathcal{S}=(S_R, \Delta_R,S_I,\Delta_I)^T$, the squared-mass matrix has the form[^10] \^2 = (
[cccc]{} \_S\^2 +\_5 v\^2+2\_ & v\^2&0&0\
v\^2& \_\^2 +v\^2&0&0\
0&0& \_S\^2 +\_5 v\^2-2\_ & v\^2\
0&0& v\^2& \_\^2 +v\^2
). Observe that the entries for the CP-even and CP-odd states are identical in the limit $\mu_\eta\rightarrow0$. This would produce degenerate states so that, in the case where the dark matter is comprised mostly of $\Delta$, the dark matter would be ruled out by XENON100 (it would be an inert complex-triplet). For non-zero $\mu_\eta$, however, the CP-even and CP-odd states are non-degenerate and viable dark-matter is achieved. When the dark matter is mostly (or completely, for model $(D)$) comprised of a real representation of $\mathcal{G}_{{\text{\tiny SM}}}$, the splitting achieved by nonzero $\mu_\eta$ also ensures direct detection signals resulting from mixing between $Z^\prime$ and $Z$ are suppressed.[^11]
There will be an additional scattering process for the dark matter due to the mixing between $\eta$ and the SM scalar, which gives a standard Higgs portal interaction. Given that the coupling for this interaction is not needed to achieve the observed dark-matter abundance, one can always choose this coupling to be small enough to comply with constraints. Thus, with this simple gauge extension, we can explain the origin of the discrete symmetry for models $(C)$, $(D)$ and $(E)$, while retaining the desirable features of radiative neutrino mass and a viable dark-matter abundance.
Note that Ma’s original proposal [@Ma:2006km], and the variant using a real triplet fermion [@Ma:2008cu], are not compatible with this minimal symmetry upgrade; in the case of scalar dark-matter, the $(S^\dagger H)^2$ term is precluded, meaning direct-detection experiments rule the model out, similar to the minimal gauge extension of models $(A)$ and $(B)$. Furthermore, one encounters gauge-anomalies given that $\mathcal{F}_R$ is a chiral field in Refs. [@Ma:2006km; @Ma:2008cu] — additional model building is therefore needed to explain the origin of the $Z_2$ symmetry in these cases. We do not pursue this matter here.
Beyond the Adjoint Representation\[sec:non-minimal\_models\]
============================================================
In the preceding sections we studied generalizations of Ma’s 2006 model with radiative neutrino mass and stable dark-matter candidates. In doing so we restricted our attention to beyond-SM multiplets no larger than the adjoint representation. As mentioned already, one can generate neutrino mass via Figure \[fig:yukawa\_loop\] and obtain dark-matter candidates in models with larger multiplets. We briefly discuss this matter in the present section.
First consider the case with a Majorana mass insertion, as in Figure \[fig:mass\_loop\]. Allowing for $SU(2)$ representations as large as the quintuplet-rep. we find two additional models. Both of these employ the quadruplet scalar $S\sim(1,4,1)$, with the real fermion being either a triplet $\mathcal{F}\sim(1,3,0)$, or a quintuplet $\mathcal{F}\sim(1,5,0)$. The latter model was detailed in Ref. [@Kumericki:2012bh].[^12] In both models we expect either scalar or fermionic dark-matter is possible, as in Ma’s original proposal; the neutral fermion does not couple to the $Z$ boson and can therefore remain consistent with direct-detection constraints.
Generalizing the models with a fermion mass insertion of the Dirac type (i.e. generalizing the models in Table \[L\_yukawa\]), we find more variants are possible. For completeness we list these in the Appendix, but here offer the following comments. As with the models in Table \[L\_yukawa\], we find that fermionic dark-matter can be ruled out for all models with larger gauge representations. The fermions remain as pseudo-Dirac particles with tiny splittings, set by the SM neutrino masses. Such small splittings permit unsuppressed tree-level scattering with SM detectors via $Z$-boson exchange, which is ruled out by XENON100. We thus rule out the parameter space in which the fermion is the lightest beyond-SM state, for the same reasons as discussed in Section \[sec:fermion\_dm\]. For the case of scalar dark-matter, one has to consider the individual models, as was needed for the models in Table \[L\_yukawa\]. Some models can be immediately ruled out for the same reasons that models $(F)$ and $(G)$ could be excluded; for example, model $(L)$ in Table \[L\_yukawa\_quadruplet\] can be excluded as it gives an inert complex-triplet model. Similarly model $(R)$ in Table \[L\_yukawa\_quintuplet\] is ruled out, as $S_2$ contains only charged components and $S_1$ has nonzero hypercharge. The other models appear to be compatible with direct-detection constraints, provided the neutral components of the scalars mix when the lightest scalar has nonzero hypercharge, much as models $(C)$ and $(E)$ were viable. For example, model $(M)$ contains $S_1\sim(1,4,1)$ as the only beyond-SM scalar with a neutral component. However, the Lagrangian allows a term $\lambda (S_1^\dagger H)^2\subset\mathcal{L}$ that can split the components of the neutral scalar, allowing one to avoid direct-detection constraints (this is analogous to the splitting obtained in an inert-doublet model).
Finally, we note that the use of larger multiplets may have an additional phenomenological benefit. Ref. [@Kopp:2013mi] shows that large multiplets that mediate interactions between dark matter and the SM can enhance loop-induced annihilation of dark matter into $2\gamma$ and $\gamma+Z$ final states, without requiring non-perturbatively large couplings. This occurs because the larger multiplets admit fields with larger electric-charges, naturally enhancing loop-processes with final-state photons. It does not appear to be possible to realize the astrophysical gamma-ray signal [@Weniger:2012tx] in the models presented in the Appendix, but simple extensions do seem compatible with this idea. For example, model $(N)$ in Table \[L\_yukawa\_quintuplet\] employs $\mathcal{F}\sim(1,4,-1)$, $S_1\sim(1,5,0)$ and $S_2\sim(1,5,2)$. When $S_1$ is the lightest beyond-SM state the dark matter is comprised (mostly) of the neutral component of $S_1$. There is a one-loop contribution to processes like $DM+DM\rightarrow 2\gamma,\gamma+Z$, containing virtual $S_2$ states in the loop, that is enhanced by the presence of the multiply charged component in $S_2$. Note that the dark-matter mass is required to be either $M_{{\text{\tiny DM}}}\sim130$ GeV or $M_{{\text{\tiny DM}}}\sim144$ GeV, in order to generate the gamma-ray excess via dark matter annihilations into either $2\gamma$ or $\gamma+Z$ final states, respectively. However, dark-matter comprised of $S_1\sim(1,5,0)$ is expected to have a mass $M_{{\text{\tiny DM}}}\gtrsim5$ TeV in order to achieve the correct relic abundance [@Cirelli:2005uq], which is too large to explain the astrophysical signal. If one adds a singlet scalar $S$, that is also odd under the $Z_2$ symmetry, to the model, then the region of parameter space where $S$ is the dark matter is compatible with $M_{{\text{\tiny DM}}}\sim130$ GeV or $M_{{\text{\tiny DM}}}\sim144$ GeV. The components of $S_1$ and/or $S_2$ can then be $\mathcal{O}(100)$ GeV and the loop-processes advocated in Ref. [@Kopp:2013mi] are present in the model, thereby enhancing the astrophysical gamma-ray signal.[^13] In this example there is a simple connection between the astrophysical signal and the mechanism of neutrino mass, with the large multiplets that enable the latter also enhancing the former. It could be interesting to take these ideas further to see if the dark matter can be realized as one of the fields in the neutrino loop-diagram, rather than an extra degree of freedom, or to study the phenomenology of the model just described.
Conclusion\[sec:conc\]
======================
We studied a class of models with radiative neutrino mass and stable dark-matter candidates. Neutrino mass was generated by a one-loop diagram with the same topography as that proposed by Ma [@Ma:2006km]. We generalized Ma’s approach, detailing all variants with beyond-SM fields no larger than the adjoint representation. In the case where the neutrino mass diagram contained a Majorana mass insertion, only two models were found, both of which were known. When the mass-insertion was of the Dirac type, such that lepton-number symmetry was broken by a vertex, we found a number of additional models. Fermionic dark-matter was excluded in all of these models, while two of the models were completely excluded due to direct-detection constraints. The remaining models allowed radiative neutrino mass and achieved a viable (scalar) dark-matter abundance. There were cases with an inert singlet, an inert doublet, and an inert triplet, providing a natural setting for inert $N$-tuplet models of dark-matter, with the additional feature of achieving radiative neutrino mass. Interestingly, some of the models allowed a simple extension, such that the (formerly imposed) discrete symmetry emerged as an accidental low-energy symmetry. We briefly discussed models with larger beyond-SM multiplets, showing that viable scenarios exist. With simple extensions, the large multiplets enabling neutrino mass can also enhance present-day astrophysical gamma-ray signals, allowing a simple connection between the mechanism of neutrino mass and the astrophysical gamma-ray signal.
Acknowledgments\[sec:ackn\] {#acknowledgmentssecackn .unnumbered}
===========================
The authors thank Y. Kajiyama, K. Nagao, H. Okada, T. Schwetz, A. Strumia, and K. Yagyu. SSCL is supported in part by the NSC under Grant No. NSC-101-2811-M-006-015 and in part by the NCTS of Taiwan. KM is supported by the Australian Research Council.
Mass Without the $\mathbf{Z_2}$ Symmetry\[app:mass\_no\_Z2\]
============================================================
Of the models presented in Table \[L\_yukawa\], only models $(A)$, $(B)$ and $(D)$ are expected to produce (dominant) radiative neutrino masses in the absence of the $Z_2$ symmetry. The other models contain the triplet scalar $S_{1,2}\sim(1,3,2)$, which Yukawa-couples to the SM leptons, and acquires a VEV due to the term $\mu HS_{1,2}H\subset V(H,S_{1},S_2)$, in the absence of the discrete symmetry. Thus, tree-level neutrino masses of the standard Type-II seesaw [@type2_seesaw] form are expected to dominate the loop effect when the $Z_2$ symmetry is discarded.[^14] For models $(A)$, $(B)$ and $(D)$, on the other hand, tree-level neutrino masses do not arise if the $Z_2$ symmetry is removed, while the loop-diagram in Figure \[fig:yukawa\_loop\] persists. Note that, if the $Z_2$ symmetry is turned off, the fermion $\mathcal{F}$ is not needed in order to generate nonzero radiative neutrino masses in model $(D)$ [@Law:2013dya]. However, the spectrum obtained without $\mathcal{F}$ is of the simplified-Zee form [@Law:2013dya], which is incompatible with the observed mixing pattern [@He:2003ih]. Thus, the fermion $\mathcal{F}\sim(1,2,-1)$ is required to obtain a *viable* mixing pattern in the absence of the $Z_2$ symmetry.
Model $(B)$ has a similar particle content to the model presented in Ref. [@Babu:2009aq], modulo the replacement $S_2\sim(1,2,3)\rightarrow S_2\sim(1,4,3)$. This difference precludes the tree-level mass found in Ref. [@Babu:2009aq] so model $(B)$ is purely a model of radiative masses, which could be studied without the discrete symmetry and dark matter.[^15]
Models with Larger Multiplets\[app:mass\_non\_minimal\]
=======================================================
In the text we found seven models that employ beyond-SM multiplets in either the fundamental or adjoint representation of $SU(2)_L$, and had an internal Dirac mass-insertion. In addition to these minimal models, one can realize radiative neutrino mass and dark-matter candidates with larger multiplets. We present the additional minimal models that arise if one permits multiplets forming the quadruplet (isospin-$3/2$) representation of $SU(2)_L$ in Table \[L\_yukawa\_quadruplet\]. The labeling scheme follows on from Table \[L\_yukawa\] in the text. If one allows for quintuplet multiplets there are additional models, shown in Table \[L\_yukawa\_quintuplet\] (also see Ref. [@Picek:2009is] for a detailed example). The first case listed as model $(M)$ was presented in Ref. [@McDonald:2013kca].
----------- -------------------- ------------------- ---------------
Model $\mathcal{F}$ $S_1$ $S_2$
$(H)$ $\ \ (1,3,-2)\ \ $ $\ \ (1,2,1)\ \ $ $(1,4,3)$
$(I)$ $\ (1,3,-2)\ $ $\ (1,4,1)\ $ $(1,2,3)$
$(J)$ $\ (1,3,-2)\ $ $\ (1,4,1)\ $ $(1,4,3)$
$(K)$ $\ (1,4,-1)\ $ $\ (1,3,0)\ $ $(1,3,2)$
$(L)$ $\ (1,4,-3)\ $ $\ (1,3,2)\ $ $(1,3,4)$
----------- -------------------- ------------------- ---------------
: \[L\_yukawa\_quadruplet\] Models with a Dirac mass insertion that employ quadruplet fields (Isospin-$3/2$).
----------- -------------------- ----------------------- -------------------
Model $\mathcal{F}$ $S_1$ $S_2$
$(M)$ $\ \ (1,4,-1)\ \ $ $\ \ (1,4\mp1,0)\ \ $ $(1,4\pm1,2)$
$(N)$ $\ (1,4,-1)\ $ $\ (1,5,0)\ $ $(1,5,2)$
$(O)$ $\ (1,4,-3)\ $ $\ (1,4\mp1,2)\ $ $(1,4\pm1,4)$
$(P)$ $\ (1,4,-3)\ $ $\ (1,5,2)\ $ $(1,5,4)$
$(Q)$ $\ (1,5,-2)\ $ $\ (1,4,1)\ $ $(1,4,3)$
$(R)$ $\ (1,5,-4)\ $ $\ (1,4,3)\ $ $(1,4,5)$
----------- -------------------- ----------------------- -------------------
: \[L\_yukawa\_quintuplet\] Models with a Dirac mass insertion that employ quintuplet fields (Isospin-$2$).
[99]{}
M. C. Gonzalez-Garcia, M. Maltoni, J. Salvado and T. Schwetz, JHEP [**1212**]{}, 123 (2012) \[arXiv:1209.3023 \[hep-ph\]\]. A. H. G. Peter, arXiv:1201.3942 \[astro-ph.CO\]. E. Ma, Phys. Rev. D [**73**]{}, 077301 (2006) \[hep-ph/0601225\]. E. Ma and D. Suematsu, Mod. Phys. Lett. A [**24**]{}, 583 (2009) \[arXiv:0809.0942 \[hep-ph\]\].
J. Kopp, E. T. Neil, R. Primulando and J. Zupan, Phys. Dark. Univ. [**2**]{} (2013) 22 \[arXiv:1301.1683 \[hep-ph\]\]. C. Weniger, JCAP [**1208**]{}, 007 (2012) \[arXiv:1204.2797 \[hep-ph\]\]. L. M. Krauss, S. Nasri and M. Trodden, Phys. Rev. D [**67**]{}, 085002 (2003) \[hep-ph/0210389\]. C. Boehm, Y. Farzan, T. Hambye, S. Palomares-Ruiz and S. Pascoli, Phys. Rev. D [**77**]{}, 043516 (2008) \[hep-ph/0612228\]; M. Lindner, D. Schmidt and T. Schwetz, Phys. Lett. B [**705**]{}, 324 (2011) \[arXiv:1105.4626 \[hep-ph\]\]; F. -X. Josse-Michaux and E. Molinaro, Phys. Rev. D [**84**]{}, 125021 (2011) \[arXiv:1108.0482 \[hep-ph\]\]; S. Kanemura, T. Nabeshima and H. Sugiyama, Phys. Rev. D [**85**]{}, 033004 (2012) \[arXiv:1111.0599 \[hep-ph\]\]; M. Gustafsson, J. M. No and M. A. Rivera, arXiv:1212.4806 \[hep-ph\]; M. Aoki, J. Kubo and H. Takano, arXiv:1302.3936 \[hep-ph\]; Y. Kajiyama, H. Okada and T. Toma, arXiv:1303.7356 \[hep-ph\]. V. Silveira and A. Zee, Phys. Lett. B [**161**]{}, 136 (1985); J. McDonald, Phys. Rev. D [**50**]{} (1994) 3637 \[hep-ph/0702143 \[HEP-PH\]\]; C. P. Burgess, M. Pospelov and T. ter Veldhuis, Nucl. Phys. B [**619**]{} (2001) 709 \[hep-ph/0011335\]. R. Barbieri, L. J. Hall and V. S. Rychkov, Phys. Rev. D [**74**]{} (2006) 015007 \[hep-ph/0603188\]; L. Lopez Honorez, E. Nezri, J. F. Oliver and M. H. G. Tytgat, JCAP [**0702**]{} (2007) 028 \[hep-ph/0612275\]. Q. -H. Cao, E. Ma and G. Rajasekaran, Phys. Rev. D [**76**]{} (2007) 095011 \[arXiv:0708.2939 \[hep-ph\]\]; S. Andreas, M. H. G. Tytgat and Q. Swillens, JCAP [**0904**]{} (2009) 004 \[arXiv:0901.1750 \[hep-ph\]\]; L. Lopez Honorez and C. E. Yaguna, JCAP [**1101**]{} (2011) 002 \[arXiv:1011.1411 \[hep-ph\]\]; M. Gustafsson, S. Rydbeck, L. Lopez-Honorez and E. Lundstrom, Phys. Rev. D [**86**]{} (2012) 075019 \[arXiv:1206.6316 \[hep-ph\]\]; S. Kashiwase and D. Suematsu, Phys. Rev. D [**86**]{} (2012) 053001 \[arXiv:1207.2594 \[hep-ph\]\]. M. Cirelli, N. Fornengo and A. Strumia, Nucl. Phys. B [**753**]{}, 178 (2006) \[hep-ph/0512090\]; M. Cirelli, A. Strumia and M. Tamburini, Nucl. Phys. B [**787**]{}, 152 (2007) \[arXiv:0706.4071 \[hep-ph\]\].
E. Ma, Phys. Rev. Lett. [**81**]{} (1998) 1171 \[hep-ph/9805219\]. F. Bonnet, M. Hirsch, T. Ota and W. Winter, JHEP [**1207**]{}, 153 (2012) \[arXiv:1204.5862 \[hep-ph\]\]. M. Fabbrichesi and S. Petcov, arXiv:1304.4001 \[hep-ph\]. K. L. McDonald, JHEP [**1307**]{}, 020 (2013) \[arXiv:1303.4573 \[hep-ph\]\].
K. Hally, H. E. Logan and T. Pilkington, Phys. Rev. D [**85**]{}, 095017 (2012) \[arXiv:1202.5073 \[hep-ph\]\]. R. Foot, H. Lew, X. G. He and G. C. Joshi, Z. Phys. C [**44**]{}, 441 (1989). S. -Y. Ho and J. Tandean, arXiv:1303.5700 \[hep-ph\].
D. Schmidt, T. Schwetz and T. Toma, Phys. Rev. D [**85**]{}, 073009 (2012) \[arXiv:1201.0906 \[hep-ph\]\]. C. -K. Chua and S. S. C. Law, Phys. Rev. D [**83**]{}, 055010 (2011) \[arXiv:1011.4730 \[hep-ph\]\]; A. Delgado, C. Garcia Cely, T. Han and Z. Wang, Phys. Rev. D [**84**]{}, 073007 (2011) \[arXiv:1105.5417 \[hep-ph\]\]; S. S. C. Law and K. L. McDonald, Phys. Lett. B [**713**]{}, 490 (2012) \[arXiv:1204.2529 \[hep-ph\]\]; I. Baldes, N. F. Bell, K. Petraki and R. R. Volkas, arXiv:1304.6162 \[hep-ph\]; G. Bambhaniya, J. Chakrabortty, S. Goswami and P. Konar, arXiv:1305.2795 \[hep-ph\]. E. Del Nobile, R. Franceschini, D. Pappadopulo and A. Strumia, Nucl. Phys. B [**826**]{}, 217 (2010) \[arXiv:0908.1567 \[hep-ph\]\]. A. Joglekar, P. Schwaller and C. E. M. Wagner, JHEP [**1212**]{} (2012) 064 \[arXiv:1207.4235 \[hep-ph\]\]. C. Arina, R. N. Mohapatra and N. Sahu, Phys. Lett. B [**720**]{} (2013) 130 \[arXiv:1211.0435 \[hep-ph\]\].
S. S. C. Law, JHEP [**1202**]{} (2012) 127 \[arXiv:1106.0375 \[hep-ph\]\].
M. Aoki, S. Kanemura and K. Yagyu, Phys. Lett. B [**702**]{}, 355 (2011) \[Erratum-ibid. B [**706**]{}, 495 (2012)\] \[arXiv:1105.2075 \[hep-ph\]\].
E. Aprile [*et al.*]{} \[XENON100 Collaboration\], Phys. Rev. Lett. [**107**]{} (2011) 131302 \[arXiv:1104.2549 \[astro-ph.CO\]\]; Phys. Rev. Lett. [**109**]{}, 181301 (2012) \[arXiv:1207.5988 \[astro-ph.CO\]\]. E. M. Dolle and S. Su, Phys. Rev. D [**80**]{} (2009) 055012 \[arXiv:0906.1609 \[hep-ph\]\].
E. Komatsu [*et al.*]{} \[WMAP Collaboration\], Astrophys. J. Suppl. [**192**]{} (2011) 18 \[arXiv:1001.4538 \[astro-ph.CO\]\].
A. Goudelis, B. Herrmann and O. Stål, arXiv:1303.3010 \[hep-ph\].
M. Klasen, C. E. Yaguna and J. D. Ruiz-Alvarez, arXiv:1302.1657 \[hep-ph\]; M. Klasen, C. E. Yaguna, J. D. Ruiz-Alvarez, D. Restrepo and O. Zapata, arXiv:1302.5298 \[hep-ph\].
P. Fileviez Perez, H. H. Patel, M. J. Ramsey-Musolf and K. Wang, Phys. Rev. D [**79**]{}, 055024 (2009) \[arXiv:0811.3957 \[hep-ph\]\];
T. Hambye, F. -S. Ling, L. Lopez Honorez and J. Rocher, JHEP [**0907**]{} (2009) 090 \[Erratum-ibid. [**1005**]{} (2010) 066\] \[arXiv:0903.4010 \[hep-ph\]\]. T. Araki, C. Q. Geng and K. I. Nagao, Phys. Rev. D [**83**]{} (2011) 075014 \[arXiv:1102.4906 \[hep-ph\]\]; T. Araki, C. Q. Geng and K. I. Nagao, Int. J. Mod. Phys. D [**20**]{} (2011) 1433 \[arXiv:1108.2753 \[hep-ph\]\]. R. Foot, A. Kobakhidze, K. L. McDonald and R. R. Volkas, Phys. Rev. D [**76**]{} (2007) 075014 \[arXiv:0706.1829 \[hep-ph\]\]; R. Foot, A. Kobakhidze, K. L. McDonald and R. R. Volkas, Phys. Rev. D [**77**]{}, 035006 (2008) \[arXiv:0709.2750 \[hep-ph\]\].
Y. Kajiyama, H. Okada and K. Yagyu, arXiv:1303.3463 \[hep-ph\]. S. Kanemura and H. Sugiyama, Phys. Rev. D [**86**]{} (2012) 073006 \[arXiv:1202.5231 \[hep-ph\]\]. A. Djouadi, O. Lebedev, Y. Mambrini and J. Quevillon, Phys. Lett. B [**709**]{}, 65 (2012) \[arXiv:1112.3299 \[hep-ph\]\].
K. Kumericki, I. Picek and B. Radovcic, Phys. Rev. D [**86**]{}, 013006 (2012) \[arXiv:1204.6599 \[hep-ph\]\]; I. Picek and B. Radovcic, Phys. Lett. B [**719**]{}, 404 (2013) \[arXiv:1210.6449 \[hep-ph\]\]. The same model is mentioned in Y. Liao, JHEP [**1106**]{}, 098 (2011) \[arXiv:1011.3633 \[hep-ph\]\]. M. Cirelli and A. Strumia, New J. Phys. [**11**]{} (2009) 105005 \[arXiv:0903.3381 \[hep-ph\]\]. W. Konetschny and W. Kummer, Phys. Lett. B [**70**]{}, 433 (1977); T. P. Cheng and L. F. Li, Phys. Rev. D [**22**]{}, 2860 (1980); M. Magg and C. Wetterich, Phys. Lett. B [**94**]{}, 61 (1980); J. Schechter and J. W. F. Valle, Phys. Rev. D [**22**]{}, 2227 (1980); G. Lazarides, Q. Shafi and C. Wetterich, Nucl. Phys. B [**181**]{}, 287 (1981); C. Wetterich, Nucl. Phys. B [**187**]{}, 343 (1981); R. N. Mohapatra and G. Senjanovic, Phys. Rev. D [**23**]{}, 165 (1981).
S. S. C. Law and K. L. McDonald, arXiv:1303.6384 \[hep-ph\]. X. -G. He, Eur. Phys. J. C [**34**]{}, 371 (2004) \[hep-ph/0307172\]. K. S. Babu, S. Nandi and Z. Tavartkiladze, Phys. Rev. D [**80**]{}, 071702 (2009) \[arXiv:0905.2710 \[hep-ph\]\]. I. Picek and B. Radovcic, Phys. Lett. B [**687**]{}, 338 (2010) \[arXiv:0911.1374 \[hep-ph\]\]; K. Kumericki, I. Picek and B. Radovcic, Phys. Rev. D [**84**]{}, 093002 (2011) \[arXiv:1106.1069 \[hep-ph\]\].
[^1]: slaw@mail.ncku.edu.tw
[^2]: klmcd@physics.usyd.edu.au
[^3]: The quoted result is for model $(B)$, and should be multiplied by an extra factor of 2 for model $(A)$.
[^4]: A single generation of exotic fermions generates nonzero masses for two SM neutrinos, which is sufficient to accommodate the experimentally observed mass and mixing spectrum. To obtain three massive neutrinos requires (at least) two generations of exotic fermions.
[^5]: A related potential was considered in Ref. [@Foot:2007ay].
[^6]: $S_-$ also interacts with the SM via the $\lambda_5$ and $\lambda_{{\text{\tiny SH}}}$ terms in Eq. . However, there is only a small admixture of $S$ in $S_-$ for the limit we discuss, so these interactions are suppressed by the small mixing angle $\theta= \mathcal{O}(\lambda_{{\text{\tiny SH}}} v^2/M_S^2)\ll1$.
[^7]: The exception being for the small resonant-region with $M_{{\text{\tiny DM}}}\approx m_h/2$, for which the dark-matter mass is smaller due to the enhanced annihilation cross section. We do not consider the resonant regions, which require tuned mass relations.
[^8]: For a recent model with radiative neutrino mass that utilizes the resonant region to obtain viable dark matter from a complex scalar triplet, see Ref. [@Kajiyama:2013zla]. If the dark-matter mass was similarly taken in the resonant region in our models, we could also take the extra exotics to be $\mathcal{O}(100)$ GeV. Also see Ref. [@Kanemura:2012rj].
[^9]: Note that real scalars must be taken as complex once they are given a nonzero $U^\prime(1)$ charge.
[^10]: In making $S$ complex, some of the coupling/mass parameters must be scaled, relative to the real-$S$ case, to obtain this form.
[^11]: The interactions with $Z^\prime$ are non-diagonal in the real and imaginary components of the beyond-SM scalars; the splitting means one of these states is not part of the present-day dark-matter abundance, thereby suppressing scattering via $Z^\prime$ exchange. Note that the Lagrangian contains a kinetic-mixing term between $U^\prime(1)$ and SM hypercharge, but even if one takes the relevant coupling to be small, mixing between $Z'$ and $Z$ is induced by loops containing exotics charged under both $U^\prime(1)$ and $\mathcal{G}_{{\text{\tiny SM}}}$.
[^12]: The real quintuplet fermion has been studied as a “Minimal Dark-Matter" candidate, due to an accidental symmetry that arises when one adds this field to the SM [@Cirelli:2009uv]. However, this symmetry is broken when an additional scalar is added to allow radiative neutrino masses; a discrete symmetry must be imposed to retain the dark-matter candidate.
[^13]: Note that whether the $2\gamma$ final state or the $\gamma+Z$ final state is dominant, and thus whether the dark matter mass is $130$ GeV or $144$ GeV, depends on whether the dark matter $S$ couples more strongly to $S_1$ or $S_2$. If the dominant coupling is to $S_2$, the $\gamma+Z$ final state is expected to dominate due to the nonzero hypercharge assignment for $S_2$, giving $M_{{\text{\tiny DM}}}\sim144$ GeV. A dominant coupling to $S_1$ instead requires $M_{{\text{\tiny DM}}}\sim130$ GeV as the $2\gamma$ final state is expected to dominate.
[^14]: The same is true for Ma’s original proposal [@Ma:2006km] and the triplet variant [@Ma:2008cu]; if the $Z_2$ symmetry is turned off one obtains tree-level neutrino masses via a Type-I or Type-III seesaw, respectively.
[^15]: One still requires a second SM-like doublet to achieve neutrino mass in this case, as the term $H^3S_2$ vanishes when there is only one SM doublet [@Law:2013dya]. The SM scalar doublet would now appear inside the loop, however.
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abstract: 'We introduce the concept of a quasiresiduated lattice and prove that every lattice effect algebra can be organized into a commutative quasiresiduated lattice with divisibility. Also conversely, every such lattice can be converted into a lattice effect algebra and every lattice effect algebra can be reconstructed form its assigned quasiresiduated lattice. We apply this method also for lattice pseudoeffect algebras introduced recently by Dvurečenskij and Vetterlein. We show that every good lattice pseudoeffect algebra can be organized into a (possibly non-commutative) quasiresiduated lattice with divisibility and conversely, every such lattice can be converted into a lattice pseudoeffect algebra. Moreover, also a good lattice pseudoeffect algebra can be reconstructed from the assigned quasiresiduated lattice.'
author:
- Ivan Chajda and Helmut Länger
title: Residuation in lattice effect algebras
---
[**AMS Subject Classification:**]{} 03G25,03G12,06D35
[**Keywords:**]{} lattice effect algebra, lattice pseudoeffect algebra, quasiresiduated lattice, quasiadjointness, divisibility
In order to axiomatize quantum logic effects in a Hilbert space, Foulis and Bennett ([@FB]) introduced the so-called effect algebras. These are partial algebras with one partial binary operation which can be converted into bounded posets in general and into lattices in particular cases. It turns out that effect algebras form a successful axiomatization of the logic of quantum mechanics, but we suppose that there exists a connection with a kind of substructural logics whose algebraic semantics is based on residuated lattices. An attempt in this direction was already done in [@CH] where the so-called conditional residuation was introduced. A disadvantage of this approach is that the axioms of residuated structures are reflected only in the case when the terms used in adjointness are defined. This is an essential restriction which prevents the development of this theory. The aim of the present paper is to introduce the more general concept of quasiresiduation and to show that lattice effect algebras and lattice pseudoeffect algebras satisfy this concept. Pseudoeffect algebras were introduced recently by Dvurečenski and Vetterlein ([@DV]).
We start with the following definition.
An [*effect algebra*]{} is a partial algebra $\mathbf E=(E,+,{}',0,1)$ of type $(2,1,0,0)$ where $(E,{}',0,1)$ is an algebra and $+$ is a partial operation satisfying the following conditions for all $x,y,z\in E$:
1. $x+y$ is defined if and only if so is $y+x$ and in this case $x+y=y+x$,
2. $(x+y)+z$ is defined if and only if so is $x+(y+z)$ and in this case $(x+y)+z=x+(y+z)$,
3. $x'$ is the unique $u\in E$ with $x+u=1$,
4. if $1+x$ is defined then $x=0$.
On $E$ a binary relation $\leq$ can be defined by $$x\leq y\text{ if there exists some }z\in E\text{ with }x+z=y$$ ($x,y\in E$). Then $(E,\leq,0,1)$ is a bounded poset and $\leq$ is called the [*induced order*]{} of $\mathbf E$. If $(E,\leq)$ is a lattice then $\mathbf E$ is called a [*lattice effect algebra*]{}.
In the sequel we will use the properties of effect algebras listed in the following lemma.
\[lem1\] [(]{}see [[@DP],[@DV])]{} If $\mathbf E=(E,+,{}',0,1)$ is an effect algebra, $\leq$ its induced order and $a,b,c,d\in E$ then the following hold:
1. $a''=a$,
2. $a\leq b$ implies $b'\leq a'$,
3. $a+b$ is defined if and only if $a\leq b'$,
4. if $a\leq b$ and $b+c$ is defined then $a+c$ is defined and $a+c\leq b+c$,
5. if $a\leq b$ then $a+(a+b')'=b$,
6. $a+0=0+a=a$,
7. $0'=1$ and $1'=0$.
A [*partial monoid*]{} is a partial algebra $\mathbf A=(A,\odot,1)$ of type $(2,0)$ where $1\in A$ and $\odot$ is a partial operation satisfying the following conditions for all $x,y,z\in A$:
1. $(x\odot y)\odot z$ is defined if and only if so is $x\odot(y\odot z)$ and in this case $(x\odot y)\odot z=x\odot(y\odot z)$,
2. $x\odot1=1\odot x=x$.
The [*partial monoid*]{} $\mathbf A$ is called [*commutative*]{} if it satisfies the following condition for all $x,y\in A$:
1. $x\odot y$ is defined if and only if so is $y\odot x$ and in this case $x\odot y=y\odot x$,
The authors already introduced a certain modification of residuation for sectionally pseudocomplemented lattices, see [@CL]. For lattice effect algebras, we introduce another version of residuation called quasiresiduation.
\[def1\] A [*commutative quasiresiduated lattice*]{} is a partial algebra $\mathbf C=(C,\vee,\wedge,$ $\odot,\rightarrow,0,1)$ of type $(2,2,2,2,0,0)$ where $(C,\vee,\wedge,0,1)$ is a bounded lattice, $\odot$ is a partial and $\rightarrow$ a full operation satisfying the following conditions for all $x,y,z\in C$:
1. $(C,\odot,1)$ is a partial commutative monoid where $x\odot y$ is defined if and only if $x'\leq y$,
2. $x''=x$, and $x\leq y$ implies $y'\leq x'$,
3. $(x\vee y')\odot y\leq y\wedge z$ if and only if $x\vee y'\leq y\rightarrow z$.
Here $x'$ is an abbreviation for $x\rightarrow0$. The [*commutative quasiresiduated lattice*]{} $\mathbf C$ is called [*divisible*]{} if $$x\leq y\text{ implies }y\odot(y\rightarrow x)=x$$ for all $x,y\in C$.
Note that the terms in (C3) are everywhere defined.
In case $y'\leq x$ and $z\leq y$ condition (C3) reduces to $$x\odot y\leq z\text{ if and only if }x\leq y\rightarrow z$$ which is usual adjointness. Therefore condition (C3) will be called [*commutative quasiadjointness*]{}. Hence, contrary to the similar concept in [@CH], in commutative quasiadjointness we have only everywhere defined terms in $\mathbf C$ although $\mathbf C$ is a partial algebra.
Our aim is to show that every lattice effect algebra can be organized into a commutative quasiresiduated lattice.
\[th1\] Let $\mathbf E=(E,+,{}',0,1)$ be a lattice effect algebra with lattice operations $\vee$ and $\wedge$ and put $$\begin{aligned}
x\odot y & :=(x'+y')'\text{ if and only if }x'\leq y, \\
x\rightarrow y & :=(x\wedge y)+x'\end{aligned}$$ [(]{}$x,y\in E$[)]{}. Then $\mathbb C(\mathbf E):=(E,\vee,\wedge,\odot,\rightarrow,0,1)$ is a divisible commutative quasiresiduated lattice.
Let $a,b,c\in E$. Obviously, $(E,\vee,\wedge,0,1)$ is a bounded lattice and (C1) and (C2) hold. If $(a\vee b')\odot b\leq b\wedge c$ then $((a\vee b')'+b')'\leq b\wedge c$ and $b'\leq a\vee b'$ and hence $$a\vee b'=b'+((a\vee b')'+b')'\leq b'+(b\wedge c)=b\rightarrow c.$$ If, conversely, $a\vee b'\leq b\rightarrow c$ then $a\vee b'\leq(b\wedge c)+b'$ and $b'\leq(b\wedge c)'$ and hence $$(a\vee b')\odot b=(b'+(a\vee b')')'\leq(b'+((b\wedge c)+b')')'=(b\wedge c)''=b\wedge c$$ proving (C3). If $a\leq b$ then $b'\leq a'$ and hence $$b\odot(b\rightarrow a)=(b'+(a+b')')'=a''=a$$ proving divisibility.
Let us mention that Definition \[def1\] can be modified in such a way that it contains only everywhere defined operations. Namely, if we put $$x\otimes y:=(x\vee y')\odot y$$ for all $x,y\in C$ then $\otimes$ is everywhere defined and satisfies the identities $x\otimes1\approx1\otimes x\approx x$, and commutative quasiadjointness can then be expressed in the form $$x\otimes y\leq y\wedge z\text{ if and only if }x\vee y'\leq y\rightarrow z.$$ This means that our definition of commutative quasiresiduation differs from that of usual residuation only in the point that $y$ occurs on the right-hand side of $x\otimes y\leq y\wedge z$ and $y'$ on the left-hand side of $x\vee y'\leq y\rightarrow z$. On the other hand, using this version, divisibility cannot be easily defined. Moreover, since in lattice effect algebras we have $$y\rightarrow z=(y\wedge z)+y'=y\rightarrow(y\wedge z),$$ commutative quasiadjointness can be rewritten in the form $$(x\vee y')\odot y\leq y\wedge z\text{ if and only if }x\vee y'\leq y\rightarrow(y\wedge z)$$ which corresponds to usual adjointness if we abbreviate $x\vee y'$ by $X$ and $y\wedge z$ by $Z$, i.e. $$X\odot y\leq Z\text{ if and only if }X\leq y\rightarrow Z.$$
We can prove also the converse.
Let $\mathbf C=(C,\vee,\wedge,\odot,\rightarrow,0,1)$ be a commutative quasiresiduated lattice and put $$\begin{aligned}
x+y & :=(x'\odot y')'\text{ if and only if }x\leq y', \\
x' & :=x\rightarrow0\end{aligned}$$ [(]{}$x,y\in C$[)]{}. Then $\mathbb E(\mathbf C):=(C,+,{}',0,1)$ is a lattice effect algebra whose order coincides with that in $\mathbf C$.
Let $a,b\in C$. It is easy to see that (E1), (E2) and (E4) hold. Since $$0\vee a'=a'\leq a'=a\rightarrow0$$ we have $$a\odot a'=a\odot(0\vee a')\leq a\wedge0=0,$$ i.e. $a\odot a'=0$. If, conversely, $a\odot b=0$ then $a'\leq b$ and hence $$a\odot(b\vee a')=a\odot b=0\leq a\wedge0$$ whence $$b=b\vee a'\leq a\rightarrow0=a'$$ showing $b=a'$. Hence $a\odot b=0$ if and only if $b=a'$. Now the following are equivalent: $$\begin{aligned}
a+b & =1, \\
a'\odot b' & =0, \\
a & =b', \\
b & =a'.\end{aligned}$$ This shows (E3). Moreover, the following are equivalent: $$\begin{aligned}
& a\leq b\text{ holds in }\mathbb E(\mathbf R), \\
& a+b'\text{ is defined}, \\
& a'\odot b\text{ is defined}, \\
& a\leq b\text{ holds in }\mathbf R.\end{aligned}$$ Since $(C,\vee,\wedge)$ is a lattice and the partial order relations in $\mathbf C$ and $\mathbb E(\mathbf C)$ coincide, $\mathbb E(\mathbf C)$ is a lattice effect algebra.
Moreover, every lattice effect algebra can be reconstructed from the assigned quasiresiduated lattice as shown in the following result.
Let $\mathbf E$ be a lattice effect algebra. Then $\mathbb E(\mathbb C(\mathbf E))=\mathbf E$.
Let $$\begin{aligned}
\mathbf E & =(E,+,{}',0,1)\text{ with lattice operations }\vee\text{ and }\wedge, \\
\mathbb C(\mathbf E) & =(E,\vee,\wedge,\odot,\rightarrow,0,1), \\
\mathbb E(\mathbb C(\mathbf E)) & =(E,\oplus,{}^*,0,1)\end{aligned}$$ and $a,b\in E$. Then $$a^*=a\rightarrow0=(a\wedge0)+a'=0+a'=a'.$$ Moreover, the following are equivalent: $$\begin{aligned}
& a\oplus b\text{ is defined}, \\
& a\leq b'\text{ in }\mathbb C(\mathbf E), \\
& a\leq b'\text{ in }\mathbf E\end{aligned}$$ and in this case $$a\oplus b=(a^*\odot b^*)^*=(a'\odot b')'=(a''+b'')''=a+b.$$
Now we turn our attention to a more general case. The following concept was introduced by Dvurečenskij and Vetterlein ([@DV]).
A [*pseudoeffect algebra*]{} is a partial algebra $\mathbf P=(P,+,\,\bar{}\,,\,\tilde{}\,,0,1)$ of type $(2,1,1,0,0)$ where $(P,\,\bar{}\,,\,\tilde{}\,,0,1)$ is an algebra and $+$ is a partial operation satisfying the following conditions for all $x,y,z\in P$:
1. If $x+y$ is defined then there exist $u,w\in P$ with $u+x=y+w=x+y$,
2. $(x+y)+z$ is defined if and only if $x+(y+z)$ is defined, and in this case $(x+y)+z=x+(y+z)$,
3. $\bar x$ is the unique $u\in P$ with $u+x=1$, and $\tilde x$ is the unique $w\in P$ with $x+w=1$,
4. if $1+x$ or $x+1$ is defined then $x=0$.
The [*pseudoeffect algebra*]{} $\mathbf P$ is called [*good*]{} if $\widetilde{\bar x+\bar y}=\overline{\tilde x+\tilde y}$ for all $x,y\in P$ with $\tilde x\leq y$.
On $P$ a binary relation $\leq$ can be defined by $$x\leq y\text{ if there exists some }z\in E\text{ with }x+z=y$$ ($x,y\in P$). Then $(P,\leq,0,1)$ is a bounded poset and $\leq$ is called the [*induced order*]{} of $\mathbf P$. If $(P,\leq)$ is a lattice then $\mathbf P$ is called a [*lattice pseudoeffect algebra*]{}.
For our investigation we need the following results taken from [@DV].
If $\mathbf P=(P,+,\,\bar{}\,,\,\tilde{}\,,0,1)$ is a pseudoeffect algebra, $\leq$ its induced order and $a,b,c\in P$ then
1. $\bar{\tilde a}=\tilde{\bar a}=a$,
2. the following are equivalent: $a\leq b$, $\bar b\leq\bar a$, $\tilde b\leq\tilde a$,
3. $a+b$ is defined if and only if $a\leq\bar b$,
4. if $a\leq b$ and $b+c$ is defined then $a+c$ is defined and $a+c\leq b+c$,
5. if $a\leq b$ and $c+b$ is defined then $c+a$ is defined and $c+a\leq c+b$,
6. if $a\leq b$ then $a+\widetilde{\bar b+a}=\overline{a+\tilde b}+a=b$,
7. $a+0=0+a=a$,
8. $\bar0=\tilde0=1$ and $\bar1=\tilde1=0$,
9. the following are equivalent: $a\leq b$, there exists some $d\in P$ with $a+d=b$, there exists some $e\in P$ with $e+a=b$.
Since pseudoeffect algebras are more general than effect algebras, we must define quasiresiduated lattice for the case when the partial operation $\odot$ is not commutative and the mapping $x\mapsto\bar x$ is not an involution.
A [*quasiresiduated lattice*]{} is a partial algebra $\mathbf Q=(Q,\vee,\wedge,\odot,\rightarrow,\leadsto,0,1)$ of type $(2,2,2,2,2,0,0)$ where $(Q,\vee,\wedge,0,1)$ is a bounded lattice, $\odot$ is a partial and $\rightarrow$ and $\leadsto$ are full operations satisfying the following conditions for all $x,y,z\in Q$:
1. $(Q,\odot,1)$ is a partial monoid where $x\odot y$ is defined if and only if $\tilde x\leq y$,
2. $\tilde{\bar x}=\bar{\tilde x}=x$, and $x\leq y$ implies $\bar y\leq\bar x$ and $\tilde y\leq\tilde x$,
3. $(x\vee\bar y)\odot y\leq y\wedge z$ if and only if $x\vee\bar y\leq y\rightarrow z$,
4. $y\odot(x\vee\tilde y)\leq y\wedge z$ if and only if $x\vee\tilde y\leq y\leadsto z$,
5. $\widetilde{\bar x\odot\bar y}=\overline{\tilde x\odot\tilde y}$.
Here $\bar x$ and $\tilde x$ are abbreviations for $x\rightarrow0$ and $x\leadsto0$, respectively. The quasiresiduated lattice $\mathbf Q$ is called [*divisible*]{} if $$x\leq y\text{ implies }(y\rightarrow x)\odot y=y\odot(y\leadsto x)=x$$ for all $x,y\in Q$.
Note that the terms in (Q3) and (Q4) are everywhere defined.
In case $\bar y\leq x$ and $z\leq y$ condition (Q3) reduces to $$x\odot y\leq z\text{ if and only if }x\leq y\rightarrow z$$ which is usual adjointness. Analogously, in case $\tilde y\leq x$ and $z\leq y$ condition (Q4) reduces to $$y\odot x\leq z\text{ if and only if }x\leq y\rightarrow z$$ which is usual adjointness if $\odot$ is commutative. Therefore conditions (Q3) and (Q4) will be called [*quasiadjointness*]{}. Hence, contrary to the similar concept in [@CH], in quasiadjointness we have only everywhere defined terms in $\mathbf Q$ although $\mathbf Q$ is a partial algebra.
Similarly as for effect algebras, we prove that every good lattice pseudoeffect algebra can be organized into a quasiresiduated lattice which, however, need not be commutative.
Let $\mathbf P=(P,+,\,\bar{}\,,\,\tilde{}\,,0,1)$ be a good lattice pseudoeffect algebra with lattice operations $\vee$ and $\wedge$ and put $$\begin{aligned}
x\odot y & :=\widetilde{\bar x+\bar y}\text{ if and only if }\tilde x\leq y, \\
x\rightarrow y & :=\bar x+(x\wedge y),\;x\leadsto y:=(x\wedge y)+\tilde x\end{aligned}$$ [(]{}$x,y\in P$[)]{}. Then $\mathbb Q(\mathbf P):=(P,\vee,\wedge,\odot,\rightarrow,\leadsto,0,1)$ is a divisible quasiresiduated lattice.
Let $a,b,c\in E$. Obviously, $(P,\vee,\wedge,0,1)$ is a bounded lattice and (Q1), (Q2) and (Q5) hold. If $(a\vee\bar b)\odot b\leq b\wedge c$ then $\widetilde{\overline{a\vee\bar b}+\bar b}\leq b\wedge c$ and $\bar b\leq a\vee\bar b$ and hence $$a\vee\bar b=\bar b+\widetilde{\overline{a\vee\bar b}+\bar b}\leq\bar b+(b\wedge c)=b\rightarrow c.$$ If, conversely, $a\vee\bar b\leq b\rightarrow c$ then $a\vee\bar b\leq\bar b+(b\wedge c)$ and $\bar b\leq\overline{b\wedge c}$ and hence $$(a\vee\bar b)\odot b=\widetilde{\overline{a\vee\bar b}+\bar b}\leq\widetilde{\overline{\bar b+(b\wedge c)}+\bar b}=\widetilde{\overline{b\wedge c}}=b\wedge c$$ roving (Q3). If $b\odot(a\vee\tilde b)\leq b\wedge c$ then $\overline{\tilde b+\widetilde{a\vee\tilde b}}\leq b\wedge c$ and $\tilde b\leq a\vee\tilde b$ and hence $$a\vee\tilde b=\overline{\tilde b+\widetilde{a\vee\tilde b}}+\tilde b\leq(b\wedge c)+\tilde b=b\leadsto c.$$ If, conversely, $a\vee\tilde b\leq b\leadsto c$ then $a\vee\tilde b\leq(b\wedge c)+\tilde b$ and $\tilde b\leq\widetilde{b\wedge c}$ and hence $$b\odot(a\vee\tilde b)=\overline{\tilde b+\widetilde{a\vee\tilde b}}\leq\overline{\tilde b+\widetilde{(b\wedge c)+\tilde b}}=\overline{\widetilde{b\wedge c}}=b\wedge c$$ proving (Q4). If $a\leq b$ then $\bar b\leq\bar a$ and $\tilde b\leq\tilde a$ and hence $$\begin{aligned}
(b\rightarrow a)\odot b & =\widetilde{\mathit{\overline{\bar b+a}+\bar b}}=\tilde{\bar a}=a, \\
b\odot(b\leadsto a) & =\overline{\tilde b+\widetilde{a+\tilde b}}=\bar{\tilde a}=a\end{aligned}$$ proving divisibility.
We can prove also the converse.
Let $\mathbf Q=(Q,\vee,\wedge,\odot,\rightarrow,\leadsto,0,1)$ be a quasiresiduated lattice and put $$\begin{aligned}
x+y & :=\widetilde{\bar x\odot\bar y}\text{ if and only if }x\leq\bar y, \\
\bar x & :=x\rightarrow0,\;\tilde x:=x\leadsto0\end{aligned}$$ [(]{}$x,y\in Q$[)]{}. Then $\mathbb P(\mathbf Q):=(Q,+,\,\bar{}\,,\,\tilde{}\,,0,1)$ is a good lattice pseudoeffect algebra whose order coincides with that in $\mathbf Q$.
Let $a,b,c\in Q$. It is easy to see that (E2) and (E4) hold. Since $$0\vee\tilde a=\tilde a\leq\tilde a=a\leadsto0$$ we have $$a\odot\tilde a=a\odot(0\vee\tilde a)\leq a\wedge0=0,$$ i.e. $a\odot\tilde a=0$. If, conversely, $a\odot b=0$ then $\tilde a\leq b$ and hence $$a\odot(b\vee\tilde a)=a\odot b=0\leq a\wedge0$$ whence $$b=b\vee\tilde a\leq a\leadsto0=\tilde a$$ showing $b=\tilde a$. Hence $a\odot b=0$ if and only if $b=\tilde a$. Since $$0\vee\bar a=\bar a\leq\bar a=a\rightarrow0$$ we have $$\bar a\odot a=(0\vee\bar a)\odot a\leq a\wedge0=0,$$ i.e. $\bar a\odot a=0$. If, conversely, $b\odot a=0$ then $\tilde b\leq a$, i.e. $\bar a\leq b$, and hence $$(b\vee\bar a)\odot a=b\odot a=0\leq a\wedge0$$ whence $$b=b\vee\bar a\leq a\rightarrow0=\bar a$$ showing $b=\bar a$. Hence $b\odot a=0$ if and only if $b=\bar a$. Now the following are equivalent: $$\begin{aligned}
a+b & =1, \\
\bar a\odot\bar b & =0, \\
a & =\bar b, \\
b & =\tilde a.\end{aligned}$$ This shows (P3). Since $$a\odot(1\vee\tilde a)=a\odot1=a\leq a\wedge a$$ we have $$1=1\vee\tilde a\leq a\leadsto a,$$ i.e. $a\leadsto a=1$. If $a\leq\bar b$ then because of $\bar b\vee a\leq\bar a\leadsto\bar a$ we have $$\bar a\odot\bar b=\bar a\odot(\bar b\vee a)\leq\bar a\wedge\bar a=\bar a$$ whence $$a=\tilde{\bar a}\leq\widetilde{\bar a\odot\bar b}=a+b$$ showing that $a+\widetilde{a+b}$ is defined. Now in case $a\leq\bar b$ the following are equivalent: $$\begin{aligned}
c & =\overline{a+\widetilde{a+b}}, \\
c+(a+\widetilde{a+b}) & =1, \\
(c+a)+\widetilde{a+b} & =1, \\
c+a & =a+b.\end{aligned}$$ Since $$(1\vee\bar a)\odot a=1\odot a=a\leq a\wedge a$$ we have $$1=1\vee\bar a\leq a\rightarrow a,$$ i.e. $a\rightarrow a=1$. If $b\leq\tilde a$ then because of $\tilde a\vee b\leq\tilde b\rightarrow\tilde b$ we have $$\tilde a\odot\tilde b=(\tilde a\vee b)\odot\tilde b\leq\tilde b\wedge\tilde b=\tilde b$$ whence $$b=\bar{\tilde b}\leq\overline{\tilde a\odot\tilde b}=\widetilde{\bar a\odot\bar b}=a+b$$ showing that $\overline{a+b}+b$ is defined. Now in case $b\leq\tilde a$, i.e. $a\leq\bar b$ the following are equivalent: $$\begin{aligned}
c & =\widetilde{\overline{a+b}+b}, \\
(\overline{a+b}+b)+c & =1, \\
\overline{a+b}+(b+c) & =1, \\
b+c & =a+b.\end{aligned}$$ This shows (P1). Now the following are equivalent: $$\begin{aligned}
& a\leq b\text{ holds in }\mathbb P(\mathbf Q), \\
& a+\tilde b\text{ is defined}, \\
& \bar a\odot b\text{ is defined}, \\
& a\leq b\text{ holds in }\mathbf Q.\end{aligned}$$ Since $(Q,\vee,\wedge)$ is a lattice and the partial order relations in $\mathbf Q$ and $\mathbb P(\mathbf Q)$ coincide, $\mathbb P(\mathbf Q)$ is a lattice pseudoeffect algebra.
As in the case of effect algebras, also every good lattice pseudoeffect algebra can be reconstructed from its assigned quasiresiduated lattice.
Let $\mathbf P$ be a good lattice pseudoeffect algebra. Then $\mathbb P(\mathbb Q(\mathbf P))=\mathbf P$.
Let $$\begin{aligned}
\mathbf P & =(P,+,\,\bar{}\,,\,\tilde{}\,,0,1)\text{ with lattice operations }\vee\text{ and }\wedge, \\
\mathbb Q(\mathbf P) & =(P,\vee,\wedge,\odot,\rightarrow,\leadsto,0,1), \\
\mathbb P(\mathbb Q(\mathbf P)) & =(P,\oplus,{}^*,{}^+,0,1)\end{aligned}$$ and $a,b\in E$. Then $$\begin{aligned}
a^* & =a\rightarrow0=\bar a+(a\wedge0)=\bar a+0=\bar a, \\
a^+ & =a\leadsto0=(a\wedge0)+\tilde a=0+\tilde a=\tilde a.\end{aligned}$$ Moreover, the following are equivalent: $$\begin{aligned}
& a\oplus b\text{ is defined}, \\
& a\leq\bar b\text{ in }\mathbb Q(\mathbf P), \\
& a\leq\bar b\text{ in }\mathbf P\end{aligned}$$ and in this case $$a\oplus b=(a^*\odot b^*)^+=\widetilde{\bar a\odot\bar b}=\overline{\tilde a\odot\tilde b}=a+b.$$
9 I. Chajda and R. Halaš, Effect algebras are conditionally residuated structures. Soft Comput. [**15**]{} (2011), 1383–1387. I. Chajda and H. Länger, Relatively residuated lattices and posets. Math. Slovaca (submitted). A. Dvurečenskij and S. Pulmannová, New Trends in Quantum Structures. Kluwer, Dordrecht 2000. ISBN 0-7923-6471-6. A. Dvurečenskij and T. Vetterlein, Pseudoeffect algebras. I. Basic properties. Internat. J. Theoret. Phys. [**40**]{} (2001), 685–701. D. J. Foulis and M. K. Bennett, Effect algebras and unsharp quantum logics. Found. Phys. [**24**]{} (1994), 1331–1352.
Authors’ addresses:
Ivan Chajda\
Palacký University Olomouc\
Faculty of Science\
Department of Algebra and Geometry\
17. listopadu 12\
771 46 Olomouc\
Czech Republic\
ivan.chajda@upol.cz
Helmut Länger\
TU Wien\
Faculty of Mathematics and Geoinformation\
Institute of Discrete Mathematics and Geometry\
Wiedner Hauptstraße 8-10\
1040 Vienna\
Austria, and\
Palacký University Olomouc\
Faculty of Science\
Department of Algebra and Geometry\
17. listopadu 12\
771 46 Olomouc\
Czech Republic\
helmut.laenger@tuwien.ac.at
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abstract: 'We propose a new method to determine magnetic fields, by using the magnetic-field induced electric dipole transition $3p^4\,3d~^4\mathrm{D}_{7/2}$ $\rightarrow$ $3p^5~^2\mathrm{P}_{3/2}$ in Fe$^{9+}$ ions. This ion has a high abundance in astrophysical plasma and is therefore well-suited for direct measurements of even rather weak fields in e.g. solar flares. This transition is induced by an external magnetic field and its rate is proportional to the square of the magnetic field strength. We present theoretical values for what we will label the reduced rate and propose that the critical energy difference between the upper level in this transition and the close to degenerate $3p^4\,3d~^4\mathrm{D}_{5/2}$ should be measured experimentally since it is required to determine the relative intensity of this magnetic line for different magnetic fields.'
author:
- Wenxian Li
- Jon Grumer
- Yang Yang
- Tomas Brage
- Ke Yao
- Chongyang Chen
- Tetsuya Watanabe
- Per Jönsson
- Henrik Lundstedt
- Roger Hutton
- Yaming Zou
title: 'A Novel Method to Determine Magnetic Fields in low-density Plasma e.g. Solar Flares Facilitated Through Accidental Degeneracy of Quantum States in Fe$^{9+}$'
---
Introduction
============
One of the underlying causes behind solar events, such as solar flares, is the conversion of magnetic to thermal energy. It is therefore vital to be able to measure the magnetic field of the corona, over hot active areas of the sun which exhibits relatively strong magnetic fields. In order to follow the evolution of a solar flare, continuous observations are required, either from space or by using a network of ground-based instruments. It is therefore unfortunate that there are no space-based coronal magnetic field measurements, but only model estimates based on extrapolations from measurements of the photospheric fields ([Schrijver et al. 2008]{}). Ground-based measurements are performed either in the radio range ([White 2004]{}) across the solar corona, or in the infrared wavelength range ([Lin et al. 2004]{}) on the solar limb. Infrared measurements of magnetic fields are limited by the fact that the spectral lines under investigation are optically thin. On the other hand, gyroresonance emission is optically thick, but refers only to a specific portion of the corona, which has a depth of around 100 km. From these measurements an absolute field strength at the base of the corona, above active regions, in the range of 0.02 - 0.2 T was obtained ([White 1997]{}).
In this work we present a completely new method to measure magnetic fields of the active corona. This method is based on an exotic category of light generation, fed by the plasma magnetic field, external to the ions, in contrast to the internal fields generated by the bound electrons. The procedure relies on radiation in the soft x-ray region of the spectrum, implying a space based method. This ¡°magnetic-field induced¡± radiation originates from atomic transitions where the lifetime of the upper energy level is sensitive to the local, external magnetic field ([Beiersdorfer et al. 2003, Li et al. 2013, Grumer et al. 2013, Grumer et al. 2014, Li et al. 2014]{}). We will show that there is a unique case where even relatively small external magnetic fields can have a striking effect on the ion, leading to resonant magnetic-field induced light, due to what is called accidental degeneracy of quantum states.
The impact of the coronal magnetic field on the ion is usually very small due to the relative weakness of these fields in comparison to the strong internal fields of the ions. The effect therefore usually only contributes very weak lines that are impossible to observe. However, sometimes the quantum states end up very close to each other in energy, they are accidentally degenerate, and the perturbation by the external field will be enhanced. If this occurs with a state that without the field has no, or only very weak, electromagnetic transitions to a lower state, a new and distinct feature in the spectrum from the ion will appear $-$ a new strong line. Unfortunately, since the magnetic fields internal to the ion and externally generated in the coronal plasma differs by about five to seven orders of magnitude, the probability of a close-enough degeneracy is small. But in this report we will discuss a striking case of accidental degeneracy in an important ion for studies of the sun and other stars, Fe$^{9+}$.
The origin of the new lines in the spectra of ions is the breaking of the atomic symmetry by the external field, which will mix atomic states with the same magnetic quantum number and parity. This will in turn introduce new decay channels from excited states ([Andrew et al. 1967, Wood et al. 1968]{}), which we will label magnetic-field induced transitions (MITs) ([Grumer et al. 2014]{}). These transitions have attracted attention recently, when accurate and systematic methods to calculate their rates have been developed ([Grumer et al. 2013, Li et al. 2013]{}).
 Schematic energy-level diagram for Chlorine-like ions with $Z~\textless~26 $ and zero nuclear spin, where $^4\mathrm{D}_{7/2}$ is the lowest level in the configuration $3s^23p^43d$. For ions with $ Z~\textgreater~26$, a level crossing has occured and $^4\mathrm{D}_{5/2}$ is lower than $^4\mathrm{D}_{7/2}$. Under the influence of an external magnetic field, an E1 transition opens up from the $^4\mathrm{D}_{7/2}$ to the ground state through mixing with the $^4\mathrm{D}_{5/2}$.](fig1.eps){width="60.00000%"}
Structure of Chlorine-like ions and MITs
========================================
The structure of the lowest levels of Chlorine-like ions is illustrated in Figure \[level\]. The important levels in the present study are the two lowest in the term $3p^43d~^4\mathrm{D}^e$, which turn out to have very different decay modes. Without external fields and ignoring the effects of the nuclear spin, they both decay to the $3p^5~^2\mathrm{P}^o_{3/2}$-level in the ground configuration, but while the $J=5/2$ has a fast electric dipole (E1) channel, the $J=7/2$ can only decay via a slow magnetic quadrupole (M2) transition. In the presence of an external magnetic field, these two states will mix and induce an E1, MIT competing transition channel from the $J=7/2$ level. For most ions the M2 transition is still the dominant decay channel, but a crossing of the fine structure levels $^4\mathrm{D}_{7/2}$ and $^4\mathrm{D}_{5/2}$ between Cobalt and Iron (see Figure. \[dEsequence\]) will change the picture. As a matter of fact, for Iron this fine-structure splitting energy is predicted to be at a minimum and the MIT-contribution to the decay of the $J=7/2$ level will be strongly enhanced.
![\[dEsequence\] The fine-structure splitting energy $\Delta E = E(^4\mathrm{D}_{7/2}) - E(^4\mathrm{D}_{5/2})$ as a function of the nuclear charge, along the isoelectronic sequence from calculations reported here. The dashed line in green marks $\Delta E = 0$.](fig2.eps){width="80.00000%"}
Unfortunately, there are large variations in the predicted value of this energy difference (see Table \[iron\]). The aim of this report is therefore to (a) make a careful theoretical study of the energy splitting between the two $^4\mathrm{D}^e$-levels along the isoelectronic sequence, to confirm the close degeneracy for iron, and (b) to make an accurate prediction of what we will label the reduced decay rate, $a^R_{MIT}$ (see next section). This reduced rate can be combined by the experimentally determined wavelength and energy splitting to give the MIT-rate for different magnetic fields.
Method $\lambda$ $\Delta E$ $A_{E1}$
------------- -- ----------------------------------------------------- -- -- -- ----------- -- -- -- ------------ -- -- -- ----------- -- -- --
Observation Solar ([Thomas et al. 1994, Brosius et al. 1998]{}) 257.25 0
Solar ([Sandlin 1979]{}) 5
present 257.7285 20.14 6.30\[6\]
MCDF ([Huang et al. 1983]{}) 246.4924 78 1.63\[6\]
MCDF ([Dong et al. 1999]{}) 256.674 108 6.27\[6\]
Theory MCDF ([Aggarwal et al. 2004]{}) 54.85
MR-MBPT ([Yasuyuki et al. 2010]{}) 257.1924 18
R-matrix ([Del Zanna et al. 2012]{}) 246.8890 109.74
CI ([Bhatia et al. 1995]{}) 256.1974 $-$58 1.21\[6\]
CI ([Deb et al. 2002]{}) 257.0846 21 2.42\[5\]
: \[iron\] ATOMIC DATA FOR FE X
Theoretical method and Computational model
==========================================
Theoretical method
------------------
The basis of our theoretical approach is described in our earlier papers on MITs ([Li et al. 2013, Grumer et al. 2013]{}). In our example the reference state is $|3p^43d~^4\mathrm{D}_{7/2}\rangle$, which we can represent by a mixture of two pure states in the presence of a magnetic field
$$\begin{aligned}
\label{WF-1}
|``3p^43d~^4\mathrm{D}_{7/2}" ~ M \rangle = d_0 | 3p^43d~^4\mathrm{D}_{7/2} ~ M \rangle + d_1(M)|3p^43d~^4\mathrm{D}_{5/2} ~ M \rangle.\end{aligned}$$
where we ignore interactions with other atomic states, since their energies are far from the reference state. The total E1 transition rate from a specific $M$ sublevel in the mixed $``3p^43d~^4\mathrm{D}_{7/2}"$ to all the $M'$ sublevels of the ground level $3p^5~^2\mathrm{P}_{3/2}$ can be expressed as:
$$\begin{aligned}
\label{MIT-2}
A_{MIT}(M) = \sum_{M'} A_{MIT}(M,M') \approx \frac{2.02613 \times 10^{18}} {3 \lambda^3} \left |\ d_1(M) \langle 3p^5~^2\mathrm{P}_{3/2} || {\bf P}^{(1)} || 3p^43d~^4\mathrm{D}_{5/2} \rangle \right |^2.\end{aligned}$$
where $\lambda$ is the transition wavelength and $d_1(M)$ depends on the magnetic quantum number $M$ of the sublevels belonging to the $3p^43d~^4\mathrm{D}_{7/2}$ level. For the $3p^43d~^4\mathrm{D}_{7/2}$ level, $d_1(M)$ is given by
$$\begin{aligned}
\label{MIT-3}
d_1(M) &=& \frac{\langle ~^4\mathrm{D}_{5/2} M | H_m | ^4\mathrm{D}_{7/2} M \rangle}{E(^4\mathrm{D}_{7/2}) - E(^4\mathrm{D}_{5/2})} \nonumber \\
&=& -B \sqrt{\frac{49-4 M^2}{63}} \frac{\langle ^4\mathrm{D}_{5/2} || {\bf N}^{(1)} + \Delta {\bf N}^{(1)} || ^4\mathrm{D}_{7/2} \rangle}{E(^4\mathrm{D}_{7/2}) - E(^4\mathrm{D}_{5/2})}.\end{aligned}$$
As a result, the total rates of the $3p^43d~^4\mathrm{D}_{7/2}\rightarrow 3p^5~^2\mathrm{P}_{3/2}$ MITs from individual sublevels can be expressed as
$$\begin{aligned}
\label{MIT4}
A_{MIT}(M) &=&a^R_{MIT}(M)\frac{B^2}{\lambda^3(\Delta E)^2}.\end{aligned}$$
where $B$ is in units of T, $\lambda$ is in units of Å, $\Delta E = E(^4\mathrm{D}_{7/2}) - E(^4\mathrm{D}_{5/2})$(in units of cm$^{-1}$) and we have defined a reduced transition rate as
$$\label{MIT5}
a^R_{MIT}(M) \approx \frac{2.02613 \times 10^{18}\cdot(49-4 M^2)}{189}
\left |\langle ^4\mathrm{D}_{5/2} || {\bf N}^{(1)} + \Delta {\bf N}^{(1)} || ~^4\mathrm{D}_{7/2} \rangle \langle ^3P_{3/2} || {\bf P}^{(1)} || ^4\mathrm{D}_{5/2} \rangle \right|^2 .$$
The reduced rate defined in this equation is independent of the transition wavelength, the magnetic field strength as well as the energy splitting. This gives us the property that relates the MIT-rates to the external magnetic field strength. To determine $A_{MIT}(M)$, we recommend to use theoretical values of $a_{MIT}^R(M)$, as reported in this work, combined with experimental values of the energy splitting and wavelength.
Correlation Model
-----------------
The calculations are based on the Multiconfiguration Dirac-Hartree-Fock (MCDHF) method, in the form of the latest version of the GRASP2K program ([Jönsson et al. 2013]{}). A single reference configuration model is adopted for the even-parity($3p^43d$) and odd-parity($3p^5$) states, and the $1s$, $2s$, $2p$ core subshells are kept closed. The set of CSFs is obtained by single and double excitations from the n=3 shell of the reference configurations to the active set. The active set is augmented layer by layer to n=7 ($l_{max}=4$) when satisfactory convergence is achieved. For each step, we optimize only the orbitals in the last added correlation layer at the time. In the final calculations, the total number of CSFs was 16490 for the odd-parity(J=3/2, 1/2) and 523421 for the even-parity (J = 1/2, 3/2, 5/2, 7/2, 9/2) cases.
The resulting excitation energies of the $^4\mathrm{D}_{7/2}$ and $^4\mathrm{D}_{5/2}$ changes by less than 0.1% in the last step of the calculation. The final excitation energies agree with experiment to within 1%. The crucial energy splitting between $^4\mathrm{D}_{7/2}$ and $^4\mathrm{D}_{5/2}$ is well-converged, except for iron where the close degeneracy occurs. To better represent this critical case we extended the calculations to include single excitations from the $2s$ and $2p$ subshells.
Results and Discussion
======================
Isoelectronic Sequence
----------------------
We present in Table \[AEH\] all the important properties, according to Eq. (\[MIT4\]), involved in computing the MIT-rates, i.e. the reduced transition rate $a^R_{MIT}(M)$, the energy splitting, $\Delta E$, between the two levels $3p^43d~^4\mathrm{D}_{5/2}$ and $^4\mathrm{D}_{7/2}$, together with the wavelength, $\lambda$, of the $3p^43d~^4\mathrm{D}_{7/2} \rightarrow 3p^5~^2\mathrm{P}_{3/2}$ transition.
------------ -- ----------- -- --------------- -- --------------- -- --------------- -- ------------ -- ----------- -- -- -- -- -- -- -- -- -- -- -- --
ions $A_{M2}$ $M=\pm {1/2}$ $M=\pm {3/2}$ $M=\pm {5/2}$ $\Delta E$ $\lambda$
Ar$^{+}$ 1.26\[0\] 7.994\[0\] 6.662\[0\] 3.997\[0\] 165.5 762.5978
K$^{2+}$ 3.20\[0\] 7.686\[0\] 6.405\[0\] 3.843\[0\] 202.87 600.4231
Ca$^{3+}$ 6.16\[0\] 7.621\[0\] 6.351\[0\] 3.810\[0\] 246.65 500.1306
Sc$^{4+}$ 1.03\[1\] 7.910\[0\] 6.592\[0\] 3.955\[0\] 283.9 430.6129
Ti$^{5+}$ 1.59\[1\] 8.443\[0\] 7.036\[0\] 4.222\[0\] 307.02 379.0310
V$^{6+}$ 2.31\[1\] 9.093\[0\] 7.577\[0\] 4.546\[0\] 306.35 338.9583
Cr$^{7+}$ 3.22\[1\] 9.807\[0\] 8.172\[0\] 4.903\[0\] 270.41 306.7765
Mn$^{8+}$ 4.33\[1\] 1.055\[1\] 8.793\[0\] 5.276\[0\] 186.08 280.2679
Fe$^{9+}$ 5.68\[1\] 1.119\[1\] 9.326\[0\] 5.594\[0\] 20.14 257.7285
Co$^{10+}$ 7.30\[1\] 1.201\[1\] 1.001\[1\] 6.006\[0\] $-$186.87 238.9746
Ni$^{11+}$ 9.20\[1\] 1.261\[1\] 1.051\[1\] 6.305\[0\] $-$505.53 222.5113
Cu$^{12+}$ 1.14\[2\] 1.309\[1\] 1.090\[1\] 6.543\[0\] $-$932.75 208.1047
Zn$^{13+}$ 1.40\[2\] 1.347\[1\] 1.123\[1\] 6.736\[0\] $-$1482.74 195.3790
------------ -- ----------- -- --------------- -- --------------- -- --------------- -- ------------ -- ----------- -- -- -- -- -- -- -- -- -- -- -- --
: \[AEH\] CALCULATIONAL RESULTS FOR CL-LIKE IONS
In the absence of an external magnetic field, magnetic quadrupole (M2) is the dominant decay channel for the $3p^43d~^4\mathrm{D}_{7/2}\rightarrow 3p^5~^2\mathrm{P}_{3/2}$ transition. When an external magnetic field is introduced, an additional decay channel is opened and we define a average transition rate $\overline{A}_{MIT}$ of the $3p^43d~^4\mathrm{D}_{7/2} \rightarrow 3p^5~^2\mathrm{P}_{3/2}$ transition,
$$\label{tau3}
\overline{A}_{MIT} = \frac{\sum_M A_{MIT}(M)}{2J+1}.$$
Rates for any field can be obtained by eq. \[MIT4\] and \[MIT5\]. We plot the transition rates $A~=~A_{M2}~+~\overline{A}_{MIT}$ along the isoelectrionic sequence in Figure \[averagetr\] (a) for some magnetic-field strengths and $\Delta E~=~20.14~\mathrm{cm}^{-1}$ for Iron. It is clear that the magnetic field influences the transition rate substantially for Iron due to the close degeneracy. To further illustrate the resonance behaviour of this effect, we also used an astrophysical value ([Sandlin 1979]{}) $\Delta E~=~5~\mathrm{cm}^{-1}$ for Iron, in Figure \[averagetr\] (b).
![\[averagetr\] (Color online) The total transition rate $A~=~A_{M2}~+~\overline{A}_{MIT}$ of the $3p^43d~^4\mathrm{D}_{7/2} \rightarrow 3p^5~^2\mathrm{P}_{3/2}$ transition along the Cl-like isoelectronic sequence for some magnetic-field strengths. We used the fine structure energy of (a) 20.14 $\mathrm{cm}^{-1}$ and (b) 5 $\mathrm{cm}^{-1}$ for iron, respectively.](fig3.eps){width="80.00000%"}
Fe X
----
Due to the close to complete cancellation for iron of the energy difference between the two $^4\mathrm{D}$-levels (see Figure \[dEsequence\]) we will pay special attention to this ion.
It is clear that some of the properties in Table \[AEH\] are more easily obtainable through theoretical calculations. We illustrate this in Table \[WS\] where we show the convergence of the calculated off-diagonal reduced matrix elements, W = $\langle ^4\mathrm{D}_{5/2} || {\bf N}^{(1)} + \Delta {\bf N}^{(1)} || ~^4\mathrm{D}_{7/2} \rangle$, representing the magnetic interaction, together with the line strength, S = $\left |\langle ^3P_{3/2} || {\bf P}^{(1)} || ^4\mathrm{D}_{5/2} \rangle \right|^2$, of the close-lying E1 transition. Since these values converges fast and are not subjected to cancellation effects, we estimate their accuracy to be well within a few percent. This will in turn imply that the prediction of the reduced transition rate $a^R_{MIT}(M)$ in Equation \[MIT4\] and \[MIT5\] is of similar accuracy.
layer W S
------- -- -- -- -------- -- -- -- ------------ -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
DF 0.5305 4.063\[4\]
n=4 0.5285 3.555\[4\]
n=5 0.5284 3.355\[4\]
n=6 0.5283 3.298\[4\]
n=7 0.5283 3.264\[4\]
: \[WS\] CONVERGENCE STUDY OF THE CALCULATIONS
We give in Figure \[deltaE\] the energy difference $\Delta E$ as a function of the maximum $n$ in the active set and thereby of the size of the CSF-expansion. It is clear that we also here reach a convergence close to a few cm$^{-1}$ for this property. The final value for the fine structure splitting is 20.14 cm$^{-1}$, in good agreement with the result from recent configuration interaction calculations ([Deb et al. 2002]{}) as well as Many-Body Perturbation Theory ([Ishikawa et al. 2010]{}). This strongly supports the prediction of the close degeneracy of the two levels for iron.
![\[deltaE\] Convergence trend of the fine-structure separation $\Delta E= E(^4\mathrm{D}_{5/2}) -E(^4\mathrm{D}_{7/2})$ for Fe$^{9+}$ as the size of the active set of orbitals is increased as defined by the maximum n-quantum number.](fig4.eps){width="80.00000%"}
The strong resonance effect for iron is especially fortunate due to its high abundance in many astrophysical plasma. As a matter of fact, the ground transition is one of the “coronal" lines used to determine the temperature of the corona and is known as the corona red line ([Swings 1943]{}). Unfortunately there is no firm experimental value for the critical energy splitting between the $^4\mathrm{D}_{7/2}$ and $^4\mathrm{D}_{5/2}$. At the same time, it is a great challenge to calculate the size of this accidental degeneracy accurately.
In the early experimental work by Smitt ([Smitt 1977]{}) the two levels were given identical excitation energies of 388708 cm$^{-1}$. Since then there has been a great deal of work by different groups to study the structure of Fe X (see Table \[iron\]). The differences between the various calculations are often much larger than the predicted fine-structure splitting, which leads to large uncertainties in the level ordering and line identifications. Huang ([Huang et al. 1983]{}) and Dong et al ([Dong et al. 1999]{}) performed multi-configuration Dirac-Fock (MCDF) calculation and predicted the $^4\mathrm{D}_{5/2,7/2}$ levels to be separated by 78 cm$^{-1}$ and 108 cm$^{-1}$ respectively. Ishikawa ([Ishikawa et al. 2010]{}) predicted 18 cm$^{-1}$ from his Multireference-MBPT method. There is a recommended value from solar observations of around 5 cm$^{-1}$, determined from short-wavelength transitions from higher levels and therefore probably quite uncertain. Predictions from The Goddard Solar Extreme Ultraviolet Rocket Telescope and Spectrograph SERTS$-$89 ([Thomas et al. 1994]{}) and SERTS$-$95 ([Brosius et al. 1998]{}) spectra give the same energy for $^4\mathrm{D}_{7/2}$ and $^4\mathrm{D}_{5/2}$, probably due to limited resolution. Finally Del Zanna ([Del Zanna et al. 2004]{}) benchmarked the atomic data for Fe X and suggested the best splitting energy to be 5 cm$^{-1}$. Although our calculations reach a convergence within the model, to the final value of around 20 cm$^{-1}$, it is clear that systematic errors, such as omitted contributions to the Hamiltonian, could be relatively important in estimating the accidental degeneracy of the two levels. We use both our theoretical value and the recommended solar spectral value to illustrate the dependence of the average rate $\overline{A}_{MIT}$ on the magnetic field in Figure \[Btr\]. It is clear that even for relatively weak magnetic fields of only a few hundreds or thousands of Gauss, the $A_{MIT}$ will be significant compared to the competing M2-rate.
![\[Btr\] A plot of the $\overline{A}_{MIT}$ as a function of magnetic fields at $\Delta E = 20.14$ cm$^{-1}$ and $\Delta E = 5$ cm$^{-1}$, and compared to $A_{M2}$. ](fig5.eps){width="80.00000%"}
Experimental Determination of The Energy Splitting
--------------------------------------------------
![\[LnRD\] The ratio of the rates for the magnetically induced $^4\mathrm{D}_{7/2}~\rightarrow~^2\mathrm{P}_{3/2}$ line and the allowed $^4\mathrm{D}_{5/2}~\rightarrow~^2\mathrm{P}_{3/2}$ line as a function of electron density and magnetic field in an EBIT. Calculations is for a monoenergetic beam energy of 250 eV and these data are displayed for some selected magnetic field strengths and over a range of densities ($10^8 - 10^{11} ~\mathrm{cm}^{-3}$) which covers the range for solar flares in the corona. Here we used the fine structure energy of 5 cm$^{-1}$.](fig6.eps){width="80.00000%"}
To improve the accuracy of the estimated rate of the MIT, we need to turn to experiment for an accurate determination of the energy splitting. For this, we need to overcome two difficulties, first enough spectral resolution, and second a light-source with a low electron density and a magnetic field. The first requirement is not impossible to fulfill since the fine structure separation can be determined using a large spectrometer with a resolution of around 80¡¯000. This is far from the highest resolution achieved, since e.g. a spectrometer at the Observatory in Meudon has a resolution of 150¡¯000. However, the line from the $^4\mathrm{D}_{7/2}$ level has not been observed due to strict requirements on the light source. Most sources used at Meudon have generated too dense plasmas in which photon transitions from long lived levels cannot be seen (collisions are destroying the population of the upper state before the photon is emitted). In addition to this, observation of the $^4\mathrm{D}_{7/2} \rightarrow ^2\mathrm{P}_{3/2}$ line requires a strong enough magnetic field of, say a, tenth of a Tesla. This leads arguably to only two possible light sources on earth: Tokamaks ([Wesson 2004]{}) and the Electron Beam Ion Traps([Levine et al. 1988]{}) (EBITs). Tokamak plasma may be too dense, but since the magnetic fields involved are higher than what we are discussing here, the line might still be observable. However, the best choice for our purposes is an EBIT, which has an inherent magnetic field to compress the electron beam and is a low density light source. Although the Meudon spectrometer demonstrates that the required resolving power can be achieved this instrument is not compatable with the EBIT operating parameters and a dedicated instrument is required.
![\[LnRB\] Ratio of rates for the magnetically induced and the allowed transition as a function of magnetic field in an EBIT. The model is for a mono-energetic electron-beam energy of 250 eV and density of $1.0 \times 10^{11} ~\mathrm{cm}^{-3}$. We used the fine structure energy of 20.14 cm$^{-1}$ and 5 cm$^{-1}$. ](fig7.eps){width="80.00000%"}
To illustrate the usability of the EBIT source, we have made several model calculations to predict the relative strength of the two involved transitions under different circumstances. It should be made clear that the EBIT is a light source with a mono-energetic beam of electrons and that these models therefore are designed to predict conditions different from those in solar flares or the corona. It is important to remember that the intermediate goal before we can proceed is to propose an experiment to determine the crucial $^4\mathrm{D}_{7/2} - ^4\mathrm{D}_{5/2}$ energy separation. We present model calculations of the line ratio as a function of magnetic field and electron density (Figure \[LnRD\]) and magnetic field (Figure \[LnRB\]) of the EBIT, based on collisional-radiative modeling using the Flexible Atomic Code ([Gu 2008]{}). We show in Figure \[LnRD\], for several magnetic fields, how the ratio of the rates of the magnetic-field induced $^4\mathrm{D}_{7/2} \rightarrow ^2\mathrm{P}_{3/2}$ and the allowed $^4\mathrm{D}_{5/2} \rightarrow ^2\mathrm{P}_{3/2}$ transitions varies as a function of the electron densities. It is clear that the magnetic-field induced line is predicted to be visible for the typical range of electron densities of an EBIT, that is $10^8 - 10^{11}$ $~\mathrm{cm}^{-3}$ (this happens to coincide with the range for solar flares). It is also clear from Figure \[LnRB\], where we show the dependence of this ratio on the magnitude of the external magnetic field for a fixed density of $10^{11}$ $~\mathrm{cm}^{-3}$, that the line ratio will be sensitive to the magnetic field strength.
Conclusion
==========
To conclude, in this paper we propose a novel and efficient tool to determine magnetic field strengths in solar flares. The method is useful for cases of low densities and small external magnetic fields (hundreds and thousands of Gauss) that have so far eluded determination. We illustrate that a spectral feature originating from the Fe$^{9+}$ ion is of special interest since it shows a strong dependence on the magnetic field strength, with two spectral lines drastically changing their relative intensities. We propose a laboratory measurement of the fine structure energy separation between the two involved excited states, a crucial parameter in the determination of the external field. When this energy separation has been established one can use our theoretical values for the reduced rate of the magnetic-field induced transition, which have an accuracy to within a few percent, to calculate the atomic response to the external magnetic field. Armed with this it is possible to design a space-based mission with a probe that could continuously observe and determine the reclusive magnetic fields of the solar flares.
Acknowledgements
================
This work was supported by the Chinese National Fusion Project for ITER No. 2015GB117000, Shanghai Leading Academic Discipline Project No. B107. We also gratefully acknowledge support from the Swedish Institute under the Visby-programme. WL and JG would like to especially thank the Nordic Centre at Fudan University for supporting their visits between Lund and Fudan Universities.
References {#references .unnumbered}
==========
Aggarwal, K. M., & Keenan, F. P. 2004, , 427, 763.
Andersson, M., & Jönsson, P. 2008, , 178(2), 156.
Andrew, K. L., Cowan, R. D., & Giacchetti, A. 1967, , 57(6), 715 .
Beiersdorfer, P., Scofield, J. H. & Osterheld, A. L. 2003, , 90, 235003.
Bhatia, A., & Doschek, G. 1995, , 60, 97.
Brosius, J., Davila, J., & Thomas, R. 1998, , 119, 255.
Cheng, K. T., & Childs, W. J. 1985, , 31(5), 2775.
Deb, N. C., Gupta, G. P., & Msezane, A. Z. 2002, , 141, 247.
Dong, C. Z., Fritzsche, S., Fricke, B., & Sepp, W.-D. 1999, , 307, 809.
Grant, I. P. 2006, Relativistic Quantum Theory of Atoms and Molecules: Theory and Computation
Grumer, J., Li, W., Bernhardt, D., et al. 2013, , 88, 022513.
Grumer J., Brage T., Andersson M., et al. 2014, , accepted.
Gu M. F. 2008, , 86, 675.
Huang K.-N., Kim K., & Cheng K.T. 1983, , 28, 355.
Ishikawa, Y., Santana, J. A., & Trabert, E. 2010, , 43, 074022.
Jönsson, P., Gaigalas, G., Biero, J., Fischer, C. F., & Grant, I. 2013, , 184(9), 2197.
Levine, M. A., Marrs R. E., Henderson J. R., Knapp D. A. & Schneider M. B., 1988, , T22, 157-163.
Li, J., Brage T., Jönsson, P. & Yang Y. 2014, .
Li, J., Grumer, J., Li, W., et al. 2013, , 88, 013416.
Lin, H., Kuhn, J. R., & Coulter, R. 2004, , 613, L177.
Sandlin G.D. 1979, , 227, L107.
Schrijver, C. J., DeRosa, M. L., Metcalf, T., et al. 2008, , 675, 1637.
Smitt, R. 1977, , 51, 113.
Stenflo, J. O. 1977, , 41(6), 865.
Swings, P. 1943, , 98, 116-128.
Thomas, R., & Neupert, W. 1994, , 91, 461.
Wesson, J., 2004,
White, S. M. 2004, Coronal Magnetic Field Measurements Through Gyroresonance Emission, Solar and Space Weather Radiophysics, (D. Gary, C. U. Keller Editors, Astrophys. And Space Science Library.).
White, S. M., & Kundu, M. R. 1997, , 174, 31-52.
Wood, D. R., Andrew, K. L., Giacchetti, A., & Cowan, R. D. 1968, , 58(6),830.
Del Zanna, G., Berrington, K. A. & Mason H. E. 2004, , 422, 731.
Del Zanna, G., Storey, P. J., Badnell, N. R., & Mason, H. E. 2012, , 541, A90.
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abstract: 'We propose a scheme to manipulate quantum correlation of output lights from two sides of a cavity by phase control. A probe laser is set to split into two beams in an interferometer with a relative phase in two arms which drive the cavity mode in opposite directions along cavity axis, individually. This phase, here named as driving-field phase, is important to build up quantum correlation in HBT (Hanbury Brown-Twiss) setup. Three control lasers propagate vertically to the cavity axis and drive the corresponding atomic transitions with a closed-loop phase. This type of closed-loop phase has been utilized to realize quantum correlation and even quantum entanglement of the atomic system in previous work \[[Phys. Rev. A 81 033836 (2010)](https://doi.org/10.1103/PhysRevA.81.033836)\]. The scheme here is useful to manipulate steady and maximum quantum correlation.'
author:
- |
[^1]\
\
*Key Lab of Coherent Light, Atomic and Molecular Spectroscopy, Ministry of Education;*\
*College of Physics, Jilin University, Changchun 130012, China*
title: '**Phase-dependent quantum correlation in cavity-atom system**'
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[1.5cm]{}[1.5cm]{}
Keywords: cavity-QED, quantum interference, optical switching, phase control
PACS number(s): 42.50.-p, 42.50.Pq, 32.80.Qk, 42.25.Bs
Introduction {#intro}
============
Electromagnetically induced transparency (EIT), one kind of quantum interference, is based on the coherent superposition of two ground-state levels which are connected with an excited level by two lasers [@OL31/2625]. This technique has been realized by increasing the intensity of one of the two lasers as a coupling field ($\Lambda$-type, V-type and ladder-type EIT configurations) [@PRA51/576; @RMP77/633; @PRL100/173602] and has various applications by introducing another level and a third laser as a controlling field [@PRL81/3611; @PRL97/063901; @OL21/1936; @OL26/548; @PRA68/041801]. Especially, it is becoming more and more important to realize and modify intracavity EIT [@OL23/295; @PRA82/033808; @PRA84/043821; @PRA87/053802; @PRA89/023806; @OL33/46; @PRA85/013840; @OC358/73] in cavity quantum electrodynamics (cavity-QED) systems [@Nature465/755; @RMP87/1379]. Comparing with intracavity EIT itself, the modified intracavity EIT has more advantages on light controlling light system, such as low light intensity of switching field and high switching efficiency [@PRA85/013814; @PRA85/013840; @OC358/73; @Annp1700427].
On the other hand, there are a lot of interesting reports on phase-dependent EIT induced by a closed loop [@PRA59/2302; @PLA324/388; @PRA71/011803; @MOP54/2459] with or without cavity, which turn out to be a more sophisticated manner in manipulating light-atom interactions. It is applied in many explorations, such as phase-control spontaneous emission [@PRL81/293], beam splitter [@PRL101/043601] and entanglement between collective fields [@PRA81/033836]. Based on phase-dependent EIT in a double four-level-atom system [@PRA81/033836], four incident fields act as two collective modes and constitute two quantum beats. As a result, entanglement occurs between two collective modes. One can also operate the nonadiabatic optical transitions, quantum mechanical superposition states, the polarization selection and even controllable phase gate [@PRA79/025401; @PRA87/013430; @PRA90/063841; @PRA92/043838] by phase control in closed-loop EIT configuration. Meanwhile, with another type of phase control in the two arms of quantum interferometer, a lot of creative applications can be realized. For an example, with a Kerr phase in one arm of quantum interferometer, the postselected measurement and the amplification of weak effect have been carried out where giant XPM (cross-phase modulation) nonlinearity is resulted from EIT technique [@PRA90/013827]. The phase difference existing in two arms of quantum interferometer takes important effect on sensitive measurement like group-delay measurement and gravitational waves detection [@PRA85/011801; @OL20/788; @PREP684/1]. Moreover, this kind of phase difference is involved in interference control of medium absorption, measurement of spatial correlation or entanglement in many literatures [@PRL105/053901; @Science331/889; @PRA92/023824; @PRA95/013841]. In the scheme here, we utilize this driving-field phase to manipulate output of cavity mode field at two ends to realize controllable quantum correlation, where dissipation of the system is controlled by the modified EIT with closed-loop phase. Here controlled quantum coherence is induced by phase coherence of the classical fields.
In the following, we firstly make an introduction to our scheme. Some four-level atoms trapped in the cavity interact with three control lasers inducing a closed-loop phase and a probe laser splits into two coherent beams which drive the cavity mode field with a driving-field phase. Then we make a theoretical analysis and obtain the analytic solutions for intracavity field and its output fields. We can make the incident probe laser perfectly absorbed in cavity or totally transmitted (reflected) out of the cavity. And the correlation between two output channels depends on the closed-loop and driving-field phases. Secondly, we move forward on the numerical results and detailed discussion in section 3. We analyze how our system is performed as a perfect photon absorber or a complete transmitter / reflector. We discuss the controlled effects of the relative phases on quantum correlation according to evolution of second-order correlation of two output channels. In the end, we make a simple summary in section 4.
Theoretical analysis {#theory}
====================
The scheme proposed here is depicted as in Fig. \[fig-system\](a). The atomic levels $|1\rangle$, $|2\rangle$, $|3\rangle$ and $|4\rangle$ as shown in figure \[fig-system\](b) correspond to $5S_{1/2}\;F=1$, $5S_{1/2}\;F=2$, $5P_{1/2}\;F=1$ and $5P_{3/2}\;F=2$ of $^{87}$Rb, respectively. Two control lasers and a terahertz (THz) wave enter into the cavity with closed-loop phase $\phi_{1}$. Those two control lasers drive the atomic transitions $|2\rangle\rightarrow|3\rangle$, $|2\rangle\rightarrow|4\rangle$ with frequency detuning $\Delta_{1}=\omega_{1c}-\omega_{32}$ and $\Delta_{2}=\omega_{2c}-\omega_{42}$, respectively. The THz wave as a third control laser couples atomic transition from level $|3\rangle$ to $|4\rangle$ with a frequency detuning $\Delta_{t}=\omega_{t}-\omega_{43}$. A probe laser ($\omega_{p}$), which has a frequency detuning $\Delta_{c}=\omega_{p}-\omega_{c}$ from cavity mode ($\omega_{c}$), is split by a beam splitter (BS). The two split beams $\alpha_{in,l}$ and $\alpha_{in,r}$ are injected into cavity from the opposite directions. With phase control device, a relative phase $\phi_{2}$, namely driving-field phase, exists between $\alpha_{in,l}$ and $\alpha_{in,r}$. Two detectors are applied to receive output signal from right and left cavity mirror. $\Delta_{ac}=\omega_{c}-\omega_{31}$ is frequency detuning of cavity mode and atomic transition $|1\rangle\rightarrow|3\rangle$. $g\sqrt{N}$ is the collective coupling coefficient of cavity-QED system. $\Omega_{1}(\omega_{1c})$, $\Omega_{2}(\omega_{2c})$ and $\Omega_{t}(\omega_{t})$ are Rabi frequency (angular frequency) of control laser 1, control laser 2 and THz wave. $\omega_{31}$, $\omega_{32}$, $\omega_{42}$ and $\omega_{43}$ are angular frequency of corresponding atomic level spacing. It is well known that EIT or modified EIT can be observed in this kind of configuration [@PLA324/388; @CPB26/074207]. The optical switch based on intensity modulation of control lasers can be realized. Different from that, however, here we utilize the phase effect to manipulate the intracavity field and output signals.
Under rotating wave approximation, the Hamiltonian is as following, $$\begin{split}
H\!\!=\!&\!-\!\hbar\Delta_{c}a^{\dag}a\!-\!\hbar\sum_{j=1}^{N}[(\Delta_{p}\!\!-\!\!\Delta_{1})\sigma_{22}^{j}\!+\!(\Delta_{p}\!\!-\!\!\Delta_{1}\!\!+\!\!\Delta_{2}\!\!-\!\!\Delta_{t})\sigma_{33}^{j}\\
&\!+\!(\Delta_{p}\!\!-\!\!\Delta_{1}\!\!+\!\!\Delta_{2})\sigma_{44}^{j}]\!-\!\hbar\sum_{j=1}^{N}(ga^{\dag}\sigma_{13}^{j}e^{i\varphi_{p}}\!+\!\Omega_{1}\sigma_{23}^{j}e^{i\varphi_{1}}\\
&\!+\!\Omega_{2}\sigma_{24}^{j}e^{i\varphi_{2}}\!+\!\Omega_{t}\sigma_{34}^{j}e^{i\varphi_{t}})\!+\!H.C.,\label{Hamiltonian}
\end{split}$$ where $H.C.$ denotes the Hermitian conjugate, $\Delta_{p}=\omega_{p}-\omega_{31}$ is frequency detuning of probe laser and atomic transition $|1\rangle\rightarrow|3\rangle$, $g=\mu_{13}\sqrt{\omega_{c}/(2\hbar\varepsilon_{0}V)}$ is cavity-QED coupling coefficient, $a^{\dag}$($a$) is creation (annihilation) operator of cavity photons, $\sigma_{mn}^{j}=|m\rangle\langle n|$ ($m,n=1,2,3,4$) is atomic operator, $\varphi_{p}$, $\varphi_{1}$, $\varphi_{2}$ and $\varphi_{t}$ are phases of probe laser, control laser 1, control laser 2 and THz wave, respectively.
For simplicity, we consider a symmetric Fabry-Perot cavity with field loss rate $\kappa_{l}$ ($\kappa_{r}$) from left (right) cavity mirror, $\kappa_{i}=T_{i}/2\tau$, where $T_{i}$ is the mirror transmission and $\tau$ is the photon round-trip time inside the cavity. The relation between input and output modes of this cavity-atom system is given by [@G.S.Agarwal2013], $$\begin{gathered}
\begin{split}
&\langle a_{out,l}\rangle+\langle a_{in,l}\rangle=\sqrt{2\kappa_{l}\tau}\;\langle a\rangle,\\
&\langle a_{out,r}\rangle+\langle a_{in,r}\rangle=\sqrt{2\kappa_{r}\tau}\;\langle a\rangle,\label{input-output relation}
\end{split}\end{gathered}$$ where $\langle{a}\rangle=\alpha$ ($\langle{a^{\dag}}\rangle=\alpha^{*}$), $\langle a_{in,l}\rangle=\alpha_{in}^{l}$ ($\langle a_{in,r}\rangle=\alpha_{in}^{r}$) and $\langle a_{out,l}\rangle=\alpha_{out}^{l}$ ($\langle a_{out,r}\rangle=\alpha_{out}^{r}$) are expectation values for the operators of intracavity field, incident probe beam and outgoing signal from left (right) mirror, respectively [@PRA93/023806]. Since the system is driven from both sides, the transmission and reflection properties of the cavity can be calculated by solving the following Heisenberg-Langevin equations of motion [@RMP87/1379], $$\begin{split}
&\langle\dot a\rangle\!\!=\!\!\frac{1}{i\,\hbar}[a,H]\!\!-\!\!(\kappa_{l}\!\!+\!\!\kappa_{r})\langle a\rangle\!\!+\!\!\sqrt{2\kappa_{l}/\tau}\langle a_{in}^{l}\rangle\!\!+\!\!\sqrt{2\kappa_{r}/\tau}\langle a_{in}^{r}\rangle,\\
&\langle\dot \sigma_{ij}\rangle\!\!=\!\!\frac{1}{i\,\hbar}[\sigma_{ij},H]\!\!-\!\!\gamma_{ij}\langle\sigma_{ij}\rangle.\label{equation of motion}
\end{split}$$ At the initial time, we assume that the populations in level $|1\rangle$ and $|2\rangle$ are both 1/2, namely $\sigma_{11}=\sigma_{22}=1/2$ and $\sigma_{33}=\sigma_{44}=0$. Then under steady-state condition, the intracavity field can be derived as, $$\alpha=\frac{\sqrt{2\kappa_{l}/\tau}\;\alpha_{in}^{l}+\sqrt{2\kappa_{r}/\tau}\;\alpha_{in}^{r}}{(\kappa_{l}+\kappa_{r})-i\Delta_{c}-i\chi},\label{intracavity field}$$ where $$\chi\!=\!\frac{g^{2}N/2(\Omega_{2}^{2}\!-\!A\!*\!B)}{2\Omega_{1}\Omega_{2}\Omega_{t}\cos{\phi_{1}}\!-\!A\!*\!\Omega_{t}^{2}\!-\!B\!*\!\Omega_{1}^{2}\!-\!C\!*\!\Omega_{2}^{2}\!+\!A\!*\!B\!*\!C}$$ is the susceptibility of atomic media and $A=\Delta_{p}-\Delta_{1}+i\,\gamma_{12}$, $B=(\Delta_{p}-\Delta_{1}+\Delta_{2})+i\,\Gamma_{4}/2$, $C=(\Delta_{p}-\Delta_{1}+\Delta_{2}-\Delta_{t})+i\,\Gamma_{3}/2$. Here $\phi_{1}=\varphi_{1}-\varphi_{2}+\varphi_{t}$ represents the relative phase of two control lasers and THz wave induced by the closed loop as shown in Fig. \[fig-system\](b), $\Gamma_{3}=\Gamma_{4}=\Gamma$ is the natural decay rate of excited states $|3\rangle$ and $|4\rangle$, and $\gamma_{12}$, much smaller than $\Gamma_{3}$ or $\Gamma_{4}$, is the decoherence rate between ground states $|1\rangle$ and $|2\rangle$.
The analytical solutions to intracavity field and output signals through right and left mirrors are, $$\begin{split}
&I_{c}=I_{in}^{r}|\frac{\sqrt{\kappa}(1+e^{i\phi_{2}})}{\kappa-i\Delta_{c}-i\chi}|^{2},\\
&I_{out}^{r}=I_{in}^{r}|\frac{\kappa(1+e^{i\phi_{2}})}{\kappa-i\Delta_{c}-i\chi}-1|^{2},\\
&I_{out}^{l}=I_{in}^{r}|\frac{\kappa(1+e^{-i\phi_{2}})}{\kappa-i\Delta_{c}-i\chi}-1|^{2}.\label{field intensity}
\end{split}$$ Here we assume that $\kappa_{l}=\kappa_{r}=\kappa/2$, $\alpha_{in}^{l}=|\alpha_{in}|e^{i\varphi_{l}}$ and $\alpha_{in}^{r}=|\alpha_{in}|e^{i\varphi_{r}}$. The driving-field phase $\phi_{2}=\varphi_{l}-\varphi_{r}$ is the relative phase of two incident probe beams, $I_{in}^{r}$ is the input field intensity from right side of cavity and $I_{c}$, $I_{out}^{r}$ ($I_{out}^{l}$) are field intensity of intracavity light and output light from right (left) mirror, respectively. The output field intensity can be manipulated either by changing media susceptibility or relative phase $\phi_{2}$. In this system, both strength / frequency of control lasers and closed-loop phase $\phi_{1}$ can be used to modulate media absorption property. As indicated in equation (\[field intensity\]) two output channels have correlation based on $\phi_{2}$, especially, when $\phi_{2}=\pi$, intracavity field intensity is always equal to zero, which means the probe beams are transmitted or reflected totally. In brief, in this scheme, light output of the two channels can be controlled by two types of phases.
Results and discussion {#result}
======================
In this section, we firstly analyze the media absorption property controlled by intracavity phase-dependent EIT induced by closed-loop phase $\phi_{1}$ since it is one of the major factors of phase-dependent correlation between two output channels. Here we emphasize the effect of initial phases of controlling lasers rather than the intensities of them. Secondly, we discuss influence by the other major factor, driving-field phase $\phi_{2}$. The two phases lead to the controllable second-order correlation and the steady and maximum value of it. Numerical results and detailed discussion are as following.
Media absorption property is presented by imaginary part of susceptibility $\chi$ as shown in Fig. \[fig-imx\](a) and \[fig-imx\](b). The parameters of Fig. \[fig-imx\] and all other figures in this section are under resonance condition $\Delta_{1}=\Delta_{2}=\Delta_{t}=\Delta_{ac}=0$. The cavity is at threshold of strong collective-coupling regime ($g^{2}N=\kappa\,\Gamma$). Without control laser 2 and THz wave, $\Lambda$-type three-level atoms interact with cavity field and control laser 1, and thus two bright polaritons are formed at $\Delta_{p}=\pm\sqrt{g^{2}N+\Omega_{1}^{2}}$ and a dark state at $\Delta_{p}=0$ which is decoupled to cavity mode. When two control lasers are applied, two new dark states because of dark state splitting and new bright states are formed at [@PRA85/013814] $$\begin{split}
&\Delta_{p}=\\
&\pm\sqrt{[\Omega_{1}^2\!+\!\Omega_{2}^2\!+\!g^2N\pm\sqrt{(\Omega_{1}^2\!+\!\Omega_{2}^2\!+\!g^2N)^2\!-\!4g^2N\Omega_{2}^2}]/2}.
\end{split}$$ Together with the strong THz wave ($\Omega_{t}$), intracavity EIT splitting will be destroyed but a closed interaction contour is formed which can lead to a phase-dependent EIT.
Fig. \[fig-imx\](a) and \[fig-imx\](b) are plotted with $\Omega_{t}=0.5\Gamma$ and $\Omega_{t}=\Gamma$ to present the harmful effect of THz wave on intracavity EIT splitting and the compensated contribution of the closed-loop phase. In figure \[fig-imx\](a) when $\Omega_{t}=0.5\Gamma$, THz wave could not completely destroy intracavity EIT splitting. This is why Im\[$\chi$\] has three peaks in absorption spectra (double EIT), but the values and locations of the three peaks are decided by closed-loop phase. As shown in Fig. \[fig-imx\](a), for $\phi_{1}$ in even order of $\pi/2$ ($\phi_{1}=0,\,\pi,\,2\pi$) and odd order of $\pi/2$ ($\phi_{1}=\pi/2,\,3\pi/2$), there are three asymmetrical and symmetrical absorption peaks with respect to $\Delta_{p}=0$, respectively. The basic principle behind this phenomenon can be explained from the expression of Im\[$\chi$\] as below, $$Im[\chi]\!=\!g^2N
\begin{cases}
\frac{X-8(\Gamma+2\gamma_{12})\Omega_{1}\Omega_{2}\Omega_{t}\Delta_{p}}{[Z_{1}-4(\Gamma+\gamma_{12})\Delta_{p}^2]^2+Y_{1}^2},&{\phi_{1}\!=\!0,\pi,2\pi}\\
\\
\frac{X}{[Z_{1}-4(\Gamma+\gamma_{12})\Delta_{p}^2]^2+Y_{2}^2},&{\phi_{1}\!=\!\pi/2,3\pi/2}
\end{cases}\label{imx}$$ where $$\begin{split}
&X=4\Gamma\Delta_{p}^4+Z_{1}(\Gamma\gamma_{12}+2\Omega_{2}^2)+Z_{2}\Delta_{p}^2,\\
&Y_{1}=Y_{2}+8\Omega_{1}\Omega_{2}\Omega_{t},\\
&Y_{2}=4\Delta_{p}^3-\Delta_{p}[\Gamma^2+4\Gamma\gamma_{12}+4(\Omega{1}^2+\Omega_{2}^2+\Omega_{t}^2)],\\
&Z_{1}=\Gamma^2\gamma_{12}+2\Gamma(\Omega_{1}^2+\Omega_{2}^2)+4\gamma_{12}\Omega_{t}^2,\\
&Z_{2}=\Gamma^3+8\gamma_{12}\Omega_{1}^2+4\Gamma(\gamma_{12}^2-2\Omega_{2}^2+\Omega_{t}^2).
\end{split}$$
When $\Omega_{t}$ is up to $\Gamma$, it stops the dark-state splitting completely for $\phi_{1}$ in the even order of $\pi/2$. While $\phi_{1}$ is equal to odd order of $\pi/2$, three symmetrical absorption peaks can exhibit as shown in Fig. \[fig-imx\](b), and thus the double EIT reappears. These phase-dependent EIT properties have also been presented in the output spectra as shown in \[fig-imx\](c) and \[fig-imx\](d) when $\phi_{2}=0$, which show that dissipation of the system can be modulated by closed-loop phase.
In the following, we explore how $\phi_{2}$ affects output fields from two channels. The output intensities of two sides of the cavity and that of intracavity field are depicted in Fig. \[fig-intensity\] under several $\phi_{2}$ when double EIT exists ($\phi_{1}=\pi/2$). Fig. \[fig-intensity\](a) shows that the output intensity from right side of cavity mirror is always the same as that from left side at any frequency detuning. It shows that, at CPA (coherent perfect absorber) resonance ($\Delta_{p}=0$), large absorption and less interference amplitudes between probe beams lead to an ideal interference “trap" for the two beams so that eventually the probe photon will be absorbed by intracavity media. With increasing relative phase $\phi_{2}$, due to interference between two probe beams, output of intracavity field is enhanced at resonance (as in Fig. \[fig-intensity\](b)) until total reflection ($I_{out}/I_{in}=1$) of the intracavity field appears at two ends of the cavity under $\phi_{2}=\pi$ as shown in \[fig-intensity\](c), which can be derived by equation (\[field intensity\]).
The physical essence of the above effect of $\phi_{2}$ is resulted from formation of a phase-dependent standing wave by two driving fields inside of cavity. When $\phi_{2}=\pi$, the standing wave (blue line in Fig. \[fig-intendis\](a)) takes concerted action with the cavity mode field (black lines in Fig. \[fig-intendis\]), thus the intracavity photon will be totally reflected by standing-wave field. When $\phi_{2}$ is equal to other values, for example $\phi_{2}=\pi/2$, the standing-wave field (red line in Fig. \[fig-intendis\](b)) will always be out of step with cavity mode field, therefore intracavity field will be partially reflected through two ends of cavity. A light switching with high contrast can be designed with different driving-field phase and closed-loop phase. With $\phi_{1}=\pi/2$, for example, the two output channels at resonance are closed when $\phi_{2}=0$. When we set $\phi_{2}=\pi/2$, the output intensity at resonant frequency is 0.5 (half-open) and at positive (negative) frequency detuning, the right (left) channel is open. The output switching is completely open when $\phi_{2}=\pi$. The switching contrast ratio at resonance between closed state and open state can be up to 1. Actually, this can be used as the transfer switching from perfect photon absorber to perfect photon transmitter or reflector.
![Field distribution in two-sided cavity with (a) $\phi_{2}=\pi$ and (b) $\phi_{2}=\pi/2$.[]{data-label="fig-intendis"}](fig4.eps){width="7cm"}
Besides as an optical switching, we notice that the output intensities from two ends of the cavity can always reach the same values under some frequencies (i.e. $\Delta_{p}=0$), while under other frequencies (such as $\Delta_{p}=\pm\Gamma$), output intensities of two ends will be different. This indicates possible enhancement or weakness of intensity correlation or quantum coherence of the two channels. As shown in Fig. \[fig-intensity\](a), the output intensities for both channels are minimum thus the intensity correlation between these two channels is at a small value under $\Delta_{p}=0$. With $\phi_{2}=\pi/2$ in figure \[fig-intensity\](b), the output intensities of two channels are increased and equal at resonance which indicates intensity correlation is enhanced at larger value for $\Delta_{p}=0$. When $\phi_{2}=\pi$ as in Fig. \[fig-intensity\](c), the output intensities are equal and maximum which indicates maximum correlation (robust entanglement) free to the environmental dissipation. This infers that quantum correlation of the two output channels can be created by classic interference between two driving fields as shown in Fig. \[fig-intendis\]. Since media absorption can be controlled by intracavity phase-dependent EIT and outputs of intracavity mode field can be manipulated by the interference of two coherent input beams induced by driving-field phase, the scheme can be used for manipulating quantum correlation and even the quantum entanglement of the two channels.
In order to show the phase-dependent quantum correlation quantitatively, we calculate the second-order correlation [@RMP87/1379] between the two channels, $$G^{(2)}=\frac{\langle a^{\dag}_{out,l}a^{\dag}_{out,r}a_{out,r}a_{out,l}\rangle}{{\langle a^{\dag}_{in,r}a_{in,r}\rangle}^{2}}.\label{G2}$$ The time evolution of $G^{(2)}$ at resonant frequency versus $\phi_{1}$ and $\phi_{2}$ are drawn as in Fig. \[fig-3d1evo\](a) and \[fig-3d1evo\](b), respectively. They are shown obviously that the initial second-order correlation decreases rapidly within $0.12\,\mu s$ (which is in accordance with system loss $\mathcal{L}=\kappa_{l}+\kappa_{r}+\kappa_{atom}=3/2\,\Gamma$, here $\Gamma=6$ MHz for $^{87}$Rb) [@RMP87/1379] then starts increasing, and finally reaches a stable value. Since dissipation which causes decoherence of system can be controlled by phase-dependent EIT, the stable value is different for different closed-loop phase $\phi_{1}$ as in figure \[fig-3d1evo\](a). $G^{(2)}$ can not get back 1 unless total reflection appears ($\phi_{2}=\pi$) as shown in Fig. \[fig-3d1evo\](b). For $\phi_{1}=0$ and $\phi_{1}=\pi$ in Fig. \[fig-3d1evo\](a), the stable correlation is around 0.47, while for $\phi_{1}=\pi/2$, the stable value decreases to 0.26, which is accordance with the media absorption property as in Fig. \[fig-imx\](b). It reveals that intracavity phase-dependent EIT with driving-field phase can be taken as one source of quantum correlation to conquer decoherence. Similarly in the scheme, quantum correlation can be modulated by standing-wave driving with $\phi_{2}$. As shown in Fig. \[fig-3d1evo\](b), although the stable correlation is much lower when $\phi_{1}=\pi/2$ and $\phi_{2}=0$, increasing $\phi_{2}$ from $\phi_{2}=0$ to $\phi_{2}=\pi$, the value of correlation function is increased. The maximum correlation can be obtained when $\phi_{2}=\pi$ where quantum correlation is free to decoherence. These results are consistent with the analysis of output spectra in Fig. \[fig-intensity\]. It shows that classical interference can also be used to control quantum correlation.
In the following, we analyze the correlation function at EIT windows ($\Delta_{p}=\pm\Gamma$) with several values of the two phases. Fig. \[fig-3d2evo\](a) shows that at $\Delta_{p}=\Gamma$, with increasing $\phi_{1}$ from 0 to $\pi$, stable quantum correlation will be improved from 0.32 to 0.70. Different from that in resonant frequency, the minimum stable correlation is obtained when $\phi_{1}=0$. This is predictable from the analysis of Fig. \[fig-imx\](c) and \[fig-imx\](d). The output intensities of two channels are equal and enhanced with $\phi_{1}$ being increased from 0 to $\pi$, therefore intensity correlation is enhanced correspondingly. While at $\Delta_{p}=-\Gamma$, the steady correlation is decreasing with changing $\phi_{1}$ from $\phi_{1}=0$ to $\phi_{1}=\pi$ (not shown).
Under $\phi_{1}=\pi/2$, quantum correlation depends on $\phi_{2}$ differently in the situations for $\Delta_{p}=0$ and $\Delta_{p}=\Gamma$ as shown in figures \[fig-3d1evo\](b) and \[fig-3d2evo\](b). This can be explained according to the analysis of Fig. \[fig-intensity\]. For $\Delta_{p}=0$, output intensities of two channels under $\phi_{2}=0$ (Fig. \[fig-intensity\](a)) are weak but equal and thus there exists minimum enhancement of intensity correlation. The second order correlation can be recovered to a steady value 0.26 under $\Delta_{p}=0$ when $\phi_{1}=\pi/2$, $\phi_{2}=0$ as shown in Fig. \[fig-3d1evo\](b). However, for $\Delta_{p}=\Gamma$ in Fig. \[fig-intensity\](b) ($\phi_{2}=\pi/2$), the intensities of two output channels are unequal, one of which is strong but the other is weak, thus the correlation can not be recovered (minimum steady correlation close to zero under $\phi_{2}=\pi/2$ at $\Delta_{p}=\Gamma$ as in Fig. \[fig-3d2evo\](b)). For $\phi_{2}=0$, $\pi/4$ and $3\pi/4$, the correlation can reach larger steady values and the maximum value for $\phi_{2}=\pi$ can be realized because of the partial or total reflection based on driving-field phases as in Fig. \[fig-intendis\].
In a word, in a cavity-atom system, with two-sided coherent driving and closed-loop EIT controlling, phase-dependent quantum correlation can be realized.
Conclusions
===========
In conclusion, we analyze media absorption and wave interference in a four-level atom-cavity system. Based on phase-dependent EIT rather than using large intensity of coupling field, optical switching with high contrast (up to 1) via phase control can be realized. Together with phase-dependent standing wave formed by two coherent driving fields, the total photon absorber or hundred-percent transmitter (reflector) can be obtained.
Due to absorption suppression by modified intracavity EIT and interference enhancement of standing-wave field in the cavity, output intensity can be controlled by closed-loop phase ($\phi_{1}$) and driving-field phase ($\phi_{2}$). Phase-dependent quantum correlation between two output channels can be manipulated. Because of the total reflection from standing-wave field under $\phi_{2}=\pi$, maximum correlation of these two channels can be obtained. By manipulation on quantum correlation, this work provides potential application in realization of controllable entangled photons in cavity system.
Acknowledgment {#acknowledgment .unnumbered}
==============
We acknowledge support from National Natural Science Foundation of China under Grant No. 11174109.
[99]{} =1.5pt plus.2pt minus.2pt Figueroa E, Vewinger F, Appel J and Lvovsky A I 2006 [*Opt. Lett.*]{} [**31** 2625-2627](https://doi.org/10.1364/OL.31.002625) Gea-Banacloche J, Li Y, Jin S and Xiao M 1995 [*Phys. Rev. A*]{} [**51** 576](https://doi.org/10.1103/PhysRevA.51.576)
Fleischhauer M, Imamoglu A and Marangos J P 2005 [*Rev. Mod. Phys.*]{} [**77** 633](https://doi.org/10.1103/RevModPhys.77.633)
Wu H, Gea-Banacloche J and Xiao M 2008 [*Phys. Rev. Lett.*]{} [**100** 173602](https://doi.org/10.1103/PhysRevLett.100.173602)
Wang Z, Marzlin K P and Sanders B C 2006 [*Phys. Rev. Lett.*]{} [**97** 063901](https://doi.org/10.1103/PhysRevLett.97.063901) Harris S E and Yamamoto Y 1998 [*Phys. Rev. Lett.*]{} [**81** 3611](https://doi.org/10.1103/PhysRevLett.81.3611) Schmidt H and Imamoglu A 1996 [*Opt. Lett.*]{} [**21** 1936-1938](https://doi.org/10.1364/OL.21.001936) Yan M, Rickey E G and Zhu Y 2001 [*Opt. Lett.*]{} [**26** 548-550](https://doi.org/10.1364/OL.26.000548) Braje D A, Balić V, Yin G Y and Harris S E 2003 [*Phys. Rev. A*]{} [**68** 041801](https://doi.org/10.1103/PhysRevA.68.041801) Lukin M D, Fleischhauer M, Scully M O and Velichansky V L 1998 [*Opt. Lett.*]{} [**23** 295-297](https://doi.org/10.1364/OL.23.000295) Zhang J, Hernandez G and Zhu Y 2008 [*Opt. Lett.*]{} [**33** 46-48](https://doi.org/10.1364/OL.33.000046) Dantan A, Albert M and Drewsen M 2012 [*Phys. Rev. A*]{} [**85** 013840](https://doi.org/10.1103/PhysRevA.85.013840) Duan Y, Lin G, Zhang S, Niu Y and Gong S 2016 [*Opt. Commun.*]{} [**358** 73-76](https://doi.org/10.1016/j.optcom.2015.09.043) Wei X, Zhang J and Zhu Y 2010 [*Phys. Rev. A*]{} [**82** 033808](https://doi.org/10.1103/PhysRevA.82.033808) Nielsen A E B and Kerckhoff J 2011 [*Phys. Rev. A*]{} [**84** 043821](https://doi.org/10.1103/PhysRevA.84.043821) Zou B and Zhu Y 2013 [*Phys. Rev. A*]{} [**87** 053802](https://doi.org/10.1103/PhysRevA.87.053802) Zou B, Tan Z, Musa M and Zhu Y 2014 [*Phys. Rev. A*]{} [**89** 023806](https://doi.org/10.1103/PhysRevA.89.023806) Mücke M, Figueroa E, Bochmann J, Hahn C, Murr K, Ritter S, Villas-Boas C J and Remp G 2010 [*Nature*]{} [**465** 755](https://doi.org/10.1038/nature09093) Reiserer A and Rempe G 2015 [*Rev. Mod. Phys.*]{} [**87** 1379](https://doi.org/10.1103/RevModPhys.87.1379) Wang Y, Zhang J and Zhu Y 2012 [*Phys. Rev. A*]{} [**85** 013814](https://doi.org/10.1103/PhysRevA.85.013814) Guo M and Su X 2018 [*Ann. Phys. (Berlin)*]{} [**530** 1700427](https://doi.org/10.1002/andp.201700427) Korsunsky E A, Leinfellner N, Huss A, Baluschev S and Windholz L 1999 [*Phys. Rev. A*]{} [**59** 2302](https://doi.org/10.1103/PhysRevA.59.2302) Xue Y, Wang G, Wu J, Xu W, Wang H, Gao J and Babin S A 2004 [*Phys. Lett. A*]{} [**324** 388-395](https://doi.org/10.1016/j.physleta.2004.02.050) Deng L and Payne M G 2005 [*Phys. Rev. A*]{} [**71** 011803](https://doi.org/10.1103/PhysRevA.71.011803) Abi-salloum T, Davis J P, Lehman C, Elliott E and Narducci F A 2007 [*J. Mod. Opt.*]{} [**54** 2459-2471](https://doi.org/10.1080/09500340701742617) Paspalakis E and Knight P L 1998 [*Phys. Rev. Lett.*]{} [**81** 293](https://doi.org/10.1103/PhysRevLett.81.293) Xiao Y, Klein M, Hohensee M, Jiang L, Phillips D F, Lukin M D and Walsworth R L 2008 [*Phys. Rev. Lett.*]{} [**101** 043601](https://doi.org/10.1103/PhysRevLett.101.043601) Wang F, Hu X, Shi W and Zhu Y 2010 [*Phys. Rev. A*]{} [**81** 033836](https://doi.org/10.1103/PhysRevA.81.033836) Basler C, Reininger K, Meinert F, Ghosh P N and Helm H 2013 [*Phys. Rev. A*]{} [**87** 013430](https://doi.org/10.1103/PhysRevA.87.013430) Hashmi F A and Bouchene M A 2009 [*Phys. Rev. A*]{} [**79** 025401](https://doi.org/10.1103/PhysRevA.79.025401) Zhu C, Deng L and Hagley E W 2014 [*Phys. Rev. A*]{} [**90** 063841](https://doi.org/10.1103/PhysRevA.90.063841) Li R B, Zhu C J, Deng L and Hagley E W 2015 [*Phys. Rev. A*]{} [**92** 043838](https://doi.org/10.1103/PhysRevA.92.043838) Li G, Wang T and Song H S 2014 [*Phys. Rev. A*]{} [**90** 013827](https://doi.org/10.1103/PhysRevA.90.013827) Jarzyna M and Demkowicz-Dobrzański R 2012 [*Phys. Rev. A*]{} [**85** 011801](https://doi.org/10.1103/PhysRevA.85.011801) Kovács A P, Osvay K, Bor Zs and Szipöcs R 1995 [*Opt. Lett.*]{} [**20** 788-790](https://doi.org/10.1364/OL.20.000788) Schnabel R 2017 [*Phys. Rep.*]{} [**684** 1-51](https://doi.org/10.1016/j.physrep.2017.04.001) Chong Y D, Ge L, Cao H and Stone A D 2010 [*Phys. Rev. Lett.*]{} [**105** 053901](https://doi.org/10.1103/PhysRevLett.105.053901) Wan W, Chong Y, Ge L, Noh H, Stone A D and Cao H 2011 [*Science*]{} [**331** 889-892](https://doi.org/10.1126/science.1200735) Agarwal G S and Zhu Y 2015 [*Phys. Rev. A*]{} [**92** 023824](https://doi.org/10.1103/PhysRevA.92.023824) Wang L, Di K, Zhu Y and Agarwal G S 2017 [*Phys. Rev. A*]{} [**95** 013841](https://doi.org/10.1103/PhysRevA.95.013841) Guo M and Su X 2017 [*Chin. Phys. B*]{} [**26** 074207](https://doi.org/10.1088/1674-1056/26/7/074207) Agarwal G S 2013 [*Quantum Optics*]{} (New York: Cambridge University Press) p 308-347 Sawant R and Rangwala S A 2016 [*Phys. Rev. A*]{} [**93** 023806](https://doi.org/10.1103/PhysRevA.93.023806)
[^1]: Corresponding author: suxm@jlu.edu.cn
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---
abstract: 'Non-interacting particles with long-range hopping are known to be delocalized in disordered systems of infinite size. It is thus natural to assume that such particles can traverse any finite-size lattice. Here, we demonstrate that this is not generally true. The delocalization mechanism is induced by resonances between distant lattice sites. The number of such resonances diverges with the system size. For a finite-size lattice the number of resonances is finite and may not be sufficient to result in delocalization. We consider particles with long-range (dipolar) hopping in three-dimensional lattices with diagonal disorder and random dilution. We compute the wavepacket dynamics of particles placed in an individual lattice site, the inverse participation ratios and the fluctuations of the wavefunctions with disorder. We show that, for certain finite-size disordered lattices, particles remain localized within a finite volume much smaller than the lattice size, and that the wavefunctions exhibit the log-normal fluctuations characteristic of Anderson localization. We characterize the localization properties as functions of dilution and diagonal disorder. We combine our results with scaling theory to obtain the size dependence of the localization–diffusion crossover. Our results indicate that particles with long-range hopping undergo exponential localization in lattices of finite size, even macroscopically finite. Our phase diagrams illustrate a rather unusual phenomenon: quantum particles can diffuse through a lattice of size $10A$, but not through a lattice of size $A$.'
author:
- 'J. T. Cantin, T. Xu, and R. V. Krems'
title: ' Localization of quantum particles with long-range hopping in 3D lattices of finite size '
---
Introduction
============
A quantum particle in a three-dimensional disordered lattice is either localized or diffusive; this depends on the strength of the disorder [@scaling-anderson]. If the particles are localized within a finite localization length $\lambda$, one observes current over length scales $\lesssim \lambda$, while no conductivity over length scales $\gg \lambda$ [@root-skinner]. If quantum transport is allowed over infinite length scales, particles are naturally assumed to diffuse over any arbitrary length scale. As was pointed out already in the original work of Anderson [@anderson], quantum particles with long-range hopping are diffusive in three-dimensional (3D) lattices of infinite size, for any disorder strength. This absence of localization was later explained by Levitov [@Levitov1989; @Levitov1990; @Levitov1990b; @Levitov1999], who showed that diffusion occurs due to resonant transitions between distant lattice sites. The number of such resonances diverges with the lattice size, leading to particle transport over infinite length scales. However, if the lattice size is finite, the number of resonances is also finite. This raises the question: If the lattice size is reduced, can the system localize to length scales smaller than the lattice size or will the localization length always remain larger than the system size?
To answer this question, we consider particles with dipolar hopping in 3D disordered lattices. Dipolar hopping is long-range in 3D. For generality, we consider lattices with random on-site energies and random dilution. We show that as the lattice size is [*reduced*]{}, delocalized particles may, depending on the dilution and amount of disorder, become localized to length scales much smaller than the lattice size. This leads to a rather unusual phenomenon: particles can diffuse through a lattice of size $10A$, but not through a lattice of size $A$. This means that quantum currents can be switched off by shrinking the size of the system.
For conventional disordered systems, the localization is typically characterized by computing certain properties as functions of the lattice size. For example, one can examine the distribution of nearest spacings of the Hamiltonian eigenvalues, which must undergo a change from the Wigner distribution to a Poisson distribution near the diffusion–localization crossover. This change becomes sharper as the lattice size increases, corresponding to a phase transition in the infinite size limit. For particles with dipolar long-range hopping in 3D lattices, such conventional analyses are not possible as the parameters of interest are functions of the system size itself and the localization length diverges with the system size, in contrast to the case with nearest neighbour hopping. Therefore, in order to examine the properties of particles with long-range hopping in finite-size lattices, we use the following approach:
- First, we compute the wavepacket expansion dynamics of particles placed in an individual site of the lattice to show that, for certain system parameters, a particle placed in the middle of the lattice does not reach the lattice edge over long times greatly exceeding the Heisenberg time;
- Second, we compute the inverse participation ratio for each eigenstate of the Hamiltonian to illustrate that, for certain system parameters, all eigenstates are localized to parts of the lattice much smaller than the lattice size;
- Third, we examine the fluctuations of the wavefunctions between instances of disorder to illustrate that these fluctuations become log-normal at strong disorder;
- Fourth, we map out the localization-diffusion crossover region as a function of dilution and on-site disorder using the wavepacket dynamics described in (i);
- Finally, we combine our results with scaling theory from Refs. [@Levitov1989; @Levitov1990; @Levitov1990b; @Levitov1999] in order to obtain the dependence of the localization - diffusion crossover region on the lattice size. Refs. [@Levitov1989; @Levitov1990; @Levitov1990b; @Levitov1999] were originally introduced to explain delocalization of particles with long-range hopping.
Wavepacket dynamics
===================
We consider particles with dipolar hopping in 3D disordered lattices, known to be diffusive over infinite length scales. We start by illustrating the absence of diffusion for such particles in finite-size lattices by computing the time evolution of wavepackets placed on an individual lattice site. For generality, we compute the wavepacket dynamics in a diluted, disordered lattice and characterize the localization-diffusion phase diagram as a function of both dilution and disorder strength, as was done for particles with short-range hopping in Ref. [@root-skinner].
The dynamics are governed by the following Hamiltonian: $$\begin{aligned}
\hat H = \sum_{i} w_i \hat c^\dagger_i \hat c_i + \sum_{i} \sum_{j \neq i} t_{ij} \hat c^\dagger_i \hat c_j,
\label{model}\end{aligned}$$ where the operator $\hat c_i$ removes the particle from site $i$, $w_i$ is the energy of the particle in site $i$, and $t_{ij}$ is the amplitude for particle tunnelling from site $j$ to site $i$. We introduce disorder by randomizing both the values of $w_i$ and $t_{ij}$, which makes Eq. (\[model\]) relevant for both disordered lattices and amorphous systems. We consider a single particle in a 3D cubic lattice with $N$ sites per dimension. We randomize the values $w_i \in [-w/2, w/2]$ drawn from a uniform distribution and the values $t_{ij}$ as in the site percolation model. For a given realization of disorder, we divide the lattice sites into two subsets $P$ and $Q$, with $p N^3$ sites in the $P$ subset and $(1-p)N^3$ in the $Q$ subset. For a given value $p$, the lattice sites are assigned to the subsets at random. The hopping amplitudes of the Hamiltonian (\[model\]) are then defined as follows: $$\begin{aligned}
t_{ij} &= \begin{cases}
\frac{\gamma}{|\vec{r}_{ij}|^{\alpha}}, & i \in P \text{ and } j \in P \\
0, & i \in Q \text{ and/or } j \in Q
\end{cases} \label{tijDef}\end{aligned}$$ where $\bm{r}_{ij} = \bm{r}_{i} - \bm{r}_j$ is the distance between sites $i$ and $j$ and $\alpha$ determines the hopping range, with $\alpha = 3$ representing dipolar hopping. The value of $\gamma$ is chosen such that $t_{ij} \equiv \tilde t = 1$ for nearest neighbour (NN) sites. With $t_{ij}$ thus defined, the Hamiltonian (\[model\]) describes a particle with dipolar hopping in a disordered, diluted lattice with $pN^3$ sites. The system parameters of interest are $p$, $w$, and $N$.
![ Time-dependence of the wavepacket size $L$ for a particle initially at the centre site. $L$ is the diameter of a sphere containing no less than 90% of the particle density. The particle has $t_{ij} \propto |{\bm r}_{ij}|^{-3}$ and is in a disordered lattice with $N^3 = 31^3$, $p=0.5$ and $w=5,10, \text{and~} 20$ (green, blue, and magenta, respectively). The horizontal dashed line shows the side-length of the cubic lattice; the grey area highlights sizes smaller than this. The inset shows the same observables, but over shorter time scales. The values of $L$ are averaged over 1050 realizations of disorder. The Heisenberg time for these systems is ${\sim} 300~\tilde t^{-1}$, as can also be seen from the inset. []{data-label="figureWavePackDyn"}](figure1a.pdf "fig:")\
We compute the wavepacket dynamics for a particle placed at the center of the cubic lattice. As described in Ref. [@tianrui], we use the eigenstates of the Hamiltonian (\[model\]) to construct the time propagator and compute the wavefunction of the particle at extremely long times ${>}10^{17} \tilde t^{-1}$, far exceeding the Heisenberg time (typically ${\sim} 300~\tilde t^{-1}$ for the systems considered here). To characterize the spread of the wavepacket in real space, we compute the diameter $L$ of the sphere enclosing at least 90% of the particle density as a function of time. The values of $L$ are averaged over 200 to 1050 realizations of disorder, as required for convergence and as specified in the figure captions. Figure \[figureWavePackDyn\] shows the dependence of $L$ on time for $p = 0.5$ and for different values of $w$. As expected, for systems with low values of $w$, the particle density expands to fill the entire lattice. However, for lattices with $w=10$ and $w=20$, the particle density, on average, does not reach the lattice size and remains localized on a scale smaller than the lattice size.
![ The distributions of the inverse participation ratios of the eigenstates of the Hamiltonian (\[model\]) computed for a point with strong disorder ($w = 17.5$; $p = 0.18$) and a point with weak disorder ($w = 4$; $p = 0.8$; $N^3 = 30^3$). The IPR distributions for the point with strong disorder are computed for three lattice sizes with $N^3=10^3$, $20^3$ and $30^3$, illustrating a slow shift towards the diffusion regime recovered in the infinite size limit (see Figure \[figureScaling\]). The distributions are averaged over 1300 ($N^3=10^3$) and 300 ($N^3=20^3$ and $N^3=30^3$) instances of disorder. []{data-label="figureIPR"}](figure3.pdf)
![ Scaled histogram of $\ln |\psi(r)|$ for $N^3 = 31^3$, $p=0.5$, $w = 80$ computed at time $T = 10^{17} \tilde t^{-1}$. The different colours refer to histograms at different values of $r$. The values of $\lambda$ and $\sigma$ are obtained by scaling the computed histograms using the formula shown on the abscissa. $\lambda$ corresponds to the localization length. The red curve shows $e^{-x^2}$, with $x$ the abscissa. The results are obtained using 3150 realizations of disorder. []{data-label="figureLogNormalFluct"}](figure1b.pdf)
![ The wavefunction spread $L$ for a particle with $t_{ij} \propto |\bm{r}_{ij}|^{-3}$ initially placed in the middle of a 3D lattice with $N^3 = 25^3$ after $10^{17}\tilde{t}^{-1}$. The circles indicate the computed values, averaged over 200 to 600 realizations of disorder, as required for convergence. For points in the black area, $L < N$, indicating the absence of transport. The white area corresponds to $L \geq N$, indicating diffusion to the lattice edges in finite time. The grey area shows the crossover regime, where some values of $(w, p)$ allow diffusion to the lattice edges and some not. The dashed line in the lower panel is the $N^3 = 25^3$ line from Figure \[figureScaling\], determined as described in the text.[]{data-label="figurePhaseDiagram"}](figure2.pdf)
Inverse participation ratios
============================
The results of Figure \[figureWavePackDyn\] illustrate that lattices with large $w$ do not permit diffusion of particles with dipolar hopping over long time scales. The slopes of the curves in Figure \[figureWavePackDyn\] are zero (within the numerical uncertainties), which indicates the absence of diffusion over infinite time. However, a numerical calculation of wavepacket dynamics cannot strictly prove the absence of diffusion at infinite time. In order to confirm localization and gain a better understanding of the eigenstates of the system, we compute the inverse participation ratio (IPR) distribution of the entire set of the eigenstates of the Hamiltonian (\[model\]) for two different dilutions and disorder strengths and at different system sizes.
Figure \[figureIPR\] shows these IPR distributions for a point with weak disorder ($w = 4$; $p = 0.8$; $N^3=30^3$) and strong disorder ($w = 17.5$; $p = 0.18$; $N^3=10^3,20^3,30^3$). For states delocalized over the entire lattice, the IPR tends to $\left(pN^3 \right)^{-1}$. For states localized on a single lattice site, the IPR is one. Figure \[figureIPR\] shows that for weak disorder the IPR is narrowly distributed near zero, meaning that nearly all of the eigenstates are delocalized over most of the lattice. The IPR for strong disorder is markedly different, with all states having a much higher IPR.
To identify the point with strong disorder as in a localized regime, it is important to prove that the distributions have absolutely no states with the IPR close to $I_d = \left(pN^3 \right)^{-1}$. To do this, we analyzed the distributions in Figure \[figureIPR\] for each instance of disorder separately, to compute the minimum value of the IPR, $I_{\rm min}$, which represents the [*most extended*]{} eigenstate. We obtained the following results: of 1300 disorders for $N^3=10^3$, the smallest ratio $I_{\rm min}/I_d = 10.6$; of 300 disorders for $N^3=20^3$, $I_{\rm min}/I_d = 25.0$; of 300 disorders for $N^3=30^3$, $I_{\rm min}/I_d = 20.1$. This means that the [*most extended*]{} eigenstate of many disorder realizations is confined to less than one tenth of all lattice sites, proving that all eigenstates are confined to a volume much smaller than the lattice size. For comparison, the same ratio for the diffusive regime distribution in Figure \[figureIPR\] is $I_{\rm min}/I_d = 1.7$. As the lattice size increases, the IPR distributions shift toward zero. These shifts should collapse the IPR distributions to a single line at zero in the limit of infinite size. However, Figure \[figureIPR\] shows that the approach of the distributions to that in the diffusive regime is very slow, suggesting that a localized regime could be expected to exist even for systems of large size.
Log-normal fluctuations
=======================
While the results of the previous section illustrate that all eigenstates of the strongly disordered Hamiltonians are confined to a fraction of the lattice sites, the IPR values do not provide information on the shape of the wavefunctions. Here, we demonstrate that the wavefunctions are exponentially localized by computing the fluctuations of the particle density between instances of disorder. We also determine the corresponding localization length.
One of the characteristic features of Anderson localization is the log-normal distribution of the particle density across disorder realizations [@kantelhardt_wave_1998]. For example, as shown in section 2 of Ref. [@mueller_disorder_2009], the transmission $T$ through a disordered region in the regime of Anderson localization follows a log-normal distribution, in contrast to the diffusive case where $T$ follows a normal distribution. In the regime of Anderson localization, $\ln T$ thus follows a normal distribution. Since $\ln T$ is proportional to the extinction coefficient $K$, the extinction coefficient also follows a normal distribution. The average of $K$ over instances of disorder is $R / \lambda$ (see Eqn. 21 of Ref. [@mueller_disorder_2009]), where $R$ is the system length and $\lambda$ is the localization length. The typical amplitude of an Anderson-localized wavefunction $|\psi|^2_{\text{Typ}} \propto \exp(-x/\lambda)$, where $x$ is distance. For a specific instance of disorder, an Anderson-localized wavefunction would then be $|\psi|^2 \propto \exp{(-Kx/R)}$. Thus, at a specific value of $x$, the magnitude of the wavefunction will vary log-normally between instances of disorder. This log-normal distribution is equivalent to the exponential decay of the wavefunction in real space.
We plot in Figure \[figureLogNormalFluct\] the histograms of $\ln |\psi(r)|$ for the lattice with $N^3 = 31^3$, $p=0.5$, and $w = 80$. $|\psi(r)|$ is obtained from wavepacket dynamics calculations at time $T = 10^{17} \tilde t^{-1}$. The histograms for different values of $r$ (distance from the site of maximum density) are scaled to lie on the same curve. The scaling is performed using the formula displayed on the abscissa of Figure \[figureLogNormalFluct\]. The parameter $\lambda$ corresponds to the localization length. The values obtained are displayed in Figure \[figureLogNormalFluct\].
Figure \[figureLogNormalFluct\] shows that these scaled histograms all follow a Gaussian distribution, illustrating the [*log-normal*]{} distribution of the particle density vs disorder realizations: $$\begin{aligned}
\nonumber
{{\cal N}(\ln{|\psi|}, r; \sigma, \lambda)} = \frac{1}{\sqrt{\pi \sigma r/\lambda}} \exp{\left [ -\frac{\left (\ln{|\psi(r)| - r/\lambda}\right)^2}{\sigma r/\lambda} \right ]}.\end{aligned}$$ This log-normal behaviour shows that particles with long-range hopping undergo exponential decay in finite-size systems. Exponential decay is one key feature of Anderson localization. Moreover, the data in Figure \[figureLogNormalFluct\] show that the localization length (see $\lambda$ in Figure \[figureLogNormalFluct\]) is much smaller than the side-length of the lattice ($N = 31$).
Diffusion–localization crossover
================================
To understand the dependence of the diffusion/localization properties on the dilution and the on-site disorder, we repeat the wavepacket dynamics calculations of Figure \[figureWavePackDyn\] for different values of $p$ and $w$. The results are presented in Figure \[figurePhaseDiagram\].
Figure \[figurePhaseDiagram\] shows two regimes: the black area corresponds to what would be observed in any experiment as the localization regime; the white area corresponds to the diffusive regime. To obtain these results, we first fix the lattice size and determine by a large number of computations whether a particle initially in the middle of the 3D lattice spreads to fill the entire lattice or forms a localized wavepacket that, after the long time $10^{17} {\tilde t}^{-1}$, stays, on average, within a sphere of size $L$ smaller than the lattice size.
We would like to emphasize that Figure \[figurePhaseDiagram\] is *not* a phase diagram as phase diagrams correspond to the thermodynamic limit and the present results are for a finite-size lattice. Similarly, what we identify is a localization-diffusion *crossover*, not a localization-diffusion *transition*.
The area of Figure \[figurePhaseDiagram\] coloured in black corresponds to disorders where the average size $L$ of the wavepacket is smaller than the lattice size; i.e., where, on average, there is no diffusion to the lattice edges over the long time $10^{17} {\tilde t}^{-1}$. We note that the absence of localization along the horizontal axis of Figure \[figurePhaseDiagram\] is consistent with the recent work of Ref. [@LevyFlights] that shows no localization for a 3D system with dipolar hopping on a diluted lattice and no on-site disorder ($w=0$). Clearly, the off-diagonal disorder caused by the diluted lattice is insufficient to cause localization; diagonal disorder is required. This also illustrates the absence of either a quantum or classical percolation threshold for a system of this size.
![ Lattice-size dependence of the localization-diffusion crossover for particles in a 3D lattice with long-range tunnelling $t_{\ij} \propto |\bm r_{ij}|^{-3}$. The solid line corresponds to the dashed line in Figure \[figurePhaseDiagram\], determined as discussed in the text. []{data-label="figureScaling"}](figure4.pdf)
Lattice-size dependence
=======================
It is important to analyze how Figure \[figurePhaseDiagram\] changes with the size of the lattice. To do this, we combine the results of wavepacket calculations in Figure \[figurePhaseDiagram\] with the scaling analysis of Levitov [@Levitov1999]. From Ref. [@Levitov1999], delocalization occurs through resonances defined by $t_{ij} \ge |w_i - w_j|$. The number of resonances diverges with the system size and delocalization must occur when $$\begin{aligned}
\frac{28 \pi}{3} \kappa \lambda \gg 1,
\label{condition}\end{aligned}$$ where $\lambda = \tilde t p / w$ and $\kappa$ is the index of a concentric sphere with radius $2^\kappa R$. Re-written as $\frac{28 \pi}{3} \kappa \lambda = a$, where $a$ is some large constant, Eq. (\[condition\]) produces a line in the $(p,w)$-phase diagram separating the diffusive and localized regimes: $$\begin{aligned}
w &= p\times \tilde t\left(A + B \ln N\right),
\label{crossover-line}\end{aligned}$$ where we assumed that $2^\kappa R \propto N$ and $A$ and $B$ are unknown constants. To determine $A$ and $B$, we first note that the probability distribution of the site energy differences is $$P(\delta) = \frac{2}{w}\left(1-\frac{\delta}{w}\right), \mbox{where }\delta = |w_i - w_j| \in [0,w],$$ and calculate the probability of resonance between sites $i$ and $j$ as $$\begin{aligned}
P( p, \gamma, w, r_{ij}) &= p P( \delta \leq t_{ij}) &= \frac{2 p \gamma }{w r^3_{ij}} - \frac{p \gamma^2}{w^2 r^6_{ij}}.
\label{probability-resonance}\end{aligned}$$ Using Eq. (\[probability-resonance\]), we calculate the probability $\tilde P$ that the central site has [*no*]{} resonance in the entire system of a given size and draw the isoprobability lines for $N^3=25^3$ on the diagram from Figure \[figurePhaseDiagram\]. The isoprobability line with $\tilde P = 1.2 \times 10^{-2}$ goes through the localization-diffusion crossover diagram inside the grey crossover regime, as shown in Figure \[figurePhaseDiagram\]. Using this value of $\tilde P$ as characterizing the diffusion-localization crossover, we obtain the crossover lines for $N^3 = 21^3; 31^3; 41^3; 101^3$ and fit Eq. (\[crossover-line\]) to the resulting curves to obtain $A$ and $B$. To verify the procedure, we compute the isoprobability lines for $N^3=301^3$ and $N^3=1001^3$, and obtain a perfect agreement with the values from Eq. (\[crossover-line\]). Figure \[figureScaling\] shows the crossover lines for different lattice sizes. Remarkably, Figure \[figureScaling\] illustrates that the localization crossover may occur at finite values of $w$ for systems with a macroscopically large size. Consider, for example, the case with $N^3=\left(10^{8}\right)^3$, which corresponds to a 3D lattice with $10^{24}$ sites, which exceeds Avogadro’s number. In the infinite size limit, the crossover line tends to the $y$-axis, which corresponds to the absence of localization and is consistent with prior work [@anderson; @Levitov1989; @Levitov1990; @Levitov1990b; @Levitov1999].
Experimental Relevance and Applications
=======================================
Our results illustrate that the detection of the crossover for particles with dipolar hopping in 3D lattices is within reach of current experiments. This can be done with ions in rf-traps, ultracold magnetic atoms, ultracold Rydberg atoms, or ultracold molecules trapped in optical lattices. An optical lattice typically contains $\approx 60$ sites per dimension. Localization can be observed with internal excitations of the trapped atoms or molecules [@dy-1; @jun-ye-0; @jun-ye-1; @jun-ye-2; @jun-ye-nature]. If an optical lattice is partially populated with atoms or molecules, the empty lattice sites serve as impurities that scatter the excitations. The parameter $p$ in the diagram shown in Figure \[figurePhaseDiagram\] can thus be controlled by simply varying the number of atoms or molecules loaded in the optical lattice. The disorder in $w_i$ can be applied using an optical speckle potential, as in [@atoms-1; @atoms-2; @atoms-3; @atoms-4]. For example, Figure \[figureScaling\] shows that for molecules on an optical lattice with $N \approx 60$ and a lattice population of 30 % [@jun-ye-0], the crossover can be observed by varying the optical speckle potential from below to above $w = 5$.
Our results have, at least, three major applications. First, they can be used for the design of miniaturized quantum devices relying on the transport of small excitons that move via dipolar hopping. Our results show how much disorder can be tolerated in such devices and what is the minimum length scale of such devices that allows high exciton mobility. Second, our results show how one can engineer exciton transport on small length scales by doping molecular crystals with impurities. This is relevant for bulk heterojunction organic solar cells and similar light-harvesting devices that rely on the high mobility of excitons [@BHJ-park; @BHJ-liang]. Third, our results are critical for the design of experiments aimed at detection of interaction-induced many-body localization (MBL). MBL is predicted to hinder the return of a perturbed quantum system to equilibrium, thus precluding thermalization, conductivity, density equilibration or energy transport. It is the subject of numerous theoretical studies, many of which propose experiments for the observation of the localized phase. Of particular interest is the localization phase transition in 3D dipolar systems, where excitations/spins hop between carrier particles due to dipolar interactions. A recent startling discovery [@long-range-effects] shows that the localized phase can be induced by interactions between the particles. In order for such a phase transition to be observable, the system must be in the diffusive regime in the limit of non-interacting particles. Our work shows that this is not guaranteed and provides the conditions required to perform these experiments.
Conclusion
==========
Particles with long-range hopping are known to be delocalized in lattices of infinite size. Here, we have presented evidence that particles with dipolar long-range hopping may undergo exponential localization in 3D disordered lattices of finite size. Our conclusions are based on three independent but mutually consistent results:
- First, we have demonstrated by rigorous quantum dynamics calculations that particles placed in an individual lattice site cannot diffuse through lattices of finite size over long time scales. By itself, this is not sufficient proof of localization as this result does not rule out the possibility that particles can fill the entire lattice at infinite time.
- To further confirm the presence of localization, we have calculated the inverse participation ratio for each eigenstate of the Hamiltonian. Our results show that no eigenstate occupies more than 10 % of the lattice for lattices of sufficiently small finite size. Our results also show how the participation ratio changes with increasing system size towards a system with delocalized states. One may argue that this is not sufficient proof of localization as it does not rule out the possibility of states that occupy a small number of lattice sites that are spread throughout the lattice (for example, every tenth site). However, the presence of such states would allow for significant transport through the lattice, which we do not observe in (i).
The results in (i) and (ii) thus provide strong evidence for localization in finite-size lattices. However, the shape of the wavefunctions is still not clear. To better understand this,
- we have computed the fluctuations of the wavefunctions between instances of disorder. Our results illustrate that the wavefunction fluctuations follow a log-normal distribution, which corresponds to exponential localization of the wavefunctions.
Our results in Figure \[figurePhaseDiagram\] illustrate the crossover between what would be observed in any experiment as the localization regime and the diffusion regime. We have demonstrated that introducing diagonal disorder can bring the system to the localization regime, even if introducing dilution alone cannot [@LevyFlights]. Furthermore, introducing diagonal disorder alone is sufficient to observe the localization regime. The crossover is shown to be within reach of current experiments.
We would like to emphasize that the absence of localization with dilution alone (x-axis of Figure \[figurePhaseDiagram\]) shows that there is no classical or quantum percolation threshold for a system of this size. The lack of classical localization means that any localization induced by diagonal disorder must be due to quantum interference effects. We have combined our wavepacket calculations in Figure \[figurePhaseDiagram\] with the scaling theory from Refs. [@Levitov1989; @Levitov1990; @Levitov1990b; @Levitov1999] to demonstrate how the crossover changes with the system size (Figure 5). Surprisingly, we find that one should expect to observe the absence of quantum transport even in systems with macroscopically finite size, providing the on-site disorder and the dilution are significant enough. These results are of practical interest as they can be used to inform the design of minituarized quantum devices, which always have finite size.
Acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge useful discussions with Andreas Buchleitner. This work is supported by NSERC of Canada.
[99]{}
E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, *Phys. Rev. Lett.* [**42**]{}, 673–676 (1979).
L. J. Root and J. L. Skinner, *J. Chem. Phys.* [**89**]{}, 3279–3284 (1988).
P. W. Anderson, *Phys. Rev.* [**109**]{}, 1492–1505 (1958).
L. S. Levitov, *Europhys. Lett.* [**9**]{}, 83 (1989).
L. S. Levitov, *Ann. N. Y. Acad. Sci.* [**581**]{}, 95–101 (1990).
L. S. Levitov, *Phys. Rev. Lett.* [**64**]{}, 547–550 (1990).
L. S. Levitov, *Ann. Phys. (Leipzig)* [**8**]{}, 697–706 (1999).
T. Xu and R. V. Krems, [*New J. Phys.*]{} [**17**]{}, 065014 (2015).
J. W. Kantelhardt and A. Bunde, [*Ann. Phys. (Leipzig)*]{} [**7**]{}, 400–405 (1998).
C. A. Müller and D. Delande, [*Disorder and interference: localization phenomena*]{} in [*Les Houches 2009 - Session XCI: Ultracold Gases and Quantum Information*]{}, edited by C. Miniatura, L.-C. Kwek, M. Ducloy, B. Gremaud, B.-G. Englert, L. Cugliandolo, A. Ekert, K. K. Phua (Oxford University Press, Oxford 2011).
X. Deng, B. L. Altshuler, G. V. Shlyapnikov, and L. Santos, [*Phys. Rev. Lett.*]{} [**117**]{} 020401 (2016).
A. Frisch, M. Mark, K. Aikawa, S. Baier, R. Grimm, A. Petrov, S. Kotochigova, G. Quéméner, M. Lepers, O. Dulieu, and F. Ferlaino, .
K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye, [*Science*]{} [**322**]{}, 231 (2008).
M. H. G. de Miranda, A. Chotia, B. Neyenhuis, D. Wang, G. Quéméner, S. Ospelkaus, J. L. Bohn, J. Ye, and D. S. Jin, [*Nat. Phys.*]{} [**7**]{} 502 (2011).
A. Chotia, B. Neyenhuis, S. A. Moses, B. Yan, J. P. Covey, M. Foss-Feig, A. M. Rey, D. S. Jin, and J. Ye, [*Phys. Rev. Lett.*]{} [**108**]{} 080405 (2012).
B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. A. Hazzard, A. M. Rey, D. S. Jin and J. Ye, [*Nature*]{} [**501**]{}, 521 (2013).
D. Clément, A. F. Varón, J. A. Retter, L. Sanchez-Palencia, A. Aspect, and P. Bouyer, [*New J. Phys.*]{} [**8**]{}, 165 (2006).
J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clément, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, [*Nature*]{} [**453**]{}, 891 (2008).
L. Fallani, J. E. Lye, V. Guarrera, C. Fort, and M. Inguscio, [*Phys. Rev. Lett.*]{} [**98**]{}, 130404 (2007).
G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, [*Nature*]{} [**453**]{}, 895 (2008).
D. E. Logan and P. G. Wolynes, .
Y. Liang, Z. Xu, J. Xia, S.-T. Tsai, Y. Wu, G. Li, C. Ray, and L. Yu, *Adv. Mater.* [**22**]{}, E135-E138 (2010).
S. H. Park, A. Roy, S. Beaupré, S. Cho, N. Coates, J. S. Moon, D. Moses, M. Leclerc, K. Lee, and A. J. Heeger. *Nat. Photon.* [**3**]{}, 297-302 (2009).
N. Y. Yao, C. R. Laumann, S. Gopalakrishnan, M. Knap, M. Müller, E. A. Demler, and M. D. Lukin, .
|
---
abstract: '[ If $A$ is a graded connected algebra then we define a new invariant, polydepth$\, A$, which is finite if $\mbox{Ext}_A^*(M,A) \neq 0$ for some $A$-module $M$ of at most polynomial growth. [**Theorem 1**]{}: If $f : X \to Y$ is a continuous map of finite category, and if the orbits of $H_*(\Omega Y)$ acting in the homology of the homotopy fibre grow at most polynomially, then $H_*(\Omega Y)$ has finite polydepth. [**Theorem 5**]{}: If $L$ is a graded Lie algebra and polydepth $UL$ is finite then either $L$ is solvable and $UL$ grows at most polynomially or else for some integer $d$ and all $r$, $\sum_{i=k+1}^{k+d} \mbox{dim}\, L_i \geq k^r$, $k\geq$ some $k(r)$.]{}'
author:
- 'Y. Felix, S. Halperin and J.-C. Thomas'
title: Graded Lie algebras with finite polydepth
---
[**AMS Classification**]{} : 55P35, 55P62, 17B55
[**Key words**]{} : Graded Lie algebras, loop space homology, depth, solvable Lie algebras.
We work over a field $\bk$ of characteristic different from 2. If $V = \{ V_k\}$ is a graded vector space we denote by $V^{\#} = \{\mbox{Hom}_{\bk}
(V_k,\bk )\}$ the dual graded vector space. A graded Lie algebra is a graded vector space $L$, equipped with a bilinear map $[\, ,\, ]$ $: L_i \times
L_j \to L_{i+j}$ satisfying $$[x,y] + (-1)^{ij} [y,x] = 0$$ and $$[x,[y,z]] = [[x,y],z] + (-1)^{ij}[y,[x,z]]$$ for $ x\in L_i$, $y\in L_j$, $z\in L$. It follows that $3[x,[x,x]] = 0$ for $x$ of odd degree, and so if char $k=3$ we further require that $[x,[x,x]] = 0$. Finally we consider only graded Lie algebras satisfying $L=\{\, L_i\,\}_{i\geq 1}$ and each $L_i$ is finite dimensional. (Any graded vector space $V$ with each $V_i$ finite dimensional is said to have [*finite type*]{}.)
The universal enveloping algebra of $L$ is denoted by $UL$ and it satisfies the classical Poincaré Birkhoff Witt Theorem (in characteristic $3$ this uses the $[x,[x,x]]=0$ requirement).
Important examples appear in topology. Let $X$ be a simply connected topological space with rational homology of finite type. Then the rational homotopy Lie algebra $L_X$ of $X$ is defined by $$L_X = \pi_*(\Omega X)\otimes {\mathbb Q}\,; \hspace{1cm} [\, ,\,] = \mbox{ Samelson
product}\,,$$ and the Hurewicz map extends to an isomorphism, [@MM], $$UL_X \stackrel{\cong }{\longrightarrow} H_*(\Omega X; {\mathbb Q})\,.$$ Analogously, if $X$ is a finite $n$-dimensional and $r$-connected CW complex, then for primes $p>n/r$, $H_*(\Omega X;F_p) = UE$ for some graded Lie algebra $E$ [@Hal].
If $M$ is a module over a (graded) algebra $A$ then the [*grade*]{} of $M$, grade$\,
M$, is the least integer $q$ (or $\infty$) such that ${\rm Ext}_A^q(M,A)\neq 0$. And if $V = \{\, V_i\,\}_{i\geq 0}$ is a graded vector space then $V$ has [*at most polynomial growth*]{} if for some constant $C$, and some non-negative integer, $d$, $\sum_{i\leq n} {\rm dim}\, V_i \leq Cn^d$, $n\geq 1$. In this case the least such $d$ is called the [*polynomial bound* ]{} for the growth of $V$ and is denoted by ${\rm polybd}\, (V)$. If $V$ does not have at most polynomial growth we put ${ \rm polybd}\, V = \infty$ and we say that $V$ grows faster than any polynomial.
In this paper we combine these two notions in the
[**Definition:**]{} the [*polygrade*]{} of an $A$-module, $M$, is the sum, ${\rm grade}\,M + {\rm polybd}\, M$, and the [*polydepth of $A$*]{} is the least integer (or $\infty$) occurring as the polygrade of an $A$-module.
In the case $A=UL$ the unique augmentation $UL \to \bk$ makes $\bk$ into a $UL$-module, and by definition, the grade of $\bk$ is the [*depth*]{} of $UL$. Since ${\rm polybd}\,\bk=0$ it follows that: $$\mbox{polydepth}\, UL \leq \mbox{depth}\, UL
\eqno{(1)}$$ Moreover (cf. Proposition 2) if ${\rm dim }\,L<\infty$ then equality holds. We shall abuse notation and refer to these invariants respectively as polydepth$\, L$ and depth$L$.
Note that ${\rm Ext}_{UL}^0(UL,UL)$ contains the identity map and so $${\rm polydepth}\, L \leq {\rm polybd}\, UL \,. \eqno{(2)}$$
Observe as well that for any graded vector space $M$, $\mbox{polybd}\, M = 0$ if and only if dim$\, M$ is finite. Thus polydepth$\, L = 0$ if and only if depth$\, L = 0$, which happens if and only if $L$ is finite dimensional and concentrated in odd degrees.
Depth has been a useful concept in topology because, on the one hand, Lusternik-Schnirelmann category satisfies [@modp] $${\rm depth}\, L_X \leq {\rm cat}\, X$$ and, on the other hand ([@LAPG], [@HAPG], [@Engel]), finite depth has important implications for the structure of a graded Lie algebra.
The purpose of this paper is to show that essentially the same implications follow from the weaker hypothesis that polydepth$\, L$ is finite, while simultaneously identifying a larger class of topological spaces and Lie algebras for which the weaker hypothesis holds.
Indeed, we have
[**Theorem 1.**]{} [*If $F\to X \stackrel{f}{\longrightarrow}Y$ is a fibration of path-connected spaces, then $${\rm polydepth}\, H_*(\Omega Y) \leq {\rm polybd} \, H_*(F) + {\rm cat}\, f\,.$$*]{}
[**proof:**]{} The fibration determines an action up to homotopy of $\Omega Y$ on $F$, which makes $H_*(F)$ into an $H_*(\Omega Y)$-module. According to [@grade], grade$\, H_*(F) \leq $ cat$\, f$. $\Box$
Our main structural theorems read:
[**Theorem 2.**]{} [*Let $E(L)$ denote the linear span of elements $x\in L_{\rm \scriptstyle even}$ such that ${\rm ad} x$ acts nilpotently on each $y \in L$. Then $${\rm dim} \, E(L) \, \leq \, {\rm polydepth}\, L\,.$$*]{}
[**Theorem 3.**]{}
*The following conditions on a graded Lie algebra $L$ are equivalent*
1. $L$ is the union of solvable ideals and ${\rm polydepth}\, L$ is finite;
2. $UL$ grows at most polynomially (${\rm polybd}\, UL$ is finite);
3. $L_{\rm \scriptstyle even}$ is finite dimensional, and for some constant $C$ $$\sum_{i\leq n} {\rm dim}\, L_i \leq C \log_2 n\,, \hspace{1cm} n\geq 1\,.$$
In this case $L$ is solvable.
[**Theorem 4.**]{} [*If $L$ is a graded Lie algebra of finite polydepth then the union of the solvable ideals of $L$ is a solvable ideal of finite polydepth.*]{}
[**Theorem 5.**]{} [*Suppose ${\rm polydepth }\, L$ is finite and $L$ is not solvable. Then there is an integer $d$ such that for all $r\geq 1$: $$\sum_{i=k+1}^{k+d}{\rm dim}\, L_i \geq k^r\,, \hspace{1cm} k \geq \mbox{\rm some}
\, k(r) \,.$$*]{}
[**Remark**]{}. In [@growth] it is shown that if $L = L_X$ where $X$ is a finite 1-connected CW complex, then we may take $d=\mbox{dim}\, X$ in Theorem 5.
Properties of polydepth
=======================
[**Lemma 1:**]{} [*If $M$ is a module for some graded algebra $A$ of finite type and if ${\rm Ext}_A^q(M,A)\neq 0$ then ${\rm Ext}_A^q(A\cdot x,A) \neq 0$ for some $x$ in a subquotient module of $M$.*]{}
[**proof:**]{} Recall that $A^{\#} = Hom_{\bk}(A,\bk)$. Then ${\rm Ext}_A^q(M,A)$ is the dual of ${\rm Tor}_q^A(M,A^{\#})$ and a direct limit argument shows that for some $x_1, \ldots ,x_n \in M$, $${\rm Tor}_q^A(A\cdot x_1 + \ldots +A\cdot x_n,A^{\#}) \neq 0\,.$$ Now use the exact sequence associated to the inclusion $A\cdot x_1 + \ldots + A\cdot x_{n-1}$ in $A\cdot x_1 + \ldots + A\cdot x_n$. $\Box$
[**Corollary:**]{} Polydepth$\, A$ is the least $m$ such that polygrade$\, N = m$ for some monogenic $A$-module $N$.
[**Remark:**]{} It follows from the Corollary that we may improve Theorem 1 to the inequality $${\rm polydepth}\, H_*(\Omega Y) \leq {\rm polybd}\,
(H_*(\Omega Y)\cdot \alpha ) + {\rm cat}\, f\,, \hspace{5mm} {\rm some}\, \alpha \in H_*(F).
\eqno{(3)}$$
[**Proposition 1.**]{}
*Let $L$ be a graded Lie algebra.*
1. Each ideal satisfies ${\rm polydepth}\, I \leq {\rm polydepth}\, L\,.$
2. Let $E$ be a Lie subalgebra of $L$. If $L$ has finite polydepth and if for each $x \in L/E$ the orbit $UE\cdot x$ has at most polynomial growth, then $E$ has finite polydepth.
3. For $n$ sufficiently large the sub Lie algebra $E$ generated by $L_{\leq n}$ satisfies ${\rm polydepth }\, E \leq {\rm polydepth}\, L$.
[**proof:**]{} (i) This follows from the Hochschild-Serre spectral sequence, converging from ${\rm Ext}_{UL/I}^p(\bk,{\rm Ext}_{UI}^q(M,UL))$ to ${\rm Ext}_{UL}^{p+q}(M,UL)$. (Note that since $UL$ is $UI$-free, grade$_{UI}(M)$ is the least $q$ such that ${\rm Ext}_{UI}^q(M,UL)\neq 0$.)
\(ii) As in Lemma 1, ${\rm Ext}_{UL}^q(M,UL)$ is dual to ${\rm Tor}_q^{UL}(M,(UL)^{\#})$, and this is the homology of the Cartan-Eilenberg-Chevalley complex $\land sL\otimes M \otimes (UL)^{\#}$. Write $L= E \oplus V$ and set $F_p =\land sE \otimes \land^{\leq p}sV\otimes M \otimes (UL)^{\#}$. This filtration determines a convergent spectral sequence, introduced by Koszul in [@Ko], and which is the Hochschild-Serre spectral sequence when $E$ is an ideal. The $E^1$-term of the spectral sequence is ${\rm Tor}_q^{UE}(\land^p sL/E \otimes M, (UL)^{\#})$, converging to ${\rm Tor}_{p+q}^{UL}(M,(UL)^{\#})$.
Each element $z\in \land^psL/E\otimes M$ is contained in a finite sum of $UE$-modules of the form $s(UE\cdot x_1) \land \cdots \land s(UE\cdot x_p) \otimes M$ and it follows that $${\rm polybd}\, (UE\cdot z) \leq p\cdot {\rm polybd}\, (UE\cdot x) + {\rm polybd}\, (M)$$ for some $x\in L/E$. Choose $M$ so that polydepth$\, L= $ polygrade$\, M$ and apply Lemma 1 with $p+q = $ grade$\, M$.
\(iii) If ${\rm Ext}_{UL}^p(M,UL)$ is non-zero and ${\rm polybd}\, (M) <\infty$ it suffices to choose $E$ so that the restriction ${\rm Ext}_{UL}^p(M,UL) \to {\rm Ext}_{UE}^p(M,UL)$ is non zero. $\Box$
[**Corollary 1.**]{} (of the proof of (ii)). [*Suppose for some $k\geq 1$ that ${\rm polybd}\,(UE\cdot x)
\leq k$, $x\in L/E$. Then ${\rm polydepth}\, E \leq k\, {\rm polydepth}\, L$.*]{}
[**Corollary 2.**]{} [*Let $E$ be a sub-Lie algebra of a graded Lie algebra $L$. If $L$ has finite polydepth and $L/E$ has at most polynomial growth, then $E$ has finite polydepth.*]{}
[**Example 1.**]{} Let ${\mathbb L}(V)$ be the free Lie algebra on a graded vector space $V$. Then for any graded Lie algebra $L$, $L\coprod {\mathbb
L}(V)$ has depth 1. Thus the injection $L \to
L\coprod L(V)$ shows that each graded Lie algebra is a sub-Lie algebra of a Lie algebra of finite polydepth. The previous corollary gives restriction on a Lie algebra $L$ for being a sub-Lie algebra of a Lie algebra of finite polydepth, $K$, when the quotient has at most polynomial growth.
[**Proposition 2.**]{} [*If $L$ is a finite dimensional graded Lie algebra then $${\rm polydepth}\, L = {\rm depth}\, L\,.$$* ]{}
[**proof:**]{} As observed in the introduction, ${\rm polydepth}\,
L \leq {\rm depth}\, L$. On the other hand, by Lemma 1, ${\rm polydepth}\, L = {\rm polygrade}\,
M$ for some monogenic module $M = UL\cdot x$. Now Theorem 3.1 in [@LAPG] asserts that ${\rm polygrade}\, M = {\rm depth}\, L$. $\Box$
Proof of Theorem 2
==================
Suppose $I\subset L$ is an ideal. If ${\rm Ext}_{UL}^m(M,UL) \neq 0$, then ${\rm Ext}_{UL/I}^p({\rm Tor}_q^{UI}(M,k),UL/I)\neq
0\,,$ some $p+q =m$. (Same proof as in: [@LAPG], Lemma 4.3, for the case $M=\bk$). By Lemma 1 there is a monogenic $UL/I$-module $N$ such that $N$ is a subquotient of $\mbox{Tor}_q^{UI}(M,\bk)$, and grade $N\leq p$.
Now suppose $L/I$ is finite dimensional. Then Theorem 3.1 in [@LAPG] asserts that $${\rm grade}\,N + {\rm polybd}\, N = {\rm dim}\, (L/I)_{\rm \scriptstyle even}\,.$$ On the other hand, write $(L/I)_{\rm \scriptstyle even} = V \oplus W$ where $V$ is the image of $E(L)$. Let $x_i\in L_{\scriptsize even}$, $y_j\in L_{\scriptsize odd}$ and $z_k\in E(L)$ represent respectively bases of $W$, $(L/I)_{\scriptsize odd}$ and $V$. Then the elements $x_1^{k_1}\cdots x_s^{k_s}y_1^{\varepsilon_1}
\cdots y_t^{\varepsilon_t}z_1^{m_1}\cdots z_u^{m_u}$, where $\varepsilon_i = 0$ or $1$, represent a basis for $UL/I$. Choose the $z_k$ to act locally nilpotently in $L$. Then this basis applied to any $\omega \in \land^qsI$, shows that ${\rm polybd}\,(UL/I)\cdot \omega \leq {\rm dim} W$. Hence if $u \in \land^qsL\otimes M$ represents a generator of $N$ then $${\rm polybd}\, N \leq {\rm polybd}\, (UL/I\cdot u)
\leq {\rm polybd}\, M+ {\rm dim} W\,.$$ Substitution in the equation above gives $$\renewcommand{\arraystretch}{1.5}
\begin{array}{ll}
{\rm dim}\,(L/I)_{\rm \scriptstyle even} & \leq {\rm grade}\, N +{\rm polybd}\, M+ {\rm dim}
\, W \\
& \leq {\rm grade}\, M + {\rm polybd}\, M + {\rm dim} \, W\,.
\end{array}
\renewcommand{\arraystretch}{1}$$ Choose $M$ so grade$\, M + {\rm polybd}\, (M) = {\rm polydepth}\, L$ and choose $I = L_{> 2k}$. Then $V \cong E(L)_{\leq 2k}$ and we have $${\rm dim}\, E(L)_{\leq 2k} \leq {\rm polydepth}\, L\,.$$ Since this holds for all $k$ the theorem is proved. $\Box$
Solvable Lie algebras
=====================
[**Lemma 2.**]{} [*Let $L$ be a Lie algebra concentrated in odd degrees. Then ${\rm Ext}_{UL}(-,UL) = {\rm Hom}_{UL}(-,UL)$. In particular $${\rm polydepth }\, L = {\rm polybd}\, UL\,.$$*]{}
[**proof:**]{} Since $L = L_{\rm \scriptstyle odd}$ it is necessarily abelian. Now ${\rm Ext}_{UL}(-,UL)$ is the dual of ${\rm Tor}^{UL}(-,(UL)^{\#})$ and this is the limit of ${\rm Tor}^{UL_{\leq n}}(-,(UL)^{\#})$, which dualizes to ${\rm Ext}_{UL_{\leq n}}(-,UL)$. Since $UL_{\leq n}$ is a finite dimensional exterior algebra and $UL$ is $UL_{\leq n}$-free it follows that ${\rm Ext}^+_{UL_{\leq n}}(-,UL) =
0$, and so $\mbox{Ext}^+_{UL}(-, UL) = 0$.
Finally, since ${\rm Ext}^0_{UL}(UL,UL)$ is non-zero, ${\rm polydepth}\, L \leq {\rm polybd}\,
UL$. On the other hand if ${\rm polydepth}\, L = m<\infty$, then for some $M$, we have ${\rm Ext}_{UL}^p(M,UL)\neq 0$ and ${\rm polybd}\, M = m-p$. By the above, $p=0$ and so there is a non zero $UL$-linear map $f : M \to UL$. Any $f(m)$ is in some $UL_{<n}$ and if $f(m)\neq 0$ it follows that $UL_{\geq n} \stackrel{\cong}{\longrightarrow} UL_{\geq n}\cdot m$. This implies ${\rm polybd}\, M
\geq {\rm polybd}\, UL$ and ${\rm polydepth}\, L \geq {\rm polybd}\, UL$. $\Box$
[**Lemma 3.**]{} [*Let $L$ be a graded Lie algebra of finite polydepth. If $I$ is an ideal in $L$ and ${\rm polybd} \, I <\infty$ then ${\rm polydepth}\, L/I < \infty$.*]{}
[**proof:**]{} Choose $M$ so that ${\rm polygrade}\, M = {\rm polydepth}\, L$. If $m = {\rm grade}\, M$ then it follows (as in [@LAPG], proof of Theorem 4.1 for the case $M=\bk$) that for some $p$, $${ \rm Ext}_{UL/I}^p({\rm Tor}_{m-p}^{UI}(M,\bk),UL/I) \neq 0\,.$$ Since ${\rm Tor}_{m-p}^{UI}(M,\bk)$ is a subquotient of $\land^{m-p}sI\otimes M$ it follows that it has polynomial growth at most equal to $(m-p) {\rm polybd}\, I $ $\Box$
[**proof of Theorem 3:**]{}
\(i) $\Rightarrow$ (ii). Let $I$ be the sum of the ideals in $L$ concentrated in odd degrees. Then $I$ is an ideal of this form, necessarily abelian, and $L/I$ has no ideals concentrated in odd degrees. Moreover ${\rm polybd}\, UI ={\rm polydepth}\, I \leq {\rm polydepth}\, L$ (Lemma 2 and Proposition 1) and hence ${\rm polydepth}\, L/I < \infty$ (Lemma 3).
Next we show that every solvable ideal $J$ in $L/I$ is finite dimensional, by induction on the solvlength. Indeed, if $J$ is abelian then $J_{\rm \scriptstyle even} = E(J)$. Since ${\rm polydepth}\, J \leq {\rm polydepth}\, L/I$ (Proposition 1), Theorem 2 asserts that $J_{\mbox{\scriptsize even}}$ is finite dimensional. Thus for some $r$, $J_{\geq r}$ is an ideal concentrated in odd degrees; i.e. $J_{\geq r} = 0$.
Now if $J$ has solvlength $k$ then its $(k+1)$st derived algebra is abelian and so finite dimensional. Thus for some $r$, $J_{\geq r}$ has solvlength $k-1$. By induction, $J$ is finite dimensional.
By hypothesis $L/I$ is the sum of its solvable ideals. Since these are finite dimensional, each $x\in (L/I)_{\rm \scriptstyle even}$ acts locally nilpotently. Thus $(L/I)_{\rm \scriptstyle even} = E(L/I)$, and this is finite dimensional by Theorem 2. But $L_{\rm \scriptstyle even} \cong (L/I)_{\rm \scriptstyle even}$ since $I$ is concentrated in odd degrees.
Suppose $L_{\rm \scriptstyle even} \subset L_{\leq 2n}$. Since $L_{>2n}$ is an ideal in odd degrees of finite polydepth, ${\rm polybd}\, UL_{>n} <\infty$, while trivially ${\rm polybd}\, UL/L_{>2n}<\infty$. Hence ${\rm polybd}\, UL < \infty$.
\(ii) $\Rightarrow$ (iii). Clearly ${\rm polybd}\, UL \geq {\rm dim}
L_{\rm \scriptstyle even}$, so the latter must be finite. It is trivial from the Poincaré Birkhoff Witt theorem that if $\sum_{i\leq n}{\rm dim}\, L_i = d(n)$ then $$\sum_{i\leq nd(n)} {\rm dim}\, (UL)_i \geq 2^{d(n)}\,.$$ Thus $2^{d(n)} \leq K{[nd(n)]^r} $ for some constant $K$ and some integer $r$, $r\geq 1$. It follows that $d(n) \leq \log_2K + r\log_2n+r\log_2d(n) \leq r\log_2n + \frac{1}{2} d(n)$, $n$ sufficiently large.
\(iii) $ \Rightarrow$ (i). Choose $N$ so that $I = L_{\geq N}$ is concentrated in odd degrees. Then $UI$ is an exterior algebra and so $$\sum_{i\leq n} {\rm dim}\, (UL)_i \leq 2^{ \sum_{i\leq n} {\rm dim}\, L_i} \leq n^C$$ for some constant $C$. Thus, since $L/I$ is finite dimensional, ${\rm polybd}\, UL$ is finite. The identity of $UL$ is in ${\rm Ext}^0_{UL}(UL,UL)$ and so ${\rm polydepth}\, L \leq {\rm polybd}\, UL$.
Finally, since $I$ is abelian and $L/I$ is finite dimensional, $L$ itself is solvable. This also proves the last assertion. $\Box$
[**proof of Theorem 4:**]{} This is immediate from Proposition 1(i) and Theorem 3. $\Box$
[**Proposition 3.**]{}
*Suppose $I$ is a solvable ideal in a Lie algebra $L$ of finite polydepth. Then*
1. ${\rm polydepth}\, L/I \leq {\rm polydepth}\, L$
2. ${\rm dim} I_{\rm \scriptstyle even} \leq {\rm polydepth}\, I \leq {\rm polybd}\, UI$.
[**proof:**]{} (i) As noted in the proof of Lemma 3, $${\rm Ext}^p_{UL/I}({\rm Tor}_{m-p}^{UI}(M,\bk),UL/I)\neq 0\,,$$ where $m + {\rm polybd}\, M = {\rm polydepth} \, L$. Also the Tor is a subquotient of $\land^{m-p}sI\otimes M$. By Theorem 3 $$\sum_{i\leq n} {\rm dim}\, (\land^{m-p}sI\otimes M)_i \leq \left( C_1\log_2 n \right)^{m-p}\cdot
C_2\cdot
n^{{\rm polybd}\,
M}\,.$$ Hence $\mbox{polybd}\, \mbox{Tor}_{m-p}^{UI}(M,\bk) \leq
\mbox{polybd}\, M + 1$, and so $\mbox{polydepth}\, L/I \leq
\mbox{polydepth}\, M + p + 1$.
If $p<m$ then this gives $\mbox{polydepth}\, L/I \leq
{\mbox{polydepth}}\, L$. If $p=m$ then $\mbox{Tor}_{m-p}^{UI}(M,\bk) = M \otimes_{UI}\bk$. Hence in this case $\mbox{polybd}\, \mbox{Tor}_{m-p}^{UI}(M,\bk ) \leq
\mbox{polybd}\, M$ and again $\mbox{polydepth}\, L/I \leq
\mbox{polydepth}\, L$.
\(ii) Since $I_{\rm \scriptstyle even} \subset I_{\leq 2n}$, some $n$ (Theorem 3) we may apply the first assertion to obtain $$\begin{array}{ll}
{\rm dim} \, I_{\rm \scriptstyle even} & = {\rm depth}\, I/I_{>2n}
\,\,\,{\rm (cf. \cite{FHJLT})}\\
&={\rm polydepth}\, I/I_{>2n}\,\,\, {\rm (Proposition \, 2)}\\
& \leq {\rm polydepth} I\,.
\end{array}$$ The second inequality has already been observed: ${\rm polybd} \, UI = {\rm polygrade}\,
UI \geq {\rm polydepth}\, I$.
$\Box$
[**Example 3.**]{} Consider a Lie algebra $L$ concentrated in odd degrees with a basis $\{\,x_i\,\, i\geq 1\,\}$ satisfying the degree relations $$\mbox{deg} x_i > \sum_{j<i} \mbox{deg} x_j \,.$$ Then for each $n$, ${\rm dim }(U L)_n \leq 1$. The identity on $UL$ shows that $\mbox{polydepth}\, L = 1$.
[**Example 4.**]{} Consider the graded Lie algebra $L = {\mathbb L}(a,x_n)_{n\geq 2}/I$, with $\mbox{deg} a = 2$, $
\mbox{deg} x_n = 2^n+1$, and where $I$ is generated by the relations $$[({\rm ad} a)^k x_r,({\rm ad} a)^lx_s]=0, \hspace{3mm} k,l\geq 0, r,s\geq 2\,, \hspace{5mm}{\rm and} \hspace{5mm} {\rm ad}^{n+1}(a)(x_n) = 0\,.$$ Then $\mbox{polybd}\, UL = 2$, so that $L$ has finite polygrade. On the other hand, $L$ is solvable but not nilpotent, and is the union of the infinite sequence of the finite dimensional Lie algebras $I_N$ generated by $a, x_2, \ldots, x_N$.
[**Proposition 4.**]{} [*Let $L $ be the direct sum of non-solvable Lie algebras $L(i)$, $i\leq n$. If polydepth$\, L(i) < \infty$ for $1\leq i\leq n$, then $$n \leq {\rm polydepth}\, L \leq \sum_i{\rm polydepth }\,
L(i)\,.$$*]{}
[**proof.**]{} We first prove by induction on $n$ that for any $UL$-module $M$ that has at most polynomial growth, we have ${\rm Ext}_{UL}^{<n}(M,UL) = 0$. Consider the Hochschild-Serre spectral sequence $${\rm Ext}_{UL(1)}^p({\rm Tor}_q^{U(L(2)\oplus \ldots \oplus L(n))}(M,(U(L(2) \oplus \ldots
L(n))^{\#}), UL(1)) \Rightarrow {\rm Ext}_{UL}^{p+q}(M,UL) \,.\eqno{(4)}$$ Since $L(1)$ commutes with the other $L(i)$ it follows that for each monogenic $UL(1)$-module $N$ that is a subquotient of $\mbox{Tor}_q^{U(L(2) \oplus \cdots \oplus L(n))} (M, U(L(2)\oplus \cdots \oplus L(n))^{\#})$ we have $\mbox{polybd}\, N \leq \mbox{polybd}\, M$.
Now since $L(1)$ is not solvable, $\mbox{polybd}\,UL(1)=\infty$ and the argument in the proof of Lemma 2 shows that $\mbox{Ext}^0_{UL(1)}
(N,UL(1)) = 0$. Thus (Lemma 1) the left hand in (4) vanishes for $p=0$. By induction on $n$ it vanishes for $q<n-1$ and so $\mbox{Ext}_{UL}^{<n}(M,UL) = 0$. Thus $\mbox{polydepth}\, L\geq n$.
On the other hand, there are $UL(i)$-modules $M(i)$ such that $\mbox{polygrade}\, M(i) =
\mbox{polydepth}\, L(i)$. Then $\otimes_{i=1}^n M(i)$ is a $UL$-module that has at most polynomial growth and whose polygrade is the sum of the polygrades of the $M(i)$. $\Box$
Growth of Lie algebras
======================
[**Proposition 5.**]{} [*Let $L$ be a non solvable graded Lie algebra of finite polydepth. Then for each integer $r\geq 1$ there is a positive integer $d(r)$ such that $$\sum_{i=k+1}^{k+d(r)} \mbox{dim}\, L_i \geq k^r\,, \hspace{1cm}
k \, \mbox{sufficiently large}\,.$$*]{}
[**proof:**]{} We distinguish two cases.
[**Case A:** ]{} $L_{\mbox{\scriptsize even}}$ contains an infinite dimensional abelian sub Lie algebra $E$.
Choose $n$ so that dim $E_{\leq n} \geq (r+3)
\mbox{polydepth}\, L$. Then there is a finite sequence $$L = I(0) \supset I(1) \supset \cdots \supset I(l)$$ in which $I(j)$ is an ideal in $I(j-1)$ and $I(l)_{\leq n} =
E_{\leq n}$.
By Proposition 1, polydepth$\, I(q) \leq$ polydepth$\, L$. Thus without loss of generality we may suppose that $L = I(l)$; i.e. that $L_{\leq n}$ is an abelian sub Lie algebra concentrated in even degrees and that dim $L_{\leq n} \geq (r+3)\mbox{polydepth}\,
L$.
Let $M$ be a $UL$-module such that $\mbox{grade}\, M +
\mbox{polybd}\, M = \mbox{polydepth}\,L$ and put $m =
\mbox{grade}\, M$. As observed in the proof of Proposition 1(ii), $\mbox{Ext}_{UL_{\leq n}}^q(\land^psL_{>n}\otimes M, UL_{\leq
n})\neq 0$, for some $p+q = m$. It follows that for some $z \in
\land^psL_{>n}\otimes M$, $$\mbox{polybd}\, UL_{\leq n}\cdot z + q \geq \mbox{dim}\, L_{\leq
n}$$ (Theorem 3.1 in [@LAPG]). Hence for some $x\in L$, $$p(\mbox{polybd} \, UL_{\leq n}\cdot x) \geq \mbox{dim}\, L_{\leq
n} - q - \mbox{polybd}\, M\,.$$
Since $p+q+ \mbox{polybd} \,M = \mbox{polydepth}\, L$ we conclude that $$(2 + \mbox{polybd}\, UL_{\leq n}\cdot
x) \cdot \mbox{polydepth}\, L \geq \mbox{dim}\, L_{\leq n}\geq (r+3)\mbox{polydepth}\, L\,.$$ As observed in the introduction, since dim$\, L = \infty$, polydepth$\, L>0$. It follows that $$\mbox{polybd}\, \left( UL_{\leq n}\cdot x\right)\geq r+1\,. \eqno{(5)}$$
On the other hand, $UL_{\leq n}$ is the polynomial algebra $\bk
[y_1, \ldots , y_s]$ on a basis $y_1, \ldots , y_s$ of $L_{\leq
n}$. Because of (5) it is easy to see (induction on $s$) that this basis can be chosen so that for some $w\in UL_{\leq n}\cdot x$, $\bk[y_1, \ldots , y_{r+1}] \to \bk [y_1, \ldots , y_{r+1}]\cdot
w$ is injective. Put $d = \prod_{i=1}^{r+1} \mbox{deg}\, y_i$ and note that $$\sum_{i=k+ \mbox{\scriptsize deg} w + 1}^{k+ \mbox{\scriptsize deg}\, w + d}
\mbox{dim}\, L_i \geq \sum_{i=k+1}^{k+d} \mbox{dim}\,\bk [y_1, \ldots ,
y_{r+1}]_i \geq \frac{1}{r!} k^r\,.$$ It follows that for any $r$, and $k$ sufficiently large, $$\sum_{i=k+1}^{k+d} \mbox{dim}\, L_i \geq k^{r-1}\,.$$ This proves Proposition 5 in case A.
[**Case B**]{} Every abelian sub Lie algebra of $L_{even}$ is finite dimensional.
Let $I$ be the sum of the solvable ideals in $L$. Then $I_{\mbox{\scriptsize even}}$ is finite dimensional and $\mbox{polydepth}\,
L/I$ is finite (Theorem 3 and Proposition 3). Thus all abelian sub Lie algebras of $L/I$ are finite dimensional. Thus it is sufficient to prove Case B when $L$ has no solvable ideals.
There are now two possibilities: either $L=L_{\mbox{\scriptsize
even}}$, or $L$ has elements of odd degree. In the latter case the sub Lie algebra generated by $L_{\mbox{\scriptsize odd}}$ is an ideal, hence non-solvable and of finite polydepth. Let $L(s)$ denote the sub Lie algebra generated by the first $s$ linearly independent elements $x_1, \ldots , x_s$ of odd degree. For $s$ sufficiently large, $\mbox{polydepth}\, L(s) \leq
\mbox{polydepth}\, L$ (Proposition 1) and $\mbox{dim}\,
L(s)_{\mbox{\scriptsize even}} > \mbox{polydepth}\, L$ (obvious). Thus $L(s)$ cannot be solvable (Theorem 3). In other words, we may assume that either $L=L_{\mbox{\scriptsize even}}$ or else $L$ is generated by finitely many elements $x_1, \ldots , x_s$ of odd degree. In either case set $E = L_{\mbox{\scriptsize even}}$, and note that $\mbox{dim}\, E$ is infinite.
Define a sequence of elements $z_i$ and sub Lie algebras $E(i)$ by setting $E(1) = E$, $z_i $ is a non-zero element in $E(i)$ and $E(i+1) \subset E(i)$ is the sub Lie algebra of elements on which ad$\, z_i$ acts nilpotently.
Since $E$ contains no infinite dimensional abelian Lie algebra some $E(N+1) = 0$ and $E(1)/E(2) \oplus \cdots \oplus E(N)/E(N+1)$ is a graded vector space isomorphic with $E$.
Put $d = \prod \mbox{deg}\, z_i$ and $d_i = d/\mbox{deg}\, z_i$. Then $(\mbox{ad}\, z_1)^{d_1} \oplus \cdots \oplus
(\mbox{ad}\,z_N)^{d_N}$ is an injective transformation of $E(1)/E(2) \oplus \cdots \oplus E(N)/E(N+1)$ of degree $d$. Since this space is isomorphic with $E$ it follows that $$\sum_{i=k+1}^{k+d}\mbox{dim}\, E_i \geq \frac{d}{k+d}
\sum_{i=1}^{k+d} \mbox{dim}\, E_i\,, \hspace{1cm} k\geq 1\,.\eqno{(6)}$$
On the other hand, choose $n$ so that $$\mbox{dim}\, E_{\leq n} \geq (r+3)\cdot\mbox{polydepth}\, L\,.$$ (This is possible because $E$ is infinite dimensional.) Set $I =
L_{>n}$. Let $M$ be an $L$-module with $\mbox{polydepth}\, L = \mbox{polygrade}\, M$. As in the proof of Theorem 4.1 in [@LAPG], $\mbox{Ext}^p_{UL/I} (\mbox{Tor}_q^{UI}(M,\bk),UL/I)\neq 0$ for $p+q = \mbox{grade}\, M$. Thus $p,q$ and $\mbox{polybd}\, M$ are all bounded above by $\mbox{polydepth}\, L$. Now Theorem 3.1 of [@LAPG] asserts that for some $\alpha
\in \mbox{Tor}_q^{UI}(M,\bk)$, $UL/I\cdot \alpha$ has polynomial growth at least equal to $\mbox{dim}(L/I)_{\mbox{\scriptsize even}}
- p $. This means that for some positive $C$, $$\sum_{i\leq k} \mbox{dim}\,(UL/I\cdot \alpha)_i \geq
Ck^{(\mbox{\scriptsize dim}\, (L/I)_{\mbox{\scriptsize even} })-p}\,,\hspace{1cm}
k\,\,
\mbox{sufficiently large}\,.$$ Since $\mbox{Tor}_*^{UI}(M,\bk)$ is the homology of $\land^*sI\otimes M$, it follows that $$\sum_{i\leq k} \mbox{dim}\, \mbox{Tor}_q^{UI}(M,\bk)_i \leq
(\sum_{i\leq k}\mbox{dim}\, L_i)^q \sum_{i\leq k}\mbox{dim}\,
M_i\,.$$ But $(L/I)_{\mbox{\scriptsize even}} \cong E_{\leq n}$ and so a quick calculation gives $$\sum_{i\leq k}\mbox{dim}\, L_i \geq Kk^{r+1}\,, \hspace{1cm}
\mbox{$k$ sufficiently large}\,. \eqno{(7)}$$
Finally, recall that either $L=L_{\mbox{\scriptsize even}}$ or else $L$ is generated by the elements of odd degree $x_i$. In the former case $L = E$ and the Proposition follows from (6) and (7). In the second case we have $L_{\mbox{\scriptsize odd}} = [x_1,E] +
\cdots + [x_s,E]+ \bk x_1 + \cdots + \bk x_s$, and hence (7) yields $$\sum_{i\leq k} E_i \geq \frac{K}{s+1} k^{r+1} + s\,,\hspace{1cm}
\mbox{$k$ sufficiently large}\,.$$ Combined with (6) this formula gives the Proposition. $\Box$
[**Proof of Theorem 5:**]{} Since $L$ is not solvable we may choose $n$ so that $\mbox{dim}\,(L_{\mbox{\scriptsize even}})_{\leq n} >
\mbox{polydepth}\, L$ (Theorem 3), and so that the sub Lie algebra generated by $L_{\leq n}$ satisfies $\mbox{polydepth}\, E\leq
\mbox{polydepth}\, L$ (Proposition 1). Then $E$ is not solvable (Proposition 3).
Let $x_1, \ldots , x_s$ generate $E$, and put $d = \mbox{max
deg}\, x_i$. Letting $UE$ act via the adjoint representation on $E$ we have that $$UE_{[0,q]}\cdot E_{[k+1,k+d]} \supset E_{[k+1,k+q]}\,.$$ For any $r\geq 1$ choose $q = q(r)$ so that $\sum_{i=k+1}^{k+q}
\mbox{dim}\, E_i \geq k^{r+1}$, $k$ sufficiently large (Proposition 5). Then $$\sum_{i=k+1}^{k+d} \mbox{dim}\, L_i \geq \sum_{i=k+1}^{k+d} \mbox{dim}\,
E_i\geq \frac{1}{\mbox{dim}\, UE_{[0,q]}}k^{r+1} \geq k^r\,,
\hspace{1cm} \mbox{$k$ sufficiently large}\,.$$ Since $d$ is independent of $r$, the Theorem is proved. $\Box$
[xx]{} Y. Felix, S. Halperin, J.-M. Lemaire and J.-C. Thomas, Mod $p$ loop space homology. [*Inventiones Math.*]{} [**95**]{} (1989), 247-262. Y. Felix, S. Halperin, C. Jacobson, C. Löfwall and J.-C. Thomas, The radical of the homoltopy Lie algebra. [*Amer. J. Math.* ]{} [**110**]{} (1988), 301-322. Y. Felix, S. Halperin and J.-C. Thomas, Lie algebras of polynomial growth. [*J. London Math. Soc.*]{} [**43**]{} (1991), 556-566. Y. Felix, S. Halperin and J.-C. Thomas, Hopf algebras of polynomial growth. [*J. Algebra* ]{} [**125**]{} (1989), 408-417. Y. Felix, S. Halperin and J.-C. Thomas, Engel Elements in the Homotopy Lie Algebra. [*Journal of Algebra*]{}, [**144**]{} (1991), 67-78. Y. Felix, S. Halperin and J.-C. Thomas, The category of a map and the grade of a module. [*Israel Journal of Mathematics*]{} [**78**]{} (1992), 177-196. Y. Felix, S. Halperin and J.-C. Thomas, [*Growth and Lie brackets in the homotopy Lie algebra*]{}. To appear. S. Halperin, Universal enveloping algebra and loop space homology. [*J. Pure and Appl. Algebra*]{} [**83**]{} (1992), 237-282. J.-L. Koszul, Homologie et cohomologie des algèbres de Lie. [*Bull. Soc. Math. France*]{} [**78**]{} (1950), 65-127. J.W. Milnor and J.C. Moore, On the structure of Hopf algebras. [*Annals of Math.*]{} [**81**]{} (1965), 211-264.
Institut de Mathématiques,
Université Catholique de Louvain
1348 Louvain-La-Neuve, Belgium
College of Computer, Mathematical and Physical Sciences
University of Maryland
College Park, MD 20742-3281, USA
Faculté des Sciences
Université d’Angers
Bd. Lavoisier, 49045 Angers, France
|
---
author:
- 'R. J. Morton and M. S. Ruderman'
date: 'Received /Accepted'
title: 'Non-axisymmetric oscillations of stratified coronal magnetic loops with elliptical cross-sections'
---
Introduction {#sec:intro}
============
The solar atmosphere is a highly dynamic and structured plasma that is able to support a wide variety of magneto-acoustic waves and oscillations. Each layer of the solar atmosphere, from the photosphere to the corona, is magnetically connected to the others via the all pervading magnetic field. The omnipresence of the waves throughout the atmosphere is becoming well documented as new and exciting techniques are being developed to help observe and study the waves (see, e.g. [@BANETAL2007]; [@TOMetal2007]).
After transverse coronal loop oscillations were first observed by TRACE in $1998$ ([@ASCetal1999]; [@NAKetal1999]), the phenomenon became one of the hot topics within solar physics. In the first theoretical interpretation of these oscillations, a coronal loop was modelled as a straight magnetic cylinder with the density constant inside and outside. Since then, a number of more complicated and realistic models have been considered. For a recent review on the theory of transverse oscillations of a coronal loop see, e.g., [@RUDERD2009].
Although the transverse coronal loop oscillations are interesting on their own, their main importance is related to the fact that they are a powerful tool of coronal seismology. [@NAKOFM2001] demonstrated this by using the observations of transverse coronal loop oscillations to estimate the magnitude of the magnetic field in the corona, while [@ANDetal2005a] suggested to use these observations to estimate the atmospheric scale height in the corona.
In this paper we continue to study the transverse oscillations of coronal loops. Coronal loops with elliptical cross-sections and a *constant* density profile have been studied previously in both cold ([@RUD2003]) and [ finite-]{}$\beta$ ([@ERDMOR2009]) plasmas. Now, we consider oscillations of loops with the density *varying* along the loop and a constant elliptic cross-section. The paper is organized as follows. In the next section we formulate the problem. In Sect. \[sec:derivation\] we derive the governing equations for non-axisymmetric oscillations of a coronal loop with an elliptic cross-section in the thin tube approximation. In Sect. \[sec:seismology\] we study the implication of our analysis on coronal seismology. Section \[sec:summary\] contains the summary of the obtained results and our conclusions.
Problem formulation {#sec:formulation}
===================
We model a coronal loop as a straight magnetic tube with an elliptical cross-section. The cold plasma approximation is used. The density varies along the tube, while the cross-section remains constant. In Cartesian coordinates $x,\,y,\,z$ the loop axis coincides with the $z$-axis. The equilibrium magnetic field is given by $\vec{B} = B\hat{\vec{z}}$, where $B$ is constant and $\hat{\vec{z}}$ is the unit vector in the $z$-direction. The plasma motion is governed by the linearised ideal MHD equations, $$\frac{\partial^2\vec{\xi}}{\partial t^2} =
\frac1{\mu_0\rho}(\nabla\times\vec{b})\times\vec{B},
\label{eq:ideal_v}$$ $$\vec{b} = \nabla\times(\vec{\xi}\times\vec{B}).
\label{eq:ideal_b}$$ Here $\vec{\xi}$ is the plasma displacement, $\vec{b}$ the magnetic field perturbation, $\rho(z)$ the equilibrium density, and $\mu_0$ the magnetic permeability of free space; $\rho(z) = \rho_{\rm i}(z)$ inside the tube and $\rho(z) = \rho_{\rm e}(z)$ outside the tube.
![Sketch showing the elliptical coordinate system used to describe the loop cross-section. [ The open and closed curves show the $s$ and $\varphi$ coordinate lines respectively. The thick closed curve shows the tube boundary.]{}[]{data-label="fig:1"}](ellipcoord2.eps)
Let us introduce the elliptic coordinates $s$ and $\varphi$ in the $xy$-plane (see Fig. \[fig:1\]). The Cartesian coordinates are expressed in terms of elliptic coordinates as $$x = \sigma\cosh s\cos\varphi, \qquad y = \sigma\sinh s\sin\varphi,
\label{eq:ellip_coord}$$ where $\sigma$ is a quantity with the dimension of length, $s$ varies from $0$ to $\infty$, and $\varphi$ from $-\pi$ to $\pi$. In the elliptic coordinates the equation of the tube boundary is $s = s_0$. Then the large and small half-axes of the tube elliptic cross-section are in the $x$ and $y$-direction, and they are given by $$a = \sigma\cosh s_0, \qquad b = \sigma\sinh s_0 .
\label{eq:axes}$$ At the tube boundary the normal component of the displacement, $\xi_s$, and the magnetic pressure perturbation, $P = \vec{b}\cdot\vec{B}/\mu_0$, has to be continuous, $$[\hspace*{-0.7mm}[\xi_s]\hspace*{-0.7mm}] = 0, \quad
[\hspace*{-0.7mm}[P]\hspace*{-0.7mm}] = 0 \quad \mbox{at} \quad s = s_0,
\label{eq:jumps}$$ where $[\hspace*{-0.7mm}[f]\hspace*{-0.7mm}]$ indicates the jumps of function $f$ across the boundary defined as $$[\hspace*{-0.7mm}[f]\hspace*{-0.7mm}] = \lim_{\varepsilon\to 0}
[f(s + \varepsilon) - f(s - \varepsilon)].
\label{eq:jump-def}$$ The magnetic field lines at the loop foot points are frozen in the dense photospheric plasma, so that $$\vec{\xi} = 0 \quad \mbox{at} \quad z = \pm L/2,
\label{eq:frozen-xi}$$ where $L$ is the loop length.
It follows from Eq. (\[eq:ellip\_coord\]) that the points with the elliptical coordinates $s = 0$, $\varphi = \varphi_0$, and $s=0$, $\varphi = -\varphi_0$ are the same point in the $xy$-plane. This implies that $P$ and $\xi_s$ have to satisfy the boundary conditions $$P(0,\varphi) = P(0,-\varphi), \qquad \xi_s(0,\varphi) = -\xi_s(0,-\varphi).
\label{eq:reg_pxi}$$ Equations (\[eq:ideal\_v\]) and (\[eq:ideal\_b\]) together with the boundary conditions (\[eq:jumps\]), (\[eq:frozen-xi\]) and (\[eq:reg\_pxi\]) will be used in the next section to derive the governing equations for non-axisymmetric oscillations in the thin tube approximation.
Derivation of governing equations {#sec:derivation}
=================================
The analysis in this section is similar to one used by [@DYMRUD2005] to derive the governing equation for a thin tube with a circular tube cross-section. We begin by noting that, in accordance with Eq. (\[eq:ideal\_v\]), $\xi_z = 0$. The system of Eqs. (\[eq:ideal\_v\]) and (\[eq:ideal\_b\]) can then be transformed to $$\frac{\partial^2\vec{\xi}}{\partial t^2} = -\frac1\rho\nabla_\perp P +
\frac B{\mu_0\rho}\frac{\partial\vec{b}_\perp}{\partial z},
\label{eq:transf_v}$$ $$\vec{b}_\perp = B\frac{\partial\vec{\xi}}{\partial z},
\label{eq:transf_b}$$ $$P = -\rho v_A^2\nabla\cdot\vec{\xi},
\label{eq:P2xi}$$ where $v_A$ is the Alfvén speed defined by $v_A^2 = B^2/\mu_0\rho$, and the operator $\nabla_\perp$ and component of the magnetic field perturbation perpendicular to the $z$-axis are given by $$\nabla_\perp = \nabla - \hat{\vec{z}}\frac\partial{\partial z}, \qquad
\vec{b}_\perp = \vec{b} - \vec{b}\cdot\hat{\vec{z}}.
\label{perp}$$ Eliminating $\vec{b}_\perp$ from Eqs. (\[eq:transf\_v\]) yields $$\frac{\partial^2\vec{\xi}}{\partial t^2} -
v_A^2\frac{\partial^2\vec{\xi}}{\partial z^2} = -\frac1\rho\nabla_\perp P.
\label{eq:momentum}$$ Taking the divergence of this equation and using Eq. (\[eq:P2xi\]) we arrive at the equation for $P$, $$\frac{\partial^2 P}{\partial t^2} -
v_A^2\frac{\partial^2 P}{\partial z^2} = v_A^2\nabla_\perp^2 P.
\label{eq:Pfull}$$ Now we use the thin tube approximation. To do this we note that the characteristic spatial scale in the $z$-direction is $L$, and the characteristic time of the problem is $L/\bar{v}_A$, where $\bar{v}_A$ is a typical value of Alfvén speed. In what follows we only consider the perturbations that decay at the distance of a few $a$ from the tube. Then the characteristic spatial scale in the $x$ and $y$-direction is $a$. It follows from this analysis that the ratio of the left-hand side of Eq. (\[eq:Pfull\]) to its right-hand side is of the order of $(a/L)^2 \ll 1$, so that we can neglect the left-hand side. Then, using the expression for $\nabla_\perp^2$ in the elliptical coordinates (e.g. [@KornKorn]), we obtain the equation for $P$ in the thin tube approximation, $$\frac{\partial^2 P}{\partial s^2} + \frac{\partial^2 P}{\partial\varphi^2} = 0.
\label{eq:Papprox}$$ The solution to this equation has to satisfy the first regularity condition in Eq. (\[eq:reg\_pxi\]), and the second boundary condition in Eq. (\[eq:jumps\]). Using Eq. (\[eq:momentum\]) we rewrite the second regularity condition in terms of $P$, $$\frac{\partial P(s,\varphi)}{\partial s}\bigg|_{s=0} =
-\frac{\partial P(s,-\varphi)}{\partial s}\bigg|_{s=0}.
\label{eq:reg_derP}$$
To derive the governing equations for non-axisymmetric tube oscillations we solve Eqs. (\[eq:momentum\]) and (\[eq:Papprox\]) inside and outside the tube, and then match the two solutions at the tube boundary. It is straightforward to obtain the general solution to Eq. (\[eq:Papprox\]) inside the tube satisfying the regularity conditions Eqs. (\[eq:reg\_pxi\]) and (\[eq:reg\_derP\]), $$P^{\rm i} = \sum_{n=1}^\infty\left[C_n^{\rm i}\cosh(ns)\cos(n\varphi) +
D_n^{\rm i}\sinh(ns)\sin(n\varphi)\right],
\label{eq:p_int}$$ where $C_n^{\rm i}$ and $D_n^{\rm i}$ are arbitrary functions of $t$ and $z$. The solution outside the tube has to decay as $s \to \infty$. Hence, its general form is $$P^{\rm e} = \sum_{n=1}^\infty e^{-ns}\left[C_n^{\rm e}\cos(n\varphi) +
D_n^{\rm e}\sin(n\varphi)\right],
\label{eq:p_ext}$$ where once again $C_n^{\rm e}$ and $D_n^{\rm e}$ are arbitrary functions of $t$ and $z$. Substituting Eqs. (\[eq:p\_int\]) and (\[eq:p\_ext\]) in Eq. (\[eq:momentum\]) and using the expression for $\nabla_\perp$ in the elliptical coordinates (e.g. [@KornKorn]), $$\nabla_\perp = \frac1{\sigma\Theta}
\left(\hat{\vec{s}}\frac\partial{\partial s} +
\hat{\vec{\varphi}}\frac\partial{\partial\varphi}\right), \quad
\Theta = (\sinh^2 s + \sin^2\varphi)^{1/2},
\label{eq:nabla2coord}$$ where $\hat{\vec{s}}$ and $\hat{\vec{\varphi}}$ are the unit vectors in the $s$ and $\varphi$-direction, we obtain the expressions for $\xi_s$ inside and outside the tube, $$\xi_s^{\rm i} = \frac1{\sigma\Theta}\sum_{n=1}^\infty
\left[F_n^{\rm i}\sinh(ns)\cos(n\varphi) +
G_n^{\rm i}\cosh(ns)\sin(n\varphi)\right],
\label{eq:xi_int}$$ $$\xi_s^{\rm e} = \frac1{\sigma\Theta}\sum_{n=1}^\infty e^{-ns}
\left[F_n^{\rm e}\cos(n\varphi) + G_n^{\rm e}\sin(n\varphi)\right].
\label{eq:xi_ext}$$ In these equations $F_n^{\rm i}$, $G_n^{\rm i}$, $F_n^{\rm e}$ and $G_n^{\rm e}$ are functions of $t$ and $z$. They are related to the functions $C_n^{\rm i}$ $D_n^{\rm i}$, $C_n^{\rm e}$ and $D_n^{\rm e}$ by $$\frac{\partial^2 F_n^{\rm i}}{\partial t^2} -
v_{A\rm i}^2\frac{\partial^2 F_n^{\rm i}}{\partial z^2} =
-\frac{C_n^{\rm i}}{\rho_{\rm i}},
\label{eq:FCint}$$ $$\frac{\partial^2 G_n^{\rm i}}{\partial t^2} -
v_{A\rm i}^2\frac{\partial^2 G_n^{\rm i}}{\partial z^2} =
-\frac{D_n^{\rm i}}{\rho_{\rm i}},
\label{eq:GDint}$$ $$\frac{\partial^2 F_n^{\rm e}}{\partial t^2} -
v_{A\rm e}^2\frac{\partial^2 F_n^{\rm e}}{\partial z^2} =
\frac{C_n^{\rm e}}{\rho_{\rm e}},
\label{eq:FCext}$$ $$\frac{\partial^2 G_n^{\rm e}}{\partial t^2} -
v_{A\rm e}^2\frac{\partial^2 G_n^{\rm e}}{\partial z^2} =
\frac{D_n^{\rm e}}{\rho_{\rm e}}.
\label{eq:GDext}$$ Substituting Eqs. (\[eq:p\_int\]) and (\[eq:p\_ext\]) in the second boundary condition in Eq. (\[eq:jumps\]) we obtain $$C_n^{\rm i}\cosh(ns_0) = e^{-ns_0}C_n^{\rm e}, \quad
D_n^{\rm i}\sinh(ns) = e^{-ns_0}D_n^{\rm e} .
\label{eq:CDint-ext}$$ Substituting Eqs. (\[eq:xi\_int\]) and (\[eq:xi\_ext\]) in the first boundary condition in Eq. (\[eq:jumps\]) yields $$F_n^{\rm i}\sinh(ns_0) = e^{-ns_0}F_n^{\rm e}, \quad
G_n^{\rm i}\cosh(ns) = e^{-ns_0}G_n^{\rm e} .
\label{eq:FGint-ext}$$ Eliminating $C_n^{\rm i}$, $C_n^{\rm e}$ and $F_n^{\rm e}$ from Eqs. (\[eq:FCint\]), (\[eq:FCext\]), (\[eq:CDint-ext\]) and (\[eq:FGint-ext\]) we obtain the equation for $F_n^{\rm i}$, $$\frac{\partial^2 F_n}{\partial t^2} -
c_{n\rm c}^2\frac{\partial^2 F_n}{\partial z^2} = 0, \quad
c_{n\rm c}^2 = \frac{B^2_0[1+\tanh(ns_0)]}
{\mu_0[\rho_{\rm i}+\rho_{\rm e}\tanh(ns_0)]},
\label{eq:govF}$$ where we have dropped the superscript ‘i’. Eliminating $D_n^{\rm
i}$, $D_n^{\rm e}$ and $G_n^{\rm e}$ from Eqs. (\[eq:GDint\]), (\[eq:GDext\]), (\[eq:CDint-ext\]) and (\[eq:FGint-ext\]) we obtain the equation for $G_n^{\rm i}$, $$\frac{\partial^2 G_n}{\partial t^2} -
c_{n\rm s}^2\frac{\partial^2 G_n}{\partial z^2} = 0, \quad
c_{n\rm s}^2 = \frac{B^2_0[1+\tanh(ns_0)]}
{\mu_0[\rho_{\rm i}\tanh(ns_0)+\rho_{\rm e}]},
\label{eq:govG}$$ where we have once again dropped the superscript ‘i’. It follows from Eqs. (\[eq:frozen-xi\]) and (\[eq:xi\_int\]) that $F_n$ and $G_n$ have to satisfy the boundary conditions $$F_n = 0, \quad G_n = 0 \quad \mbox{at} \quad z = \pm L/2.
\label{eq:frozen-FG}$$ In Eqs. (\[eq:govF\]) and (\[eq:govG\]) $n = 1$ corresponds to kink modes, and $n > 1$ to fluting modes.
In the elliptical coordinates the loop axis ($x=y=0$) is defined by $s = 0$ and $\varphi = \pi/2$. It follows from Eq. (\[eq:xi\_int\]) that the kink mode described by Eq. (\[eq:govF\]) does not displace the loop axis in the $s$-direction which, at the loop axis, coincides with the $y$-direction. Hence, the loop axis displacement is in the $x$-direction, i.e. this mode is polarised in the direction of the larger axis of the tube cross-section. The kink mode described by Eq. (\[eq:govG\]) displaces the loop axis in the $s$-direction. It is straightforward to show that it does not displace it in the $\varphi$-direction which, at the loop axis, coincides with the $x$-direction. Hence, the loop axis displacement is in the $y$-direction, i.e. this mode is polarised in the direction of the smaller axis of the tube cross-section.
When the density is constant, we can use Eqs. (\[eq:govF\]) and (\[eq:govG\]) with the boundary conditions Eq. (\[eq:frozen-FG\]) to recover the results obtained by [@RUD2003]. Let us look for the eigenmodes and restrict the analysis to the fundamental modes in the $z$-direction. This implies that we take $F_n$ and $G_n$ proportional to $e^{-i\omega
t}\cos(\pi z/L)$. Then we immediately obtain that the eigenfrequencies of the boundary value problem defined by Eq. (\[eq:govF\]) and the boundary conditions (\[eq:frozen-FG\]) are given by $$\omega_{n\rm c}^2 = \frac{\pi^2 c_{n\rm c}^2}{L^2} =
\frac{\pi^2 B^2_0[1+\tanh(ns_0)]}
{\mu_0 L^2[\rho_{\rm i}+\rho_{\rm e}\tanh(ns_0)]},
\quad n = 1,2,\dots,
\label{eq:Feigen}$$ and the eigenfrequencies of the boundary value problem defined by Eq. (\[eq:govG\]) and the boundary conditions (\[eq:frozen-FG\]) are given by $$\omega_{n\rm s}^2 = \frac{\pi^2 c_{n\rm s}^2}{L^2} =
\frac{\pi^2 B^2_0[1+\tanh(ns_0)]}
{\mu_0 L^2[\rho_{\rm i}\tanh(ns_0)+\rho_{\rm e}]}.
\quad n = 1,2,\dots
\label{eq:Geigen}$$ In particular, the squares of eigenfrequencies of the kink modes are given by $$\omega_{1\rm c}^2 = \frac{\pi^2 B^2_0(a + b)}
{\mu_0 L^2(a\rho_{\rm i} + b\rho_{\rm e})}, \quad
\omega_{1\rm s}^2 = \frac{\pi^2 B^2_0(a + b)}
{\mu_0 L^2(b\rho_{\rm i} + a\rho_{\rm e})}.
\label{eq:kink-eigen}$$ It is straightforward to see that the eigenfrequencies satisfy $$\omega_{1\rm c} < \omega_{2\rm c} < \dots < \omega_{2\rm s} < \omega_{1\rm s}.
\label{eq:order}$$
Implication on coronal seismology {#sec:seismology}
=================================
After [@VERetal2004] reported two cases of observations of the transverse coronal loop oscillations where, in addition to the fundamental harmonic, the first overtone was also observed, [@ANDetal2005a] suggested observations of this nature could be used to estimate the scale height in the solar corona. [@ANDetal2005a] assumed that an oscillating loop has a half-circle shape and a circular cross-section, and it is in the vertical plane. They also assumed that the atmosphere is isothermal. In that case, the dependence of the plasma density on $z$ is given by $$\rho_{\rm e} = \rho_{\rm f}\exp\left(-\frac L{\pi H}\cos\frac{\pi z}L\right),
\quad \rho_{\rm i} = \zeta\rho_{\rm e},
\label{eq:density}$$ where $H$ is the atmospheric scale height, $\rho_{\rm f}$ the plasma density at the loop foot points outside the loop, and $\zeta > 1$ a constant. [@ANDetal2005a] calculated the ratio of frequencies of the first overtone and fundamental mode and found that this ratio is a monotonically decreasing function of the parameter $L/H$. Hence, if we know the ratio of frequencies and $L$, we can determine $H$. For a recent review of coronal seismology using kink oscillation overtones see [@ANDetal2009].
A very important question is how robust is this method. [@DYMRUD2006b] and [@MORERD2009] have found that the account of the loop shape can moderately affect the estimates of the atmospheric scale height. [@RUD2007] has shown that the twist of magnetic field lines in the loop can be safely neglected when estimating the atmospheric scale height in the corona. [@ROBetal2010] found that the estimates of the atmospheric scale height obtained using the two-thread model are exactly the same as those obtained using the model of a monolithic coronal loop with a circular cross-section of constant radius. Recently [@RUD2010] showed that the account of stationary time independent siphon flows in coronal loops have little influence [on]{} the estimates of the coronal scale height found using the frequency ratio. On the other hand, [@RUDetal2008] and [@VERERDJES2008] found that the account of the loop expansion can strongly affect these estimates.
In this section we study what the effect the elliptic cross-section has on the estimates of the coronal scale height. As we have already seen, when a loop has an elliptic cross-section, its kink oscillations are polarised along the axes of the cross-section. The kink mode polarised in the direction of the larger axis is described by Eq. (\[eq:govF\]) with $n = 1$, while the kink mode polarised in the direction of the smaller axis is described by Eq. (\[eq:govG\]) with $n = 1$. Let us consider the solutions to these equations in the form of eigenmodes and take $F_1$ and $G_1$ proportional to $\exp(-i\omega t)$. Using Eq. (\[eq:density\]) we obtain $$c_{1\rm c}^2 = \frac{B^2_0(a+b)}{\mu_0\rho_{\rm f}(a\zeta + b)}
\exp\left(\frac L{\pi H}\cos\frac{\pi z}L\right),
\label{eq:phase-speed-c}$$ $$c_{1\rm s}^2 = \frac{B^2_0(a+b)}{\mu_0\rho_{\rm f}(b\zeta + a)}
\exp\left(\frac L{\pi H}\cos\frac{\pi z}L\right)
\label{eq:phase-speed-s}$$ Then, introducing $$\Omega_{\rm c}^2 = \frac{\mu_0\rho_{\rm f}(a\zeta + b)\omega^2}{B^2_0(a+b)},
\quad
\Omega_{\rm s}^2 = \frac{\mu_0\rho_{\rm f}(b\zeta + a)\omega^2}{B^2_0(a+b)},
\label{eq:omega-scale}$$ we reduce Eqs. (\[eq:govF\]) and (\[eq:govG\]) with $n = 1$ to $$\frac{d^2 U}{dz^2} + \Omega^2 U
\exp\left(\frac L{\pi H}\cos\frac{\pi z}L\right) = 0,
\label{eq:govern-scale}$$ where either $U = F_1$ and $\Omega = \Omega_{\rm c}$, or $U = G_1$ and $\Omega = \Omega_{\rm s}$, and $U$ satisfies the boundary conditions $U = 0$ at $z = \pm L/2$. Since Eq. (\[eq:govern-scale\]) does not contain $a$ and $b$, the eigenvalues of the boundary value problem for $U$ are independent of $a$ and $b$. In particular, they are the same as those for a loop with the circular cross-section. Since $$\frac{\Omega_{2\rm c}}{\Omega_{1\rm c}} = \frac{\Omega_2}{\Omega_1}, \qquad
\frac{\Omega_{2\rm s}}{\Omega_{1\rm s}} = \frac{\Omega_2}{\Omega_1},$$ it follows that we obtain the same estimates of the atmospheric scale height no matter if we use the observation of the kink oscillations polarised in the direction of the larger or smaller axis. The estimates are also independent of $a$ and $b$ and are the same as those obtained for a loop with the circular cross-section.
Summary and conclusions {#sec:summary}
=======================
In this paper we have studied non-axisymmetric oscillations of straight magnetic loops with a constant elliptic cross-section and density varying along the loop. We derived the governing equations for kink and fluting modes in the thin tube approximation. All these equations are similar to the equation describing kink oscillations of a straight tube with the circular cross-section. We found that there are two kink modes, one polarised in the direction of larger axis of the elliptic cross-section, and the other polarised in the direction of smaller axis. The frequencies of fundamental mode and overtones of these two kinds of kink oscillation are different. However, the ratio of frequencies of the first overtone and the fundamental mode is the same for both kink oscillations, and it is independent of the ratio of the ellipse half-axes $a/b$. This result implies that we obtain the same estimates of the atmospheric scale height no matter if we use the observation of the kink oscillations polarised in the direction of larger or smaller axis. The estimates are also the same as those obtained for a loop with the circular cross-section. This demonstrates that the model shows a very robust nature when considering a static plasma. However, if the plasma in the loops is dynamic (i.e. time dependent) then the ability of the static model to provide accurate estimates may become questionable (see e.g. [@MORERD2009b]).
The authors thank the Science and Technology Facilities Council (STFC), UK for the financial support they received.
[21]{} natexlab\#1[\#1]{}
, J., [Arregui]{}, I., & [Goossens]{}, M. 2005, , 624, L57
, J., [van Doorsselaere]{}, T., [Roberts]{}, B., [et al.]{} 2009, Space Science Reviews, 149, 3
, M. J., [Fletcher]{}, L., [Schrijver]{}, C. J., & [Alexander]{}, D. 1999, , 520, 880
, D., [Erd[é]{}lyi]{}, R., [Oliver]{}, R., & [O’Shea]{}, E. 2007, , 246, 3
, M. V. & [Ruderman]{}, M. S. 2005, , 229, 79
, M. V. & [Ruderman]{}, M. S. 2006, , 459, 241
, R. & [Morton]{}, R. J. 2009, , 494, 295
, G. & [Korn]{}, T. 1961, [Mathematical Handbook for Scientists and Engineers]{} (New York: McGraw-Hill)
, R. & [Erd[é]{}lyi]{}, R. 2009, , 605, 493
, R. J. & [Erd[é]{}lyi]{}, R. 2009, , 707, 750
, V. M. & [Ofman]{}, L. 2001, , 372, L53
, V. M., [Ofman]{}, L., [Deluca]{}, E. E., [Roberts]{}, B., & [Davila]{}, J. M. 1999, Science, 285, 862
, D., [Ruderman]{}, M. S., & [Taroyan]{}, Y. 2010, , 515
, M. S. 2003, , 409, 287
, M. S. 2007, , 246, 119
, M. S. 2010, , submitted
, M. S. & [Erd[é]{}lyi]{}, R. 2009, Space Science Reviews, 149, 199
, M. S., [Verth]{}, G., & [Erd[é]{}lyi]{}, R. 2008, , 686, 694
, S., [McIntosh]{}, S. W., [Keil]{}, S. L., [et al.]{} 2007, Science, 317, 1192
, G., [Erd[é]{}lyi]{}, R., & [Jess]{}, D. B. 2008, , 687, L45
, E., [Nakariakov]{}, V. M., [Ofman]{}, L., & [Deluca]{}, E. E. 2004, , 223, 77
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---
abstract: 'How the prepared states and Unruh effect affect Measurement-Induced-Nonlocality (MIN) is studied. We show that, as the Unruh temperature increases, the MIN between modes $\mathrm{A}$ and $\mathrm{I}$ decreases but the MIN between modes $\mathrm{A}$ and $\mathrm{II}$ increases. We prove that the parameters $c_i$ which decide initial prepared states affect not only the values of the MIN, but also the dynamical behavior of it. By comparing the MIN with the maximal expectation values of CHSH inequality and geometric discord between modes $\mathrm{A}$ and $\mathrm{I}$, we also find that the MIN is more general than the quantum nonlocality related to violation of Bell’s inequalities, and its values is always equal or bigger than that of the geometric discord.'
author:
- 'Zehua Tian and Jiliang Jing[^1]'
title: 'Effects of prepared states and Unruh temperature on Measurement-Induced-Nonlocality'
---
introduction
============
The investigation of relativistic quantum information not only supplies the gap of interdiscipline refer to quantum information and relativity theory, but also has a positive promotion on the development of them. As a result of that, this domain has been paid much attention in the last decade [@Alsing1; @Alsing2; @Fuentes; @Yi; @Ling; @Shahpoor; @Pan; @Montero; @Wang; @Datta; @Han; @David; @Mann]. Among them, most papers have focused on quantum resource, e. g., quantum entanglement[@Alsing1; @Alsing2; @Fuentes; @Yi; @Ling; @Shahpoor; @Pan; @Montero; @Mann] and discord [@Wang; @Datta], because quantum resource plays an very important role in the quantum information tasks such as teleportation [@Bennett] and computation [@Horodecki; @Bouwmeester], and studying it in a relativistic setting is very closely related to the implementation of quantum tasks with observers in arbitrary relative motion. In addition, extending this work to the black hole background is very helpful for us to understand the entropy and paradox [@Lee; @Hawking] of the black hole.
Despite much effort has been paid to extend quantum information theory to the relativistic setting, another important foundation of quantum mechanics—nonlocality is barely considered. Recently, Nicolai Friis $\emph{et al}$ firstly studied the nonlocality in the noninertial frame, and they pointed out that residual entanglement of accelerated fermions is not nonlocal [@Nicolai]. Following them Alexander Smith $\emph{et al}$ studied the tripartite nonlocality in the noninertial frames [@Smith], and DaeKil Park considered tripartite entanglement-dependence of tripartite nonlocality [@Park]. Generally, most researchers analyzed the quantum nonlocality by means of Bell’s inequalities [@J.; @F.; @Clauser] for bipartite system and Svetlichny inequality for tripartite system [@G.; @Syetlichny], respectively. Because, these inequalities are satisfied by any local hidden variable theory, but they may be violated by quantum mechanics. However, Shunlong Luo and Shuangshuang Fu have introduced a new way to quantify nonlocality by measurement, which is called the MIN [@Luo], and following their paper, a number of papers emerged to perfect its definition [@Xi; @Sayyed] and discussed its properties [@Sen; @Guo]. In addition, some authors have analyzed its dynamical behavior and compared it with other quantum correlation measurements such as the geometric discord [@Hu; @Zhang]. However, all of these studies don’t involve the effect on the MIN resulting from relativistic effect. In fact, the study that how the Unruh effect [@Unruh] affects the MIN not only is very significant to theory study, but also plays a key role in practice, which can help us to implement the quantum task preferably and more efficiently. Inspired by that, we analyze how the Unruh effect and prepared states affect the MIN in this paper and we will show some new properties.
Our paper is constructed as follows. In section II we introduce the different vacuums for relativistic observers and the definition of the MIN. In section III how the Unruh effect and prepared states affect the MIN are studied. And in the last section we summarize and discuss our conclusions.
Vacuums, excited sates and definition of MIN
=============================================
It is well known that an uniformly accelerated observer will detect a thermal particle distribution in the Minkowski vacuum. The Minkowski vacuum can be factorized as a product of the vacua of all different Unruh modes $$\begin{aligned}
\label{Minkowsik vacuum}
|0\rangle_\mathrm{M}=\otimes_w|0_w\rangle_\mathrm{U}.\end{aligned}$$
For the Dirac field, the Unruh monochromatic mode $|0_w\rangle_\mathrm{U}$, from the noninertial observers’ perspective, can be expressed as [@Alsing2; @Montero; @Wang] $$\begin{aligned}
\label{vacuum}
|0_w\rangle_\mathrm{U}=(e^{-w/T}+1)^{-\frac{1}{2}}|0_w\rangle_\mathrm{I}|0_w\rangle_\mathrm{II}
+(e^{w/T}+1)^{-\frac{1}{2}}|1_w\rangle_\mathrm{I}|1_w\rangle_\mathrm{II},\end{aligned}$$ where $|m\rangle_\mathrm{I}$ $(|n\rangle_\mathrm{II})$ denote Rindler mode in region $\mathrm{I}$ (region $\mathrm{II}$), and $\mathrm{T}=a/2\pi$ is the Unruh temperature in which $a$ denotes the proper acceleration of the noninertial observer. Likewise, the particle state of Unruh mode $w$ in the Rindler basis is found to be $$\begin{aligned}
\label{particle state}
|1_w\rangle_\mathrm{U}=|1_w\rangle_\mathrm{I}|0_w\rangle_\mathrm{II}.\end{aligned}$$
Recently, Luo $\emph{et al}$ [@Luo] have introduced a way to quantify nonlocality from a geometric perspective in terms of measurements, which is named the MIN. For a bipartite quantum state $\rho$ shared by subsystem $A$ and $B$ with respective to Hilbert space $H^A$ and $H^B$, we can find the difference between the overall pre-measurement and post-measurement states by performing a local von Neumann measurements on part $A$. To capture the genuine nonlocal effect of the measurements on the state, the key point is that the measurements do not disturb the local state $\rho^A=tr_B\rho$. Based on this idea, the MIN can be defined by $\emph{et al}$ [@Luo] $$\begin{aligned}
\label{intial MIN}
N(\rho)=\max_{\Pi^A}\parallel\rho-\Pi^A(\rho)\parallel^2.\end{aligned}$$ For a general $2\times2$ dimensional system $$\begin{aligned}
\label{Bloch representation}
\rho=\frac{1}{2}\frac{\mathbf{1}^A}{\sqrt{2}}\otimes\frac{\mathbf{1}^B}{\sqrt{2}}
+\sum^3_{i=1}x_iX_i\otimes\frac{\mathbf{1}^B}{\sqrt{2}}
+\frac{\mathbf{1}^A}{\sqrt{2}}\otimes\sum^3_{j=1}y_jY_j
+\sum^3_{i=1}\sum^3_{j=1}t_{ij}X_i\otimes Y_j,\end{aligned}$$ its MIN is given by [@Luo] $$\begin{aligned}
\label{MIN}
N(\rho)=
\left\{
\begin{array}{lr}
trTT^t-\frac{1}{\|\mathbf{x}\|^2}\mathbf{x}^tTT^t\mathbf{x} & ~~{\text{if}}~~ \mathbf{x}\neq0,\\
trTT^t-\lambda_3 & ~~{\text{if}}~~ \mathbf{x}=0,
\end{array}
\right.\end{aligned}$$ where $TT^t(T=(t_{ij}))$ is a $3\times3$ dimensional matrix, $\lambda_3$ is its minimum eigenvalue, and $\|\mathbf{x}\|^2=\sum_ix^2_i$ with $\mathbf{x}=(x_1,x_2,x_3)^t$.
MIN of X-type initial states
============================
We now assume that Alice and Rob share a X-type initial state $$\label{initial states}
\rho_{AB}=\frac{1}{4}\left(I_{AB}+ \sum_{i=1}^{3}c_{i}\sigma_{i}%
^{(A)}\otimes\sigma_{i}^{(B)}\right),$$ where $I_{A(B)}$ is the identity operator in subspace $A(B)$, and $\sigma_{i}^{(n)}$ is the Pauli operator in direction $i$ acting on the subspace $n=A,B$, $c_{i} \in\mathfrak{R}$ such that $0\leq \mid
c_{i}\mid\leq1$ for $i=1,2,3$. Obviously, Eq. (\[initial states\]) represents a class of states including the well-known initial states, such as the Werner initial state ($\left\vert
c_{1}\right\vert =\left\vert c_{2}\right\vert =\left\vert
c_{3}\right\vert =\alpha$), and Bell basis state ($\left\vert
c_{1}\right\vert =\left\vert c_{2}\right\vert =\left\vert
c_{3}\right\vert =1$).
After the coincidence of Alice and Rob, Alice stays stationary while Rob moves with an uniform acceleration $a$. To describe the states shared by these two relatively accelerated observers in detail, we must use Eqs.(\[vacuum\]) and (\[particle state\]) to rewrite Eq.(\[initial states\]) in terms of Minkowski modes for Alice, Rindler modes I for Rob and Rindler modes II for Anti-Rob, which implies that Rob and Anti-Rob are respectively confined in region $\mathrm{I}$ and $\mathrm{II}$. The regions $\mathrm{I}$ and $\mathrm{II}$ are causally disconnected, and the information which is physically accessible to the observers is encoded in the Minkowski modes $A$ and Rindler modes $\mathrm{I}$, but the physically unaccessible information is encoded in the Minkowski modes $A$ and Rindler modes $\mathrm{II}$. So we must trace over the Rindler modes $\mathrm{II}$ (modes $\mathrm{I}$) when we only consider the Physically accessible (unaccessible) information.
MIN shared by Alice and Rob
---------------------------
We first consider the MIN between modes $A$ and $\mathrm{I}$. By taking the trace over the states of region $\mathrm{II}$, we obtain $$\begin{aligned}
\label{state of Alice and Rob}
\nonumber\rho_{A,I}=\frac{1}{4}
\left(
\begin{array}{cccc}
\frac{1+c_3}{e^{-w/T}+1} & 0 & 0 & \frac{c_1-c_2}{(e^{-w/T}+1)^{\frac{1}{2}}} \\
0 & (1-c_3)+\frac{1+c_3}{e^{w/T}+1} & \frac{c_1+c_2}{(e^{-w/T}+1)^{\frac{1}{2}}} & 0 \\
0 & \frac{c_1+c_2}{(e^{-w/T}+1)^{\frac{1}{2}}} & \frac{1-c_3}{e^{-w/T}+1} & 0 \\
\frac{c_1-c_2}{(e^{-w/T}+1)^{\frac{1}{2}}} & 0 & 0 & (1+c_3)+\frac{1-c_3}{e^{w/T}+1}\\
\end{array}
\right)
,\end{aligned}$$ where $|mn\rangle=|m\rangle_{\mathrm{A}}|n\rangle_{\mathrm{I}}$. For convenience to calculate the MIN, we rewrite the state $\rho_{A,\mathrm{I}}$ in terms of Bloch representation, which is given by $$\begin{aligned}
\label{AI Bloch representation}
\rho_{A,I}=\frac{1}{4}\left( \mathbf{1}_A\otimes\mathbf{1}_I
+c'_0\mathbf{1}_A\otimes\sigma^{(I)}_3
+\sum^3_{i=1}c'_i\sigma^{(A)}_i\otimes\sigma^{(I)}_i \right),\end{aligned}$$ where $c'_0=\frac{-1}{(e^{w/T}+1)}$, $c'_1=\frac{c_1}{(e^{-w/T}
+1)^{\frac{1}{2}}}$, $c'_2=\frac{c_2}{(e^{-w/T}+1)^{\frac{1}{2}}}$ and $c'_3=\frac{c_3}{(e^{-w/T}+1)}$. From Eq.(\[MIN\]), we find that the MIN for the state $\rho_{A,I}$ is $$\begin{aligned}
\label{MIN for AI}
N(\rho_{A,I})=\frac{1}{4}\Big\{\frac{(c_1)^2}{(e^{-w/T}+1)}
+\frac{(c_2)^2}{(e^{-w/T}+1)}+\frac{(c_3)^2}{(e^{-w/T}+1)^2}
\nonumber \\
-\min[\frac{(c_1)^2}{(e^{-w/T}+1)},\frac{(c_2)^2}{(e^{-w/T}+1)},
\frac{(c_3)^2}{(e^{-w/T}+1)^2}]\Big\}.\end{aligned}$$ Obviously, $\min\big[\frac{(c_1)^2}{(e^{-w/T}+1)},\frac{(c_2)^2}{
(e^{-w/T}+1)}, \frac{(c_3)^2}{(e^{-w/T}+1)^2}\big]$ depends on both the coefficients $c_i$ of the states in Eq.(\[initial states\]) and the Unruh temperature.
\(i) If $|c_1|,|c_2|\geq|c_3|$ in Eq.(\[initial states\]), the minimum term in Eq.(\[MIN for AI\]) is $\frac{(c_3)^2}{(e^{-w/T}+1)^2}$. In this case, the MIN, provided taking fixed $c_i$, decreases monotonously as the Unruh temperature increases.
\(ii) For the case of $|c_3|>\min\{|c_1|,|c_2|\}$ and both $c_1$ and $c_2$ don’t equals to 0 at the same time, if $\min\{|c_1|,|c_2|\}\geq\frac{\sqrt{2}}{2}|c_3|$, the MIN has a peculiar dynamics with a sudden change as the Unruh temperature increases, i.e., $N(\rho_{A,I})$ decays quickly until $$\begin{aligned}
\label{TSC}
T_{sc}=\frac{-w}{\ln(\frac{|c_3|^2}{min\{|c_1|^2,|c_2|^2\}}-1)},\end{aligned}$$ and then $N(\rho_{A,I})$ decays relatively slowly. Otherwise, the MIN decays monotonously as the temperature increases.
\(iii) Finally, if $|c_1|=|c_2|=0$, we have a monotonic decay of $N(\rho_{A,I})$ as the temperature increases.
The decrease of MIN means that the difference between the pre- and post-measurement states becomes smaller, i.e., the disturbance induced by local measurement weaken. If we understand the MIN as some kind of correlations, this decrease means that the quantum correlation shared by two relatively accelerated observers decreases, i.e., less quantum resource can be used for the quantum information task by these two observers. So the Unruh effect affects quantum communication process by inducing the decrease of quantum resource.
By taking $w=1$, the dynamical behavior of $N(\rho_{A,I})$ is shown in Fig. \[MIN1\]. We find from the figure that the MIN, as the Unruh temperature approaches to the infinite, has a limit $$\begin{aligned}
\label{limit of N}
\lim_{T\rightarrow\infty}N(\rho_{A,I})= \frac{1}{16}\{2(c_1)^2
+2(c_2)^2+(c_3)^2-\min[2(c_1)^2,2(c_2)^2,(c_3)^2]\}.\end{aligned}$$ That is to say, as long as the initial MIN does not equal to zero, it can persist for arbitrary Unruh temperature.
![(Color online) The MIN of state $\rho_{A,I}$ as a function of Unruh temperature $T$. We take parameters $c_1=1$, $c_2=0.9$ and $|c_3|\leq|c_1|,|c_2|$ for red dashed line; $c_1=0.9$, $c_2=0.85$ and $c_3=1$ for blue solid line. The insert shows the detail of sudden change.[]{data-label="MIN1"}](MIN1-1.eps){width="50.00000%"}
We study $T_{sc}$ of Eq.(\[TSC\]), if $|c_1|\leq|c_2|$, by taking fixed $c_3$, we plot how the parameter $c_1$ affects it in Fig. \[Tsc\], which shows that it decreases monotonously as $c_1$ increases. That is to say, the bigger $c_1$ is, the sudden change behavior occurs earlier. And when $|c_2|\leq|c_1|$, it is interesting to note that with the increase of $c_2$, it decreases monotonously too.
![(Color online) The $T_{sc}$ as a function of $c_1$, here we take $|c_1|\leq|c_2|$ and $c_3=0.9$.[]{data-label="Tsc"}](Tsc-2.eps){width="50.00000%"}
In Fig. \[MINc\], we study how the prepared states affect the MIN for case (i). It is found that the $N(\rho_{A,I})$ increases monotonously as $|c_i|$ (i=1,2) increases. And for the case that $|c_3|>\min\{|c_1|,|c_2|\}\geq\frac{\sqrt{2}}{2}|c_3|$, the MIN depends only on $|c_3|$ and $\max\{|c_1|,|c_2|\}$ before $T_{sc}$, while after $T_{sc}$ it is independent of $|c_3|$ but dependent of $|c_1|$, $|c_2|$. However, no matter which case, the MIN increases with the increase of $|c_i|$ it depends on.
![(Color online) The MIN of state $\rho_{A,I}$ as function of $|c_1|$ and $|c_2|$ with $|c_1|,|c_2|\geq|c_3|$. Here we take fixed $T=0.1,1,20$ from top to bottom, respectively.[]{data-label="MINc"}](MINc-3.eps){width="54.00000%"}
Because the MIN is introduced to describe non-locality, its definition is very similar to that of geometric discord. For further understanding it, we will compare it with the maximal possible value $\langle B_{max}\rangle$ of the Bell-CHSH expectation value and geometric discord.
As shown in Ref.[@Nicolai], the $\langle B_{max}\rangle$ for a given state $\rho$ is determined by $$\begin{aligned}
\label{Bell expectation value}
\langle B_{max}\rangle_\rho=2\sqrt{\mu_1+\mu_2},\end{aligned}$$ where $\mu_1$, $\mu_2$ are the two largest eigenvalues of $U(\rho)=TT^t$, the matrix $T=(t_{ij})$ with $t_{ij}=Tr[\rho\sigma_i\otimes\sigma_j]$. And the geometric discord is defined as [@Bor; @Adesso] $$\begin{aligned}
\label{geometric discord}
D(\rho)=\frac{1}{4}(||\overrightarrow{x}||^2+||T||^2-k_{max}),\end{aligned}$$ where $k_{max}$ is the largest eigenvalue of matrix $K=\overrightarrow{x}\overrightarrow{x}^t+TT^t$, where $\overrightarrow{x}=(x_i)^t$ with $x_i=Tr[\rho\sigma_i\otimes\mathbf{1}]$ and $T$ have the same definitions with Eq.(\[Bell expectation value\]).
Using Eqs.(\[AI Bloch representation\]), (\[Bell expectation value\]) and (\[geometric discord\]), $\langle
B_{max}\rangle_{\rho_{A,I}}$ and $D(\rho_{A,I})$ are given by $$\begin{aligned}
\label{Bmax}
\langle B_{max}\rangle_{\rho_{A,I}}=2\{\frac{(c_1)^2}{(e^{-w/T}+1)}
+\frac{(c_2)^2}{(e^{-w/T}+1)}+\frac{(c_3)^2}{(e^{-w/T}+1)^2}
\nonumber \\
-\min[\frac{(c_1)^2}{(e^{-w/T}+1)},\frac{(c_2)^2}{(e^{-w/T}+1)},
\frac{(c_3)^2}{(e^{-w/T}+1)^2}]\}^{1/2},\end{aligned}$$ and $$\begin{aligned}
\label{D}
D(\rho_{A,I})=\frac{1}{4}\{\frac{(c_1)^2}{(e^{-w/T}+1)}
+\frac{(c_2)^2}{(e^{-w/T}+1)}+\frac{(c_3)^2}{(e^{-w/T}+1)^2}
\nonumber \\
-\max[\frac{(c_1)^2}{(e^{-w/T}+1)},\frac{(c_2)^2}{(e^{-w/T}+1)},
\frac{(c_3)^2}{(e^{-w/T}+1)^2}]\},\end{aligned}$$ respectively.
It is interesting to note that $$\begin{aligned}
N(\rho_{A,I})=\frac{1}{16}\langle B_{max}\rangle_{\rho_{A,I}}^2.\end{aligned}$$ We plot $N(\rho_{A,I})$ versus $\langle B_{max}\rangle_{\rho_{A,I}}$ in Fig.\[NB relation\], which shows that $N(\rho_{A,I})$ increases monotonously as $\langle B_{max}\rangle_{\rho_{A,I}}$ increases and it vanishes at zero point of $\langle B_{max}\rangle_{\rho_{A,I}}$. It is well known that Bell inequality must be obeyed by local realism theory, but may be violated by quantum mechanics. If we get $\langle B_{max}\rangle_{\rho_{A,I}}>2$, it means that the violation of Bell-CHSH inequality, which tells us that there exists nonlocal quantum correlation. But when $\langle
B_{max}\rangle_{\rho_{A,I}}\leq2$, it doesn’t mean that no quantum correlation exists, at leat for some mixed states, which have quantum correlation but obey the Bell inequality. So we can’t be sure that whether quantum correlation exists or not when $\langle
B_{max}\rangle_{\rho_{A,I}}\leq2$. However, the MIN, which is an indicator of the global effect caused by locally invariant measurement, is introduced to quantify nonlocality, and nonzero MIN means existence of nonlocality. And form Fig.\[NB relation\] we see that the MIN persists for all $\langle
B_{max}\rangle_{\rho_{A,I}}$ except for zero. Thus, the MIN, understood as some kind of correlations, is more general than the quantum nonlocality related to violation of the Bell’s inequalities.
![(Color online) The MIN of state $\rho_{A,I}$ as function of the maximally possible value of the Bell-CHSH expectation value.[]{data-label="NB relation"}](NandB-4.eps){width="50.00000%"}
From Eqs.(\[MIN for AI\]) and (\[D\]), we can see that the MIN is proportional to the two largest eigenvalues of the matrix $TT^t$, while the geometric discord is proportional to the two smallest eigenvalues of it, so we know that the MIN should be always equal or larger than the geometric discord. In Fig. \[ND relation\] we plot the MIN versus the geometric discord for the Werner ($|c_1|=|c_2|=|c_3|=c$) states. It is shown that the MIN increases monotonously as the geometric discord increases, and it is always equal or larger than the geometric discord. So as the quantum resource, the MIN is more robust than geometric discord.
![(Color online) The MIN of state $\rho_{A,I}$ as a function of geometric discord $D(\rho_{A,I})$ for the Werner sates, and the red solid line represents $N(\rho_{A,I})=D(\rho_{A,I})$.[]{data-label="ND relation"}](NandD-5.eps){width="55.00000%"}
MIN shared by Alice and Anti-Rob
--------------------------------
Now we consider the MIN between modes $\mathrm{A}$ and $\mathrm{II}$. By tracing over all modes in region $\mathrm{I}$, we get $$\begin{aligned}
\label{AII Bloch representation}
\rho_{A,II}=\frac{1}{4}\left( \mathbf{1}_A\otimes\mathbf{1}_{II}
+c'_0\mathbf{1}_A\otimes\sigma^{(II)}_3
+\sum^3_{i=1}c'_i\sigma^{(A)}_i\otimes\sigma^{(II)}_i \right),\end{aligned}$$ where $c'_0=\frac{1}{(e^{-w/T}+1)}$, $c'_1=\frac{c_1}{(e^{w/T}
+1)^{\frac{1}{2}}}$, $c'_2=\frac{-c_2}{(e^{w/T}+1)^{\frac{1}{2}}}$ and $c'_3=\frac{-c_3}{(e^{w/T}+1)}$. Similarly, the MIN of state $\rho_{A,II}$ can be obtained according to Eq.(\[MIN\]), which is $$\begin{aligned}
\label{MIN for AII}
N(\rho_{A,II})=\frac{1}{4}\{\frac{(c_1)^2}{(e^{w/T}+1)}
+\frac{(c_2)^2}{(e^{w/T}+1)}+\frac{(c_3)^2}{(e^{w/T}+1)^2}
\nonumber \\
-\min[\frac{(c_1)^2}{(e^{w/T}+1)},\frac{(c_2)^2}{(e^{w/T}+1)},
\frac{(c_3)^2}{(e^{w/T}+1)^2}]\}.\end{aligned}$$
\(i) If $|c_1|,|c_2|\geq|c_3|$, the MIN increases monotonously as the Unruh temperature increases provided taking fixed $c_i$.
\(ii) For the case of $|c_3|>\min\{|c_1|,|c_2|\}$ and both $c_1$ and $c_2$ don’t equal to 0 at the same time, if $\min\{|c_1|,|c_2|\}\leq\frac{\sqrt{2}}{2}|c_3|$, the MIN has a peculiar dynamics with a sudden change at $T_{sc}$ $$\begin{aligned}
\label{TSC2}
T_{sc}=\frac{w}{\ln(\frac{|c_3|^2}{min\{|c_1|^2,|c_2|^2\}}-1)}.\end{aligned}$$ Otherwise, the MIN increases monotonously with the increase of the Unruh temperature.
\(iii) Finally, if $|c_1|=|c_2|=0$, we have a monotonic increase of $N(\rho_{A,II})$ as the Unruh temperature increases.
We plot $N(\rho_{A,II})$ versus the Unruh temperature in Fig. \[MIN2\]. It is found that the MIN, as the Unruh temperature approaches to the infinite, is close to $$\begin{aligned}
\lim_{T\rightarrow\infty}N(\rho_{A,II})=\frac{1}{16}\{2(c_1)^2
+2(c_2)^2+(c_3)^2-\min[2(c_1)^2,2(c_2)^2,(c_3)^2]\},\end{aligned}$$ which is the same as $\lim_{T\rightarrow\infty}N(\rho_{A,I})$. In addition, as $T=0$ the MIN vanishes, which means that the correlation between $\mathrm{A}$ and $\mathrm{II}$ is local when the observers are inertial.
![(Color online) The MIN of state $\rho_{A,II}$ as a function of Unruh temperature $T$. We take parameters $c_1=1$, $c_2=0.9$ and $|c_3|\leq|c_1|,|c_2|$ for red dashed line; $c_1=0.9$,$c_2=0.55$ and $c_3=1$ for blue solid line. The insert shows the detail of sudden change.[]{data-label="MIN2"}](MIN2-6.eps){width="50.00000%"}
When $|c_1|<|c_2|$, by taking fixed $c_3$, we plot $T_{sc}$ as a function of $c_1$ in Fig. \[Tsc1\]. We learn from the figure that, unlike the Fig.\[Tsc\], $T_{sc}$ increases monotonously with the increase of $c_1$. That is to say, the bigger $c_1$ is, the sudden change behavior occurs latter. And when $|c_2|<|c_1|$, it is also important to note that as $|c_2|$ increases $T_{sc}$ increases monotonously too.
![(Color online) The $T_{sc}$ as a function of $c_1$, here we take $|c_1|\leq|c_2|$ and $c_3=0.9$.[]{data-label="Tsc1"}](TSC1-7.eps){width="50.00000%"}
How the prepared states affect the MIN for case (i) is shown in Fig. \[MINc1\], which tells us that $N(\rho_{A,II})$ increases monotonously as $|c_i|$ ($i=1,2$) increases. And for the case that $|c_3|>\min\{|c_1|,|c_2|\}$, the MIN is independent of $|c_3|$ but dependent of $|c_1|$, $|c_2|$ before $T_{sc}$, while after $T_{sc}$ it depends on $|c_3|$ and $\max\{|c_1|,|c_2|\}$. However, no matter which case, the MIN increases with the increase of $|c_i|$.
![(color online) The MIN of state $\rho_{A,II}$ as function of $c_1$ and $c_2$ with $|c_1|,|c_2|\geq|c_3|$. Here we take fixed $T=\infty,2,1$ from top to bottom, respectively.[]{data-label="MINc1"}](MINc1-8.eps){width="55.00000%"}
From the above discussions, we know that the Unruh effect can induce the degradation for $N(\rho_{A,I})$, but the increase for $N(\rho_{A,II})$. However, $N(\rho_{A,I})+N(\rho_{A,II})$ has different dynamics for different classes of states: (i) When $|c_1|,|c_2|\geq|c_3|$, $N(\rho_{A,I})+N(\rho_{A,II})$ is independent of the Unruh temperature. That is to say, $N(\rho_{A,I})+N(\rho_{A,II})$ is a constant versus the Unruh temperature for this class of states; (ii) When $|c_3|>\min\{|c_1|,|c_2|\}\geq\frac{\sqrt{2}}{2}|c_3|$, with the increase of the Unruh temperature $N(\rho_{A,I})+N(\rho_{A,II})$ decreases monotonously until $$\begin{aligned}
T_{sc}=\frac{-w}{\ln(\frac{|c_3|^2}{min\{|c_1|^2,|c_2|^2\}}-1)},\end{aligned}$$ and from then on it remains constant; And (iii) when $\min\{|c_1|,|c_2|\}<\frac{\sqrt{2}}{2}|c_3|$, $N(\rho_{A,I})+N(\rho_{A,II})$ decays quickly until $$\begin{aligned}
T_{sc}=\frac{w}{\ln(\frac{|c_3|^2}{min\{|c_1|^2,|c_2|^2\}}-1)},\end{aligned}$$ and after that it decays relatively slowly. We plot these dynamical behaviors in Fig. \[SUMN\].
![(Color online) The sum of $N(\rho_{A,I})$ and $N(\rho_{A,II})$ as a function of the Unruh temperature. We take $c_1=1,c_2=0.9$ and $|c_3|\leq|c_1|,|c_2|$ for the red dashed line; $c_3=1,c_1=0.9$ and $c_2=0.85$ for the blue solid line, and $c_3=1,c_1=0.9$ and $c_2=0.5$ for the yellow solid line.[]{data-label="SUMN"}](SUMN-9.eps){width="50.00000%"}
conclusions
===========
The effect of the prepared states and Unruh temperature on the MIN of Dirac fields was investigated and the following new properties were found: (i) The MIN $N(\rho_{A,I})$ for the X-type states decreases as the Unruh temperature increases, but $N(\rho_{A,II})$ increases with the increase of the Unruh temperature. (ii) For fixed Unruh temperature, it is found that the MIN always increases as $|c_i|$ $(i=1,2,3)$ increases, and it takes the maximal value for the Bell basis states. (iii) Both $N(\rho_{A,I})$ and $N(\rho_{A,II})$ have a peculiar dynamics with a sudden change at $T_{sc}$ provided $c_i$ appropriately chosen, the $T_{sc}$ for $N(\rho_{A,I})$ decreases as $c_i$ increases, while it is contrary for $N(\rho_{A,II})$. (iv) The MIN is more general than the quantum nonlocality related to violation of Bell’s inequalities. Besides, it increases as the geometric discord increases, and it is always equal or larger than the geometric discord. And (v) $N(\rho_{A,I})+N(\rho_{A,II})$ has three kinds of dynamics: (a) When $|c_1|,|c_2|\geq|c_3|$, it is independent of the Unruh temperature; (b) When $|c_3|>\min\{|c_1|,|c_2|\}\geq\frac{\sqrt{2}}{2}|c_3|$, with the increase of the Unruh temperature it decreases monotonously until $T_{sc}=\frac{-w}{\ln(\frac{|c_3|^2}{min\{|c_1|^2,|c_2|^2\}}-1)}$, and from then on it remains constant; And (c) when $\min\{|c_1|,|c_2|\}<\frac{\sqrt{2}}{2}|c_3|$, with the increase of the Unruh temperature, it decays quickly until $T_{sc}=\frac{w}{\ln(\frac{|c_3|^2}{min\{|c_1|^2,|c_2|^2\}}-1)}$, and after that it decays relatively slowly.
Here we just simply discuss the relation between the MIN and the maximal expectation values of CHSH inequality and geometric discord. More detailed study of this relation can help us to ont only understand the MIN more clearly, but also distinguish difference of quantum resource based on different correlation measurements. Such topics are left for future research.
This work was supported by the National Natural Science Foundation of China under Grant No. 11175065, 10935013; the National Basic Research of China under Grant No. 2010CB833004; the SRFDP under Grant No. 20114306110003; PCSIRT, No. IRT0964; the Hunan Provincial Natural Science Foundation of China under Grant No 11JJ7001; and Construct Program of the National Key Discipline.
[99]{}
P. M. Alsing, and G. J. Milburn, Phys. Rev. Lett. [**91**]{}, 180404 (2003).
P. M. Alsing I. Fuentes-Schuller, R. B. Mann, and T. E. Tessier, Phys. Rev. A. [**74**]{}, 032326 (2006).
I. Fuentes-Schuller, and R. B. Mann, Phys. Rev. Lett. [**95**]{}, 120404 (2005).
Y. Ling, S. He, W. G. Qiu, and H. B. Zh, Journal of Physics A Mathematical and Theoretical. [**40**]{}, 9025 (2007).
S. Moradi, Phys. Rev. A. [**79**]{}, 064301 (2009).
Q. Y. Pan, and J. L. Jing, Phys. Rev. A. [**77**]{}, 024302 (2008); Q. Y. Pan, and J. L. Jing, Phys. Rev. D. [**78**]{}, 065015 (2008); J. C. Wang, and J. L. Jing, Phys. Rev. A. [**83**]{}, 022314 (2011); J. C. Wang, and J. L. Jing, Annals of Physics. [**2**]{}, 327, 283-291 (2011).
M. Montero, and E. Martín-Martínez, Phys. Rev. A. [**84**]{}, 012337 (2011); D. E. Bruschi, J. Louko, E. Martín-Martínez, A. Dragan, and I. Fuentes, Phys. Rev. A. [**82**]{}, 042332 (2010); J. Leon, and E. Martín-Martínez, Phys. Rev. A. [**80**]{}, 012314 (2009).
J. C. Wang, J. F. Deng, and J. L. Jing, Phys. Rev. A. [**81**]{}, 052120 (2010); Z. H. Tian, and J. L. Jing, Phys. Lett. B. [**707**]{}, 264-271 (2012).
A. Datta, Phys. Rev. A. [**80**]{}, 052304 (2009).
M. X. Han, S. J. Olson, and J. P. Dowling, Phys. Rev. A. [**78**]{}, 022302 (2008).
David C. M. Ostapchuk, and R. B. Mann, Phys. Rev. A. [**79**]{}, 042333 (2009).
R. B. Mann, and V. M. Villalba, Phys. Rev. A. [**80**]{}, 022305 (2009).
C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. [**70**]{}, 1895 (1993).
R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki. Rev. Mod. Phys. [**81**]{}, 865 (2009).
*The Physics of Quantum Information*, edited by D. Bouwmeester, A. Ekert, A. Zeilinger (Springer-Verlag, Berlin, 2000).
L. Bombelli, R. K. Koul, J. Lee, and R. Sorkin. Phys. Rev. D [**34**]{}, 373 (1986).
S. W. Hawking, Commun. Math. Phys. [**43**]{}, 199 (1975); Phys. Rev. D [**14**]{}, 2460 (1976); H. Terashima, Phys. Rev. D [**61**]{}, 104016 (2000).
N. Friis, P. Köhler, E. Martín-Martínez, and R. A. Bertlmann. Phys. Rev. A. [**84**]{}, 062111 (2011).
A. Smith, and R. B. Mann. arxiv: 1107. 4633.
D. Park, arXiv:1201. 1335.
J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. [**23**]{}, 880-884 (1969).
G. Syetlichny, Phys. Rev. D. [**35**]{}, 3066 (1987).
Sh. L. Luo, and Sh. Sh. Fu, Phys. Rev. Lett. [**106**]{}, 120401 (2011).
Zh. J. Xi, X. G. Wang, and Y. M. Li, arxiv:1112. 0736.
S. Y. Mirafzali, I. Sargolzahi, A. Ahanj, K. Javidan, and M. Sarbishaei, arxiv:1110. 3499.
A. Sen, D. Sarkar, and A. Bhar, arxiv:1201. 2041.
Y. Guo, and J. Ch. Hou, arxiv:1107. 0355.
M. L. Hu, and H. Fan, arxiv:1201. 6430.
G. F. Zhang, H. Fan, Ai. L. Ji, and W. M. Liu, arxiv:1201. 1949.
W. G. Unruh, Phys. Rev. D. [**14**]{}, 870 (1976).
B. Dakić, V. Vedral, and Č. Brukner, Phys. Rev. Lett. [**105**]{}, 190502 (2010).
D. Girolami, and G. Adesso, Phys. Rev. A. [**83**]{}, 052108 (2011).
[^1]: Corresponding author, Email: jljing@hunnu.edu.cn
|
---
abstract: 'This paper is devoted to study the modified holographic dark energy model by taking its different aspects in the flat Kaluza-Klein universe. We construct the equation of state parameter which evolutes the universe from quintessence region towards the vacuum. It is found that the modified holographic model exhibits instability against small perturbations in the early epoch of the universe but becomes stable in the later times. We also develop its correspondence with some scalar field dark energy models. It is interesting to mention here that all the results are consistent with the present observations.'
author:
- |
M. Sharif [^1] and Abdul Jawad[^2]\
Department of Mathematics, University of the Punjab,\
Quaid-e-Azam Campus, Lahore-54590, Pakistan.
title: '**Reconstruction of Scalar Field Dark Energy Models in Kaluza-Klein Universe**'
---
**Keywords:** Kaluza-Klein cosmology; Modified holographic dark energy; Scalar field models.\
**PACS:** 04.50.Cd; 95.36.+x.
Introduction
============
During the last decades, astrophysicists and astronomers have made remarkable predictions about the gap of more than $70$ percent of the overall energy density in the universe. Based on recent observations [@1; @2], it has been made consensus that dark energy (DE) fills this gap. It is the unknown force having large negative pressure and referred to be responsible for accelerated expansion of the universe. Many efforts have been made about its identity but its nature is still unknown. The most obvious candidate of DE is the cosmological constant (or vacuum energy) but it suffers two major problems [@2a]. It is an interesting as well as the most challenging problem to find the best fit model of DE.
There are different dynamical DE models out of which holographic dark energy (HDE) model is the most prominent. It is proposed in the scenario of quantum gravity on the basis of holographic principle [@3]. In fact, the derivation of this model is based on the argument [@4] that *the vacuum energy (or the quantum zero-point energy) of a system with size $L$ should always remain less than the mass of a black hole with the same size due to the formation of black hole in quantum field theory*. In mathematical form, we have $\rho_{\Lambda}=\frac{3c^{2}}{8\pi GL^{2}}$, known as HDE density [@5; @6]. Here the constant $3c^2$ is used for convenience, $L$ represents the infrared (IR) cutoff and $G$ is the gravitational constant.
The choice of IR cutoff in the HDE model is very crucial. Li [@6] remarked that instead of Hubble or particle horizons, the future event horizon should be the IR cutoff which shows compatibility with the present evolution of the universe. Later on, it was pointed out [@7] that this choice of IR cutoff suffers the causality problem and proposed the age of the universe as IR cutoff, called agegraphic DE model. Some other proposals have also been given for the choice of IR cutoff. Granda and Oliveros [@8] proposed that IR cutoff should be the function of square of the Hubble parameter and its derivative. This type of IR cutoff is motivated from the Ricci scalar of the FRW universe [@8a].
The scalar field DE models also belong to the family of dynamical DE models which explain the DE phenomenon. A wide variety of these models exists in literature including quintessence, K-essence, tachyon, phantom, ghost condensates and dilaton [@9; @10]. Also, the well-known theories such as the supersymmetric, string and M theories cannot describe potential of the scalar field independently. It would be interesting to reconstruct the potential of DE models so that the scalar fields may describe the cosmological behavior of the quantum gravity.
The modification in the gravitational part and enhancement of dimensions in the original general relativity is another way to handle the DE puzzle. In higher dimensional theories, Kaluza-Klein (KK) theory [@11] has been used extensively for the purpose of cosmological implications. This theory has been described into ways, i.e., compact and non-compact forms depending on its fifth dimension. In its compact form, fifth dimension is like a circle having very small radius while in non-compact form, it behaves as a vacuum of 4D geometry. Moreover, the HDE has also been derived in higher dimensions with the help of the $N$-dimensional mass of the Schwarzschild black hole [@12] known as modified holographic dark energy (MHDE) model [@13; @14]. Sharif et al. [@15]-[@17] have investigated the evolution as well as generalized second law of thermodynamics of MHDE with Hubble and future event horizons as IR cutoffs in the flat and non-flat KK universe models. In a recent paper [@18], the scalar field models are constructed for HDE with Hubble horizon and Granda and Oliveros cutoff as IR cutoff in flat and non-flat universes.
Here we use the Granda and Oliveros [@8] IR cutoff for MHDE in flat KK universe. We discuss the evolution, instability and scalar field DE models in this scenario. The paper is organized as follows. Section **2** contains discussion of the evolution and instability of MHDE in flat KK universe. In section **3**, we reconstruct scalar field models of MHDE. The last section summarizes the results.
Modified Holographic Dark Energy
================================
In this section, we make analysis of the equation of state (EoS) parameter and instability of MHDE in compact flat KK universe [@19] whose metric is given by $$\label{1}
ds^{2}=-dt^{2}+a^{2}(t)[dr^{2}
+r^{2}(d\theta^{2}+\sin\theta^{2}d\phi^{2})+d\psi^{2}],$$ where $a(t)$ indicates the cosmic scale factor. The corresponding field equations are $$\begin{aligned}
\label{2}
H^2&=&\frac{1}{6}\rho_{\Lambda},\\\label{3}
\dot{H}+2H^2&=&-\frac{1}{3} p_{\Lambda},\end{aligned}$$ where $H$ is the Hubble parameter, dot indicates differentiation with respect to time and $8\pi G=1$ for the sake of simplicity. Also, $p_{\Lambda}$ and $\rho_{\Lambda}$ are the pressure and energy density due to DE respectively. In order to derive MHDE, we use the formula of the mass of the Schwarzschild black hole in $N$ dimensions $$\begin{aligned}
M=\frac{(N-1)A_{N-1}}{16\pi G_{N}}r^{N-2}_{H},\end{aligned}$$ where $8\pi G_{N}\equiv M_{*}$ is Planck mass in higher dimensions, $A_{N-1}$ is the area of $N$ unit spheres and $r_{H}$ represents the size of the black hole. For KK universe, we take $N=4$ and use the formula of area, it follows that $$\begin{aligned}
M=\frac{3\pi^2 M^3_{*}}{2}r^{5}_{H}.\end{aligned}$$ Here, we assume $r_{H}=L=(\mu H^2+\lambda\dot{H})^{-\frac{1}{2}}$ ($\mu$ and $\lambda$ are positive constants, this IR cutoff is proposed by Granda and Oliveros [@8]) and $M_{*}$ should be less than the Planck length. This shows that horizon scale does not make the mass of five dimensional black hole larger than compactification scale of KK universe (i.e., of the order of Planck length), while it reduces its mass. With the help of Cohen et al. [@4] relation, we can derive MHDE in the following form [@15] $$\label{4}
\rho_{\Lambda}=3\pi^{2}L^2=\frac{3\pi^{2}}{\mu H^2+\lambda\dot{H}},$$ here we take $M_{*}$ to be unity.
The equation of continuity for the MHDE become $$\begin{aligned}
\label{5}
\dot{\rho}_{\Lambda}+4H(\rho_{\Lambda}+p_{\Lambda})=0.\end{aligned}$$ Equations (\[2\]) and (\[4\]) lead to $$\label{6}
\frac{dH^4}{dx}+\frac{4\mu}{\lambda}H^4=\frac{2\pi^2}{\lambda},$$ where $x=\ln a$. Solving the above equation, we obtain $$\label{7}
H^4=\frac{\pi^2}{2\mu}+be^{\frac{-4\mu}{\lambda}x},$$ where $b$ is an integration constant. The EoS parameter for MHDE can be obtained by using Eqs.(\[4\]), (\[5\]) and (\[7\]) $$\begin{aligned}
\label{8}
\omega_{\Lambda}=-1+\frac{\mu^2
be^{\frac{-4\mu}{\lambda}x}}{\lambda(\pi^2+2\mu
be^{\frac{-4\mu}{\lambda}x})}.\end{aligned}$$
We plot $\omega_{\Lambda}$ versus redshift parameter $z$ by using $a=a_{0}(1+z)^{-1}$ as shown in Figure $\textbf{1}$ by taking $\mu=0.7,~b=0.5,~a_{0}=1$ and different values of $\lambda=0.4,~0.5,~0.6$. It is observed that the EoS parameter evolutes the universe from quintessence DE era towards vacuum DE era. This also shows that the parameter $\omega_{\Lambda}$ always remains in the quintessence region for $\lambda\geq0.5$. However, it corresponds to the early inflation era for $\lambda<0.5$ in the early time.
Now we explore the linear perturbation in order to examine the instability of MHDE. For this purpose, the square of the speed of sound ($\upsilon^2_s$) is evaluated which characterizes the stability of DE models. The speed of sound has the form [@2a] $$\label{9}
\upsilon_{s}^2=\frac{\dot{p}}{\dot{\rho}}=\frac{p'}{\rho'},$$ where prime shows differentiation with respect to $x$. We would like to mention here that DE models are classically unstable as $\upsilon^2_s<0$ and vice versa. Equations (\[4\]), (\[5\]), (\[7\]) and (\[9\]) yield $$\begin{aligned}
\label{10}
\upsilon^2_{s}&=&\frac{\pi^2(\mu-\lambda)-\mu
b(2\lambda-\mu)e^{\frac{-4\mu}{\lambda}x}}{\lambda(\pi^2+2\mu
be^{\frac{-4\mu}{\lambda}x})}.\end{aligned}$$ We plot the speed of sound versus $z$ as shown in Figure **2** by keeping the same values of constant parameters as given above. This shows that the MHDE remains unstable in the early epoch and stable for the present and later time.
Correspondence of MHDE with Scalar Field Models
===============================================
The current prediction of accelerated expansion of the universe has roots in the early inflationary epoch. It is argued that the inflation field has similar properties as the cosmological constant. Moreover, inspired by the inflation field, scalar field models have been proposed. Scalar field models are used in explaining the DE phenomenon due to their dynamical nature. These models describe the quintessence behavior of the universe and also provide the effective description of DE models through reconstruction scenario. In our case, the EoS for MHDE obeys this argument and hence it would be interesting to reconstruct the scalar field DE models in this scenario. We discuss all the results graphically with the same values of the constant parameters as given in the previous section.
Quintessence Dark Energy Model
------------------------------
This model is originated in the light of scalar field $\phi$ which is minimally coupled with gravity. The energy density and pressure of the quintessence DE model are given by [@10] $$\begin{aligned}
\label{11}
\rho_{q}=\frac{1}{2}\dot{\phi}^{2}+V(\phi),\quad
p_{q}=\frac{1}{2}\dot{\phi}^{2}-V(\phi),\end{aligned}$$ where $\dot{\phi}^{2}$ is the kinetic energy and $V(\phi)$ is the potential of scalar field. This equation gives the EoS parameter as $$\omega_{q}=\frac{\dot{\phi}^{2}-2V(\phi)}{\dot{\phi}^{2}+2V(\phi)}.$$ We identify $\rho_{q}=\rho_{\Lambda}$ and $p_{q}=p_{\Lambda}$ to establish the correspondence between MHDE and quintessence scalar field. Thus, it follows from Eq.(\[11\]) that $$\begin{aligned}
\label{12}
\dot{\phi}^{2}&=&\frac{\mu\sqrt{6b}e^{\frac{-2\mu}{\lambda}x}}{\sqrt{\lambda(\pi^2+2\mu
be^{\frac{-4\mu}{\lambda}x})}},\\\label{13}
V(\phi)&=&\frac{6\pi^2\lambda-3b\mu(\mu-4\lambda)e^{\frac{-4\mu}{\lambda}x}}
{\lambda\sqrt{2\mu(\pi^2+2\mu ce^{\frac{-4\mu}{\lambda}x})}}.\end{aligned}$$ The value of scalar field turns out to be $$\begin{aligned}
\label{14}
\phi=\int^{x}_{0}\sqrt{\frac{6\mu^2be^{\frac{-4\mu}{\lambda}x}}{\lambda(\pi^2+2\mu
be^{\frac{-4\mu}{\lambda}x})}}dx.\end{aligned}$$ The evolutionary trajectories of the scalar field $\phi$ and the corresponding scalar potential are shown in Figures **3** and **4**. Here we use the initial condition of the scalar field $\phi(0)=0$. We notice that the quintessence field increases initially but becomes flat at high redshift. This shows that the field decreases gradually with the expansion of the universe. Also, we observe that the quintessence potential becomes more steeper with the decrease of $\lambda$ and tends to flat in the later epoch.
Tachyon Dark Energy Model
-------------------------
The energy and pressure of the tachyon DE model has the form [@10] $$\begin{aligned}
\label{15}
\rho_{t}=\frac{V(\phi)}{\sqrt{1-\dot{\phi}^{2}}},\quad
p_{t}=-V(\phi)\sqrt{1-\dot{\phi}^{2}}.\end{aligned}$$ The corresponding EoS parameter is $$\label{16}
\omega_{t}=\dot{\phi}^{2}-1.$$ The correspondence between MHDE and tachyon model is obtained for $\rho_{t}=\rho_{\Lambda}$ and $p_{t}=p_{\Lambda}$, which leads to $$\begin{aligned}
\label{17}
\dot{\phi}^{2}&=&\frac{\mu^2be^{\frac{-4\mu}{\lambda}x}}{\lambda(\pi^2+2\mu
ce^{\frac{-4\mu}{\lambda}x})},\\\label{18}
V(\phi)&=&\sqrt{\frac{18(\pi^2\lambda+\mu
b(2\lambda-\mu)e^{\frac{-4\mu}{\lambda}x})}{\mu\lambda}}.\end{aligned}$$ Equations (\[7\]) and (\[17\]) give $$\begin{aligned}
\label{19}
\phi'(x)=\left(\frac{2\mu}{\pi^2+2\mu
ce^{\frac{-4\mu}{\lambda}x}}\right)^{\frac{1}{4}}
\sqrt{\frac{\mu^2be^{\frac{-4\mu}{\lambda}x}}{\lambda(\pi^2+2\mu
be^{\frac{-4\mu}{\lambda}x})}}.\end{aligned}$$ We solve and plot it against $z$ as shown in Figure $\textbf{5}$. The evolution of tachyon field is very much similar to the quintessence field. The tachyon field attains the maximum value at early epoch and then decreases and goes towards zero at the present epoch. Also, we remark that the tachyon field rolls down potential slowly with the expansion of the universe as shown in Figure **6**.
K-essence Dark Energy Model
---------------------------
The energy density and pressure of this model are [@10] $$\begin{aligned}
\label{20}
\rho_{k}=V(\phi)(-\chi+3\chi^{2}),\quad
p_{k}=V(\phi)(-\chi+\chi^{2}),\end{aligned}$$ where $\chi=\frac{1}{2}~\dot{\phi}^2$ and $V(\phi)$ represents the scalar potential of K-essence model. This leads to the following EoS parameter for tachyon DE model $$\label{21}
\omega_{k}=\frac{1-\chi}{1-3\chi}.$$
We set $\rho_{k}=\rho_{\Lambda}$ and $p_{k}=p_{\Lambda}$ for the correspondence between MHDE and K-essence model and obtain $$\begin{aligned}
\label{22}
\chi&=&\frac{2\pi^2\lambda+\mu
b(4\lambda-\mu)e^{\frac{-4\mu}{\lambda}x}}{4\pi^2\lambda+\mu
b(8\lambda-3\mu)e^{\frac{-4\mu}{\lambda}x}},\\\label{23}
V(\phi)&=&\frac{3(1-3\omega_{\Lambda})^2}{1-\omega_{\Lambda}}\left(\frac{2\mu}{\pi^2+2\mu
ce^{\frac{-4\mu}{\lambda}x}}\right)^{\frac{1}{2}}.\end{aligned}$$
Finally, the expression $\chi=\frac{1}{2}\dot{\phi}^2$ leads to $$\begin{aligned}
\label{24}
\phi'(x)=\left(\frac{2\mu}{\pi^2+2\mu
ce^{\frac{-4\mu}{\lambda}x}}\right)^{\frac{1}{4}}\sqrt{\frac{2(2\pi^2\lambda+\mu
b(4\lambda-\mu)e^{\frac{-4\mu}{\lambda}x})}{4\pi^2\lambda+\mu
b(8\lambda-3\mu)e^{\frac{-4\mu}{\lambda}x}}}.\end{aligned}$$ Figure $\textbf{7}$ shows that the K-essence scalar field decreases with the increment of MHDE parameter $\lambda$. The EoS of tachyon DE model is compatible with the accelerated expansion of the universe in the range $\frac{1}{3}<\omega_{\Lambda}<\frac{2}{3}$. It is noted that the kinetic term is consistent with this EoS for $\lambda=0.5,~0.6$, but it is inconsistent for $\lambda\leq0.4$ as shown in Figure $\textbf{8}$. Also, the plot $\textbf{9}$ indicates that the K-essence potential increases with the increase of the field $\phi$. It rolls down the potential because K-essence scalar field decreases with the expansion of the universe.
Dilaton Dark Energy Model
-------------------------
The Lagrangian of dilaton field describes the pressure of scalar field given by [@9] $$\label{25}
p_d=-\chi+me^{n\phi}\chi^2,$$ where $m$ and $n$ are taken as positive constants. The corresponding energy density is $$\begin{aligned}
\label{26}
\rho_{d}&=&-\chi+3me^{n\phi}\chi^{2}.\end{aligned}$$ This model has the EoS parameter $$\label{27}
\omega_{d}=\frac{-1+me^{n\phi}\chi}{-1+3m~e^{n\phi}\chi}.$$ The replacement of $\rho_{d}$ by $\rho_{\Lambda}$ and $p_{d}$ by $p_{\Lambda}$ (for correspondence) gives $$\begin{aligned}
\label{28}
e^{n\phi}\chi&=&\frac{2\pi^2\lambda+\mu
b(4\lambda-\mu)e^{\frac{-4\mu}{\lambda}x}}{m(4\pi^2\lambda+\mu
b(8\lambda-3\mu)e^{\frac{-4\mu}{\lambda}x})}.\end{aligned}$$ Its plot against $z$ with $m=1.5$ and $n=0.05$ is shown in Figure **10**. We observe that $e^{n\phi}\chi $ shows consistency with the expanding scenario of the universe predicated by EoS of this model. The solution of the above equation follows $$\begin{aligned}
\nonumber
\phi(x)&=&\frac{2}{n}\ln\left[1+\frac{n}{\sqrt{2m}}\int^{x}_{0}
\sqrt{\frac{2\pi^2\lambda+\mu
b(4\lambda-\mu)e^{\frac{-4\mu}{\lambda}x}}{4\pi^2\lambda+\mu
b(8\lambda-3\mu)e^{\frac{-4\mu}{\lambda}x}}}\right.\\\label{29}
&\times&\left.\left(\frac{2\mu}{\pi^2+2\mu
be^{\frac{-4\mu}{\lambda}x}}\right)^{\frac{1}{4}}dx\right].\end{aligned}$$ Its graphical presentation is shown in Figure **11** which exhibits direct proportionality with respect to $z$ leading to the scaling solutions for dilaton model [@9].
Summary
=======
It is believed that our universe expands with accelerated expansion and the scalar field DE models act as an effective theories of an underlying phenomenon of DE. We investigate the evolution of MHDE, instability and reconstructed scalar field DE models. We consider MHDE with IR cutoff as a function of the Hubble parameter and its derivative in the flat Kaluza-Klein universe. This type of DE density avoids the causality problem due to dependence on the local quantities. The MHDE parameter $\lambda$ plays the crucial role in evaluating the results. We have used the best fit values of $\mu=0.7$ and $\lambda=0.4,~0.5,~0.6$ obtained by Wang and Xu [@21] through type Ia supernovae, baryon acoustic oscillations, CMBR and the observational Hubble data. The results are summarized as follows.
- The EoS parameter $\omega_{\Lambda}$ shows transition from quintessence DE era ($-\frac{1}{3}<\omega<-1$) towards vacuum DE era ($\omega=-1$) as displayed in Figure $\textbf{1}$. This type of behavior has led to reconstruct the scalar field DE models. Also, our result of the present value of $\omega_{\Lambda}=-0.95$ at $z=0$ (for $\lambda=0.4,~0.5,~0.6$) is almost closer to the value obtained for HDE in GR [@6; @8; @22; @23].
- It is interesting to check the viability of the MHDE model due to its dependence upon local quantities. Using the squared speed of sound, we have found that the MHDE with new IR cutoff is unstable for early epoch but stable for the later time as shown in Figure $\textbf{2}$. It is mentioned here that the MHDE is always stable for $\lambda<0.35$, while in GR, the new HDE is unstable [@24]. In addition, the Chaplygin and tachyon Chaplygin gases are stable as $\upsilon^2_s>0$, but the holographic [@25], agegraphic [@26] and QCD ghost DE [@27] models are classically unstable as $\upsilon^2_s<0$.
- Finally, we have provided the correspondence of MHDE with scalar field DE models including quintessence, tachyon, K-essence and dilaton models. In all these models, the scalar field shows the decreasing behavior with the expansion of the universe. The scalar potential increases and becomes steeper with the increase of scalar field $\phi$ for quintessence, tachyon and K-essence DE models. The plots of scalar field DE models are given in Figures **3**$-$**11**. In K-essence and dilaton DE models, the kinetic terms exactly lie in the required region (where EoS parameter predicts the accelerated expansion of the universe). We would like to remark here that the results of scalar field and corresponding potential are consistent with the current status of the universe. These are also consistent with the results of Zhang [@22] for quintessence HDE, for tachyon HDE [@23; @28; @29] and Rozas-Fern$\acute{a}$ndez [@30] for dilaton HDE models.
[43]{}
Bennett, C. L. et al.: Astrophys. J. **148**(2003)1.
Tegmark, M. et al.: Phys. Rev. D **69**(2004)103501.
Peebles, P.J.E.: Rev. Mod. Phys. **75**(2003)559.
Susskind, L.: J. Math. Phys. **36**(1995)6377.
Cohen, A., Kaplan, D. and Nelson, A.: Phys. Rev. Lett. **82**(1999)4971.
Hsu, S.D.H.: Phys. Lett. B **594**(2004)13.
Li, M.: Phys. Lett. B **603**(2004)1.
Wei, H. and Cai, R.G.: Phys. Lett. B **660**(2008)113.
Granda, L. and Oliveros, A.: Phys. Lett. B **669**(2008)275.
Gao, C., Chen, X. and Shen, Y.G.: Phys. Rev. D **79**(2009)043511.
Piazza, F. and Tsujikawa, S.: JCAP **07**(2004)004.
Copeland, E.J., Sami, M. and Tsujikawa, S.: Int. J. Mod. Phys. D **15**(2006)1753.
Kaluza, T.: *Zum Unitatsproblem der Physik* Sitz. Press. Akad. Wiss. Phys. Math. k1 (1921)966; Klein, O.: Zeits. Phys. **37**(1926)895.
Myers, R.C.: Phys. Rev. D **35**(1987)455.
Gong, Y. and Li, T.: Phys. Lett. B **683**(2010)241.
Liu, D.J., Wang, H. and Yang, B.: Phys. Lett. B **694**(2010)6.
Sharif, M. and Khanum, F.: Gen. Relativ. Gravit. **43**(2011)2885.
Sharif, M. and Jawad, A.: Astrophys. Space Sci. **337**(2012)789.
Sharif, M. and Jawad, A.: Eur. Phys. J. C **72**(2012)1901.
Granda, L. and Oliveros, A.: Phys. Lett. B **671**(2009)199; Karami, K. and Fehri, J.: Phys. Lett. B **684**(2010)61; Sheykhi, A.: Phys. Rev. D **84**(2011)107302.
Ozel, C., Kayhan, H. and Khadekar, G.S.: Ad. Studies. Theor. Phys. **4**(2010)117.
Wang, Y and Xu, L.: Phys. Rev. D **81**(2010)083523.
Zhang, X.: Phys. Lett. B **648**(2007)1.
Setare, M.R.: Phys. Lett. B **653**(2007)116.
Malekjani, M and Khodam-Mohammadi, A.: Int. J. Mod. Phys. D **20**(2011)281.
Myung, Y.S.: Phys. Lett. B **652**(2007)223.
Kim, K.Y., Lee, H.W. and Myung, Y.S.: Phys. Lett. B **660**(2008)118.
Ebrahimi, E. and Sheykhi, A.: Int. J. Mod. Phys. D **20**(2011)2369.
Zhang, J., Zhang, X. and Liu, H.: Phys. Lett. B **651**(2007)84.
Rozas-Fern$\acute{a}$ndez, A., Brizuela, D. and Cruz, N.: Int. J. Mod. Phys. D **19**(2010)573.
Rozas-Fern$\acute{a}$ndez, A.: Eur. Phys. J. C **71**(2011)1536.
[^1]: msharif.math@pu.edu.pk
[^2]: jawadab181@yahoo.com
|
---
abstract: 'This paper reports the first measurements of solid-phase vacuum-ultraviolet (VUV) absorption cross sections of heavy isotopologues present in icy dust grain mantles of dense interstellar clouds and cold circumstellar environments. Pure ices composed of D$_2$O, CD$_{3}$OD, $^{13}$CO$_{2}$, and $^{15}$N$^{15}$N were deposited at 8 K, a value similar to the coldest dust temperatures in space. The column density of the ice samples was measured *in situ* by infrared spectroscopy in transmittance. VUV spectra of the ice samples were collected in the 120-160 nm (10.33-7.74 eV) range using a commercial microwave discharged hydrogen flow lamp as the VUV source. Prior to this work, we have recently submitted a similar study of the light isotopologues (Cruz-Diaz et al. 2013a; Cruz-Diaz et al. 2013b). The VUV spectra are compared to those of the light isotopologues in the solid phase, and to the gas phase spectra of the same molecules. Our study is expected to improve very significantly the models that estimate the VUV absorption of ice mantles in space, which have often used the available gas phase data as an approximation of the absorption cross sections of the molecular ice components. We will show that this work has also important implications for the estimation of the photodesorption rates per absorbed photon in the ice.'
author:
- |
G. A. Cruz-Diaz$^{1}$[^1], G. M. Muñoz Caro$^{1}$, and Y.-J. Chen$^{2,3}$\
$^{1}$Carretera de Ajalvir, km 4, Torrejon de Ardoz, 28850 Madrid, Spain\
$^{2}$Space Sciences Center and Department of Physics and Astronomy,\
University of Southern California, Los Angeles, CA 90089-1341, USA\
$^{3}$Department of Physics, National Central University, Jhongli City, Taoyuan Country 32054, Taiwan
date: 'Accepted 2013 ——–. Received 2013 ——–; in original form 2013 ——–'
title: 'Vacuum-UV absorption spectroscopy of interstellar ice analogs. Isotopic effects.'
---
\[firstpage\]
interstellar ice analogs – VUV-absorption cross section.
Introduction
============
After molecular hydrogen (H$_2$), the molecules H$_2$O, CO, CO$_2$, and CH$_3$OH, are among the most abundant in the interstellar medium, as it has been inferred from observations of the gas and solid phase (Mumma & Charnley 2011, and references therein). The main elements, and their corresponding isotopes, which compose most volatile molecules in the interstellar medium, are H:D, $^{12}$C:$^{13}$C, $^{14}$N:$^{15}$N, and $^{16}$O:$^{17}$O:$^{18}$O.
Deuterium enrichment can be the result of low temperature gas-grain reactions because of the differences in zero-point energies between deuterated and non-deuterated species (Wilson et al. 1973). Observation toward prestellar cores indicates, in gas phase, that abundances of singly deuterated molecules are typically higher than the cosmic atomic D/H ratio of 1.5 $\times$ 10$^{-5}$(Linsky 2003), also, doubly and triply deuterated molecules have been observed with D/H ratios reaching $\sim$ 30 % for D$_2$CO and $\sim$ 3 % for CD$_3$OH (see, Ceccarelli et al. 1998; Loinard et al. 2002; Parise et al. 2004; Ratajczak et al. 2009). Deuterated methanol molecules were detected in the gas phase toward low-mass class 0 protostars with abundances up to about 60 % relative to CH$_3$OH (Parise et al. 2006). D$_2$O has been detected toward the solar-type protostar IRAS 16293-2422 (Butner et al. 2007; Vastel et al. 2010). [@Roberts] showed that the multiply deuterated isotopologues of H$_3$$^+$ can efficiently transfer deuterons to other neutral molecules in very cold ($\leq$ 20 K) gas depleted of its CO (because the CO molecules are frozen onto refractory dust grain mantles).
Isotopic substitution often alters the chemical and physical properties of atoms and molecules, resulting in differences in absorption spectra and reaction rates. Therefore, measurements of the isotopic compositions of various species can be used to interpret the physico-chemical histories and the chemical reaction pathways in these environments. In particular, isotope effects in the non-dissociative photoionization region of molecular nitrogen play an important role in isotopic fractionation in planetary atmospheres and other environments (e.g., interstellar molecular clouds, the solar nebula, and in the atmospheres of Earth, Mars, and Titan) in which N$_2$ and VUV radiation are present (see, Croteau et al. 2011, and references therein).
Carbon dioxide is an important constituent of quiescent and star forming molecular clouds (Gerakines et al. 1999, and references therein). It is primarily present in the solid state (van Dishoeck et al. 1996). The $^{13}$CO$_2$ isotope has been detected with a two orders of magnitude lower abundance with respect to CO$_2$ (d’Hendecourt et al. 1996). The stretching band of $^{13}$CO$_2$ is an independent and sensitive probe of the ice mantle composition (Boogert et al. 2000, and references therein). Studies in the gas phase have shown that the $^{12}$C/$^{13}$C ratio increases with Galacto-centric radius (Wilson & Rood 1994; Keene et al. 1998). In the solid phase, the behavior of this ratio agrees with the gas phase studies (Boogert et al. 2000). Also, determination of this ratio is an important input for evolutionary models of the Galaxy, since $^{12}$C is produced by Helium burning by massive stars, which can be converted to $^{13}$C in the CNO cycle of low- and intermediate-mass stars at later times (Boogert et al. 2000).
The average cross section for a certain spectral range are useful when there is no information (flux, photodesorption rate) for each specific wavelength within that range. An example are the photodesorption rate values reported in the literature (e.g., Öberg et al. 2007, 2009; Muñoz Caro et al. 2010), which correspond to the full continuum emission spectrum of the hydrogen VUV lamps (an analog of the secondary UV emission in dense clouds). To estimate the photodesorption rate per absorbed photon in that case, what is used is the average photon energy and the average VUV absorption cross section in the same range. In addition, these average VUV absorption cross sections allow comparison with previous works that estimated those average values in similar spectral ranges in an indirect way (i.e., with no use of VUV spectroscopy), see e.g, Cottin et al. (2003).
Heavy isotopologues, in the context of laboratory astrophysics, are often used to study ice photoprocessing and, in particular ice photodesorption experimentally, to avoid problems with contamination (e.g., $^{13}$CO, $^{15}$N$_2$; see Oberg et al. 2007, 2009; Fayolle et al. 2013). In addition, it is interesting to search for differences in the absorption of these ices compared to those made of light isotopologues.
The estimation of the VUV-absorption cross sections of molecular ice components allows to calculate the photon absorption of icy grains in that range. In addition, the VUV-absorption spectrum as a function of photon wavelength is required to study the photo-desorption processes over the full photon emission energy range.
It is therefore important to study the physical and chemical properties of molecules containing heavy isotopes. This study focusses on the isotopic effects observed in the vacuum-ultraviolet (VUV) absorption spectra of three of the most abundant inter- and circumstellar species in the solid phase: H$_2$O, CH$_3$OH, and CO$_2$. Among the possible isotopologues, the fully deuterated ones were selected, D$_2$O and CD$_3$OD, in addition to $^{13}$CO$_2$. Also $^{15}$N$^{15}$N, henceforth abbreviated as $^{15}$N$_2$, was included in our study to explore the isotopic effects in a homonuclear diatomic molecule that was also observed in space (Bergin et al. 2002; Belloche & André 2003). Two recent papers report the VUV absorption cross sections of the light isotopologues in the ice (Cruz-Diaz et al. 2013a, 2013b), henceforth referred to as Papers I and II, respectively). The data in Paper I and II were used for comparison to this work. In addition, the VUV absorption spectra of the same molecules in the gas phase were also adapted to illustrate the differences between the gas and the solid phase samples.
Experimental protocol
=====================
The measurements were conducted using the Interstellar Astrochemistry Chamber (ISAC). This set-up and the standard experimental protocol were described in detail in [@Munoz]. ISAC mainly consists of an ultra-high-vacuum (UHV) chamber, with pressure typically in the range P = 3-4.0 $\times$ 10$^{-11}$ mbar at room temperature, where an ice layer is made by deposition of a gas species onto a cold finger at 8 K. The low temperature is achieved by means of a closed-cycle helium cryostat. The ice sample can be either UV-irradiated or warmed up to room temperature. The evolution of the solid sample was monitored with *in situ* Fourier transform infrared (FTIR) spectroscopy in transmittance and VUV spectroscopy. The chemical compounds used for the experiments described in this paper were: D$_2$O(liquid), Cambridge Isotope Laboratories, Inc (C.I.L.) 99.9%; CD$_3$OD(liquid), C.I.L. 99.8%; $^{13}$CO$_2$(gas), C.I.L. 99.0%; and $^{15}$N$_2$(gas), C.I.L. 98.0%.
The deposited ice layer was VUV irradiated using a microwave discharged hydrogen flow lamp (MDHL), from Opthos Instruments. The source has a UV-flux of $\approx 2 \times 10^{14}$ cm$^{-2}$ s$^{-1}$ at the sample position, measured by CO$_{2}$ $\to$ CO actinometry, see [@Munoz]. The Evenson cavity of the MDHL is refrigerated with air. The VUV spectrum was measured routinely [*in situ*]{} during the experiments with the use of a McPherson 0.2 meter focal length VUV monochromator (Model 234/302) with a photomultiplier tube (PMT) detector equipped with a sodium salicylate window, optimized to operate from 100-500 nm (11.27-2.47 eV), with a spectral resolution of 0.4 nm. The characterization of the MDHL spectrum was previously reported (Chen et al. 2010; Paper I) and was discussed in more detail by [@Chen2].
VUV spectroscopy {#VUV}
================
VUV absorption cross sections were obtained for pure ices composed of D$_{2}$O, CD$_{3}$OD, $^{13}$CO$_2$, and $^{15}$N$_{2}$. These measurements were performed following the procedure described in Paper I and summarized below. The column density of the deposited ice was calculated using FTIR spectroscopy in transmittance, according to the formula
$$%\hspace{3cm}
N = \frac{1}{\mathcal{A}} \int_{band} \tau_{\nu}d{\nu}
\label{1}$$
where $N$ is the column density of the ice, $\tau_{\nu}$ the optical depth of the band, $d\nu$ the wavenumber differential in cm$^{-1}$, and $\mathcal{A}$ is the band strength in cm molecule$^{-1}$. The VUV absorption cross section was estimated according to the Beer-Lambert law
$$% \hspace{3cm}
I_t(\lambda)=I_0(\lambda) {e}^{-\sigma(\lambda) N}
\label{2}$$
where $I_{t}(\lambda)$ is the transmitted intensity for a given wavelength $\lambda$, $I_{0}(\lambda)$ the incident intensity, $N$ is the column density in cm$^{-2}$ obtained using eq. \[1\], and $\sigma$ is the VUV absorption cross section in cm$^{2}$.
For each ice spectrum a series of three measurements was performed: i) the emission spectrum of the VUV-lamp was measured, to monitor the intensity of the main emission bands, ii) the emission spectrum transmitted by the MgF$_{2}$ substrate window was measured, to monitor its transmittance, and iii) the emission spectrum transmitted by the substrate window with the deposited ice on top was measured. The absorption spectrum of the ice corresponds to the spectrum of the substrate with the ice after subtraction of the bare MgF$_2$ substrate spectrum.
A priori, the VUV absorption cross section of the ice was not known. Therefore, several measurements for different values of the initial ice column density were performed to improve the spectroscopy. Table \[table1\] provides the infrared band positions and band strengths of D$_{2}$O, CD$_{3}$OD, and $^{13}$CO$_2$ used to estimate the column density. Solid $^{15}$N$_{2}$ does not display absorption features in the mid-infrared, therefore the column density of $^{15}$N$_{2}$ was thus measured using the expression
$$N = \frac{\rho_{N_2} \; d_H}{N_A \; m_{N_2}}
\label{3}$$
where $\rho_{N_2}$ is the density of the $N_2$ ice, see Table \[table1\], $m_{N_2}$ is the molar mass of the $N_2$ molecule, $N_A$ is the Avogadro constant (6.022 $\times$ 10$^{23}$ mol$^{-1}$), and $d_H$ is the ice thickness in cm. The latter was estimated following the classical interfringe relation
$$% \hspace{3cm}
d_H= \frac{1}{2n_H \Delta \nu}
\label{4}$$
where $n_H$ is the refractive index of the ice at deposition temperature, and $\Delta \nu$ is the wavenumber difference between two adjacent maxima or minima of the fringes observed in the infrared spectrum of the ice.
-- -- -- -- --
-- -- -- -- --
: Infrared band positions, infrared band strengths ($\mathcal{A}$), column density ($N$) in ML (as in previous works, one ML is here defined as 10$^{15}$ molecules cm$^{-2}$), and refractive index ($n_H$) of the samples used in this work. Pure $^{15}$N$_{2}$ ice does not display any features in the mid-infrared.
\
[$^{a,b}$ Calculated by the us, see Section \[VUV\], $^c$ [@Gerakines]]{}\
\[table1\]
No IR band strength values were found in the literature for the D$_2$O and CD$_3$OD species. These values were therefore calculated using eqs. \[4\], \[3\], and \[1\]. Refractive indices of solid H$_2$O, CH$_3$OH, and N$_2$ were used as an approximation (1.30, 1.39, and 1.21, respectively, see Mason et al. 2006, Hudgins et al. 1993, and Satorre et al. 2008). Error values for the column density in Table \[table1\] have been estimated taking into account the error in the calculation of the column density and the column density decrease by UV irradiation during the VUV spectral acquisition.
The main emission peaks of the MDHL fall at 121.6 nm (Lyman-$\alpha$), 157.8 nm, and 160.8 nm (molecular H$_2$ bands). These peaks are thus also present in the secondary VUV photon spectrum generated by cosmic rays in dense interstellar clouds and circumstellar regions where molecular hydrogen is abundant (Gredel et al. 1989). For this reason, the VUV absorption cross section values measured at these wavelengths are provided for each molecule in the following sections.
The VUV absorption cross section spectra of D$_2$O, CD$_3$OD, $^{13}$CO$_2$, and $^{15}$N$_2$ ices were fitted using the sum of two or more Gaussian profiles using an in-house IDL code. These fits correspond to the lowest $\chi^2$ values. Table \[tableGauss\] summarizes the Gaussian profile parameters used to fit the spectra of the different ice compositions deposited at 8 K.
----------------- ------------- ---------- ----------
[Molecule]{} [Centre]{} [FWHM]{} [Area]{}
D$_{2}$O $\sim$120.0 17.6 7.9
141.5 16.2 9.5
151.2 9.9 1.2
CD$_{3}$OD $\sim$120.2 25.9 26.7
145.7 20.9 11.3
160.5 11.5 1.4
$^{13}$CO$_{2}$ 115.3 4.2 1.8
126.4 9.9 2.1
$^{15}$N$_{2}$ 115.5 0.5 0.8
117.0 1.1 1.5
119.2 1.1 1.7
120.8 1.1 2.8
123.0 1.1 3.4
123.5 1.6 1.0
125.0 0.7 3.2
126.1 1.6 0.6
127.4 0.7 3.4
128.5 0.9 1.3
129.9 0.8 1.9
130.8 0.8 2.4
132.1 0.7 0.7
133.2 0.8 3.2
134.8 1.1 0.3
136.2 0.5 4.9
138.0 2.1 0.1
139.0 0.8 2.4
142.2 0.6 4.6
145.4 1.8 3.8
----------------- ------------- ---------- ----------
: Gaussian parameter values used to fit the spectra of the different molecular ices deposited at 8 K.
\[tableGauss\]
Solid deuterium oxide
---------------------
The VUV absorption cross section spectrum of D$_{2}$O ice (black trace) and H$_2$O ice (blue trace) are displayed in Fig. \[D2O\]. [@Cheng1] and [@Chung] report the VUV absorption cross sections of D$_2$O and H$_2$O in the gas phase, depicted in Fig. \[D2O\] as red and violet traces, respectively. The transition 4a$_{1}$:Ã$^{1}$B$_{1} \leftarrow $ 1b$_{1}$:X$^{1}$A$_{1}$ accounts for the absorption in the 145-180 nm region, which reaches its maximum at 166.0 nm for D$_2$O and at 167.0 nm for H$_2$O in the gas phase, this accounting for a shift of $\sim$ 1 nm. The same transition was observed for both solid D$_2$O and H$_2$O, with bands centered at 141.4 nm and 142.6 nm, respectively. This corresponds to a shift of 24.6 $\pm$ 0.4 nm for D$_2$O and 24.4 $\pm$ 0.4 nm for H$_2$O ices compared to the gas phase. Solid D$_2$O presents a maximum in the VUV absorption cross section with a value of 5.8$^{+0.2}_{-0.2}$ $\times$ 10$^{-18}$ cm$^{-2}$, a value close to the one estimated for solid H$_2$O, 6.0$^{+0.4}_{-0.4}$ $\times$ 10$^{-18}$ cm$^{-2}$. The portion of the band in the 120-132 nm range (attributed to the transition B$^{1}$A$_{1}$ $\leftarrow$ X$^{1}$A$_{1}$, according to [@Lu]) is present in the four spectra, but due to the MgF$_2$ window cutoff in our set-up it was not possible to determine the position of the maximum for this band in the solid samples.
![VUV absorption cross section as a function of photon wavelength (bottom X-axis) and photon energy (top X-axis) of D$_{2}$O ice deposited at 8 K, black trace. Blue trace is the VUV absorption cross section spectrum of solid phase H$_{2}$O, adapted from Paper I. Red and violet traces are the VUV absorption cross section spectra of gas phase D$_{2}$O and H$_2$O, respectively, adapted from Cheng et al. (2004) and Chung et al. (2001).[]{data-label="D2O"}](Cross_D2O.ps){width="\columnwidth"}
The average VUV absorption cross section of solid D$_2$O has a value of 2.7$^{+0.1}_{-0.1}$ $\times$ 10$^{-18}$ cm$^{2}$ in the 120-165 nm (10.35-7.51 eV) spectral region, i.e. lower than the 3.4$^{+0.2}_{-0.2}$ $\times$ 10$^{-18}$ cm$^{2}$ value of solid H$_2$O. The total integrated VUV absorption cross section of solid D$_2$O is 1.2$^{+0.3}_{-0.3}$ $\times$ 10$^{-16}$ cm$^{2}$ nm (8.6$^{+0.1}_{-0.1}$ $\times$ 10$^{-18}$ cm$^{2}$ eV) in the same spectral region, which again is low compared to solid H$_2$O, 1.8$^{+0.1}_{-0.1}$ $\times$ 10$^{-16}$ cm$^{2}$ nm. The VUV absorption cross sections of D$_{2}$O ice at 121.6 nm, 157.8 nm, and 160.8 nm are, respectively, 4.4$^{+0.1}_{-0.1}$ $\times$ 10$^{-18}$ cm$^{2}$, 0.8$^{+0.1}_{-0.1}$ $\times$ 10$^{-18}$ cm$^{2}$, and 0.3$^{+0.05}_{-0.05}$ $\times$ 10$^{-18}$ cm$^{2}$, i.e. lower than the values for H$_2$O, respectively, 5.2$^{+0.4}_{-0.4}$ $\times$ 10$^{-18}$ cm$^{2}$, 1.7$^{+0.1}_{-0.1}$ $\times$ 10$^{-18}$ cm$^{2}$, and 0.7$^{+0.05}_{-0.05}$ $\times$ 10$^{-18}$ cm$^{2}$. The VUV absorption cross section of D$_{2}$O in the gas phase has an average value of 3.4 $\times$ 10$^{-18}$ cm$^{2}$. D$_{2}$O gas data have been also integrated in the 120-180 nm range, giving a value of 1.9 $\times$ 10$^{-16}$ cm$^{2}$ nm (1.1 $\times$ 10$^{-17}$ cm$^{2}$ eV). Both of them, the average and the integrated values, are larger than the ones obtained for solid D$_{2}$O. The VUV absorption cross sections of gas phase D$_{2}$O at 157.8 nm and 160.8 nm are, respectively, 4.0 $\times$ 10$^{-18}$ cm$^{2}$ and 5.5 $\times$ 10$^{-18}$ cm$^{2}$, also larger than the solid phase measurements. No gas phase data was found for the Ly-$\alpha$ wavelength (121.6 nm).
Solid deuterated methanol
-------------------------
Fig. \[CD3OD\] shows the VUV absorption cross section of solid CD$_{3}$OD, black trace, and solid CH$_3$OH, blue trace, as a function of wavelength and photon energy. [@Cheng2] reported the VUV absorption cross section spectra of CD$_3$OD (red trace) and CH$_3$OH (violet trace) in the gas phase, see Fig. \[CD3OD\]. The VUV spectra corresponding to the gas phase contain plenty of features, while solid VUV spectra are very smooth, with no distinct local maxima. Paper I reports a bump centered at 146.9 nm (associated to the 2$^{1}$A” $\leftarrow$ X$^{1}$A’ molecular transition) for solid CH$_3$OH; this band is centered at 145.7 nm for solid CD$_{3}$OD. These maxima were estimated using Gaussian fits of the bands. The peaks centered at 146.5 nm and 159.3 nm for gas phase CD$_3$OD are shifted to shorter wavelengths with respect to gas phase CH$_3$OH (peaks centered at 146.8 nm and 160.4 nm, respectively). The MgF$_2$ window cutoff in our setup, near 114 nm, only allowed the detecting of a fraction of the broad band corresponding to the 3$^{1}$A” $\leftarrow$ X$^{1}$A’ molecular transition. This band is present in the four spectra of Fig. \[CD3OD\].
![VUV absorption cross section as a function of photon wavelength (bottom X-axis) and UV-photon energy (top X-axis) of CD$_{3}$OD ice deposited at 8 K, black trace. Blue trace is the VUV absorption cross section spectrum of solid phase CH$_{3}$OH adapted from Paper I. Red and violet traces are the VUV absorption cross section spectra of gas phase CD$_{3}$OD and CH$_3$OH, respectively, adapted from Cheng et al. (2002). []{data-label="CD3OD"}](Cross_CD3OD.ps){width="\columnwidth"}
The average VUV absorption cross section of solid CD$_3$OD has a value of 4.6$^{+0.2}_{-0.4}$ $\times$ 10$^{-18}$ cm$^{2}$ in the 120-175 nm (10.33-7.04 eV) spectral region, quite similar to the solid CH$_3$OH value of 4.4$^{+0.4}_{-0.7}$ $\times$ 10$^{-18}$ cm$^{2}$. The total integrated VUV absorption cross section of solid CD$_3$OD is 2.6$^{+0.1}_{-0.3}$ $\times$ 10$^{-16}$ cm$^{2}$ nm (1.7$^{+0.1}_{-0.2}$ $\times$ 10$^{-17}$ cm$^{2}$ eV) in the same spectral region, very close to the solid CH$_3$OH value, 2.7$^{+0.2}_{-0.4}$ $\times$ 10$^{-16}$ cm$^{2}$ nm, reported in Paper I. At the Ly-$\alpha$ wavelength, 121.6 nm, the VUV absorption cross section of CD$_3$OD ice is higher than the value corresponding to CH$_3$OH ice (9.7$^{+0.8}_{-1.1}$ $\times$ 10$^{-18}$ cm$^{2}$ and 8.6$^{+0.7}_{-1.3}$ $\times$ 10$^{-18}$ cm$^{2}$, respectively). For the H$_2$ molecular transitions at 157.8 nm and 160.8 nm, the VUV absorption cross sections of CD$_3$OD ice (2.9$^{+0.2}_{0.6}$ $\times$ 10$^{-18}$ cm$^{2}$ and 2.2$^{+0.2}_{0.5}$ $\times$ 10$^{-18}$ cm$^{2}$) are lower than the values corresponding to CH$_3$OH ice (3.8$^{+0.3}_{-0.6}$ $\times$ 10$^{-18}$ cm$^{2}$ and 2.9$^{+0.2}_{-0.4}$ $\times$ 10$^{-18}$ cm$^{2}$). The VUV absorption cross section of CD$_{3}$OD in the gas phase has an average value of 8.6 $\times$ 10$^{-18}$ cm$^{2}$, almost twice larger than the value measured for the solid phase. CD$_{3}$OD gas data were integrated in the 120-175 nm range giving a value of 3.4 $\times$ 10$^{-16}$ cm$^{2}$ nm (2.2 $\times$ 10$^{-17}$ cm$^{2}$ eV), i.e, larger than the VUV absorption cross section (2.6 $\times$ 10$^{-16}$ cm$^{2}$ nm) of solid CD$_{3}$OD. The VUV absorption cross sections of CD$_{3}$OD gas at 121.6 nm, 157.8, and 160.8 nm are, respectively, 13.4 $\times$ 10$^{-18}$ cm$^{2}$, 10.8 $\times$ 10$^{-18}$ cm$^{2}$, and 0.6 $\times$ 10$^{-18}$ cm$^{2}$, which are also larger than the ice phase measurements provided above, except for the 160.8 nm value.
Solid carbon-13C dioxide
------------------------
The VUV absorption cross section of $^{13}$CO$_{2}$ ice as a function of the wavelength and photon energy is shown in Fig. \[13CO2\], black trace. It is similar to the one reported for CO$_2$ ice in Paper II, depicted as a blue trace in Fig. \[13CO2\]. A broad band centered at 9.8 eV, assigned to the $^{1}\Pi_{g} \leftarrow ^{1}\Sigma^{+}_{g}$ transition, is observed in these spectra. Paper II reports a vibrational structure in the 120.0-133.0 nm range for CO$_2$; these weak features were poorly resolved in our spectrometer. Fig. \[13CO2\] inlet shows the same bands for $^{13}$CO$_2$ ice. For comparison, the dotted lines in the inlet represent the five band positions reported in Paper II for solid CO$_2$.
![VUV absorption cross section as a function of photon wavelength (bottom X-axis) and VUV photon energy (top X-axis) of $^{13}$CO$_{2}$ ice deposited at 8 K, black trace. Blue trace is the VUV absorption cross section spectrum of solid CO$_{2}$ adapted from Paper II. Inlet figure is a close-up of the $^{13}$CO$_{2}$ VUV absorption cross section in the 124-129 nm range. The spectrum of solid CO$_2$ was offset by 1 $\times$ 10$^{-18}$ cm$^{2}$ for clarity.[]{data-label="13CO2"}](Cross_13CO2.ps){width="\columnwidth"}
All the discrete bands observed beyond 130 nm correspond to the absorption of photo-produced CO in the CO$_2$ ice matrix. It was therefore not possible to measure the spectrum of pure CO$_2$ ice with our experimental configuration. For this, a synchrotron radiation source is required, see Paper II and ref. therein. Similarly, the bands of photo-produced $^{13}$CO are present in the spectrum of $^{13}$CO$_2$ ice, see Fig. \[13CO2\]. The proportion of $^{13}$CO relative to the deposited $^{13}$CO$_2$ is around 14% in this experiment, as it was inferred from integration of the infrared absorption features of $^{13}$CO$_2$ at 2283 cm$^{-1}$ and $^{13}$CO at 2092 cm$^{-1}$. Features centered at 155.0 nm, 151.4 nm, 148.2 nm, 145.2 nm, 142.4 nm, 139.6 nm, 137.2 nm, and 135.0 nm correspond to the photoproduced $^{13}$CO. The measured VUV spectrum corresponds therefore to a mixture of $^{13}$CO$_{2}$ and $^{13}$CO. An important effect is the shift of the $^{13}$CO features in this experiment with respect to those of pure CO ice, but the spectrum of $^{13}$CO was not available for comparison to our results. This issue was discussed in Paper II for CO in the CO$_2$ ice matrix, which was compared to pure CO ice. Such ice mixture effects have important implications for the VUV absorption of ice in space, where the molecular components are either mixed or layered in the ice mantles.
Upper limits for the average and the total integrated VUV absorption cross sections were calculated after subtraction of the $^{13}$CO spectrum; they are, respectively, 6.9 $^{+0.6}_{-1.0}$ $\times$ 10$^{-19}$ cm$^{2}$ and 3.1$^{+0.3}_{-0.4}$ $\times$ 10$^{-17}$ cm$^{2}$ nm (2.3$^{+0.1}_{-0.2}$ $\times$ 10$^{-18}$ cm$^{2}$ eV). These values are comparable with the CO$_2$ values reported in Paper II, 6.7$^{+0.5}_{-0.9}$ $\times$ 10$^{-19}$ cm$^{2}$ and 2.6$^{+0.2}_{-0.3}$ $\times$ 10$^{-17}$ cm$^{2}$ nm for the average and the total integrated VUV absorption cross sections, respectively. The VUV absorption cross section of $^{13}$CO$_{2}$ ice at 121.6 nm is 1.1$^{+0.2}_{-0.3}$ $\times$ 10$^{-18}$ cm$^{2}$, very close to the 1.0$^{+0.1}_{-0.2}$ $\times$ 10$^{-18}$ cm$^{2}$ value for CO$_2$. No previous gas or solid phase VUV spectra of $^{13}$CO$_2$ were found in the literature.
Solid nitrogen-$^{15}$N$_2$
---------------------------
The VUV absorption cross section of $^{15}$N$_2$ ice, black trace in Fig. \[15N2\], analogous to that of N$_2$, blue trace, is very low. For this reason, a deposition of about $N$ = 4009 $\pm$ 410 $\times$ 10$^{15}$ molecules cm$^{-2}$ was required to detect the absorption features. Paper II summarizes the complete study of solid and gas phase N$_2$. Solid $^{15}$N$_2$ should present the same vibrational structure as solid N$_2$ in the 114-147 nm (10.87-8.43 eV) region. The two systems (attributed to a$^{1}\Pi_{g}$ $\leftarrow$ X$^{1}\Sigma_{g}^{+}$ and w$^{1}\Delta_{u}$ $\leftarrow$ X$^{1}\Sigma_{g}^{+}$ transitions) can be appreciated in Fig. \[15N2\]. The noise level in these measurements was high compared to the other ices studied, due to the low intensity of the bands and the detection limit of the VUV spectrometer. Some features are shifted to a shorter wavelengths above the 0.4 nm resolution of our measurements, but these shifts did not exceed 0.8 nm.
The average VUV absorption cross section of $^{15}$N$_2$ ice has a value of 8.7 $\times$ 10$^{-21}$ cm$^{2}$, i.e. higher than 7.0 $\times$ 10$^{-21}$ cm$^{2}$ calculated for N$_2$ in Paper II. The total integrated VUV absorption cross section of $^{15}$N$_2$ ice has a value of 3.0 $\times$ 10$^{-19}$ cm$^{2}$ nm (2.2 $\times$ 10$^{-20}$ cm$^{2}$ eV) in the 114.6-146.8 nm (10.82-8.44 eV) spectral region, which can be compared to 2.3 $\times$ 10$^{-19}$ cm$^{2}$ nm for N$_2$ ice. The VUV-absorption cross section spectrum of $^{15}$N$_{2}$ is not as well resolved as in other works but this should not affect the VUV-absorption cross section scale that we provide, because no integration of the band area is involved, see Paper II. The VUV absorption cross section at Ly-$\alpha$ (121.6 nm) is very low, we estimated an upper limit value of 1.5 $\times$ 10$^{-21}$ cm$^{2}$. There are no observable VUV absorption features at the molecular hydrogen band wavelengths (157.8 and 160.8 nm). No gas or solid phase VUV spectra of $^{15}$N$_2$ were found in the literature.
![VUV absorption cross section as a function of photon wavelength (bottom X-axis) and VUV photon energy (top X-axis) of $^{15}$N$_{2}$ ice deposited at 8 K, black trace. Blue trace is the VUV absorption cross section spectrum of solid phase N$_{2}$ adapted from Paper II. The spectrum of solid N$_2$ was offset by 4 $\times$ 10$^{-20}$ cm$^{2}$ for clarity.[]{data-label="15N2"}](Cross_15N2.ps){width="\columnwidth"}
Astrophysical implications and final conclusions
================================================
The absorption of energetic photons by gas phase molecules and dust grains in various space environments is a key issue in astrophysics. If the absorption cross sections are known for the photon wavelength range of interest, a quantitative estimation of the photon absorption and the photon penetration depth in the absorbing material can be attained.
The absorbing ice column density of a species in the solid phase, can be calculated from the VUV absorption cross section following
$$% \hspace{3cm}
N(\lambda)= \frac{- 1}{\sigma(\lambda)} \ln \left( \frac{I_t(\lambda)}{I_0(\lambda)} \right)$$
where $I_{t}(\lambda)$ is the transmitted intensity for a given wavelength $\lambda$, $I_{0}(\lambda)$ the incident intensity, $N(\lambda)$ is the absorbing column density in cm$^{-2}$, and $\sigma(\lambda)$ is the cross section in cm$^{2}$. Table \[penetration\] summarizes the absorbing column densities of the ice species for an absorbed photon flux of 95% and 99% using the cross section value at Ly-$\alpha$, the average cross section in the 120-160 nm range, and the maximum cross section in the same range. The values corresponding to the lighter isotopologues are reported in Papers I and II.
------------------- ----------------- ---------- ---------- ----------------- ---------- ----------
[species]{} [Ly-$\alpha$]{} [Avg.]{} [Max.]{} [Ly-$\alpha$]{} [Avg.]{} [Max.]{}
[D$_2$O]{} [6.8]{} [11.1]{} [5.3]{} [10.5]{} [17.1]{} [8.1]{}
[CD$_{3}$OD]{} [3.1]{} [6.5]{} [3.1]{} [4.7]{} [10.0]{} [4.7]{}
[$^{13}$CO$_2$]{} [27.2]{} [43.7]{} [12.0]{} [41.8]{} [67.1]{} [18.4]{}
[$^{15}$N$_2$]{} [19971]{} [3443]{} [749]{} [30701]{} [5293]{} [1151]{}
[H$_{2}$O]{} [5.8]{} [8.3]{} [4.9]{} [8.9]{} [13.0]{} [7.7]{}
[CH$_3$OH]{} [3.5]{} [5.7]{} [3.4]{} [5.4]{} [8.7]{} [5.3]{}
[CO$_2$]{} [29.3]{} [44.5]{} [15.1]{} [45.1]{} [68.4]{} [23.3]{}
[N$_2$]{} [29957]{} [4280]{} [881]{} [46052]{} [6579]{} [1354]{}
------------------- ----------------- ---------- ---------- ----------------- ---------- ----------
: Absorbing column densities, of the different ice species, corresponding to an absorbed photon flux of 95% and 99%. “Ly-$\alpha$” corresponds to the cross section at the Ly-$\alpha$ wavelength, 121.6 nm. “Avg.” corresponds to the average cross section in the 120-160 nm range. “Max.” corresponds to the maximum cross section in the same wavelength range.
\[penetration\]
A larger column density of solid D$_2$O ice is needed to reach 95 and 99% of the total photon absorption with respect to the solid H$_{2}$O values, due to its lower VUV absorption cross section. The same holds for solid CD$_3$OD and CH$_3$OH, with the exception of the Avg. value, which is higher in the solid CD$_3$OD sample. For solid $^{13}$CO$_2$ a larger column density is needed to reach 95 and 99% of the total photon absorption with respect to solid CO$_{2}$. In the other hand, within the significant errors associated to their VUV absorption cross section measurements, $^{15}$N$_2$ ice seems to be similar or slightly more absorbing than N$_2$. For the first time, we report the VUV absorption cross section as a function of photon energy for D$_2$O, CD$_3$OD, $^{13}$CO$_2$, and $^{15}$N$_2$ in the solid phase at 8 K.
All four molecules present a shift to shorter wavelengths in their VUV spectrum with respect to their corresponding light isotopologues. Deuterated species experience the largest blue-shift among the molecules studied. This could be expected from previous works on deuterated species in the gas phase, but the shifts measured in the solid phase were larger in comparison. The average and the integrated VUV absorption cross section values are close for the different isotopologues. The relatively small variations between isotopologues may only play a minor role in the absorption of VUV radiation in space.
Large differences were found between the VUV absorption cross section spectra of solid and gas phase species (Papers I, II, and ref. therein; this work). This has important implications for the absorption of VUV photons in dense clouds and circumstellar regions.
There is a clear correspondence between the photodesorption rates measured at different photon energies and the VUV absorption spectrum for the same photon energies. This indicates that photodesorption of some ice species like N$_2$ and CO is mainly driven by a desorption induced by electronic transition (DIET) process (Fayolle et al. 2011, 2013). Unfortunately, the N$_2$ and $^{15}$N$_2$ ice absorption spectra at photon energies higher than 12.4 eV, where photodesorption is efficient, have not been measured. But the low photodesorption rates measured at energies below 12 eV by [@Fayolle2] (no more than 4 $\times$ 10$^{-3}$ molecules per incident photon for $^{15}$N$_2$ ice) is compatible with its low ice absorption cross section, reported here for the same spectral range. In addition, the observed photodesorption occurs in the same spectral range where the absorption bands of Fig. \[15N2\] are present. The lower photodesorption reported by Fayolle et al. 2013 for the Ly-$\alpha$ wavelength at 121.6 nm, 1.5 $\times$ 10$^{-3}$ molecules per incident photon, coincides with a low absorption in Fig. \[15N2\], and the maximum in the photodesorption occurs approximately at $\sim$ 135 nm, where the most intense absorption band is present, see Fig. \[15N2\]. The photodesorption rate per absorbed photon in that range, $R^{\rm abs}_{\rm ph-des}$, can be estimated as follows $$R^{\rm abs}_{\rm ph-des} = \frac{I_0}{I_{abs}} \; R^{\rm inc}_{\rm ph-des}$$ where $$\begin{aligned}
I_{abs} &=& \displaystyle\sum\limits_{\lambda_i}^{\lambda_f} \quad I_0(\lambda) - I(\lambda) = \displaystyle\sum\limits_{\lambda_i}^{\lambda_f} \quad I_0(\lambda)(1 - e^{- \sigma(\lambda) N}) \nonumber\end{aligned}$$ and $I_0$ is the total photon flux emitted (Fayolle et al. 2013 reports 3-11.5 $\times$ 10$^{12}$ photons cm$^{-2}$ s$^{-1}$, in our experiments this flux is about 2.0 $\times$ 10$^{14}$ photons cm$^{-2}$ s$^{-1}$), $I_{abs}$ is the total photon flux absorbed by the ice, $I_0(\lambda)$ is the photon flux emitted at wavelength $\lambda$, $\sigma(\lambda)$ is the VUV absorption cross section at the same wavelength, and $N$ is the column density of the ice sample. $R^{\rm inc}_{\rm ph-des}$ corresponds to a photodesorption rate of $\leq$ 4 $\times$ 10$^{-3}$ molecules per incident photon in the spectral range below 12.4 eV for $N$ = 60 $\times$ 10$^{15}$ cm$^{-2}$ (60 monolayers) from [@Fayolle2], while the average absorption cross section for $^{15}$N$_2$ ice that we measured in that range is $\sigma$ = 8.7 $\pm$ 1.9 $\times$ 10$^{-21}$ cm$^{2}$. The resulting photodesorption rate is thus quite high, $R^{\rm abs}_{\rm ph-des}$ $\leq$ 7.7 molecules per absorbed photon, meaning that a very small fraction of the incident photons are absorbed in the ice but each absorbed photon led to the photodesorption of about 7.7 molecules on average (this in fact is the maximum value because [@Fayolle2] measured photodesorption rates, $R^{\rm inc}_{\rm ph-des}$, that *do not exceed* 4 $\times$ 10$^{-3}$ molecules per incident photon).
In the case of CO ice deposited also at 15 K, it was found that only the photons absorbed in the top 5 monolayers led to photodesorption with a rate of 2.5 CO molecules per absorbed photon in those 5 monolayers (based on Muñoz Caro et al. 2010, but using an average cross section of CO ice of 4.7 $\pm$ 0.4 $\times$ 10$^{-18}$ cm$^2$ adapted from Paper I). This value for CO ice is about 3.1 times lower than the maximum estimated above for the 60 ML of the $^{15}$N$_2$ ice experiment of [@Fayolle2]. A more direct comparison between N$_2$ and CO ice photodesorption could be made if the number of N$_2$ monolayers closer to the ice surface that truly contribute to the photodesorption was known (in the case of CO ice these are $\sim$ 5 monolayers, this value has not been estimated for N$_2$ ice and therefore the values of $R^{\rm inc}_{\rm ph-des}$ and $R^{\rm abs}_{\rm ph-des}$ correspond to the total ice column density of 60 monolayers in the experiment of Fayolle et al. 2013). With this uncertainty still remaining, we can conclude that if the VUV absorption cross section of each specific ice composition is taken into account, it is possible to know what is the efficiency of the photodesorption per absorbed photon; in the case of N$_2$ and CO, for VUV photon energies that do not lead to direct dissociation of the molecules in the ice, these values are higher than unity. The values of $R^{\rm abs}_{\rm ph-des}$ >1 and the fact that the photons absorbed in ice monolayers deeper than the top monolayers (up to 5 for CO) can lead to a photodesorption event, indicate that the excess photon energy is transmitted to neighboring molecules in the ice within a certain range (this range may correspond to about 5 monolayers in the case of CO ice, e.g., Rakhovskaia et al. 1995; Öberg et al. 2007, 2009; Muñoz Caro et al. 2010); if a molecule on the ice surface receives sufficient energy, it may photodesorbs (Muñoz Caro et al. 2010).
It should also be noted that ice photodesorption experiments performed with a continuum emission source (like the MDHL), mimicking the secondary VUV field in dense cloud interiors, can lead to photodesorption rates that are intrinsically different from those obtained in experiments using a monochromatic source (generally provided by a synchrotron beam), we refer to [@Chen2] for the case of CO ice photodesorption.
This work, along with Papers I and II, provides essential data to attempt a more quantitative study of VUV absorption of molecules forming ice mantles, and the photon processes involved: photo-processing leading to destruction of molecules and formation of new species, and photo-desorption of molecules in the ice that are ejected to the gas phase.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was financed by the Spanish MICINN under projects AYA2011-29375 and CONSOLIDER grant CSD2009-00038. This work was partially supported by NSC grants NSC99-2112-M-008-011-MY3 and NSC99-2923-M-008-011-MY3, and the NSF Planetary Astronomy Program under Grant AST-1108898.
[99]{} Belloche A., André P., 2003, ApJ, 593, 906. Bergin E.A., Langer W.D., Goldsmith, P.F., 2002, ApJ, 570, L101. Boogert A.C.A., Ehrenfreund P., Gerakines P.A., Tielens A.G.G.M., Whittet D.C.B. et al., 2000, A&A, 353, 349. Butner H.M., Charnley S.B., Ceccarelli C., Rodgers S.D., Pardo J.R. et al., 2007, ApJ, 659, L137. Ceccarelli, C., Castets, A., Loinard, L., Caux, E., & Tielens, A. G. M.. A&A, 1998, 338, L43. Chen Y.-J., Chu C.C., Lin Y.C., Yih T.S., Wu C.Y.R. et al., Adv. Geosci., 2010, 25, 259. Chen Y.-J., Chuang K.Y., & Muñoz Caro G.M., et al., ApJ, 2013, 781, 15 Cheng B.-M., Chung C.-Y., Bahou M., Lee Y.-P., Lee L.C. et al., 2004, Journal of Chemical Physics, 120, 224. Cheng B.-M., Bahou M., Chen W.C., Yui C.H., Lee Y.-P., Lee, L.C., 2002, Journal of Chemical Physics, 117, 1633. Chung C.-Y., Chew E.P., Cheng B.-M., Bahoub M., Lee Y.-P., 2001, Nuclear Instruments and Methods in Physics Research A, 467-468, 1572. Cottin, H., Moore, M.H., & Bénilan, Y., 2003, ApJ, 590, 874. Cruz-Diaz G.A., Muñoz Caro G.M., Chen Y. J., 2013a, A&A, in press. Cruz-Diaz G.A., Muñoz Caro G.M., Chen Y. J., 2013b, A&A, in press. Ehrenfreund, P. & van Dishoeck, E. F., 1998, Adv. Space Res., 21, 15 Fayolle E.C., Bertin M., Romanzin C., Poderoso H.A.M., Philippe L. et al., 2013, A&A, 556, A122. Fayolle E.C., Bertin M., Romanzin C., Michaut X., Öberg K.I. et al., 2011, ApJ Letters, 739, L36. Fuchs, G. W., Acharyya, K., Bisschop, S. E., et al., 2006, Faraday Discussions, 133, 331. Gerakines P., Schutte W.A., Greenberg J.M., van Dishoeck E.F., 1995, A&A, 296, 810. Goldsmith, P. F., Liseau, R., Bell, T. A., et al., 2011, ApJ, 737, 96. Gredel R., Lepp S., Dalgarno A., 1989, ApJ, 347, 289. Hudgins, D. M., Sandford, S. A., Allamandola, L. J., & Tielens, A. G. G. M., 1993, ApJSS, 86, 713. Hudson, R. D., & Carter, V. L. 1968, J. Opt. Soc. Am., 58, 227. Linsky, J. L., Space. Sci. Rev., 2003, 106, 49. Liseau, R., Goldsmith, P. F., Larsson, B., Pagani, L., & Bergman, P., 2012., A&A, 541, A73. Loinard, L., Castets, A., Ceccarelli, C., et al., Planet. Space Sci., 2002, 50, 1205. Lu H.C., Chen H.K., Cheng B.-M., Ogilvie J.F., 2008, Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy, 71, 1485. Mason, N. J., Dawes, A., Holton, P. D., et al. 2006, Faraday Discussions, 133, 311. Mumma M.J., Charnley S.B., 2011, Annu. Rev. Astro. Astrophys, 49, 471. Meyer, D. M., Jura, M., & Cardelli, J. A. 1998, ApJ, 493, 222-. Muñoz Caro G.M., Jiménez-Escobar A., Martin-Gago J.A., Rogero C., Atienza C. et al., 2010, A&A, 522, A108. Öberg K.I., Fuchs G.W., Awad Z. et al., 2007, ApJ, 662, L23. Öberg K.I., van Dishoeck E.F., Linnartz H., 2009, A&A, 496, 281. Okabe, H. 1978, Photochemistry of small molecules, ed. John Wiley & Sons, New York. Parise B., Castets, A., Herbst, E., et al. A&A, 2004, 416, 159. Parise B., Ceccarelli C., Tielens A.G.G.M., Castets A., Caux E. et al., 2006, A&A, 453, 949. Pontoppidan, K. M., van Dishoeck, E. F., & Dartois, E. 2004, A&A, 426, 925 Rakhovskaia O., Wiethoff P., Feulner P., 1995, NIM B, 101, 169 Ratajczak, A., Quirico, E., Faure, A., Schmitt, B., & Ceccarelli, C., 2009, A&A, 496, L21. Roberts F., 2003, Space Science Reviews, 106, 87. Samson, J. A. R., & Ederer, D. L. 2000, Vacuum Ultaviolet Spectroscopy, ed. Elsevier Inc. Satorre, M. Á., Domingo, M., Millán, C., et al., 2008, Planetary and Space Science, 56, 1748. Vastel C., Ceccarelli C., Caux E., Coutens A., Cernicharo J. et al., 2010, A&A, 521, L31. Wilson R.W., Penzias A.A., Jefferts K.B., Solomon P.M., 1973, ApJ, 179, L107.
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[^1]: E-mail: cruzdga@cab.inta-csic.es
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author:
- 'B. Husemann'
- 'G. Worseck'
- 'F. Arrigoni Battaia'
- 'T. Shanks'
bibliography:
- 'references.bib'
date: 'Received ...; accepted ...'
title: 'Discovery of a dual AGN at $z\simeq 3.3$ with 20 kpc separation[^1]'
---
Introduction
============
It has long been suggested that the circumgalactic medium (CGM) of QSOs may be detectable in emission via the Ly$\alpha$ line that is powered by recombination radiation, collisional excitation, and Ly$\alpha$ scattering [@Rees:1988; @Haiman:2001; @Cantalupo:2005; @Kollmeier:2010]. Early narrow-band imaging and longslit spectroscopic surveys revealed extended ($\sim 100$kpc) Ly$\alpha$ nebulae almost exclusively around radio-loud $2<z<4$ QSOs [e.g., @Hu:1991; @Heckman:1991b], suggesting an origin in radio jets, as commonly observed in radio galaxies [e.g., @McCarthy:1990; @Reuland:2003; @Humphrey:2006; @Villar-Martin:2007]. Subsequent surveys focusing on radio-quiet QSOs found smaller ($\la 70$kpc) and fainter ($\sim 10\times$) Ly$\alpha$ nebulae around $\sim 50$% of the targets [@Christensen:2006; @North:2012], but only recent campaigns have ubiquitously detected them and captured their diverse morphologies [e.g., @Hennawi:2013; @Borisova:2016; @ArrigoniBattaia:2016].
Species other than hydrogen enable studies of the ionization conditions and the gas density. Extended $\lambda$1640 and $\lambda$1549 emission is common around radio galaxies and radio-loud QSOs [e.g., @Villar-Martin:2007], but only $\sim 6$% of the nebulae around radio-quiet QSOs show these lines [@Borisova:2016]. In giant (300–460kpc) Ly$\alpha$ nebulae, multiple AGN with separations of several tens of kpc have been discovered via isolated and metal lines [@Cantalupo:2014; @Hennawi:2015; @Cai:2017; @ArrigoniBattaia:2018].
In this *Letter*, we analyze the environment of the radio-quiet QSO LBQS 0302$-$0019 at $z=3.2859$ [@Shen:2016] that has been intensely targeted for studies of the intergalactic medium [IGM, e.g., @Hu:1995] and the impact of foreground galaxies and QSOs on the CGM and IGM [e.g., @Steidel:2003; @Jakobsen:2003; @Tummuangpak:2014; @Schmidt:2017]. In particular, LBQS 0302$-$0019 is one of the few UV-transparent $z>3$ sight lines that allow for *Hubble Space Telescope* UV spectroscopy of intergalactic Ly$\alpha$ absorption [e.g., @Jakobsen:1994; @Syphers:2014]. Here we discuss the detection of various high-ionization lines in its surrounding Ly$\alpha$ nebula, which shows that LBQS 0302$-$0019 is actually an unobscured/obscured dual AGN system with only 20kpc projected separation.
We adopt a flat cosmology with $\Omega_\mathrm{m}=0.3$, $\Omega_\Lambda=0.7$, and $H_0=70$kms$^{-1}$Mpc$^{-1}$. The physical scale at $z=3.286$ is $7.48\,\mathrm{kpc}\,\mathrm{arcsec}^{-1}$.
Observations and results
========================
Observations and data reduction
-------------------------------
{width="85.00000%"}
Observations of LBQS 0302$-$0019 were taken between October 2014 and January 2015 with the MUSE instrument [@Bacon:2010] at the Very Large Telescope. MUSE covers a $\sim1\arcmin\times1\arcmin$ field of view (FoV) with a sampling of 02 and spectral coverage from 4750Å to 9300Å at a spectral resolution of $1800<R<3600$. The observations were split into $11\times 1450$s exposures subsequently rotated by 90 with some small dithering. The median seeing was $\simeq 0\farcs9$. We reduced the data with the latest MUSE data reduction pipeline [v2.0.3. @Weilbacher:2012], which performs all major tasks, i.e., bias subtraction, wavelength calibration, flat-fielding, flux calibration based on photometric standards, and reconstruction of the data cube. While the sky-dominated regions of the FoV are used for an initial sky subtraction, prominent skyline residuals are further suppressed using our own PCA software [@Husemann:2016a; @Peroux:2017]. The deep reconstructed $r$-band image and the coadded spectrum of LBQS 0302$-$0019 are shown in the top panels of Fig. \[fig:overview\].
QSO subtraction and extended Ly$\alpha$ nebula
----------------------------------------------
To study the extended nebular emission around bright QSOs it is crucial to subtract the point-like QSO emission that is smeared out due to the seeing, as characterized by the point-spread function (PSF). Various studies have used empirical PSF estimates from the data as a function of wavelength [e.g., @Christensen:2006; @Husemann:2014; @Herenz:2015; @Borisova:2016]. Here we follow the empirical method described in @Borisova:2016. We constructed a PSF from a median image (150Å wide in the observed frame) at each monochromatic slice of the data cube, which is subsequently subtracted after matching the central $0\farcs6\times0\farcs6$. The subtraction of the QSO reveals a Ly$\alpha$ nebula with a maximum diameter of $\simeq 16\arcsec$ (120kpc) as shown in the bottom left panel of Fig. \[fig:overview\].
The Ly$\alpha$ flux integrated over an aperture of 8 radius is $f_{\mathrm{Ly}\alpha}=18.1\times10^{-16}\,\mathrm{erg}\,\mathrm{s}^{-1}\,\mathrm{cm}^{-2}$ which corresponds to a luminosity $L_{\mathrm{Ly}\alpha}=1.7\times 10^{44}\,\mathrm{erg}\,\mathrm{s}^{-1}$. The size and luminosity of this Ly$\alpha$ nebula are similar to those of other radio-quiet QSOs [@Borisova:2016]. In this case the Ly$\alpha$ surface brightness distribution is asymmetric, with a bright knot about 2.9 ($\sim$20kpc) northeast of the QSO. We refer to this source as Jil, Klingon for neighbor, with coordinates $\alpha$=03:04:50.03, $\delta$=-00:08:12.5 (J2000), and a peak surface brightness of $\Sigma_{\mathrm{Ly}\alpha}=1.05\times 10^{-16}\,\mathrm{erg}\,\mathrm{s}^{-1}\,\mathrm{cm}^{-2}\,\mathrm{arcsec}^{-2}$.
Emission-line diagnostics and photoionization modeling
------------------------------------------------------
{width="90.00000%"}
The coadded spectrum within a circular aperture of $1\arcsec$ radius around Jil is presented in the bottom right panel of Fig. \[fig:overview\]. We clearly detect and $\lambda\lambda 1548,1550$ at $>10\sigma$ significance. Coupling the kinematics to we also detect \[\]$\lambda1907$ and \]$\lambda1909$ at $3\sigma$ significance. All lines are well fit with single Gaussian profiles whose parameters are listed in Table \[tab:lines\]. Ly$\alpha$ is redshifted by 35$\mathrm{km}\,\mathrm{s}^{-1}$ (rest frame) compared to the other lines and also shows a significantly larger velocity dispersion after correcting for the wavelength-dependent spectral resolution of MUSE [@Bacon:2017]. Both effects are likely caused by resonant scattering of Ly$\alpha$ photons.
--------------------- -------------------------------------------------------------- ------------------------------------------------ -------- ----------------------------------------------- --
Line $f_\mathrm{line}$ $\log(L_\mathrm{line})$ $z$ $\sigma$
$\left[10^{-16}\frac{\mathrm{erg}}{\mathrm{s\,cm}^2}\right]$ $\left[\frac{\mathrm{erg}}{\mathrm{s}}\right]$ $\left[\frac{\mathrm{km}}{\mathrm{s}}\right]$
Ly$\alpha$ $2.15\pm0.04$ $43.32\pm0.04$ 3.2887 $261\pm7$
$\lambda1548$ $0.22\pm0.01$ $42.33\pm0.05$ 3.2882 $171\pm10$
$\lambda1550$ $0.15\pm0.01$ $42.16\pm0.05$ 3.2882 $171\pm10$
$\lambda1640$ $0.18\pm0.01$ $42.24\pm0.05$ 3.2882 $126\pm10$
$[$$]$$\lambda1907$ $0.029\pm0.007$ $41.45\pm0.13$ 3.2882 $126\pm10$
\]$\lambda1909$ $0.028\pm0.007$ $41.43\pm0.13$ 3.2882 $126\pm10$
--------------------- -------------------------------------------------------------- ------------------------------------------------ -------- ----------------------------------------------- --
: Emission-line measurements for the source Jil.[]{data-label="tab:lines"}
At high redshifts, has mainly been detected around radio AGN [e.g., @Heckman:1991b; @Villar-Martin:2007], and to date only a few dedicated searches have been performed to detect and in the nebulae around radio-quiet QSOs [e.g., @ArrigoniBattaia:2015] and for bright high-redshift galaxies in the re-ionization era [e.g., @Laporte:2017]. @Borisova:2016 detected at $2\sigma$ in 1 out of 17 nebulae around radio-quiet QSOs. Isolated emitters have been found within two of the four known giant Ly$\alpha$ nebulae, 71 and 86kpc from the primary unobscured radio-quiet QSO [@Hennawi:2015; @ArrigoniBattaia:2018]. The presence of an obscured AGN was invoked in both cases based on the narrow velocity width, the line ratios, and the compactness of the emitting region. In Fig. \[fig:line\_ratios\] we show /Ly$\alpha$ vs. /Ly$\alpha$ and / vs. \]/ for Jil in comparison to various individual nebulae of radio galaxies, unobscured QSOs, and obscured AGN. We also plot the line ratios of a composite spectrum of obscured AGN [@Alexandroff:2013]. The nebular line ratios are inconsistent with the limits for radio-quiet QSOs obtained by @Borisova:2016, but agree with those of radio-loud QSOs and most obscured AGN. Due to the high surface brightness, we can derive proper line ratios in a matched aperture.
The detection of several lines allows us to explore ionization properties through a grid of photoionization models with the `CLOUDY` code [v10.01, @Ferland:2013] using the following assumptions and input parameters: (1) a power-law AGN spectral energy distribution $f_\nu\propto\nu^{\alpha_\nu}$ with $\alpha_\nu=-1.7$ at $\lambda_\mathrm{rest}<912$Å [@Lusso:2015]; (2) three different ionizing luminosities $L_\mathrm{912\AA}^\mathrm{AGN}=L_\mathrm{912\AA}^\mathrm{QSO}/250$, $L_\mathrm{912\AA}^\mathrm{AGN}=L_\mathrm{912\AA}^\mathrm{QSO}/600$, and $L_\mathrm{912\AA}^\mathrm{AGN}=L_\mathrm{912\AA}^\mathrm{QSO}/1000$, where $L_\mathrm{912\AA}^\mathrm{QSO}$ is estimated by scaling the @Lusso:2015 QSO template to the observed SDSS $i$-band magnitude of LBQS 0302$-$0019; (3) a plane-parallel geometry with an inner distance of 100pc from the AGN; (4) a constant volume number density $n_{\rm H}$ in the range $10^2$–$10^5$cm$^{-3}$; (5) three different metallicities $Z=0.1 Z_{\odot}$, $0.5 Z_{\odot}$, and $1 Z_{\odot}$; (6) a column density $N_{\rm H}$ determined by the stopping criterion of the calculations at $T=4000$K[^2]. For each $L_\mathrm{912\AA}^\mathrm{AGN}$, the ionization parameter $U\equiv\Phi_{912\AA}/(cn_{\rm H})$ results from the $n_{\rm H}$ variation, and ranges from $-2.6\lesssim\log U\lesssim2.4$. Our parameter space is similar to works modeling narrow-line regions (NLRs) of obscured AGN [e.g., @Groves:2004; @Nagao:2006; @Nakajima:2017].
From the output of the `CLOUDY` calculations we extract the predictions for the relevant emission-line fluxes and the radius of the emitting region calculated as the ratio between the column density of and $n_{\rm H}$. In Fig. \[fig:line\_ratios\], we show the predictions of our photoionization models as a function of $U$ for our three AGN luminosities and our three metallicities. We find that an obscured AGN with a luminosity $1000\times$ fainter than the QSO is sufficient to produce the observed luminosity within an emitting region of $R_{\ion{He}{ii}}<200$pc. Our simple models cover the region defined by the observed line ratios, implying $Z<Z_{\odot}$ for the gas around the obscured AGN. At fixed metallicity, models with different $\left(L_\mathrm{912\AA}^\mathrm{AGN}, n_{\rm H}\right)$ yielding the same $U$ parameter are expected to give very similar results (Fig. \[fig:line\_ratios\]).
Intrinsic vs. external AGN ionization source
--------------------------------------------
Although an obscured AGN appears to be able to power Jil, we also checked whether the QSO can power the emission. We tested this hypothesis by comparing $L_{\ion{He}{ii}}$ with the incident -ionizing flux 20kpc from the QSO intercepted by a homogeneously filled sphere of radius $R_{\ion{He}{ii}}$. Scaling the @Lusso:2015 broken power-law spectrum to the dereddened SDSS $i$-band magnitude $m_i=17.34$mag leads to an extrapolated absolute monochromatic magnitude at the edge of $M_{228\AA}=-25.78$mag. This corresponds to a photon flux of $\Phi(\mathrm{He}^+)=1.85\times10^{10}\,\mathrm{photons}\,\mathrm{s}^{-1}\mathrm{cm}^{-2}$ at a distance of 20kpc. Assuming that every emitted $\lambda$1640 photon requires at least one -ionizing photon, we can predict the maximum number of emitted $\lambda$1640 photons from a sphere with radius $R_\ion{He}{ii}$, $L_{\ion{He}{ii}}(\lambda1640) = \pi R_\ion{He}{ii}^2 \Phi(\mathrm{He}^+)\times h\nu_{1640\AA}\times\frac{\alpha_\mathrm{eff}}{\alpha_\mathrm{B}}$, where we assumed case B recombination and that all incident ionizing photons passing through the sphere are absorbed.
The results of this computation are shown in the right panel of Fig. \[fig:line\_ratios\], indicating $R_\ion{He}{ii}\gtrapprox$1kpc. This size of the emitting region is a hard lower limit given our simple and very conservative assumption for the QSO ionization scenario. A diameter of $>$2kpc would correspond to $>$03 projected on the sky. This size is borderline consistent with the observations at our spatial resolution. A low-luminosity obscured AGN is sufficient to power the observed compact emission with a much smaller $R_\ion{He}{ii}$, and we do not detect even coadding the rest of the larger Ly$\alpha$ nebula. We argue that the embedded obscured AGN scenario is much more likely also considering the asymmetry of the nebula. This scenario would naturally explain the Ly$\alpha$ velocity shift due to scattering in the NLR a few 100pc away from a highly dust-obscured source, which would not be the case if directly illuminated by the QSO. Hence, LBQS 0302$-$0019 and Jil form a close dual AGN system with 20kpc projected separation.
Discussion
==========
At low redshifts ($z<1$) numerous dual AGN with kpc-scale separation have been identified through high-resolution X-ray imaging with *Chandra* [e.g., @Koss:2012] or through radio interferometry [e.g., @Fu:2015; @MuellerSanchez:2015]. Both methods probe the core emission and are robust in detecting AGN. However, the sensitivity of *Chandra* is limited and radio-jets can mimic dual AGN signatures in the radio, which makes the methods difficult to apply at high redshifts. Alternatively, the high-ionization \[\] $\lambda\lambda 4960,5007$ lines of the NLR have been employed to search for dual AGN. In particular, double-peaked \[\] emitters were considered a parent sample for dual AGN candidates [e.g., @Liu:2010a], but spatially resolved spectroscopy revealed that rotating disks, AGN outflow, jet-cloud interactions are the origin of the double-peaked lines in most cases [e.g., @Fu:2012; @Nevin:2016]. The most robust kpc-scale dual AGN systems are always associated with the nuclei of two merging galaxies that are spatially coincident with AGN signature from the NLR [e.g., @Woo:2014], from X-rays [e.g., @Liu:2013c; @Ellison:2017], or from radio cores [e.g., @MuellerSanchez:2015].
Multiple AGN systems at high redshifts are mainly identified as independent bright QSOs in large imaging and spectroscopic surveys [e.g., @Hennawi:2006; @Myers:2008]. While most of the known nearby QSOs have separations of several 100kpc, a few dual QSOs at $\sim$10kpc are identified in the redshift range $0.5<z<2.5$ @Gregg:2002 [@Pindor:2006; @Hennawi:2006; @Eftekharzadeh:2017], and only one QSO pair with $<$20kpc separation was reported at $z>3$ by [@Hennawi:2010]. Overall the statistics for QSOs at high redshift indicates an excess of QSO clustering at very small separations [e.g., @Hennawi:2006; @Myers:2007]. This is somewhat expected as the rapid growth of massive SMBH in the early Universe is related to overdensities as inferred from QSO clustering studies [e.g., @Shen:2007b]. While the prevalence of AGN in major galaxy mergers is highly controversial at low and intermediate redshifts [e.g., @Cisternas:2011; @Treister:2012; @Villforth:2017], the role of major mergers for BH growth may be more important at early cosmic times $z>3$.
In case major mergers at high redshifts are more prevalent in triggering AGN, it is possible that many close dual AGN are currently missed because AGN in gas-rich major mergers may often be highly obscured [e.g., @Kocevski:2015; @Ricci:2017]. To detect close dual AGN with at least one obscured companion is challenging at high redshift given the lack of spatial resolution and sensitivity at hard X-rays, and the limited diagnostic power of optical emission lines due to an increasing ionization parameter in high-redshift galaxies [e.g., @Kewley:2013]. Instead, the sensitivity of MUSE allows us to detect the rest-frame far-UV high-ionization emission lines of AGN from which already several obscured AGN at $>$50kpc were identified in giant Ly$\alpha$ nebulae around bright QSOs at $z>2$ [e.g., @Hennawi:2015; @ArrigoniBattaia:2018] and around the radio-loud QSO PKS 1614$+$051 [@Djorgovski:1985; @Husband:2015]. Our dual AGN system is detected with the same method, but at a much smaller projected separation of 20kpc, which is already in a regime where the PSF of the bright QSO needs to be subtracted properly for a detection. In the dual AGN scenario we expect two strongly interacting massive host galaxies to be associated with the two nuclei. This major merger scenario is testable with deep high-resolution rest-frame optical imaging with *Hubble* and mapping the molecular gas at high angular resolution with ALMA in the sub-mm.
Conclusions
===========
We report the detection of a emission-line source, named Jil, at $z=3.28$ that is close to the luminous radio-quiet QSO LBQS 0302-0019. Based on emission-line ratio diagnostics we verified that Jil is ionized most likely by an embedded obscured AGN. With a projected separation of only $\sim$20kpc to the QSO, this system represents the tightest unobscured/obscured dual AGN system reported at $z>3$.
High-redshift rest-frame far-UV line diagnostics supersede the classical rest-frame optical line ratios when H$\alpha$ is shifted out of the $K$ band at $z>3$. Furthermore, current X-ray observatories lack the sensitivity and spatial resolution to systematically detect small separation obscured dual AGN at high redshifts. Hence, VLT-MUSE is the ideal instrument to look for tight dual AGN candidates at high-redshift that would be missed otherwise. The ground-layer adaptive optics system of MUSE will further enhance the detectability of these dual AGN, due to a significant increase in spatial resolution and point-source sensitivity.
[^1]: Based on observations collected at the European Southern Observatory, Paranal, Chile under program ID 094.A-0767(A) (PI: T. Shanks)
[^2]: For the predictions of interest we found similar results for calculations with a stopping threshold of $T=100$K.
|
---
abstract: 'We propose and study the use of photon-mediated interactions for the generation of long-range steady-state entanglement between $N$ atoms. Through the judicious use of coherent drives and the placement of the atoms in a network of Cavity QED systems, a balance between their unitary and dissipative dynamics can be precisely engineered to stabilize a long-range correlated state of qubits in the steady state. We discuss the general theory behind such a scheme, and present an example of how it can be used to drive a register of $N$ atoms to a generalized W state and how the entanglement can be sustained indefinitely. The achievable steady-state fidelities for entanglement and its scaling with the number of qubits are discussed for presently existing superconducting quantum circuits. While the protocol is primarily discussed for a superconducting circuit architecture, it is ideally realized in any Cavity QED platform that permits controllable delivery of coherent electromagnetic radiation to specified locations.'
author:
- Camille Aron
- Manas Kulkarni
- 'Hakan E. Türeci'
bibliography:
- 'SpinChain.bib'
title: |
Photon-mediated interactions: a scalable tool\
to create and sustain entangled states of $N$ atoms
---
Introduction {#sec:intro}
============
Photon-mediated interactions are ubiquitous in nature. While the traditional formulation of quantum electrodynamics places equal emphasis on fields and sources, it is possible to take a point of view where the electromagnetic degrees of freedom are integrated out to reach an effective nonlocal field theory for matter only [@feynman_mathematical_1950]. A particularly beautiful example of this point of view, most closely related to the phenomena investigated here, is Schwinger’s formulation of the Casimir effect [@schwinger_casimir_1975]. Here, the electromagnetic degrees of freedom are integrated out and result in a photon-mediated retarded and nonlocal interaction between two conducting surfaces.
A classic example where photon-mediated interactions are at play is the superradiance (and subradiance) of a cluster of dipoles, first pointed out by Dicke [@dicke_coherence_1954]. Here, photon-mediated interactions ultimately lead to generation of transient coherence between dipoles [@gross_superradiance_1982; @brandes_coherent_2005], which results in the emission of a powerful pulse whose intensity scales with $N^2$, where $N$ is the number of dipoles within a volume $V \sim \lambda^3$ ($\lambda$ is the wavelength of radiation). In free space, however, such interactions decay fast with interdipole distance, and it is challenging to engineer such interactions in a controlled way [@devoe_observation_1996; @eschner_light_2001]. Photon-mediated interactions are also at play in the self-organization transition of optically driven cold atoms in a cavity [@ritsch_cold_2013]. In these systems, cavity-mediated long-range interactions between atoms, tunable by the drive strength, lead to softening of a motional excitation mode recently observed in experiments [@baumann_dicke_2010; @mottl_roton-type_2012]. Certain aspects of the underlying critical behavior of this non-equilibrium phase transition can be described through photon-mediated interactions between the atoms constituting the condensate [@brennecke_real-time_2013; @strack_dicke_2011; @kulkarni_cavity-mediated_2013; @konya_damping_2014].
In recent years, we have seen the first attempts to use such photon-mediated interactions to generate strong coupling and possibly entanglement between artificial atoms in solid-state cavity QED systems. These approaches capitalize on strong light-matter interactions that can be generated in confined geometries such as resonators [@majer_coupling_2007; @filipp_multimode_2011] and waveguides [@loo_photon-mediated_2013]. In particular, Ref. [@loo_photon-mediated_2013] demonstrated coherent exchange interactions between two superconducting qubits separated by as much as a full wavelength (in that case $\lambda \sim 18.6$ mm) in an open quasi-1D transmission line. More recently, superradiance of two artificial atoms was observed and characterized in a controlled setting in a superconducting quantum circuit [@mlynek_observation_2014].
![(color online) (a) One-dimensional array of cavity-qubit systems coupled by photon exchange and subject to one or several ac microwave drives, cavity decay at rate $\kappa$, qubit relaxation $\gamma$ and pure dephasing $\gamma_\phi$. (b) Implementation with superconducting transmon qubits embedded in interconnected microwave cavities and driven by external continuous-wave generators.[]{data-label="figschem1d"}](many1d.eps "fig:"){width="8.5cm"} ![(color online) (a) One-dimensional array of cavity-qubit systems coupled by photon exchange and subject to one or several ac microwave drives, cavity decay at rate $\kappa$, qubit relaxation $\gamma$ and pure dephasing $\gamma_\phi$. (b) Implementation with superconducting transmon qubits embedded in interconnected microwave cavities and driven by external continuous-wave generators.[]{data-label="figschem1d"}](irf.eps "fig:"){width="7.0cm"}
The goal of this paper is to show how photon-mediated interactions between qubits embedded in an engineered electromagnetic environment can be harnessed to controllably generate a certain large-scale entangled state of $N$ qubits [*in the steady state*]{}. In the present work, the role of photons is twofold. First, they mediate a coherent coupling between distant qubits. Second, they provide a controllable dissipative mechanism which can be used to stabilize a long-range correlated many-body state of the qubits. By driving the system at a suitable frequency, one can achieve a transition that produces the desired many-body state while dumping energy into one of the electromagnetic modes. The dissipative mechanisms are key to make the scheme steady-state, in contrast to better-known alternatives, such as those based on Rabi cycling between a ground state and a desired excited state. We show how the balance of the unitary and dissipative contributions can be precisely tuned by the placement of qubits in an engineered photonic environment, and a set of coherent electromagnetic drives with specified amplitudes and frequencies.
Engineered dissipative dynamics has recently been employed in superconducting circuits to cool a single qubit to a desired state on the Bloch sphere [@murch_cavity-assisted_2012], and two qubits that reside in a single cavity to a Bell state [@shankar_autonomously_2013]. Furthermore, a number of recent theoretical works have focused on generation of high-fidelity steady-state entanglement between two superconducting qubits [@leghtas_stabilizing_2013; @reiter_steady-state_2013; @huang_generation_2013; @aron_steady-state_2014]. We present the general theory underlying these phenomena, deriving and solving the dynamics of $N$ qubits residing in an arbitrary open electromagnetic environment, subject to coherent driving and losses. [The underlying new principle is based on using collective electromagnetic (EM) modes of a large structure (such as Bloch modes of a lattice of cavities) to dissipatively stabilize a collective state of spins. By “scalability” we refer to the fact that, for the specific protocol worked out here for W states, that the same protocol used for $N$ qubits can be used for $N+1$ qubits. Generally it is to be expected that the fidelity of stabilization degrades as $N$ is increased because of spectral crowding, which we discuss for the particular protocol we investigate here, and provide an estimate for the maximum number of qubits that can be stabilized reliably. Interestingly, spatial symmetries in the system (such as discrete translational invariance) can be used along with a [*spatially*]{} modulated coherent drive, to greatly enhance the spectral resolution. In one of the protocols we propose, this allows us to resolve W states in a way that the state selectivity is limited by only the finesse of the EM system, instead of the mean level spacing of the collective spin states.]{}
In Sec. \[sec:1dmodel\], we lay out a particular architecture that we have in mind for a proof-of-principle demonstration: we consider of model of $N$ qubits residing in a one-dimensional array of cavities. We present the different layers of approximations that allow us to have an analytic handle over the problem. First, we show how cavity photons mediate effective qubit-qubit interactions through the collective electromagnetic modes of the array. In Sec. \[sec:cooling\], we show how the [*full Liouvillian*]{} describing the evolution of the reduced density matrix of the $N$ qubits can be engineered to drive the qubit subsystem to generalized W states, and the entanglement sustained as long as the drives are on. The fabrication tolerances of such a cavity array has recently been studied experimentally [@underwood_low-disorder_2012]. Thus, the controlled fabrication of such arrays is well within reach of the current superconducting circuit technology both in coplanar and 3D configurations [@underwood_thesis_2015]. We present the fidelities that can be expected in recently fabricated systems and analyze the fault tolerance of the method to phase and amplitude noise of the drive parameters, as well as the nonuniformity of qubit and cavity parameters. Finally, in Sec. \[sec:ph-mediated\], we generalize our scheme to arbitrary arrangements of qubits coupled to an engineered photonic backbone, going beyond the tight-binding approximation for the EM system used in previous sections.
One-dimensional lattice model {#sec:1dmodel}
=============================
Consider a one-dimensional array of $N$ identical microwave cavities with nominally equal frequencies $\omega_{{\rm c}}$, coupled to each other capacitively described by a nearest-neighbor tunneling matrix element $J$. Each cavity houses a superconducting qubit with splitting $\omega_{{\rm q}}$. We shall consider both the cases of open (*i.e.* non-periodic) and periodic boundary conditions (identifying site $N$ with site 0). Later, we will relax our assumptions on identical cavity resonance frequencies and consider the most general case, showing that the general approach to the dissipative stabilization of a generalized W state stands.
Furthermore, each cavity shall be driven by a coherent monochromatic microwave source with frequency $\omega_{{\rm d}}$ and a site-dependent amplitude $\epsilon^{{{\rm d}}}_i$ and phase $\Phi_i$; see Fig. \[figschem1d\](b) for a 3D superconducting circuit architecture of the system Fig. \[figschem1d\](a). We work in a regime where $\omega_{{\rm c}}$, $\omega_{{\rm q}}$ and $\omega_{{\rm d}}$ are mutually far detuned from each other (typically on the order of GHz). The starting Hamiltonian is that of a one-dimensional driven Jaynes-Cummings lattice model studied before in Refs. [@knap_emission_2011; @nissen_nonequilibrium_2012; @grujic_non-equilibrium_2012; @hur_many-body_2015], $$\begin{aligned}
\label{eq:H}
H(t) = H_\sigma + H_{\sigma a} + H_a(t) \,,
\end{aligned}$$ where $H_\sigma$, $H_{\sigma a}$, and $H_a(t)$ are respectively the qubit, the Jaynes-Cummings light-matter coupling, and the driven cavity Hamiltonians, $$\begin{aligned}
H_\sigma =& \! \sum_i \omega_{{\rm q}}\frac{\sigma_i^z}{2},\, H_{\sigma a} = g \! \sum_i\! \left[ a_i^\dagger \sigma^-_i + \mbox{H.c.} \right] \,,
\\
H_a(t) =& \! \sum_i\! \left[ \omega_{{\rm c}}a_i^\dagger a_i
- J ( a_{i}^\dagger a_{i+1} + \mbox{H.c.} ) \right.
\nonumber \\
& \qquad \left. + 2 \epsilon^{{{\rm d}}}_i \cos(\omega_{{\rm d}}t + \Phi_i) \left( a_i + a^\dagger_i \right)\right] \,. \label{eq:Ha}\end{aligned}$$ Here, $i$ runs from site $0$ to $N-1$. The qubits are two-level systems described by the usual Pauli pseudo-spin operators $\sigma_i^{x,y,z}$ and $\sigma_i^\pm \equiv (\sigma^x_i \pm {{\rm i}}\sigma_i^y)/2$. $\epsilon^{{\rm d}}_i$ and $\Phi_i$ are, respectively, the cavity-dependent amplitude and phase of the ac microwave drives. Thermal equilibrium is achieved by setting all drive amplitudes to zero, $\epsilon^{{\rm d}}_i=0$. Without loss of generality, we assume that the detuning between cavity and qubit frequency $\Delta \equiv \omega_{{\rm q}}- \omega_{{\rm c}}> 0$, and $J>0$. We operate in the dispersive regime corresponding to $g/\Delta \sim 10^{-1}$ and at sufficiently weak drive amplitudes to ensure the presence of very few photons in the cavities, so the Schrieffer-Wolff perturbation theory to be applied shortly is well justified. Below, we entirely integrate out the photonic degrees of freedom, resulting in (a) an effective qubit-qubit interaction as discussed on more general grounds in Sec. \[sec:ph-mediated\], (b) local Zeeman fields of the form $\epsilon^{{\rm d}}_i \sigma_i^{x}$ (for $\Phi_i=0$), and (c) a non-unitary evolution characterized by controllable transition rates.
#### Intrinsic dissipation in the system. {#intrinsic-dissipation-in-the-system. .unnumbered}
In addition to the unitary dynamics described by Eq. (\[eq:Ha\]), we assume the individual qubits are coupled to uncontrolled environmental degrees of freedom that give rise to single qubit spin-flip rate ($\gamma$), a single-qubit pure dephasing rate $\gamma_\phi$, and a cavity decay rate $\kappa$.
#### Typical system parameters. {#typical-system-parameters. .unnumbered}
In a recent experiment studying the $N=2$ case of the protocol described here [@ucbpaper] in a 3D superconducting circuit architecture, typical system parameters were: $\omega_{{\rm c}}\simeq 7, \omega_{{\rm q}}\simeq 6, g \simeq J \simeq 10^{-1}, \kappa \simeq 7 \times 10^{-4}, \gamma \simeq 4 \times 10^{-5}$ all in units of $2\pi$ GHz. Below, our analytical approach is performed assuming the hierarchy of energy scales $\Delta \gg g, J \gg \kappa \gg \gamma \gg \gamma_\phi$.
#### Rotating wave approximation. {#rotating-wave-approximation. .unnumbered}
We eliminate the time dependence in $H_a(t)$ by working in the frame rotating at $\omega_{{\rm d}}$ and dropping the non-secular terms. In the rest of the Hamiltonian (\[eq:H\]), this also amounts to replacing $\omega_{{\rm q}}$ by $\Delta_{{\rm q}}\equiv \omega_{{\rm q}}- \omega_{{\rm d}}$ and $\omega_{{\rm c}}$ by $-\Delta_{{\rm c}}\equiv \omega_{{\rm c}}- \omega_{{\rm d}}$. Once expressed in the eigenbasis of the (undriven) coupled cavity system, $H_{a}$ is given by $$\begin{aligned}
\label{eq:Hak}
H_{a} = \sum_k (\omega_k -\omega_{{\rm d}}) a^\dagger_k a_k + ( \epsilon^{{\rm d}}_k a^\dagger_k + \mbox{H.c.} )\,.\end{aligned}$$ Here, $k$ is the discrete Bloch wavevector, $a_k^\dagger = \sum_j \varphi_{k}^*(j) \, a_j^\dagger$ creates a photon in the $k^{\mathrm{th}}$ mode. The specific discrete values of $k$ and the corresponding mode profiles $\varphi_k(j)$ are to be fixed by the boundary conditions. For periodic boundary conditions, the set of quasi-momenta are $k= 2\pi\, n/N$, with $n = 0,\ldots, N-1$ and $\varphi_{k}(j) = {{\rm e}}^{-{{\rm i}}k j} / \sqrt{N}$. For open boundary conditions, $ k = \pi (n+1)/(N+1)$ and $\varphi_k(j) = \sqrt{2/(N+1)} \sin\left(k(j+1)\right)$. In Eq. (\[eq:Hak\]), we incorporated the phase $\Phi_j$ in $\epsilon^{{\rm d}}_j$, which can henceforth be complex and $\epsilon^{{\rm d}}_k = \sum_j \varphi_{k}(j) \, \epsilon_j^{{\rm d}}$. The eigenfrequencies are given by the photonic dispersion relation $$\begin{aligned}
\omega_k = \omega_{{\rm c}}- 2 J \cos(k)\,.\end{aligned}$$
![(color online) Engineered qubit many-body spectrum. The non-equilibrium drive and the resulting photon fluctuation bath are used to create a dominant transition from the trivial ground state to a target entangled W state in the 1-excitation manifold. The existence of a non-zero spontaneous decay $\gamma$ is critical to avoid populating higher-excited state manifolds.[]{data-label="fig:mediated"}](mechaN.eps){width="3.5cm"}
#### Effective dissipative $XY$ model. {#effective-dissipative-xymodel. .unnumbered}
We eliminate the light-matter interaction to second-order perturbation theory in $g/\Delta$ by means of a Schrieffer-Wolf (SW) transformation which maps $H \mapsto {{\rm e}}^{X} H {{\rm e}}^{X^\dagger}$ where $$\begin{aligned}
X \equiv g \sum_k \left[ \frac{a_k \sigma_k^\dagger}{\omega_{{\rm q}}- \omega_k}
-\mathrm{H.c.}
\right]\,.\end{aligned}$$ Collecting all the terms, we obtain an isotropic $XY$ model subject to a magnetic field for the qubit subsystem ($H_\sigma$), weakly coupled to the photon fluctuations of the EM backbone ($H_{\sigma a}$): $$\begin{aligned}
H_\sigma =& \! \sum_i
\boldsymbol{h_i} \cdot \frac{\boldsymbol{\sigma}_i}{2} - \frac{J}{2} \left(\frac{g}{\Delta}\right)^2 \left[ \sigma^x_{i} \sigma^x_{i+1} + \sigma^y_{i} \sigma^y_{i+1}\right]\,, \label{eq:Hspinchain} \\
H_{\sigma a} &= \sum_i \left(\frac{g}{\Delta}\right)^2 \sigma_i^z( \Delta a_i^\dagger a_i + \epsilon^{{\rm d}}_i a_i^\dagger +{\epsilon^{{\rm d}}_i}^* a_i)\,.
\label{eqtransXY}\end{aligned}$$ Here, ${h}^x_i = 2 \mbox{Re}(\epsilon^{{\rm d}}_i) (g/\Delta)$, $h^y_i= -2 \mbox{Im}(\epsilon^{{\rm d}}_i) (g/\Delta)$, $h^z_i = \Delta_{{\rm q}}+ \Delta (g/\Delta)^2 $. The effective magnetic field $\boldsymbol{h}_i$ is mainly oriented along the $z$ direction but we shall see that, while they break the integrability of the model, the $x$ and $y$ components of the emergent Zeeman field play a crucial role for the scheme below.
We note that the SW transformation is also responsible for subleading corrections to the strength of the dissipative terms (*e.g.* the qubit decay $\gamma$ acquires a contribution from the cavity decay $\kappa$, corresponding to a Purcell contribution, and *vice versa*) the local contributions of which can be simply included by working with renormalized parameters $\kappa$, $\gamma$ and $\gamma_\phi$ at low photon occupations [@boissonneault_dispersive_2009], which is the regime studied here. These effective parameters can be extracted from experimental measurements.
When the drives are off, $\epsilon^{{\rm d}}_i = 0$, the system simply thermalizes with its environment – this is just the physics of blackbody radiation in a coupled cavity system (including $N$ dipoles). Viewed from the perspective of the qubits, while $H_\sigma$ describes a quantum phase transition from a paramagnetic to a ferromagnetic phase when the magnitude of the transverse field is on the order of the nearest-neighbor coupling $J ( g/\Delta)^2$, this regime is never reached for realistic system parameters. For Raman-driven qubits the situation is more interesting, and the phase diagram was recently studied displaying various exotic attractors [@schiro_exotic_2015]. Henceforth, we only work in the experimentally achievable regime $\omega_{{\rm q}}\gg J(g/\Delta)^2$ and, to leading order, the ground state of $H_\sigma$ is simply the separable state $|\boldsymbol{0} \rangle \equiv | \! \downarrow \ldots \downarrow \rangle$. The low-energy spectral content of the qubit sector is depicted in Fig. \[fig:mediated\]: the first $N$-qubit excited manifolds are roughly separated by $\Delta_{{\rm q}}$ while the lifting of degeneracy within each manifold is controlled by $J (g/\Delta)^2$
From the point of view of equilibrium statistical mechanics, the many-body system of qubits described by $H_\sigma$ is not interesting because it would thermalize with the radiative reservoir, reaching a steady state $\rho_\sigma (t \rightarrow \infty) \propto {{\rm e}}^{-\beta H_\sigma} \sim | \boldsymbol{0} \rangle \langle \boldsymbol{0}|$, a collection of uncorrelated qubits in their ground state. We shall see that turning on the drives will change this situation dramatically. A careful choice of drive parameters (even for small amplitudes of drives) will be shown to enable the stabilization of a particular many-body state of qubits in the excited-state manifold.
Stabilization of generalized W states {#sec:cooling}
=====================================
Our goal in this section is to identify non-trivial entangled eigenstates of the spin chain $H_\sigma$ and design a protocol which, starting from the ground state $|\boldsymbol{0}\rangle$ that can be straightforwardly prepared, achieves the stabilization of an interesting excited state of choice. Below we discuss the details of a robust protocol for the stabilization of a generalized W state of qubits with [*minimal*]{} resources.
#### Linearized photon spectrum. {#linearized-photon-spectrum. .unnumbered}
We first address the non-linearities in $H_{\sigma a}$ by decomposing the photonic field into mean-field plus bosonic fluctuations: $$\begin{aligned}
a_k \equiv \bar a_k + d_k \mbox{ with } \bar a_k = \frac{\epsilon^{{\rm d}}_k}{\omega_{{\rm d}}- \omega_k + {{\rm i}}\kappa/2}\;. \label{eqsmallfluct}\end{aligned}$$ We assume here that all Bloch modes have the same loss $\kappa$. This can be made more precise but it will not qualitatively change the results we present. Neglecting those terms that are quadratic in the fluctuations and that couple to the qubits, the light sector reduces to $$\begin{aligned}
H_a \rightarrow H_d &= \sum_k (\omega_k - \omega_{{\rm d}}) d_k^\dagger d_k \,, \label{eq:bath1} \\
H_{\sigma a} \rightarrow H_{\sigma d} &= \left(\frac{g}{\Delta}\right)^2 \sum_i \sigma^z_i [( \Delta \bar{a}_i + \epsilon^{{\rm d}}_i ) d_i^\dagger + \mbox{H.c.} ]\,. \label{eq:bath2}\end{aligned}$$
#### Diagonalization of the matter sector. {#diagonalization-of-the-matter-sector. .unnumbered}
The Hamiltonian Eq. (\[eq:Hspinchain\]) in the presence of non-zero drive terms is non-integrable. We therefore proceed by projecting it into the low-energy sector with a maximum one excitation: $$\begin{aligned}
\label{eq:Hstrunc}
H_\sigma =\sum_k E_k |k \rangle \langle k | + \left( \frac{g}{\Delta} \right) \left( \epsilon^{{\rm d}}_k |k\rangle\langle \boldsymbol{0} | + \mbox{H.c.} \right),\end{aligned}$$ We have set the energy of the ground state $|\boldsymbol{0} \rangle \equiv | \downarrow \ldots \downarrow \rangle$ to zero ($E_{\boldsymbol{0}} = 0$). Here the states $$|k\rangle = \sum_{i=0}^{N-1} \varphi_k^* (i) \ |i\rangle\,,$$ with $|i\rangle \equiv |\downarrow_0 \ldots\downarrow_{i-1} \, \uparrow_i \, \downarrow_{i+1} \ldots \downarrow_{N-1} \rangle$ indicating one excitation located at site $i$, are states that carry a single qubit excitation of quasi-momentum $k$, entangled over the entire chain. These are the eigenstates of the [*undriven*]{} spin chain (*i.e.* for $\epsilon^{{\rm d}}_i =0 \, \forall\, i$) with a dispersion relation $$\begin{aligned}
E_k = \epsilon_k - \omega_{{\rm d}}, \; \epsilon_k = \omega_{{\rm q}}+ \delta\omega_{{\rm q}}- 2 J \left(\frac{g}{\Delta}\right)^2 \! \cos(k)\;.\end{aligned}$$ Here $\delta\omega_{{\rm q}}\simeq g^2/\Delta$ is the cavity-induced Stark shift. This truncation of the Hamiltonian holds if the higher-excitation manifolds are not significantly occupied during the dynamics. This can be checked a posteriori and we do so. Let us first discuss the case of open boundary conditions for which the absence of translational and space-reversal symmetry generically yields a fully non-degenerate spectrum. The effect of the drive term in Eq. (\[eq:Hstrunc\]), assumed to be small as stated before, can be taken into account through a perturbation theory and yields the following eigenstates of $H_\sigma$ to lowest order in $({g}/{\Delta}) (\epsilon_k^{{\rm d}}/\Delta_{{\rm q}})$: $$\begin{aligned}
|\widetilde{\boldsymbol{0}} \rangle
&\simeq
|\boldsymbol{0} \rangle \!-\! \left(\frac{g}{\Delta}\right) \! \sum_{k} \! \frac{\epsilon^{{\rm d}}_k}{\Delta_{{\rm q}}} |k \rangle,\,
\widetilde{E}_{\boldsymbol{0}} \simeq \!-\! \left(\frac{g}{\Delta}\right)^2 \! \sum_k \! \frac{|\epsilon^{{\rm d}}_k|^2}{\Delta_{{\rm q}}},\\
|\widetilde{k} \rangle & \simeq
|k \rangle + \left(\frac{g}{\Delta}\right) \frac{{\epsilon^{{\rm d}}_k}^*}{\Delta_{{\rm q}}} |\boldsymbol{0} \rangle,\,
\widetilde{E}_k \simeq E_k + \left(\frac{g}{\Delta}\right)^2 \frac{|\epsilon^{{\rm d}}_k|^2}{\Delta_{{\rm q}}}.
\label{eqqbitcollen}\end{aligned}$$ The above corrections to the undriven eigenstates are crucial for the success of the two-photon cooling mechanism presented below.
#### Transition rates. {#transition-rates. .unnumbered}
By virtue of Eq. (\[eqsmallfluct\]), the coupling of the photonic fluctuations (on top of a classical part) to the spin-chain Eq. (\[eqtransXY\]) can be treated in perturbation theory. This permits us to integrate them out arriving at an effective master equation for qubits only. We note that the fluctuations of the collective photon modes of the lattice, described by spectral function per mode $q$ $$\begin{aligned}
\rho_q(\omega) = - \frac{1}{\pi} \mbox{Im } \frac{1}{\omega -\omega_q+{{\rm i}}\kappa/2}\;,\end{aligned}$$ can be easily manipulated by the design of the cavity lattice. Assuming that the photon fluctuations, which couple to the spin degrees of freedom via [Eq. (\[eq:bath2\])]{}, thermalize with the radiative environment which in turn is taken to be at very low temperature, we arrive at the effective master equation for spins only $\rho_\sigma$ at steady state, $\rho_\sigma^{\rm NESS} \equiv \lim\limits_{t\to\infty} \rho_\sigma$: $$\begin{aligned}
\label{eq:master}
\partial_t \rho^{\rm NESS}_\sigma & \! \! = 0 = -{{\rm i}}\left[ H_\sigma,\rho^{\rm NESS}_\sigma \right]
+
\sum_k \Gamma_{\boldsymbol{0}\to k} \mathcal{D}[| \widetilde{k} \rangle \langle \widetilde{\boldsymbol{0}} | ] \rho^{\rm NESS}_\sigma \nonumber \\
& \hspace{-3em} + \gamma \sum_k \mathcal{D}[| \widetilde{\boldsymbol{0}} \rangle \langle \widetilde{k} | ] \rho^{\rm NESS}_\sigma
+ \frac{2\gamma_\phi}{N} \sum_{k\,q} \mathcal{D}[| \widetilde{q} \rangle \langle \widetilde{k} | ] \rho^{\rm NESS}_\sigma.\end{aligned}$$ The derivation of this master equation relies on the separation of time scales: the relaxation time scale of cavity fluctuations $d_k$, on the order of $1/\kappa$, is much shorter than that of the reduced density matrix of the spins $\rho_\sigma(t)$. This separation of time scales is perfect in the steady state [@afoot; @gsc; @amitra] which we are interested in here.
The Lindblad-type operators are defined as $\mathcal{D}[X] \rho \equiv \left( X \rho X^\dagger - X^\dagger X \rho + \rm{H.c.} \right)/2$ and the $\Gamma_{\boldsymbol{0} \to k} $’s correspond to non-equilibrium transition rates between the ground state $|\widetilde{\boldsymbol{0}} \rangle$ and a given excited many-body state $| \widetilde{k} \rangle$ \[see Eq. (\[eqqbitcollen\])\] in the one-excitation manifold of $H_\sigma$, given in Eq. (\[eq:Hspinchain\]). These are found to be $$\begin{aligned}
\label{eq:rate}
\Gamma_{\boldsymbol{0} \to k} = 2 \pi \sum_q \Lambda_{kq}^2\rho_q(\omega_{{\rm d}}+\widetilde{E}_{\boldsymbol{0}} -\widetilde{E}_k)\,,\end{aligned}$$ where the transition matrix element is given by $$\begin{aligned}
\Lambda_{kq} =& \left|
\left(1+2\frac{\Delta}{\Delta_{{\rm c}}} \right) \!\!
\frac{1}{\Delta_{{\rm q}}} \!\! \left(\frac{g}{\Delta}\right)^3 \!\!\!
\sum_{k' k'' q'} \!\!\! f_{k k' k''} f_{q' q k'}^* \epsilon^{{\rm d}}_{k''} \epsilon^{{\rm d}}_{q'} \right|\,,\end{aligned}$$ with the tensor $f_{k k' k''} \equiv \sum_i \varphi_k(i) \varphi_{k'}^*(i) \varphi_{k''}^*(i)$. The integration over the photon-fluctuation degrees of freedom also yields Lamb-shift corrections of the energy levels, but this does not play any substantial role in our scheme, see discussion below Eq. (\[eq:wdopt\]).
#### Dynamics. {#dynamics. .unnumbered}
By virtue of having written the steady-state master equation Eq. (\[eq:master\]) in the eigenbasis of $H_{\sigma}$, all off-diagonal matrix elements of $\rho_\sigma^{\rm NESS}$ by construction vanish as the steady state is approached. Consequently, the dynamics can be faithfully described by effective rate equations for the populations of eigenstates, $n_{\boldsymbol{0}}$ and $n_k$: $$\begin{aligned}
\frac{{{\rm d}}n_{\boldsymbol{0}}}{{{\rm d}}t} &= \gamma \, \sum_q n_q - \Gamma_{\boldsymbol{0}\to q} \, n_{\boldsymbol{0}} \label{eq:eom1} \\
\frac{{{\rm d}}n_k}{{{\rm d}}t} &= - \gamma \, n_k + \Gamma_{\boldsymbol{0}\to k} \, n_{\boldsymbol{0}}
+ \frac{2 \gamma_\phi}{N} \sum_{q} (n_q -n_k\, ). \label{eq:eom2}\end{aligned}$$ The terms in $\gamma$ correspond to qubit decay, flipping down the pseudo spins and relaxing the energy by $\Delta_{{\rm q}}$ (in the rotating frame). The terms in $\gamma_\phi$ correspond to pure dephasing processes, the action of which is to equalize the populations of the states in the one-excitation manifold. The emergent level structure and rates are summarized in Fig. \[fig:mediated\].
We note that, while the full dynamical evolution of the qubit-EM system (viz. Eq. (\[eq:H\]) in the presence of qubit and cavity decay) is clearly non-Markovian, the proper secularization of the equations around the operation frequency $\omega_d$ allows us to describe the qubit dynamics through the relatively transparent rate equations (\[eq:eom1\] and \[eq:eom2\]).
Irrespective of the initial conditions, Eqs. (\[eq:eom1\] and \[eq:eom2\]) have a unique non-equilibrium steady-state solution and, after transient dynamics, the occupation of the state $|k\rangle$ is given by $$\begin{aligned}
n_k^{\rm NESS} &= \frac{1}{1+2\gamma_\phi/\gamma} \frac{\Gamma_{\boldsymbol{0} \to k} + (2\gamma_\phi/N\gamma) \sum_q \Gamma_{\boldsymbol{0} \to q} }{\gamma + \sum_q \Gamma_{\boldsymbol{0} \to q}}
\,. \label{eq:nqness}\end{aligned}$$
#### Stabilization protocol. {#stabilization-protocol. .unnumbered}
Equation (\[eq:nqness\]), together with Eq. (\[eq:rate\]), transparently elucidates how to stabilize a given pure entangled state of qubits $|k\rangle$ in the steady state. The protocol requires the maximization of $\Gamma_{\boldsymbol{0}\to k}$, given in Eq. (\[eq:rate\]), to make it the largest of all rates among ($\{ \Gamma_{\boldsymbol{0}\to q} \}$, $\gamma$, $\gamma_\phi$). This is performed by optimally tuning the drive frequency $\omega_{{\rm d}}$ such that the sharply peaked photonic spectrum $\rho_q(\omega_{{\rm d}}+\widetilde{E}_{\boldsymbol{0}} -\widetilde{E}_k)$ in Eq. (\[eq:rate\]) reaches the maximum amplitude of the Lorentzian, which is on the order of $1/\kappa$. This is possible whenever there is at least one mode $q_0$ with $\Lambda_{k q_0} \neq 0$ and the optimum $\omega_{{\rm d}}$ is the solution of the energy-conservation equation $\omega_{{\rm d}}+\widetilde{E}_{\boldsymbol{0}} -\widetilde{E}_k = \omega_{q_0}$, *i.e.* $$\begin{aligned}
\omega_{{\rm d}}= & \frac{\omega_{{\rm q}}+ \delta\omega_{{\rm q}}+\omega_{{\rm c}}}{2} - J \cos(q_0) \nonumber \\
& \quad + \left(\frac{g}{\Delta}\right)^2 \left[ - J \cos(k) + \frac12 \sum_{q\neq k} \frac{|\epsilon^{{\rm d}}_q|^2}{\Delta_{{\rm q}}} \right]
\,.
\label{eq:wdopt}\end{aligned}$$ This energy-matching condition describes a one-photon process in the rotating frame equivalent to a two-photon process in the laboratory frame. The corresponding Raman inelastic scattering process uses the energy of the two incoming drive photons to perform the qubit transition while simultaneously dumping a photon in one of the cavity modes [@ucbpaper]. We note that when Eq. (\[eq:wdopt\]) is satisfied, the Lamb-shift correction of $\widetilde{E}_q$ vanishes.
![\[fig:gamma\] (color online) Schematics of the pumping rate $\Gamma_{0 \to k}$ as a function of the drive frequency $\omega_{{\rm d}}$ for $N=5$. (a) Driving first cavity only: any of the five photon-fluctuation modes (responsible for the five peaks) can be used for dissipative stabilization. Driving to $|k\rangle$ while avoiding populating the nearest state $|k'\rangle$ necessitates $\kappa < \Delta \widetilde{E} \equiv |\widetilde{E}_k - \widetilde{E}_{k'}| \sim 2\pi J (g/\Delta)^2 /N$. (b) Driving all cavities equally: only one of the five modes can channel the mechanism and driving to the nearest state $|k'\rangle$ is avoided if $\kappa < \Delta \omega \equiv |\omega_k - \omega_{k'}| \sim 2\pi J / N$. ](Gamma2.eps){width="8.4cm"}
#### [Scalability and limitations.]{} {#scalability-and-limitations. .unnumbered}
Equation (\[eq:nqness\]) sets an upper bound on the fidelities, $n_{k}^{\rm NESS} \leq n^{\rm max} =
(\gamma+2\gamma_\phi/N)/(\gamma+ 2 \gamma_\phi)$, which highlights the necessity of working with qubits that have a pure dephasing rate $\gamma_\phi$ much smaller than their relaxation rate $\gamma$. This upper bound is not tight but allows us to highlight the role of the pure dephasing mechanisms. The success of the protocol, [and its scalability to large N]{}, also relies, via the numerator of Eq. (\[eq:nqness\]), on the resolving power of the spectral width of the photon density of states ($\sim \kappa$) *i.e.* the precision with which the photon fluctuations can target the spin-chain state $|k\rangle$ without exciting other eigenstates close in energy. The limitations on the resolving power depend strongly on the drive spatial profile. This is illustrated here considering two extreme cases, one where only one cavity is driven ($\epsilon^{{\rm d}}_i = \epsilon^{{\rm d}}\delta_{i1}$), the other corresponding to a case where all cavities are driven with equal amplitude ($\epsilon^{{\rm d}}_i = \epsilon^{{\rm d}}\, \forall i$). In the first case \[Fig. \[fig:gamma\](a)\], the transition rates for every $0 \to k$ have multiple peaks. Thus the stabilization of $|k\rangle$ at the optimal frequency given by Eq. (\[eq:wdopt\]) while avoiding the population of the nearest state $|k'\rangle$ requires the condition $\kappa < \Delta \widetilde{E} \equiv |\widetilde{E}_k - \widetilde{E}_{k'}| \sim 2\pi J (g/\Delta)^2 /N$. On the other hand, the second case, driving each site identically yields rates that have a single peak, that for neighboring spin chain states $k$ and $k'$ are separated by the free spectral range (of the collective EM modes) of the cavity chain, $\Delta \omega \equiv |\omega_k - \omega_{k'}| \sim 2\pi J / N$. [This indicates that the protocol with uniform driving can be scaled up to a number of qubits on the order of $N_{\rm max} \lesssim 2 \pi J / \kappa$.]{}
This situation is easily understood: the rates given in Eq. (\[eq:rate\]) contain the fulfillment of an energy-conservation condition, which can, in principle, be satisfied picking any collective EM mode $q$. The set of non-zero transition matrix elements in the sum Eq. (\[eq:rate\]) can, however, be substantially narrowed down by choosing a drive amplitude profile ($\epsilon^{{\rm d}}_{i}$) that is narrow in the momentum domain. For example, $\epsilon^{{\rm d}}_{p} = \delta_{pp_0}$, collapses the sum to a single term by imposing a quasi-momentum conservation condition between $p_0$, the target spin-chain state with momentum $k$, and $q_0$, the quasi-momentum of the collective EM mode picked for stabilization (note that conservation of momentum is strictly valid in periodic systems). This is given by $k = 2p_0 - q_0$ and is consistent with the interpretation of stabilization via a two-photon process. For such driving, the optimal frequency of the drive $\omega_{{\rm d}}$ is then given by Eq. (\[eq:wdopt\]). We note that this transparent criterion was used in Ref. [@schwartz_toward_2015] to selectively stabilize either the triplet or the singlet state of two transmon qubits.
#### Effective master equation simulations. {#effective-master-equation-simulations. .unnumbered}
To complement the analytic approach, we have performed full numerical simulations of the effective master equation (\[eq:master\]) where we (i) compute exactly the full spectrum of the spin chain $H_\sigma$ in Eq. (\[eq:Hspinchain\]) with $N=5$ qubits and open boundary conditions (ii) determine the rates between all the eigenstates and (iii) solve for the steady-state populations. In these calculations, sufficient number of higher-excitation manifolds of the spin chain were included to achieve convergence.
Because the parameter space is fairly large, we performed our simulations for a presently existing fabricated system for $N=2$ [@schwartz_toward_2015]. These parameters are quoted in the caption of Fig. \[fig:firstsiteopen\], where the fidelities to achieve various spin-chain states ($k$) for $N=5$ are compared for a localized drive \[Fig. \[fig:firstsiteopen\](a)\] and a spatially uniform drive \[Fig. \[fig:firstsiteopen\](b)\]. We note that compared to the current state of the art [@shankar_autonomously_2013], these are remarkable fidelities. These fidelities can be significantly improved by reducing the ratio of the dephasing over the qubit relaxation rate.
We have also tested the robustness of our protocol against site-to-site inhomogeneities of the different parameters and found no qualitative difference for $\delta\omega_{{\rm c}}/\omega_{{\rm c}}\sim 10^{-2}$, $\delta\omega_{{\rm q}}/\omega_{{\rm q}}\sim 10^{-4}$, $\delta g/g \sim 10^{-4}$ and $\delta J/J \sim 10^{-2}$. A more extensive analysis of achievable fidelities in the presence of site-to-site inhomogeneities will be presented in future work.
#### Periodic boundary conditions. {#periodic-boundary-conditions. .unnumbered}
Let us now consider the case of periodic boundary conditions for which the [*undriven*]{} system is space-translational and space-reversal invariant. Such a symmetry results in degeneracies between the eigenstates $|k\rangle$ and $|2\pi -k \rangle$ of the undriven spin chain (except for $k=0$ and $k=\pi$). A symmetry-breaking drive profile will generically lift the degeneracy of the spectrum and, importantly, the emergent eigenstates will strongly depend on the particular drive profile. To exemplify this point, let us start by driving the first cavity only: $\epsilon^{{\rm d}}_i = \epsilon^{{\rm d}}\delta_{i,0}$. Second-order degenerate perturbation theory in $({g}/{\Delta})(\epsilon^{{\rm d}}/\sqrt{N}\Delta_{{\rm q}})$ lifts the degeneracy in the subspaces spanned by $|k\rangle$ and $|2\pi - k \rangle$. To lowest order, the eigenstates are $$\begin{aligned}
\label{eq:kpm}
|k_\pm \rangle \equiv \frac{| k \rangle \pm | 2\pi - k \rangle}{\sqrt{2}}\end{aligned}$$ for all $k \in\, ]0,\pi[$ complemented with the W state $|0_+\rangle \equiv | 0 \rangle$ and $|\pi_+\rangle \equiv | \pi \rangle$ (for $N$ even), one obtains the rates $\Gamma_{\boldsymbol{0} \to k_\pm} = 2 \pi \Lambda_{k_\pm}^2 \sum_q \rho_{q}(\omega_{{\rm d}}+\widetilde{E}_{\boldsymbol{0}} -\widetilde{E}_{k_\pm})$ with $$\begin{aligned}
\Lambda_{k_-} \!= 0 \mbox{ and } \Lambda_{k_+} \! = \!\left| \frac{\sqrt{2}}{N^2} \left(1+2\frac{\Delta}{\Delta_{{\rm c}}} \right)
\left(\frac{g}{\Delta}\right)^3 \frac{{(\epsilon^{{\rm d}}})^2}{\Delta_{{\rm q}}} \right|
\end{aligned}$$ for all $|k_-\rangle$ and $|k_+\rangle$ except for $|0\rangle$ or $|\pi\rangle$ in which case $\Lambda_{k_+}$ is reduced by a factor $\sqrt{2}$. Figure \[fig:firstsite\] shows that the $|k_+\rangle$ states can be obtained with substantial fidelities.
It is worth noting that, in the case of a generic driving profile $\epsilon^{{\rm d}}_i$, instead of Eq. (\[eq:kpm\]), the emergent eigenstates are given by $$\begin{aligned}
|k_\pm \rangle \equiv \frac{| k \rangle \pm \alpha_k | 2\pi - k \rangle}{\sqrt{1+\alpha_k^2}}\end{aligned}$$ where $\alpha_k$, the relative weight of $|k\rangle$ and $|2\pi-k\rangle$, is now controlled by the ratio $\epsilon^{{\rm d}}_k/\epsilon^{{\rm d}}_{2\pi-k}$. Therefore, such a non-equilibrium symmetry-breaking scenario offers highly flexible control over the target entangled state by simply engineering the drive profile $\epsilon^{{\rm d}}_k$.
Photon-mediated interactions: General formulation
=================================================
![\[fig:3dEM\] (color online) Light-mediated interactions offer a highly versatile platform to design and control networks of interacting qubits. (a) The qubits (blue arrows) are embedded in an electromagnetic environment, which is the solution of the Maxwell equations in a given scattering geometry. (b) Integrating out the EM degrees of freedom yields effective interactions between the qubits, forming a network that can sustain large-scale entangled many-body states.](3dEM2.eps){width="7.5cm"}
\[sec:ph-mediated\] In the previous sections, we focused on a particular geometry of the cavity-qubit system, namely a one-dimensional tight-binding lattice of photons. In this section, we show that our dissipative stabilization scheme via photon-mediated interactions is broadly applicable to any engineered EM environment. The situation we consider here is depicted in Fig. 6 where the two-level systems are now placed at arbitrary locations and interact with a general EM environment \[Fig. 6(a)\]. In practice, the latter is described by the solution of Maxwell equations in an arbitrary scattering geometry, characterized by a spectral problem with certain continuity and boundary conditions. In the simplest case, this would be a single resonator or a waveguide or, as in the specific example discussed previously, an array of evanescently (or capacitively) coupled cavities (transmission line cavities). After the derivation of the most general result, we show how the effective tight-binding result can be derived from first principles.
The qubits and their coupling to the EM fluctuations are described by the Hamiltonian $$\begin{aligned}
H = H_{\sigma} + H_{\sigma-{\rm EM}} + H_{\rm EM}\,,\end{aligned}$$ with ($\hbar = 1$) $$\begin{aligned}
& H_{\sigma} = \sum_i \omega^{{\rm q}}_i \frac{\sigma^z_i}{2}, \quad
H_{\rm EM} = \frac{\epsilon_0}{2} \int_\mathcal{V} \!\! {{\rm d}}^3 \boldsymbol{x} \, \left( \boldsymbol{E}^2 + c^2 \boldsymbol{B}^2 \right), \\
& H_{\sigma-{\rm EM}} = - \int_\mathcal{V} \!\! {{\rm d}}^3 \boldsymbol{x} \,\boldsymbol{P} \cdot \boldsymbol{E}\,.\end{aligned}$$ The integrals above run over the entire volume $\mathcal{V}$ of the scattering structure. We note that such a Hamiltonian can indeed be obtained for sub-gap electrodynamics in a general superconducting circuit architecture by a proper choice of normal modes [@malekakhlagh_origin_2015], starting from the parameters (position-dependent capacitances and inductances per unit length and the parameters of the qubits) of the underlying electrical circuit.
For qubits residing at $\boldsymbol{x}_i$, each with a dipole moment strength $\mu$ (projected along a particular eigenpolarization of the electromagnetic medium), the collective atomic polarization operator can be written as $$\begin{aligned}
P(\boldsymbol{x}) = \mu \sum_i \delta^3(\boldsymbol{x}-\boldsymbol{x}_i) \, \sigma_i^x\,.\end{aligned}$$ The electric field can generally be written in terms of a complete set of modes and corresponding eigenfrequencies $\{ \varphi_n , \omega_n \}$ specific to the chosen architecture $$\begin{aligned}
E(\boldsymbol{x}) = \sum_{n} \varepsilon_n \varphi_n(\boldsymbol{x}) a_{n} + \mbox{H.c.}\;,\end{aligned}$$ so that $$\begin{aligned}
\label{eq:Ha1}
H_{\rm EM} = \sum_{n} \omega_n a_{n}^\dagger a_{n}\,.\end{aligned}$$ Here, $a^\dagger_n$ ($a_n$) creates (annihilates) a photon in the spatial mode $\varphi_n(\boldsymbol{x})$ with frequency $\omega_n$ and corresponding zero-point electric field $\varepsilon_n$. The modes are assumed to satisfy the completeness and orthogonality conditions $\sum_n \varphi_n(\boldsymbol{x}) \varphi_n ^* (\boldsymbol{x}') = \delta^3(\boldsymbol{x}-\boldsymbol{x}')$ and $ \int_\mathcal{V} \! {{\rm d}}^3 \boldsymbol{x} \, \varphi_n(\boldsymbol{x}) \varphi_m^*(\boldsymbol{x}) = \delta_{nm} $. With these normalization conditions, the zero-point fields are given by $\varepsilon_n \equiv \sqrt{{\omega_n}/{2\epsilon_0}}$. Neglecting counter-rotating terms, the light-matter coupling becomes $$\begin{aligned}
H_{\sigma-{\rm EM}} = & \sum_{i,n} g_n \varphi_n(\boldsymbol{x}_i)\, a_n^\dagger \sigma_i^- + \mbox{H.c.}\,,\end{aligned}$$ where $\sigma_i^\pm \equiv (\sigma^x_i \pm {{\rm i}}\sigma_i^y)/2$ and $g_n \equiv - \mu \varepsilon_n$.
We shall consider the regime in which the qubit frequencies $\omega^{{\rm q}}_i$ are far detuned from the photonic modes $\omega_n$ such that the light-matter coupling can be treated via second-order perturbation theory in $g_n/(\omega^{{\rm q}}_i - \omega_n)$. This is achieved by means of a Schrieffer-Wolf (SW) transformation [@schrieffer_relation_1966] which maps $H \mapsto {{\rm e}}^{X} H {{\rm e}}^{X^\dagger}$ where $$\begin{aligned}
\label{eq:SWX}
X \equiv \sum_{n,i} \left[ \frac{g_n \varphi_n(\boldsymbol{x}_i)}{\omega_i^{{\rm q}}- \omega_n} \sigma_i^+ a_n
-\mathrm{H.c.}
\right]. \end{aligned}$$ This yields the following Hamiltonian $H = H_\sigma + H_{\sigma a} + H_a$ to $\mathcal{O}(g^2/\Delta^2)$, with $$\begin{aligned}
H_{\sigma} =& \sum_i \omega^{{\rm q}}_i \frac{\sigma^z_i}{2} + \frac{1}{2} \sum_{ij} \Sigma_{ij}(\omega^{{\rm q}}_i) \sigma_j^- \sigma_i^+ + \mbox{H.c.}\,, \label{eq:hsigma} \\
H_{\sigma a} =& \sum_{i, mn} \lambda_{i, mn}(\omega_i^{{\rm q}}) \,\frac{\sigma^z_i}{2} \, a_n a^\dagger_m + \mbox{H.c.} \,, \label{eq:sigmaa}\end{aligned}$$ and $H_a$ is still given by Eq. (\[eq:Ha1\]). In this low-energy Hamiltonian, the second term in $H_\sigma$ describes the qubit-qubit interactions mediated by virtual photons. These photons can be emitted into and absorbed from photonic channels at frequencies $\omega_n$ with spatial distribution $\varphi_n(\boldsymbol{x})$. This is precisely the story told by the coefficients $\Sigma_{ij}(\omega) = \sum_n |g_n|^2 \varphi_n(\boldsymbol{x}_i) \varphi_n^* (\boldsymbol{x}_j)/(\omega - \omega_n)$, which appears as a self-energy correction to the qubit sector. $\Sigma_{ij}(\omega) \sigma_j^x$ can be seen as the electric field generated at $\boldsymbol{x}_i$ by a dipole at $\boldsymbol{x}_j$, oscillating harmonically at frequency $\omega$. Note that the bare electromagnetic retarded Green’s function is given by $G^{\rm R} (\boldsymbol{x},\boldsymbol{x}'; \omega) = \sum_n \varphi_n(\boldsymbol{x}) \varphi_n^*(\boldsymbol{x}')/(\omega - \omega_n)$. This immediately implies that, in principle, all qubits interact with each other, to the extent that they can radiate EM radiation to each other \[see schematic in Fig. 6(b)\]. We note that realization-specific and restricted versions of this interaction vertex have been derived before [@majer_coupling_2007; @loo_photon-mediated_2013].
For what is proposed here, however, an equally important role is played by the term $H_{\sigma a}$ in Eq. (\[eq:sigmaa\]). This is the generalized version of the ac Stark-shift contribution to a qubit’s frequency that is well known in the dispersive regime of single-mode Cavity QED [@boissonneault_dispersive_2009], which can also be interpreted as a scattering term for photons generated by the interaction of the radiation field with qubits. This term expresses the spatial fluctuations of the effective index of refraction of the electromagnetic medium through the dynamically generated polarization fluctuations \[*i.e.* $P(\boldsymbol{x})$\] of the qubits. The interaction vertex here is again given by the resonant modes and their frequencies: $\lambda_{i,mn}(\omega) = g_n g_m^* \varphi_n (\boldsymbol{x}_i) \varphi_m^*(\boldsymbol{x}_i)/(\omega - \omega_n)$. We note that for a 1D tight-binding model of a cavity array with nearest-neighbor hopping, $\Sigma_{ij} (\omega_{{\rm q}}) \simeq (g/\Delta)^2 [ \Delta \delta_{i,j} - J\delta_{i,j\pm 1}] $ and $\lambda_{i,kq} (\omega_{{\rm q}}) \simeq \Delta (g/\Delta)^2 \varphi_k^*(i) \varphi_q(i)$ to lowest relevant order in $J/\Delta$ and $g/\Delta$. This result agrees with our direct derivation in Sec. \[sec:cooling\]. These results can be extended to a full-fledged stabilization protocol for a generalized W state $| W_n \rangle = \sum_{i=0}^{N-1} \varphi_n^* (\boldsymbol{x}_i) |\downarrow_0 \ldots\downarrow_{i-1} \, \uparrow_i \, \downarrow_{i+1} \ldots \downarrow_{N-1} \rangle$.
Discussion and Conclusions {#sec:end}
==========================
We proposed a general and scalable method based on photon-mediated interactions to drive a set of $N$ qubits to a desired generalized W state. The particular protocol discussed here for qubits embedded in a cavity array, amounts to the dissipative stabilization of a particular [*excited state*]{} of a many-body system (in the present case, a non-integrable variant of the XY model). This approach stands in contrast to cooling techniques employed for condensed-matter and cold atomic systems that, at least in principle, target the stabilization of the ground state.
An interesting feature of the present approach compared to earlier approaches to dissipative engineering of entanglement [@kraus_preparation_2008; @diehl_quantum_2008; @cormick_dissipative_2013; @lee_emergence_2013; @rao_deterministic_2014; @reiter_scalable_2015] is the fact that [*both*]{} the unitary and the dissipative parts of the dynamics are adjusted through coupling to a common photonic bath. The Hamiltonian part provides the set of pure many-body states that can be reached in the steady state, while the dissipative part determines the occupation of those states. For cavities that are high-Q, the transitions can be made very selective. In the case of the dissipative stabilization of a W state of $N$ qubits, we discussed the scaling of the fidelity with the system size $N$.
The fact that various properties of multi-qubit dynamics can be precisely adjusted by drive parameters provides a suitable platform for quantum simulation [@rotondo_dicke_2015] and computation [@verstraete_quantum_2009]. In scaling up CQED-based simulators [[@PhysRevLett.115.240501]]{} to larger architectures, one of the main obstacles is the uncontrolled site-to-site fluctuation of system parameters [@underwood_low-disorder_2012]. In the presented scheme, the [*dynamical tuning*]{} of effective spin-chain parameters, in fact both the unitary and the dissipative parameters, through the drive frequency and amplitude provides a promising route to realize large-scale quantum simulators.
More generally, the method proposed here and its possible generalization to higher dimensional lattices holds promise for various quantum information applications, such as deterministic teleportation [@agrawal_perfect_2006; @wang_simple_2009]. The reduction of the collective dephasing mechanisms and the generation of entangled states in higher-excitation manifolds are important goals. Another interesting open question is the adaptation and extension of our protocol for targeted many-body state preparation in the photonic sector, scaling up recent approaches [@leghtas_confining_2015].
Acknowledgements {#sec:ack}
================
We are grateful to Mollie Schwartz, Leigh Martin, Emmanuel Flurin, Irfan Siddiqi, Liang Jiang, and Alexandre Blais for helpful discussions. This work has been supported by ARO Grant No. W911NF-15-1-0299 and NSF Grant No. DMR-1151810. M.K gratefully acknowledges support from the Professional Staff Congress of the City University of New York award No. 68193-0046.
|
---
abstract: 'Quantum control is valuable for various quantum technologies such as high-fidelity gates for universal quantum computing, adaptive quantum-enhanced metrology, and ultra-cold atom manipulation. Although supervised machine learning and reinforcement learning are widely used for optimizing control parameters in classical systems, quantum control for parameter optimization is mainly pursued via gradient-based greedy algorithms. Although the quantum fitness landscape is often compatible with greedy algorithms, sometimes greedy algorithms yield poor results, especially for large-dimensional quantum systems. We employ differential evolution algorithms to circumvent the stagnation problem of non-convex optimization. We improve quantum control fidelity for noisy system by averaging over the objective function. To reduce computational cost, we introduce heuristics for early termination of runs and for adaptive selection of search subspaces. Our implementation is massively parallel and vectorized to reduce run time even further. We demonstrate our methods with two examples, namely quantum phase estimation and quantum gate design, for which we achieve superior fidelity and scalability than obtained using greedy algorithms.'
author:
- |
Pantita Palittapongarnpim$^1$,Peter Wittek$^{2,3}$,Ehsan Zahedinejad$^{1}$,\
Shakib Vedaie$^{1}$ and Barry C. Sanders$^{1,4,5,6}$
bibliography:
- 'noisyde.bib'
title: 'Learning in Quantum Control: High-Dimensional Global Optimization for Noisy Quantum Dynamics'
---
1- Institute for Quantum Science and Technology, University of Calgary\
Calgary, Alberta T2N 1N4 Canada 2- ICFO-The Institute of Photonic Sciences\
Castelldefels (Barcelona), 08860 Spain\
3- University of Borås\
Borås, 501 90 Sweden\
4- Program in Quantum Information Science\
Canadian Institute for Advanced Research\
Toronto, Ontario M5G 1Z8 Canada\
5-Hefei National Laboratory for Physical Sciences at Microscale,\
University of Science and Technology of China,\
Hefei, Anhui 230026, People<span style="font-variant:small-caps;">13</span>s Republic of China\
6-Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center\
in Quantum Information and Quantum Physics,\
University of Science and Technology of China,\
Shanghai 201315, People<span style="font-variant:small-caps;">13</span>s Republic of China
Introduction {#sec:intro}
============
Quantum mechanics has been recognized as a superior foundation for performing computation [@Cav15; @Shor97; @Grov96], secure communication [@BB84; @SBC+09] and metrology [@GLM11; @TA14], also leading to technological advancements such as nuclear magnetic resonance and other resonators [@NMR1; @KRK+05; @RSK05; @MVT98; @HJH+03], femtosecond lasers [@ABB+98; @MS98] and laser-driven molecular reactions [@BS92; @TR85]. Central to these applications is the ability to steer quantum dynamics towards closely realizing specific quantum states or operations; i.e., the ability to control the system [@DP10].
Control theory of classical systems has a long history and is extremely well developed. Control theory most often relies on a mathematical model of a physical system. The primary goal of control is to make the system’s dynamics follow a reference trajectory or optimize the dynamics according to an objective function if a reference trajectory is not available. A mathematical model, however, can be difficult to specify exactly or solve analytically. Reinforcement learning is an alternative approach to control that does not necessarily have an explicit mathematical model of the underlying physical system, but rather it optimizes system’s behavior by studying responses given a set of inputs [@SBW92; @KLM96]. If the control provided by the reinforcement learning algorithm is discrete in time, we can view the algorithm as a form of machine learning where the typical assumption of sampling from independent and identical distributions is replaced by that of a Markov decision process.
The purpose of quantum control is identical to the classical case: generate a feasible [*policy*]{} for the given control problem. A policy is a set of instructions that determine the control parameters, and hence the effectiveness of the control scheme. This task is complicated by the quantum mechanical nature of the system, which allows non-classical correlations and noncontinuous jumps of the system’s state [@DP10]. For control tasks that involve continuous control over quantum states, usually through applying control pulses, algorithms such as GRadient-Ascent Pulse Engineering (GRAPE) have been applied to generate policies. These tasks are found in spectroscopy [@KRK+05], ultracold-atom research [@KPK+04; @JRG+14], and implementation of quantum computation [@RJ12]. When feedback is included, such as for adaptive parameter estimation [@AJSD+02; @WBB+09; @Cap12] and for stabilization of a quantum state [@MV07; @VMS+12], the dynamics of the state becomes nonlinear and noncontinuous. In this case, the optimization methods account for the quantum state’s allowed trajectories [@WK98; @BW00; @WMW02; @RPH2015].
As in the case of classical control, if the mathematical model of the quantum physical system is overly complex or elusive altogether, we can turn to reinforcement learning. For instance, in quantum control problems that involve measurement and feedback, reinforcement learning is gaining attention. Examples include an agent-based model in measurement-based quantum computation [@TGB15], mapping quantum gates on a spin system [@BPB15], suppressing errors in quantum memory [@AN16], optimization in ultra-cold-atom experiments [@WEH+15], and earlier work on the adaptive quantum phase estimation problem using heuristic optimization [@HS10; @LCPS13]. Whereas optimization methods such as Bayesian and Markovian feedback require the knowledge of the system dynamics [@WMW02], the machine learning approach [@Bis06] enables us to treat the quantum system as a black box. The policy is generated in response to the outcome that closely approximates the target channel of the procedure irrespective of the dynamics involved. This approach has been used successfully to find policies for quantum control problems, such as the classification of qubits and trajectories [@MGM+15; @MW10].
Greedy algorithms are used for finding successful policies because the algorithms are fast in converging on successful solutions when performing local searches. Greedy algorithms are not guaranteed to succeed (i) when the search domain is non-convex or (ii) when the computational resource or the time for performing the control task is constrained [@ZSS14]. Early quantum-control schemes employ standard heuristics, such as genetic algorithms, to find successful policies [@SD94; @BYW+97]. These algorithms fail when the number of control parameters increases or when decoherence and loss are included.
As greedy algorithms fail to provide successful policies for the quantum control problems at hand, and simple heuristics for global optimizations also fail under realistic conditions of quantum systems, we develop new variants of optimization algorithms. We consider in particular differential evolution (DE) as a basis. Our decision is due to the algorithm’s superior performance to particle swarm optimization (PSO) and other evolutionary algorithms for high-dimensional optimization problems [@SP97; @VT04; @PSL05].
We demonstrate learning-based quantum-control schemes to find successful policies for two topics relevant to quantum control: phase estimation via adaptive quantum-enhanced metrology (AQEM) and designing fast quantum logic gates. Adaptive phase estimation aims to estimate an unknown phase shift on a light field such that the precision is enhanced by the use of a quantum state of light [@BW00; @RPH2015; @Wis95; @WK97; @OIO+12]. Quantum metrology has many applications, such as in atomic clocks and gravitational wave detection. Quantum gate design tackles the problem of performing a specific gate operation on a quantum bit by steering its evolution [@BPB15; @ZGS15; @ZGS16]. Fast quantum logic gates are required for designing fast quantum processing units as the timescales are limited by the decoherence time of the qubits. Both problems require optimization of the procedure over the limited resource and time.
Building on the DE-based reinforcement learning and machine learning algorithm, we address critical issues for applying quantum control in realistic physical systems in the following way.
1. We develop a scheme that can operate when practical imperfections such as noise and loss are included. The primary means to this is the way in which the objective function is evaluated.
2. We improve scalability to a higher dimensional search space. For the problem of adaptive phase estimation, scalability is achieved by the accept-reject criterion that allows an early or a late termination of calculations. For the problem of gate design, we devise a subspace-selective self-adaptive DE (SuSSADE) that alternates between a search in the subspace and the overall space while adaptively updates the algorithmic constants of the standard DE algorithm during the search process.
3. Furthermore, we vectorize the time-critical operations to use the parallel resources available efficiently in contemporary CPUs and GPUs.
This article is structured as follows. In Sec. \[sec:quantum control\], we introduce the relevant concepts in quantum mechanics and quantum control. We also explain the control procedures in our two examples: adaptive phase estimation and quantum-gate design. In Sec. \[sec:machine learning\], we describe the connection between machine learning, reinforcement learning, and control. In Sec. \[sec:method\] we formulate adaptive phase estimation and quantum gate design as learning problems and show the methods for creating noise-resistant DE and increasing the scalability of our learning algorithms. The results for both control problems are in Sec. \[sec:results\].
Quantum control framework {#sec:quantum control}
=========================
Quantum control concerns the application of control procedures to systems whose dynamics are governed by quantum mechanics [@DP10]. In this regime, behaviors that are associated with classical dynamics are violated, leading to challenges in applying classical control theory directly to the system [@DHJ+00; @AT12]. In this section, we explain the key concepts in quantum mechanics that are necessary to understand quantum control, especially to adaptive phase estimation and quantum gate design. Readers interested in the complete formalism of quantum mechanics are referred to the many publications on this subject [@Wat11; @Hol12].
Elements of Quantum Mechanics
-----------------------------
In the regime of quantum mechanics, the state of an isolated particle A is given by a vector $\ket{\psi}_{\text{A}}$ of norm one in a Hilbert space $\mathscr{H}_\text{A}$. We restrict our attention to the finite dimensional case where the Hilbert space is $\mathbb{C}^n$. More generally, the state of a particle is a self-adjoint trace class operator of trace one [@DFM-Z08], given in the case of the adjoint vector $\ket{\psi}_{\text{A}}$ as $\hat{\rho}=\ket{\psi}_{\text{A}}\bra{\psi}$, called a density operator. The density operator can be represented by a matrix given for a chosen basis of the Hilbert space.
Whereas a classical particle has definite values for its characteristics such as position and momentum, a characteristic of a quantum particle can only be described in terms of its probability distribution. Given a chosen basis $\{\ket{i}_\text{A}\}$, the state can be represented by $\ket{\psi}_{\text{A}}=\sum\limits_{i} c_i\ket{i}_\text{A}$, where $c_i\in\mathbb{C}$. The absolute square of $c_i$ determines the probability distribution on the chosen basis and, hence, must satisfy the condition $\sum\limits_{i}\left|c_i\right|^2=1$. In the matrix representation of the density matrix, the distribution is determined by the diagonal elements, which leads to the trace of one. When two or more particles exist in the system, correlations can exist that cannot be expressed by any local hidden variable model. For instance, a qubit is a state in $\mathbb{C}^2$, and a two-qubit state of particles A and B is in $\mathbb{C}^2_\text{A}\otimes\mathbb{C}^2_\text{B}$. A quantum correlation between the qubits leads to the state $\ket{\psi}_\text{AB}$ that cannot be expressed as $ \ket{\psi}_{\text{A}}\otimes\ket{\psi}_\text{B}$ and a phenomenon known as entanglement where a local operation on the subsystem $A$ affects the state of the subsystem $B$ regardless of the space separation between $A$ and $B$. The connection between state and probability distribution becomes apparent when we consider the measurement of the state. Measurements are described by a positive operator-valued measure (POVM) $\{\hat{E}_x\}$ [@Bra03] acting on system A, which delivers outcome $x$ with probability $\operatorname{tr}\left(\hat{E}_x\ket{\psi}_\text{A}\bra{\psi}\right)$. POVMs satisfies the condition $\sum\limits_x \hat{E}_x^\dagger \hat{E}_x=\mathds{1}$, and $\hat{E}_x$ is positive for all $x$. The state after the measurement is $$\ket{\psi,x}_\text{A}=\frac{\hat{E}_x\ket{\psi}_\text{A}}{\sqrt{\left|\hat{E}_x\ket{\psi}_\text{A}\right|^2}}.$$ As the outcome $x$ is random, the state after the measurement is a random jump unless the state is an eigenstate of $\hat{E}_x$.
Manipulation of the state is accomplished by providing outside interaction to the system. This interaction is described by a completely positive trace-preserving (CPTP) map [@Hol12], ensuring that the vector remains a quantum state. We refer to these maps as quantum channels $\{\hat{\mathcal{C}}_j\}$ that satisfy $\sum\limits_{j}\hat{\mathcal{C}}_j\hat{\mathcal{C}}^\dagger_j=\mathds{1}$. If these linear operators are unitary, the system does not interact with its environment, and its dynamics remains reversible.
As particles, in reality, are not perfectly isolated, the system can be considered as having a constant weak interaction with a bath, which is another quantum system in a larger Hilbert space. The state of the bath is not accessible through measurement, and this loss of information leads to the decoherence of the system’s state. Quantum correlations suffer from this kind of interaction because the state turns into a convex combination of the resulting states of all possible interactions between bath and system, $\hat{\rho}_\text{A}=\sum\limits_{i}P_i\ket{\psi^{(i)}}_\text{A}\bra{\psi^{(i)}}$, where $\{P_i\}$ is a classical probability distribution. The state is no longer pure, but forms a mixed state. In the case of strong system-bath interaction, the system loses the quantum-mechanical characteristics altogether [@Zur07].
The essence of quantum control is to steer a quantum channel towards the desired operator. One way to test the channel resulting from the control is to monitor the channel through process tomography [@DPS03]. A known quantum state is injected as the input, and the output state is measured. This random measurement outcome is then used to infer the operation performed by the channel. In the process of obtaining a successful policy, the outcome is monitored, and the policy that is used to adjust the control parameters is updated accordingly. The difference between the channel and the target is determined using an objective function. However, if the goal of the control does not explicitly involve the channel, other methods and objective functions can be selected.
In this work, we consider two examples of quantum-control procedures. The first example is the adaptive quantum-enhanced metrology, which utilizes a multiparticle entangled state to attain quantum-enhanced estimation of an unknown parameter and consolidates previous material [@PWS16]. The second case study is quantum gate design, which uses the control to apply logic gates on three quantum bits (qubits) [@ZSS14].
Adaptive quantum-enhanced metrology {#AQEM}
-----------------------------------
The task of a quantum-enhanced metrology (QEM) scheme is to infer an unknown parameter $\phi$ using entangled states of $N$ particles such that the scaling in uncertainty surpasses $\Delta\phi\propto1/\sqrt{N}$ obtained using a classical strategy [@TA14]. This scaling is known as the standard quantum limit (SQL). The use of quantum resources enables a QEM scheme to approach the Heisenberg limit (HL) corresponding to the scaling of $\Delta\phi\propto1/N$ [@ZPK12]. This quadratic improvement in precision is valuable for applications where the measurement is operating at the limit of $N$ that can be safely produced or detected.
AQEM is one strategy for performing QEM that involves splitting the input state into a sequence of single-particle bundles [@BWB01]. A bundle is injected into the channel and measured at the output. The measurement outcome is used to update the control parameter in preparation for the next bundle. The value of the control parameter after the $N^\text{th}$ measurement is then taken to be the estimate.
The evolution of the quantum state over the course of the measurement process is noncontinuous, and so the policy that can achieve quantum-enhanced precision is non-trivial to find. For this reason, machine learning has been introduced [@BW00; @HS10; @LCPS13]. In our work, we focus on the channel that includes noise and loss, which are imperfections present in every practical measurement setup.
\[fig:MZIControl\] 
We consider the problem of optical interferometric-phase estimation (Fig. \[fig:MZIControl\]), which is well-studied due to its connection to the detection of gravitational wave [@Cav81; @AAA+16] and atomic clocks [@BS13]. The interferometer has two input modes and two output modes. The input state containing $N$ entangled photons is injected into the interferometer one photon at a time.
Neglecting loss, the $m^\text{th}$ photon comes out from either of the output modes with a probability that depends on $\phi-\Phi_{m-1}$. Our interferometer model includes Gaussian noise on the phase shift with standard deviation $\sigma$. We label the outcome by $x_m\in\{0,1\}$, where 0 refers to the photon exiting the first port and 1 to the photon exiting the second port. The sequence of outcomes from the first to the $m^{\text{th}}$ photon is given by $\bm{x}_m=(x_1x_2\cdots x_m)$.
The exit path of the photon is used to determine the value of $\Phi_m$ for the next round of measurement. Once all photons are put to use in the $M^{\text{th}}$ measurement, allowing for the loss of photons such that $1\leq M\leq N$, the estimate $\widetilde{\phi}$ of $\phi$ is inferred from $\Phi_M$ to be $\widetilde{\phi}\equiv\Phi_M$. As the measurement outcome $\bm{x}_{M}$ is a string of discrete random variables, the estimate of phase from this scheme, which is a function of $\bm{x}_M$, is also discrete.
Because the distribution of the estimate $\widetilde{\phi}$ is periodic, the standard deviation is not an appropriate choice to quantify the imprecision $\Delta\widetilde{\phi}$. Unless the domain is bounded, the standard deviation is skewed by the existing peak appearing in the distribution outside of the domain $\left[0,2\pi\right)$ [@Hil02]. The imprecision is instead quantified by the Holevo variance [@BW00], $$\begin{aligned}
\label{eq:Holevo variance}
V_H & = S^{-2}-1,\\
\label{eq:Sharpness}
S & = \left|\sum\limits_{\widetilde{\phi}} P(\widetilde{\phi}|\phi) \text{e}^{\text{i}(\phi-\widetilde{\phi})}\right|,
\end{aligned}$$ which is one possible choice for a periodic distribution [@FA01]. Our goal is to generate a feedback policy such that $V_H$ is minimized and the power-law scaling with $N$ exceeds $1/\sqrt{N}$.
Quantum gate design {#QGD}
-------------------
Quantum computing employs quantum mechanics to perform computation and promises speed-up in computational time for algorithms such as factorization [@Shor97] and database search [@Grov96]. Implementing quantum computing has been challenging due to the interaction between the quantum system and the environment, which introduces errors and may even nullify the advantage of using quantum resources altogether [@Shor96]. If the error rate can be reduced to a value that is less than a specified threshold, error correction can be introduced, and quantum information is thereby protected [@SWS16].
Information is encoded on qubits, for each qubit has the state spanned by the basis $\left\{\ket{0},\ket{1}\right\}$. Therefore, unlike a classical bit, a qubit can exist in any superposition $a\ket{0}+b\ket{1}$, where $a,b\in\mathbb{C}$ and $\left|a\right|^2+\left|b\right|^2=1$. Quantum algorithms use these qubits as resources to perform computations. Just as in classical computation, the quantum algorithm operating on a large number of qubits can be decomposed into gates acting on a few qubits at a time. An ideal quantum gate is reversible and is represented by a unitary transformation $U$. For each of these gates, error threshold can be assigned such that fault-tolerant quantum computing is attained.
Quantum computing operations can be decomposed efficiently down to a set of one- and two-qubit gates [@BBC+95; @MVBS04]. However, this decomposition results in an increased processing time and lower overall fidelity. One way to increase the efficiency is to convert these operations to gates that act on more than two qubits. The Toffoli gate is one such gate. This gate is a controlled-controlled-not gate acting on three qubits (Fig. \[fig:Toffoli\_CZ\]), whose action is summarized in Table \[table:CCNOTtruth\].
![The quantum circuit representation of the Toffoli (CCNOT) gate. The horizontal solid black lines are circuit wires with each wire representing a qubit. [${\bullet}$]{} shows the control qubits and $\bigoplus$ denotes the NOT operator acting on the target qubit. $\ket{C_1},\ket{C_1},\ket{C_\text{T}}$ refer to the states of the first, second and target qubits. The gate accepts the input from the left side and output a new state on the right side.[]{data-label="fig:Toffoli_CZ"}](Toffoli_CZ.pdf)
----------- ----------- ----------- ----------- ----------- -----------
$C_1$ $C_2$ $C_T$ $C_1$ $C_2$ $C_T$
$\ket{0}$ $\ket{0}$ $\ket{0}$ $\ket{0}$ $\ket{0}$ $\ket{0}$
$\ket{0}$ $\ket{0}$ $\ket{1}$ $\ket{0}$ $\ket{0}$ $\ket{1}$
$\ket{0}$ $\ket{1}$ $\ket{0}$ $\ket{0}$ $\ket{1}$ $\ket{0}$
$\ket{0}$ $\ket{1}$ $\ket{1}$ $\ket{0}$ $\ket{1}$ $\ket{1}$
$\ket{1}$ $\ket{0}$ $\ket{0}$ $\ket{1}$ $\ket{0}$ $\ket{0}$
$\ket{1}$ $\ket{0}$ $\ket{1}$ $\ket{1}$ $\ket{0}$ $\ket{1}$
$\ket{1}$ $\ket{1}$ $\ket{0}$ $\ket{1}$ $\ket{1}$ $\ket{1}$
$\ket{1}$ $\ket{1}$ $\ket{1}$ $\ket{1}$ $\ket{1}$ $\ket{0}$
----------- ----------- ----------- ----------- ----------- -----------
: The truth table representation of Toffoli gate. $C_1$ and $C_2$ denote the control qubits, and $C_\text{T}$ represents the target qubit. The Input and Output columns show the states of the three qubits before and after applying Toffoli.[]{data-label="table:CCNOTtruth"}
The current experimental schemes to design fast Toffoli gate are limited to decomposition approaches, with the fidelity limited to 68.5% in a three-qubit circuit QED system [@FSB+12], 71% in an ion-trap system [@MKH+09], 78% in a four-qubit circuit QED system [@RDN+12] and 81% in a post-selected photonic circuit [@LBA+09]. Here our goal is to devise a machine learning algorithm to design a single-shot threshold-fidelity Toffoli gate without any need to resort to decomposition approach. Our approach to creating a high-fidelity Toffoli gate is not restricted to a specific physical model for quantum computation. We choose to design a Toffoli gate for an architecture of three nearest-neighbour capacitively coupled superconducting transmons [@SHK+08].
In a transmon system, the computational basis $\left\{\ket{0},\ket{1}\right\}$ is assigned to the two lowest energy levels, although the transmon system consists of multiple discrete levels. The evolution of the quantum states for the three transmons are controlled through electrical pulses changing the frequency of each of the transmons. Because the three transmons are coupled, these changes tune their interactions and allow us to perform gate evolution on the computational bases of the three qubits.
The unitary transformation of the Toffoli gate is approximated using a sequence of constant control pulses lasting $\delta t=1~\text{ns}$ applied over time $\tau$. At time $t$, the control pulses $\bm{\varepsilon}(t)=\left(\varepsilon_1(t),\varepsilon_2(t),\varepsilon_3(t)\right)$ is applied to the three transmons, effectively subjecting the system to a unitary transformation $U(\bm{\varepsilon}(t),\delta t)$. The approximate unitary for the gate is, therefore, $$\tilde{U}(\bm{\varepsilon}_\tau,\tau)=U(\bm{\varepsilon}(\tau-\delta t),\delta t)U(\bm{\varepsilon}(\tau-2\delta t),\delta t))\cdots U(\bm{\varepsilon}(0),\delta t),$$ where $\bm{\varepsilon}_\tau$ denotes the entire sequence of the three control pulses.
The gate’s performance is quantified using the intrinsic fidelity [@ZGS16], $$\mathcal{F}=\frac{1}{8}\left|\text{tr}\left(U_{T}^{\dagger}\tilde{U}(\bf\varepsilon(\tau),\tau)\right)\right|,
\label{eq:fidelity}$$ where $U_T$ is the unitary transformation of the Toffoli gate. The fidelity $\mathcal{F}$ has a value between 0 and 1, where $\mathcal{F}=1$ is attained when $\tilde{U}=U_T$. We aim for an intrinsic fidelity beyond 0.9999 for the design of the Toffoli gate as required by fault-tolerant quantum computing.
In the experimental realization, non ideal behavior arises due to electronic imperfections. One is the distortion of the control pulses, which we treat as sequences of piecewise constant functions. A more realistic pulse shape takes into account the response time of the electronics, which acts as a Gaussian filter. Another source of imperfection is the disturbance of the pulses associated with the thermal noise in the electronics. Although this noise is not included in the optimization, we test the robustness of our control pulses by adding random noise $\delta\varepsilon \cdot \text{rand}(-1,1)$, where $\text{rand}(-1,1)$ uniformly generates a random number in $(-1,1)$ to the control parameters at each time bin. The value of $\delta\varepsilon$ is varied from 0 to 300 kHz. We then use the distorted pulse to calculate the intrinsic fidelity for each value of $\delta \varepsilon$.
Note that we have already devised a quantum control scheme to design a high-fidelity quantum gate [@ZGS15; @ZGS16]. Our main goal in this article is to present the problem in a different framework, namely machine learning. In particular, we formulate the problem for supervised learning. Expressing the problem in the machine learning context can provide a new perspective on finding control pulses for when the transformations, $U(\varepsilon(t),\delta t)$, are no longer unitary.
Machine Learning and Evolutionary Algorithms for Control {#sec:machine learning}
========================================================
In this section, we explain how machine learning and reinforcement learning can be used as tools for control. A control problem involves optimization of a control policy such that a target performance is met [@Leigh04]. Optimizing based on a model is not useful if the model is incorrect, for example by not properly incorporating environmental interaction. As a satisfactory noise model might not exist, the alternative is to implement optimization procedures that are independent of the underlying interaction, with evolutionary algorithms being one such example.
Control theory concerns finding a way to steer a system to achieve a target [@Leigh04]. The method of control may involve monitoring of a system, in which case the control may involve a feedback loop adjusting the control signal following a policy. Such a control procedure is known as closed-loop control [@Haid13]. For a system that cannot be monitored, or for a system whose monitoring is deemed unnecessary, an open-loop strategy is applied, and the control signal is predetermined using a model of the evolution of the system [@Leigh04; @AM08].
For the cases where the system models are not known, not accurate, or too complicated to be used for generating feasible control policies, machine learning and reinforcement learning can be used to generate the policy through trial and error. Machine learning typically assumes sampling from random variables that are independent and identically distributed (i.i.d.). Concentration theorems based on this assumption give guarantees of estimation accuracy or generalization performance [@Vap95; @DGL96]. A variant of machine learning takes training instances one by one and updates its model accordingly. This scheme is known as online learning. Some results from statistical learning theory extend to this case if the samples remain i.i.d. [@Long99]. In reinforcement learning, an agent responds to a random outcome from its interaction with an environment [@SB98]. The agent is equipped with a set of possible actions and a scoring function with which it can evaluate its performance. The agent can either greedily optimize the scoring function or aim to maximize long-term performance quantified after the task is completed. In this case, due to the interaction between the learning agent and the environment, the sampling loses the i.i.d. assumption. The most we can assume is that the interaction is modeled by a Markov decision process. In both machine learning and reinforcement learning, we often boil down the learning problem to a constrained optimization problem with an objective function that can be regularized. The function is seldom convex, making it hard to find the global optimum of the problem. We can use a relaxation and a convex approximation or substitution. For instance, support vector machines replace the optimal 0-1 loss with the hinge-loss to have a convex optimization problem to solve [@EBG11]. Another option is to use a non-convex optimization algorithm. This approach works if a convex relaxation is not easy to derive or when we are not satisfied with a local optimum. Sometimes even a greedy algorithm works well for the non-convex case, depending on the topology of the space.
Derivative-free heuristics are common in global optimization. Two widely used algorithms are PSO [@KE95], which is a form of swarm intelligence, and DE [@SP97], which is an evolutionary algorithm. In comparative studies, DE has been shown to be the most powerful in that it converges on a near-optimal solution quickly and can find solutions without stagnation in higher dimensional search spaces than other algorithms [@VT04].
Quantum control tasks also employ closed-loop and open-loop controls to accomplish the desired goals. A key difference between quantum and classical control lies in the quantum system’s response to the measurement. Whereas classical systems are unaffected by the measurement procedure, the quantum states response to a measurement with a random jump depending on the measurement outcome. For applications where the quantum states are important to the goal, such as gate design, measurements are to be avoided, and the control procedure for the task is designed based on open-loop control. On the other hand, when a quantum state is used as a resource for accomplishing the desired goal, or if measurements are possible, closed-loop control can be applied.
To implement quantum control in the experimental settings, the control procedures must be designed to be resilient to imperfections, including the nonideal evolution of the quantum state due to its interaction with the environment. Although noise models can be incorporated into computer simulations of the control procedures, the models might not match the noises in the real-world system. One approach is to design a procedure that can learn the control policies directly from experiments, including as few assumptions about the noise as possible.
Reinforcement learning is a valuable tool for applying control as it provides a method to generate a policy that does not rely on the knowledge of the system [@SBW92; @KLM96; @CDL+2014]. Learning is performed through trial and error, and the system’s dynamics is treated as a black box. In fact, this black box treatment of a process is not restricted to a control scenario: if we only want to approximate the output of the process and we have a certain number of disposal of the black box, we can employ a supervised learning algorithm with the assumption of the sampling being i.i.d. For quantum control, this feature is valuable because the knowledge of the dynamics of a real quantum system is never exact. Furthermore, in systems involving measurements, the back-action leads to an exponentially-growing number of state trajectories with the number of possible measurement outcomes $\left|\{x\} \right|$, making the problem difficult to solve analytically.
Method {#sec:method}
======
We now explain how to construct quantum-control procedures for AQEM and quantum gate design as learning problems and the challenges they pose to learning algorithms. We then explain how we create noise-resistant differential evolution and methods for generating policies in high-dimensional problems. Here we assign a fixed mutation rule, although a new DE variant created by random selection of the mutation is possible [@MCD2015].
Quantum control procedures as learning problems
-----------------------------------------------
In this subsection, we discuss how the control procedures are performed in an adaptive phase estimation scheme and on the superconducting circuit to create a Toffoli gate. The feedback control applied in the adaptive measurement fits into the framework of reinforcement learning, for which we discuss the policy and the training set. Quantum gate design, on the other hand, fits with open-loop control as direct monitoring of quantum states interfere with their evolution. To move beyond treating the problem as optimization, in which the noise models are required, we formulate the gate design as a learning problem fit for supervised learning.
### Generating feedback policy for AQEM
The goal of adaptive phase estimation is to infer the value of an unknown parameter $\phi$ following a sequence of measurements performed on a string of $N$ single-photon pulses. In order to deliver quantum-enhanced precision, not only must the particles be entangled, but a policy must be generated that can attain the desired precision. In principle, the policy must be optimized for all sequences $\bm{x}_M$ and all value of $\phi\in\left[0,2\pi\right)$, which is impossible as $\phi$ is in a continuous domain. For this reason, we employ the reinforcement learning approach to generate a feasible policy using a training set selected from the domain of $\phi$.
Reinforcement learning is the suitable framework because the approach is designed to optimize a decision-making process given random inputs. The adaptive phase measurement involves decision making by the processing unit in response to the random measurement outcomes, which fits in this framework. In the learning algorithm, the adaptive procedure is simulated many times with a fixed input quantum state and treated by the learning algorithm as a black box. The fitness values of the policies are computed over a training set of randomly generated $\phi$ and are used by the algorithm for the optimization.
In AQEM, the feedback policy is a set of rules that determines how the controllable phase shifter $\Phi$ is adjusted. In the $m^{\text{th}}$ round of measurement, the policy is a function of the sequence of previous outcomes $\bm{x}_{m-1}\in\{0,1\}^{m-1}$. The process is better understood by representing a policy as a binary decision tree, where each branch from the root to the leaf corresponds to a sequence of $\bm{x}_N$. An advantage of using this representation is that the size of a policy is readily determined from the number of branches, and its size scales as $2^N-1$.
The exponential scaling of the policy size makes generating the policy increasingly expensive in computational time, which limits the number of particles $N$ in which the application of reinforcement learning is practical. To reduce the policy’s size and make searching for the policy tractable, we restrict to Markov feedback [@WO12], in which only the current outcome $x_m$ is used to determined the value of $\Phi_m$. In particular, we impose the update rule $$\Phi_m=\Phi_{m-1}+(-1)^{x_m}\Delta_m;
\label{eq:update}$$ thus the decision tree is parameterized by a vector $\bm{\Delta}=\left(\Delta_1,\dots,\Delta_N\right)$. Hence, the size of the policy $\varrho=\bm{\Delta}$ is reduced to scale linearly with $N$. The space in which the policy can be searched is therefore restricted to $[0,2\pi)^N$. Through the implementation of this rule in previous work [@HS11b; @HS10; @LCPS13], we found that the rule leads to feasible policies for adaptive phase estimation.
The update rule (\[eq:update\]) restricts the estimate $\widetilde{\phi}\equiv\Phi_M(\bm{x}_M)$ to a discrete value even though $\phi$ is continuous in $\left[0,2\pi\right)$. In other words, the AQEM scheme delivers a discrete approximation of $\phi$ based on the discrete outcomes $\bm{x}_M$. As $N$ increases, the approximation becomes refined, and an increasingly precise estimate may be obtained.
To determine the performance of a policy, the imprecision of all possible value of $\phi\in\left[0,2\pi\right)$ has to be taken into account. This task poses an additional challenge to calculating Eq. (\[eq:Sharpness\]), which is computationally expensive. We instead estimate the value of $S$ from $$S:=\left|\sum\limits_{k=1}^{K}\frac{\text{e}^{\text{i}\theta_k}}{K}\right|
\label{eq:Sharpness_est}$$ where $\theta_k=\phi_k-\widetilde{\phi}_k$ and $K=10N^2$. This particular choice of $K$ has been shown to deliver the estimate of $S$ that converges in the previous work [@HS10]. The samples are also training data $\{\phi_k\}$ chosen uniformly from $[0,2\pi)$ to avoid the problem of overfitting.
The adaptive phase estimation scheme works by refining the estimate $\widetilde{\phi}$ through tweaking $\Phi$ according to the outcome. In the event of small probability $\eta$ that the photon is lost, the automated system is instructed to do nothing in the absence of information to update the estimate. We assume a small loss in which case we optimize without loss but test the performance accounting for loss. If the test succeeds, we adopt the policy; if the test fails, we repeat the optimization process.
### Shaping the frequencies of transmon system
The goal for quantum gate design is to generate a set of pulse sequences $\bm{\varepsilon}_\tau$ such that the transformation of the three transmons over time $\tau$ is approximately the Toffoli gate. The pulses are affected by Gaussian distortion, thereby smoothing out the constant piecewise functions. Disturbances are included by adding frequency noises to the pulses after the optimization to test for robustness. For both noises considered, the assumption of unitary transformation holds, and the problem of finding the control pulses can be treated as an optimization problem. In this work, however, we formulate this problem for supervised learning as the learning approach can be implemented regardless of whether the transformation is unitary or not.
When the transformations of the quantum state are unitary, i.e., the quantum system is perfectly isolated from the environment, the control pulses can be optimized over a set of basis states. That is because the quantum state remains a pure state, and due to linearity in quantum mechanics, any transformation on the superposition of the basis states are equal to the superposition of the transformed basis. Hence, optimizing the gate’s performance over the basis states is equivalent to the optimization of the problem over the entire state space.
This observation is no longer true when the state becomes mixed, as is the case when the quantum system is coupled to the environment, and the transformations are represented by quantum channels. In this case, optimization over the basis states would not guarantee that the gate’s performance is maximized over the entire space of all input states. Hence, the optimization process has to take into account other input states as well. Finding a feasible policy, in this case, is non-trivial as there are infinitely many superposition states. Machine learning becomes a promising approach for generating the control pulses.
The task of supervised learning is to discover a model given a set of input-output data (a training set), which approximates the true function that maps the inputs data to their corresponding outputs. By casting the Toffoli gate as a model for we wish to learn the parameters $\bm{\varepsilon}_{\tau}$, we formulate the Toffoli gate design as a regression problem. We use the basis states and their corresponding output states in Table \[table:CCNOTtruth\] as the training set. The time $\tau$ for the gate operation is found through trial and error such that the confidence in the model, quantified by the intrinsic fidelity (\[eq:fidelity\]) exceeds 0.9999 as required by fault-tolerant quantum computing. The learning procedure output a successful policy.
Noise-resistant global optimization heuristics
----------------------------------------------
DE is able to find feasible solutions in high-dimensional search space for a set of test problems [@VT04] and for adaptive phase estimation [@LCPS13]. However, when DE is employed for the problem of noisy phase estimation for $N$ up to 100, we observe that DE does not perform as well as PSO and, in fact, fails to deliver better than SQL scaling. To devise a noise-resistant global optimization algorithm for our scheme, we use the mean value $\bar{S}$ instead of $S$ (\[eq:Sharpness\_est\]) to determine the performance of a policy. This strategy is one of the many strategies proposed in the literature to create noise-resistant DE [@DKC05; @BPW04] and is found to work best for our problem.
The principle behind the use of mean objective value is as follows: if noise is added to the fitness function, the process of averaging recovers the true objective value. The optimization using this value is, therefore, a close approximation to the noiseless optimization. The major drawback of this approach is that computing the objective function multiple times makes the procedure computationally expensive. Therefore, determining the smallest sample size of $\{S\}$ necessary to recover $S$ is crucial. To this end, we employ the heuristic applied to PSO in the previous work [@HS11b]. The method updates $\bar{S}$ by computing one new sample of $S$ every iteration until a better offspring is generated.
The sample size for computing $S$ is then determined by the probability for DE to generate an offspring that is better than the parent. This probability decreases as the candidates converge on the optimal value. As a result, the sample size grows automatically as the optimization progresses. The computational resources are allocated towards candidates that are close to optimal, which is a favorable strategy as the differences between objective values are dominated by noise in this case. Large sample size enables accurate estimations of the candidates’ objective values.
In the particular case of adaptive phase estimation, the phase noise is not additive in $S$ due to the exponential dependence. The mean value computed from $J$ samples, therefore, does not converge on the objective value of the noiseless case but an estimate of $S$ using sample size $JK$. This method, thereby, provides a better estimate of imprecision than for the sample size of $K$ for adaptive phase estimation including phase noise.
Improving scalability
---------------------
In this subsection, we explain two techniques for achieving scalable learning algorithms. One of the technique is to create an accept-reject criterion, allowing the algorithm to run for as long as it is necessary to generate a feasible policy. This technique is applied to the adaptive phase estimation, where successful policies from small values of $N$ are used to identify a region with a feasible policy for $N+1$. Another algorithm is devised for quantum gate design that alternates between optimizing in subspaces and overall space with self-adaptive DE.
### Adaptive phase estimation at $N>90$
In this subsubsection, we discuss accept-reject criteria and how this technique leads to DE delivering successful policies up to $N=100$. Although DE can generate successful policies for $N>45$, which is the limitation observed when PSO is used [@LCPS13], the variances also display stagnation when $N>90$. To generate policies from a search space that scales up to 100 dimensions, we implement a criterion to the noise-resistant DE to ensure that only successful policies are accepted.
The stagnation is a manifestation of the algorithm not being able to converge to a successful solution in the time limit imposed. Previously the algorithm accepted a policy after a fixed number of iterations regardless of whether the population converges. However, as the dimension of the search space increases, so does the time for the population to converge. Eventually, the algorithm fails to deliver a policy that passes the test. We change the criterion for accepting a policy from a fixed number of iterations to only if $V_{\rm H}$ is within a distance corresponding to a confidence interval of 0.98 from the inverse power-law line. Thus, we guarantee that the policy from our algorithm always delivers a power-law scaling better than SQL.
The acceptable error $\delta_y$ for $N>93$ is calculated from the statistics of $V_{\rm H}$. The Holevo variance $V_{\rm H}$ are collected from $N=\{4,5,\dots,93\}$, in which we accept the policies after a fixed number of iterations. A linear equation is determined from $\{y_i\}=\{\log V_{\rm H}(N)\}$ and $\{x_i\}=\{\log N\}$, and is used to predict the next data point $y'$. The acceptable error from this predicted value is calculated using the previously stored data and the best value of $V_{\rm H}$ at iteration $G$ from a statistical formula, namely [@YS09; @BF10] where $n'$ is the number of data points, $x'=\log N$ for which the error is calculated, and $\bar{x}$ is the average of $\{x_i\}$. The value $t^*_{n'-2}$ is the quantile of the Student’s $t$ distribution for $n'-2$ data points, which we approximate using a normal distribution. The policy is accepted if $$\left|\log V_{\rm H}(N)-y'\right|\leq\delta_y,$$ or the optimization continues.
The noise-resistant DE variant, including accept-reject criterion, works as follows.
**Step 1** Initialize the population of size $N_P$ randomly.
**Step 2** Evaluate the objective function for each candidate *twice*, and store the mean objective value and the sample size.
**Step 3** Generate a donor $\bm{D}_i(G)$ for each of candidate $\bm{V}_i(G)$, where $G$ is the iterative time step, from three other candidates $\{\bm{V}_{i,1}(G),\bm{V}_{i,2}(G),\bm{V}_{i,3}(G)\}$ chosen randomly. For element $j$ of the donor $D_i(G)^{(j)}$, $$\begin{aligned}
D_i(G)^{(j)}= \left\{
\begin{array}{l l}
V_{i,1}^{(j)}(G)+F\cdot(V_{i,2}^{(j)}(G)-V_{i,3}^{(j)}(G)),& \mathrm{if~}r\leq C_r,\\
V_i(G)^{(j)},& \text{else},
\end{array}\right.
\end{aligned}$$ where $F$ is the mutation rate, $C_r\in[0,1]$ is the crossover rate, and $r\in[0,1]$ is a random number.
**Step 4** Evaluate the mean objective value for each of the new candidates from two samples.
**Step 5** Compare and select the candidate for $G+1$ using the mean objective value, $$\begin{aligned}
\bm{V}_i(G+1)= \left\{
\begin{array}{l l}
\bm{D}_i(G)& \text{if~} \bar{S}(\bm{D}_i(G))>\bar{S}(\bm{V}_i(G)),\\
\bm{V}_i(G)& \text{else}.
\end{array}
\right.
\end{aligned}$$
**Step 6** Evaluate the objecting function once, and update the mean value and the sample size.
**Step 7** Repeat steps 3 to 6 until the criterion to terminate the algorithm is met.
**Step 8** Compute the objective value of the entire population 10 more times before selecting the candidate with the highest mean objective value as the solution.
The computational complexity of the algorithm is polynomial [@LCPS13], but it has a high degree, and therefore it is important to identify the performance critical parts of the implementation. We establish that over 90% of the execution time is spent on generating random numbers one by one. The random number generation is primarily used in estimating the Holevo variance as the computation involves simulations of the adaptive measurement procedure. Generating random numbers as they are needed is not efficient on contemporary hardware. The operations can be vectorized to use the single-instruction multiple-data architectures of the central and the graphics processing units (GPUs). Abstracting the random number generation routines and introducing a buffer, we are able to vectorize the respective operations. We study two approaches: one relies on the CPU, using the Intel Vector Statistical Library (VSL), the other on graphics processing units. Eventually, the VSL-based vectorized solution proves to be scalable.
### SuSSADE for quantum gate design
In this subsubsection, we turn to another technique to improve DE’s scalability and devise the subspace-selective self-adaptive differential evolution (SuSSADE). SuSSADE combines two technique to improve convergence of the algorithm to a successful policy: the self-adaptive heuristic search and the reduction of the search space size. The algorithm alternates between optimizing over the entire search space and one of the subspaces randomly selected for each iteration.
The efficacy of the algorithm for a particular landscape and dimension is determined by the search parameters ($F$, $C_r$), set to constant values in traditional DE. Instead of determining optimal values of the parameters through trial and error, which is infeasible when the problem has a large size such as in quantum gate design, we implemented an algorithm to adapt the value of $F$ and $C_r$ between iterations. This self-adaptive approach [@BZM06] has also been used to enhance the performance of DE for the high-dimensional optimization problems and reads as follows.
At iteration $G$, the algorithm determines the mutation rate $F$ and crossover rate $C_r$ for $G+1$ from $$\label{Frate}
F_{G+1}= \begin{cases}
F_l+r_1\cdot F_u &\text{if $r_2<\kappa_1$}\\
F_{G} &\text{otherwise}\\
\end{cases}$$ and $$\label{Crate}
C_{r_{G+1}}= \begin{cases}
r_3 &\text{if $r_4<\kappa_2$}\\
C_{r_{G}} &\text{otherwise,}\\
\end{cases}$$ where $r_j, j\in{1,2,3,4}$ are random numbers uniformly sampled from $\left(0,1\right]$. The value of $F_l$ and $F_u$ are predetermined to be $0.1$ and $0.9$ respectively. The adaptive rate $\kappa_1$ and $\kappa_2$ are both set to $0.1$.
In addition to adapting the search parameters, the algorithm also randomly decides whether the optimization is performed on the entire search space or a subspace. The switching rate $\mathcal{S}$ determines how often this switch occurs and is set by the user. DE is applied on the whole domain if a random number $r<\mathcal{S}$ for $r\in[0,1]$. Otherwise, the optimization is applied to a smaller domain of the problem. Further details of the SuSSADE optimization algorithm can be found in [@ZGS15].
Results {#sec:results}
=======
AQEM
----
![Logarithm of Holevo variance from adaptive interferometric-phase estimation. The interferometer includes small phase noise of width $\sigma$ and loss rate $\eta$. Three algorithms are used to generate the feedback policies: DE, PSO, and stochastic hill-climbing. This image is a rescaled version of Figure 1 in Ref. [@PWS16].[]{data-label="fig:logHV"}](logHV_twocolumn2)
By applying the method of creating noise-resistant to DE, we are able the obtain a policy that delivers the scaling of $V_H \propto N^{-1.421}$ when the width of the Gaussian distribution $\sigma=0.2$ rad, and the probability of losing a photon $\eta = 0.2$ are included. This result shows a scaling exceeding $N^{-1}$ expected from SQL, which is given for the ideal interferometer as a benchmark in Figure \[fig:logHV\]. The SQL data is generated using a non-entangled $N$-particle state. The HL shown is an extrapolation using the intercept from the SQL data and is included for the purpose to providing a possible benchmark for the scheme.
Although both the SQL and the HL are reported in the literature for the mean-squared error $\Delta\widetilde{\phi}$ [@GLM11], we use the same benchmark for Holevo variance $V_{\rm H}$. This follows from the approximation of $V_{\rm H}$ at low error $\left|\phi_k-\widetilde{\phi}_k\right|\ll 1$. Under this condition, the sharpness in Eq. (\[eq:Sharpness\_est\]) is approximated by a series expansion, and through this approximation, $V_{\rm H}$ is found to be the mean-squared error.
The accept-reject criterion applied to $N>93$ enables the scheme to show the attain the power-law up to $N=100$ (Figure \[fig:logHV\]). The limitation at 100 photons is due to the computational time and the rounding error in the generation of the large multiparticle entangled state. The time required to find a policy under the accept-reject criterion from 94 to 100 photons is between 1.5 to 3 hours per data point.
Policies that are found using stochastic hill climbing break down at 20 photons even for ideal phase estimation. The noise-resistant PSO shows the breakdown at 45 photons, consistent with the previous result [@LCPS13]. We did not apply the accept-reject criterion to PSO as the computational time would have exceeded the time used by DE at the same number of $N$ and hence not considered worth an investment.
Quantum Gate Design {#quantum-gate-design}
-------------------
Our machine learning approach to designing a three-qubit gate succeeds in generating policies for the design of a high-fidelity Toffoli gate. The resultant fidelity exceeds 0.9999, which exceeds the threshold fidelity for the fault-tolerant quantum computing. The gate operation time is found to be $26~\text{ns}$. Therefore, the number of learning parameters add to the total of 81. Although our machine learning technique has optimized the shape of the tranmons’ frequencies over a piecewise-error-function, we have shown [@ZGS15] that the algorithm does not rely on the shape of the pulse but on the number of learning parameters to generate successful policy for the gate design. Our quantum Toffoli gate operates as fast as a two-qubit entangling quantum gate under the same experimental conditions. The policies are also robust against the random uniform noise on the control pulses. The threshold of the frequency $\delta\varepsilon$ of which the intrinsic fidelity remains above 0.9999 is well above the practical noise (up to 100kHz) of the control devices (Fig. \[fig:T\_robustness\]).
![Intrinsic fidelity $\mathcal{F}$ of the Toffoli gate as a function of $\delta\varepsilon_{1,2,3}$, corresponding to the noise level on each of the three transmons. The vertical red dotted line denotes the threshold, such that $\mathcal{F}>0.9999$ on the left of the line.[]{data-label="fig:T_robustness"}](robust)
Conclusion {#sec:conclusion}
==========
In this work, we report on two examples of applying machine-learning algorithms to quantum control, namely the adaptive phase estimation and quantum gate design. We employ reinforcement learning to adaptive phase estimation including noise and loss. We are able to attain enhanced precision better than SQL up to 100 photons using a noise-resistant variant of DE and accept-reject criterion. The supervised-learning technique using SuSSADE enables us to perform single-shot high-fidelity three-qubit gates that are as fast as an entangling two-qubit gate under the same experimental constraints.
The methods we employed do not require explicit knowledge of the system’s dynamics, although the convexity of the objective functions, the dimension of the problems and the presence of noise have to be taken into account in order to generate a feasible policy. We minimize the runtime of the algorithms by vectorizing the random number generation and employing GPUs and VSL. This technique mostly affects the simulation of the quantum system as the simulations are the most time- and resource-consuming components of the current algorithms.
In principle, the simulation in the learning algorithms can be replaced by signals from experimental setup or simulations of other quantum control schemes. This work can be used as the basis to develop learning algorithms for solving other quantum control problems, such as estimating more than one unknown parameters, which has an application in the characterization of quantum information processing devices, controlled quantum-state transfer in a spin chain [@GB07], and quantum error correction [@ADL02].
Acknowledgement {#sec:ackowledgement}
===============
This work is financially supported by NSERC and AITF. P.W. acknowledges financial support from the ERC (Consolidator Grant QITBOX), MINECO (Severo Ochoa grant SEV-2015-0522 and FOQUS), Generalitat de Catalunya (SGR 875), and Fundació Privada Cellex. B.C.S. also acknowledges support from the 1000 Talent Plan. The computational work is enabled by the support of WestGrid (www.westgrid.ca) and Calcul Québec (www.calculquebec.ca) through Compute Canada Calcul Canada (www.computecanada.ca).
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---
abstract: 'External magnetic field, temperature, and spin-polarized current are usually employed to create and control nanoscale vortex-like spin configurations such as magnetic skyrmions. Although these methods have proven successful, they are not energy-efficient due to high power consumption and dissipation. Coupling between magnetic properties and mechanical deformation, the magnetoelastic (MEL) effect, offers a novel approach to energy-efficient control of magnetism at the nanoscale. It is of great interest in the context of ever-decreasing length scales of electronic and spintronic devices. Therefore, it is desirable to establish a comprehensive framework capable of predicting effects of mechanical stress and enabling deterministic control of magnetic textures and skyrmions. In this work, using an advanced scheme of multiscale simulations and Lorentz transmission electron microscopy measurements we demonstrate deterministic control of topological magnetic textures and skyrmion creation in thin films and racetracks of chiral magnets. Our investigation considers not only uniaxial but also biaxial stress, which is ubiquitous in thin-film devices. The biaxial stress, rather than the uniaxial one, was shown to be more efficient to create or annihilate skyrmions when the MEL coefficient and strain have the same or opposite signs, respectively. It was also demonstrated to be a viable way to stabilize skyrmions and to control their current-induced motion in racetrack memory. Our results open prospects for deployment of mechanical stress to create novel topological spin textures, including merons, and in control and optimization of skyrmion-based devices.'
author:
- 'Phuong-Vu Ong'
- 'Tae-Hoon Kim'
- Haijun Zhao
- 'Brandt A. Jensen'
- Lin Zhou
- Liqin Ke
bibliography:
- 'strain\_sk\_ref\_aps.bib'
title: 'Deterministic strain-control of stability and current-induced motion of skyrmions in chiral magnets'
---
Introduction
============
Nanoscale vortex-like spin configurations, *i.e.*, magnetic skyrmions[@Braun21], are promising information carriers for future magnetic memories[@Kang16; @Fert13]. Investigations on these real-space topological states are usually devoted to their creation or annihilation via external magnetic field, temperature [@Muhlbauer09; @Li13; @Karube16; @Tokunaga15; @Peng18; @Yu10; @Guoqiang19; @Anjan17], and spin-polarized current control[@Jiang15]. Although these methods have proven successful, they require high power consumption with high power dissipation in the form of heat. Coupling between magnetic properties and mechanical deformation, or magnetoelastic (MEL) effect, promises energy-efficient control of magnetism at the nanoscale. Moreover, the integration of skyrmions into current semiconductor technology is highly likely to rely on thin-film heterostructures, in which large stresses are often presented due to the lattice mismatch between substrate and component layers. Therefore, it is necessary to understand the effects of MEL coupling and strain on magnetization dynamics, in general, and on skyrmions, in particular, and the mechanism which ensures their stability in strained systems.
Studies have shown that uniaxial stress can deform skyrmions from circular to elliptic shape in FeGe thin plates[@Shibata15]. It can also fine-tune skyrmion crystals (SkXs) phase region in MnSi[@Chacon15; @Nii15] and Cu$_2$OSeO$_3$ crystals [@Seki17]. Different mechanisms have been proposed to explain the observed phenomena, including strained-induced magnetic anisotropy (MA) along the stress direction[@Butenko10; @Nii15; @JWang18] and strained-induced anisotropy of Dzyaloshinskii–Moriya interaction(DMI) [@Shibata15]. It is worth mentioning that even for the uniaxial stress, the concomitant strain is induced in all three orthogonal directions. As a result, an effective MEL field is induced not only in the direction along the stress axis, but also perpendicular to it. This is a unique feature of stress-induced MA, and it is important to consider the combined effect of the effective MEL fields in all directions. Moreover, these studies indicated that different types of stress, *i.e.*, tensile or compressive is required to stabilize skyrmions in different materials. Therefore, it is crucial to establish a principle that allows deterministic control of topological spin textures in materials with given physical properties. Lastly, biaxial stress, as opposed to the uniaxial one, is ubiquitous in modern electronic devices, which are based on stacking of multilayers. Therefore, a complete and practical understanding of the stress effect should include the biaxial stress.
Here, we propose a theoretical framework for study of the stress effects, which includes the second-order (in strain) MEL effect and enables a deterministic control of topological spin textures and their dynamics. Based on this framework, we report results of advanced multiscale simulations and Lorentz transmission electron microscopy observation on the effects of both uniaxial and biaxial stress on magnetization dynamics and skyrmions in thin films and racetracks of chiral magnets. MnSi and $\beta$-Mn-type Co-Zn-Mn were employed as two typical model materials. Our results elucidate the relationship between MEL properties, types of stress, and their effects. It was found that the biaxial stress, rather than the uniaxial one, is more efficient to stabilize skyrmions and to control their dynamics in thin films and racetracks. The results open prospects for deployment of mechanical stress to create novel topological spin textures, including merons, and in control and optimization of skyrmion-based devices.
Theoretical and experimental details
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Landau-Lifchitz-Gilbert equation with magnetoelastic interaction
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Suppose that a three dimensional magnetic sample occupies some domain $\Omega\subset\mathbb{R}^3$ and let $\mathbf{m}$ be normalized magnetization ($\mathbf{m} = \mathbf{M}/|\mathbf{M}|$, where $|\mathbf{M}|=M_s$ is saturated magnetization). For systems with bulk Dzyaloshinskii–Moriya interaction(DMI), the initial boundary value problem for the Landau-Lifchitz-Gilbert (LLG) equation which we are interested takes the form
$$\begin{aligned}
% \nonumber % Remove numbering (before each equation)
\begin{cases}
\label{eq:ibvp1}
\partial_t\mathbf{m} - \alpha \mathbf{m} \times \partial_t\mathbf{m}
= - \mathbf{m} \times \mathbf{H}_{eff} \\
\quad \text{in $\Omega\times(0,\infty)$,}\\
\label{eq:ibvp2}
2\ell_{ex}^2 \partial_{\mathbf{n}}\mathbf{m}
= - \ell_{dm}\mathbf{m} \times \mathbf{n} \quad \text{in $\Gamma\times(0,\infty)$,} \\
\label{eq:ibvp3}
\mathbf{m}(0)= \mathbf{m}^0 \quad \text{in $\Omega$.}
\end{cases}\end{aligned}$$
where $\ell_{ex} =\sqrt{\frac{2A}{\mu_0M_s^2}}$ and $\ell_{dm} = \frac{2D}{\mu_0M_s^2}$ are exchange and DMI lengths, respectively, in which $A$ and $D$ are exchange interaction and DMI parameter, respectively; $\alpha$ is the Gilbert damping parameter; $\mu_0$ is the vacuum magnetic permeability. $\Gamma$ is the domain boundary and $\mathbf{m}^0$ is initial magnetization. Note that the LLG equation in is in the dimensionless form. The effective field, $\mathbf{H}_{eff}$, is determined by the first derivative of total energy functional, $\mathcal{E}[\mathbf{M}]$, with respect to the magnetization vector $\mathbf{M}$, more precisely
$$\label{eq:Heff}
\begin{split}
\mathbf{H}_{eff} = & -\frac{1}{\mu_0M_s}\frac{\delta\mathcal{E}[\mathbf{M}]}{\delta\mathbf{M}}
= \ell_{ex}^2\triangle\mathbf{m} + \mathbf{H}_d + \mathbf{H}_{ext} \\
& + Q(\mathbf{e}.\mathbf{m})\mathbf{e} - \ell_{dm}\mathbf{\nabla}\times\mathbf{m} + \mathbf{H}_{mel}
\end{split}$$
where $\mathbf{H}_d$ and $\mathbf{H}_{ext}$ are effective demagnetizing and external fields, respectively. The descriptions of these conventional terms can be found in Refs. [-@Praetorius18; -@Alouges2014]; $Q=\frac{2K}{\mu_0M_s^2}$ is the MA coefficient, in which $K$ is uniaxial MA energy. We focus on the new term, $\mathbf{H}_{mel}$, which describes the MEL effect and can be determined from the MEL energy functional[@Gilbert04; @Eastman66]:
$$\label{eq:melenergy}
\begin{split}
U_{mel}[\textbf{M}] = & \int\limits_{\Omega}\left( B_{1}\sum\limits_{i}m_i^2e_{ii}
+ \frac{1}{2}B_{2}\sum\limits_{i\neq j}m_im_je_{ij} \right. \\
& \left. + \frac{1}{2}D_{11}\sum\limits_{i}m_i^2e_{ii}^2 \right)d\textbf{r}
\end{split}$$
where $m_i$, $i=$ $x$, $y$, or $z$, are the Cartesian component of the normalized magnetization $\mathbf{m}$, $B_1$ and $B_2$ are the first-order, and $D_{11}$ is the second-order MEL coefficients, and $e_{ij}$ is strain tensor. Mechanical deformation induced by lattice mismatch usually does not involve changes in crystallographic angles. Therefore, shear strain will not be considered in the present work, *i.e.*, $e_{ij}\approx 0$ for $i\neq j$. With this in mind, we calculated the first functional derivative of the equation with respect to **M** and obtained the effective MEL field:
$$\label{eq:melfield}
\begin{split}
\textbf{H}_{mel}[\textbf{M}]
= & -\frac{1}{\mu_0M_s^2}\left [ ( 2B_1 + D_{11}e_{xx} ) m_x e_{xx}\hat{\textbf{x}} \right. \\
& \left. + ( 2B_1 + D_{11}e_{yy} ) m_y e_{yy}\hat{\textbf{y}} \right. \\
& \left. + ( 2B_1 + D_{11}e_{zz} ) m_z e_{zz}\hat{\textbf{z}} \right ]
\end{split}$$
where $\hat{\textbf{x}}$, $\hat{\textbf{y}}$, and $\hat{\textbf{z}}$ are the unit basis vectors of the Cartesian coordinate system.
For the FEA of MnSi skyrmion racetrack with the current-in-plane configuration, the Zhang-Li spin torque term [@ZhangLi04] will be added to the effective field :
$$\begin{aligned}
\mathbf{H}_{ZL}[\textbf{M}] &=& [ \textbf{m} \times (\textbf{u}. \boldsymbol{\nabla}) \textbf{m} ] +
\xi (\textbf{u}. \nabla) \textbf{m} \\
\textbf{u} &=& \frac{1} {1+\xi^2} \frac{Pg_e\mu_B} {|e|\mu_0|\gamma| M_s^2} \textbf{j}_e\end{aligned}$$
where $e$, $g_e$, and $\gamma$ are electron charge, $g-$factor, and gyromagnetic ratio, respectively, $\mu_B$ is Bohr magneton. Nonadiabatic spin torque parameter, $\xi$, is typically an order of magnitude larger than the damping parameter $\alpha$ [@Garate09].
Uniaxial and biaxial strain in isotropic materials
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For an isotropic materials, we have the following relations between stress tensor ($e_{ij}$) and strain tensor ($\sigma_{ij}$)[@Nye04],
$$\begin{aligned}
% \nonumber % Remove numbering (before each equation)
\label{eq:stressstrainrel1}
e_{xx} &=& \frac{1}{E}\left[ \sigma_{xx}-\nu(\sigma_{yy} + \sigma_{zz}) \right] \\
\label{eq:stressstrainrel2}
e_{yy} &=& \frac{1}{E}\left[ \sigma_{yy}-\nu(\sigma_{zz} + \sigma_{xx}) \right] \\
\label{eq:stressstrainrel3}
e_{zz} &=& \frac{1}{E}\left[ \sigma_{zz}-\nu(\sigma_{xx} + \sigma_{yy}) \right]\end{aligned}$$
where $E$ is Young’s modulus, $\nu$ is Poisson’s ratio, $\sigma_{xx}$, $\sigma_{yy}$, $\sigma_{zz}$ are stress along, $x$, $y$, and $z$ directions.
In the case of *uniaxial* stress, $\sigma_{xx}=\sigma$ and all other stress components vanish. From -, we have
$$\begin{aligned}
% \nonumber % Remove numbering (before each equation)
\label{eq:ustressstrainrel1}
e_{xx} &=& \frac{\sigma}{E} \\
\label{eq:ustressstrainrel2}
e_{yy} &=& -\nu e_{xx} \\
\label{eq:ustressstrainrel3}
e_{zz} &=& -\nu e_{xx}\end{aligned}$$
In the case of *biaxial* stress, $\sigma_{xx}=\sigma_{yy}=\sigma$ and $\sigma_{zz}=0$, we have,
$$\begin{aligned}
% \nonumber % Remove numbering (before each equation)
\label{eq:bistressstrainrel1}
e_{xx} &=& (1-\nu)\frac{\sigma}{E} \\
\label{eq:bistressstrainrel2}
e_{yy} &=& e_{xx} \\
\label{eq:bistressstrainrel3}
e_{zz} &=& -\frac{2\nu}{1-\nu} e_{xx}\end{aligned}$$
For metals, typical value of Poisson’s ratio is $\nu\approx1/3$[@Gercek07]. Therefore, from - we have $e_{xx} = e_{yy}\approx -e_{zz}$ for the biaxial stress.
For materials with nonlinear MEL property such as MnSi, we performed finite element analysis (FEA) of the problem using the *Commics* (COmputational MicroMagnetICS) code[@Pfeiler18], modified to include the MEL field . We are not aware of any micromagnetic codes, which are capable of calculating the nonlinear MEL effects. The FEA was performed using the (almost) second-order tangent plane scheme. Within this scheme, numerical solution of the LLG has been proved to converge toward the weak solution within an error of (almost) second-order in time-step size[@Alouges2014; @Fratta17]. The FEA implementation is based on the multiphysics finite element software *Netgen/NGSolve*[@ngsolve].
For MnSi thin films and racetracks, the FEA was performed using Delaunay tetrahedralization with maximal global mesh-size of 6.0 nm was employed. Time-step size was set to 0.1 ps. Thickness of the thin films and racetracks was fixed at 18.0 nm. Damping parameter was $\alpha$ = 0.02. Material physical parameters are obtained from the experiments for a 17.6-nm thick MnSi thin film[@Karhu12]. In particular, exchange stiffness $A$ = $0.76\times10^{-12}$ J/m; bulk DMI coefficient $D_M$ = $0.34\times10^{-3}$ J/m$^2$; saturation magnetization $M_s$ = $0.16\times10^{6}$ A/m; uniaxial MA $K$ = $0.9\times10^{4}$ J/m$^3$; the first order and second order MEL coefficients were $B_1$ = $1.0\times10^{6}$ J/m$^3$ and $D_{11} = -7.8\times10^{7}$ J/m$^3$, respectively. It was found that most of these parameters are independent of or weakly dependent on thickness, except to uniaxial MA[@Karhu12].
For simulations of MnSi racetrack with an applied in-plane spin-polarized current, we set $\xi$ = 0.2. Degree of spin-polarization $P$ was set to the experimental value of 0.1 [@Neubauer09]. Electron current density $j_e$ = 5.0$\times$10$^{10}$ A/m$^2$ was employed. We note that $\textbf{j}_e$ denotes direction of electron motion, which is opposite to motion of positive charge.
On the other hand, the alloy Co$_8$Zn$_{8.5}$Mn$_{3.5}$ is a linear MEL material (as shown below). Moreover, it hosts skyrmions with relatively large radii and thus requires a large simulation area ($\sim\mu m$). Therefore, micromagnetic simulations were carried out using the finite-difference code *mumax*$^3$ [@Vansteenkiste14]. The simulations were performed on a sample of area 2.5$\times$2.5 $\mu m^2$ and thickness 190 nm with discretization cell size of 5$\times$5$\times$190 nm$^3$. External magnetic field was $B_{ext}$ = 0.15 T and damping parameter was $\alpha$ = 0.02. Cubic MA constant was set to $K$ = $0.5\times10^{4}$ J/m$^3$. Dulk DMI coefficient was set to $D_M$ = $0.7\times10^{-3}$ J/m$^2$, which is larger than the experimental value for the Co$_8$Zn$_{8}$Mn$_{4}$ bulk ($0.53\times10^{-3}$ J/m$^2$)[@Takagi17], but found to give the simulated period of the helical stripes in agreement with our experimental value for the Co$_8$Zn$_{8.5}$Mn$_3$ thin plate. For other material physical parameters, we adopted the experimental values for Co$_8$Zn$_{8}$Mn$_{4}$ alloy. In particular, exchange stiffness $A$ = $9.2\times10^{-12}$ J/m[@Takagi17] and saturation magnetization $M_s$ = $0.35\times10^{6}$ A/m[@Bocarsly19].
Density functional theory (DFT) calculation of the MEL coefficient
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To calculate the MEL coefficients and Poisson’s ratio $\nu$ of the Co-Zn-Mn alloy, we performed DFT calculations based on the projector augmented-wave formalism for electron-ion potential[@Bloch94], as implemented in the Vienna *Ab initio* Simulation Package[@KRESSE199615]. Exchange-correlation interaction was treated within the Perdew-Burke-Ernzerhoff functional form of the generalized gradient approximation (GGA)[@Perdew96]. The binary parent alloy Co$_{10}$Zn$_{10}$ has a simple cubic structure (space group number 213), in which the 8$c$ sites are mainly occupied by Co atoms and the 12$d$ sites are mainly occupied by Zn and also randomly occupied by the remaining Co atoms (Fig. \[DFTresult\]a)[@Xie13]. In a derived alloy Co$_x$Zn$_y$Mn$_z$ alloys ($x+y+z=20$), Mn atoms mainly occupy the 12$d$ sites, but also share the 8c sites with Co atoms [@Nakajima19; @Hori07]. To avoid complications due to the randomness of site occupancies in the Co$_x$Zn$_y$Mn$_z$ alloys, we employed a simplified model based on the binary alloy Co$_{10}$Zn$_{10}$. The model has 10 Co atoms (8 at the 8$c$ sites and 2 at two arbitrary 12$d$ sites) and 10 Zn atoms at the 12$d$ sites.
Uniaxial stress was applied along the $x$ axis. At each strain $e_{xx}$ $=\varepsilon_u$, geometry optimization, magnetic, and atomic relaxation were performed until the change in the total energy between two ionic relaxation steps is smaller than $10^{-6}$ eV. Then, the corresponding strain $e_{yy}$ and $e_{zz}$ along the $y$ and $z$ directions, respectively, can be determined as a function of $e_{xx}$ (Fig. \[DFTresult\]b, right ordinate). The MA was calculated as the total-energy difference between the two magnetic states in which the magnetization is aligned along the \[100\] or \[100\] directions, respectively. The obtained data is shown in Fig. \[DFTresult\]b (left ordinate).
In the phenomenological theory, MA can be determined by the following expression: $$\label{eq:MAdiff}
\text{MA} = f_M(e_{ij}, m_x=1) - f_M(e_{ij}, m_z=1)$$
where the magnetic energy density functional $f_M$ is expressed by:
$$\label{eq:magenergy}
\begin{split}
f_M(e_{ij}, m_i) = & K(1-m_x^2)+ B_1 \sum \limits_{i=x,y,z} e_{ii}m_i^2 \\
& + B_2\sum\limits_{i\neq j}e_{ij}m_im_j
\end{split}$$
Since shear strain will not be considered, *i.e.*, $e_{ij}=0$ for $i\neq j$. Substituting into we obtain:
$$\label{eq:MAdiff}
\text{MA} = -K + B_1(e_{xx}-e_{zz})$$
For uniaxial stress, $e_{zz} = -\nu e_{xx}$. Therefore, with putting $e_{xx} = \varepsilon_u$ we obtain
$$\label{eq:MAfit}
\text{MA} = -K + B_1(1+\nu)\varepsilon_u$$
By fitting the $e_{yy(zz)}$ data (Fig. \[DFTresult\]b, right ordinate) with the linear relations and , $\nu$ was found to be 0.34, which agrees with the typical Poisson’s ratio for metals ($\approx1/3$). By fitting the calculated MA data (Fig. \[DFTresult\]b, left ordinate) to , we obtained a magnetoelastic coefficient value $B_1$ = $-1.65\times10^{6}$ J/m$^3$, which is negative, as opposed to the positive value for MnSi. As shown below, this result will be confirmed by our experimental results.
![**TEM sample preparation for in-situ tensile testing.** (a) SEM image of push-to-pull (PTP) device. By pushing semi-circular part on top, tensile strain can be applied to the loaded thin plate. (b) (100)Co$_8$Zn$_{8.5}$Mn$_3$ thin plate loaded on the PTP device. Both ends of the plate were welded by carbon deposition. (c) Thickness map of the thin plate obtained by electron energy loss spectroscopy showing that the thin plate has a uniform thickness of $\sim$190 nm.[]{data-label="exptfigS1"}](exptfigS1.pdf)
Sample preparation and measurements
-----------------------------------
The bulk Co$_8$Zn$_{8.5}$Mn$_{3.5}$ sample was prepared by first sealing individual metals (all $>$ 99.9% metals basis) in a quartz ampoule backfilled with ultra-high purity argon. The ampoule was placed into a furnace and heated to 1000 $^\circ$C for 12 hours, then cooled at 1 $\circ$C/hr to 925 $^\circ$C and held for 96 hours before quenching into water. Magnetic measurements were performed using a Quantum Design VersaLabTM vibrating sample magnetometer. A thin, polycrystalline piece was polished so the sample dimensions were greater than 5:1 aspect ratio. The sample was field cooled at a rate of 2 K/min under an applied field of $H =$ 20 Oe. The polycrystalline sample has a Curie temperature of 330 K and magnetization of 0.2 $\mu_B$/f.u. under an applied field of $H =$ 20 Oe. A Co$_8$Zn$_{8.5}$Mn$_{3.5}$ (100) plate with \[110\] lateral orientation was fabricated using the focused ion beam system FEI Helios NanoLab G3. The crystal orientation was examined by electron backscattered diffraction analysis before lift-out. The plate was thinned to approximately 190 nm thick and transferred on a push-to-pull (PTP) device with 150 N/m$^2$ stiffness. Carbon deposition was conducted to ensure that the plate was clearly adhered to the PTP device. To prevent Ga-ion-induced beam damage during the transfer, low-kV Ga ion imaging was performed. Thickness map of the sample (Fig. \[exptfigS1\]c) was obtained using electron energy loss spectroscopy with Gatan Quantum ER 965.
In-situ Lorentz transmission electron microscopy (LTEM) observation was carried out on an FEI Tecnai G2-F20 operating at 200 kV of accelerating voltage. A Hysitron PI 95 TEM PicoIndenter enables quantitative uniaxial tensile testing in the LTEM experiments. Real-time applied force and displacement were measured. By subtracting the PTP device portion from the measured force, actual force applied to the specimen was calculated. As a result, a linear stress-strain relation with Young’s modulus of 85.52 GPa was obtained. An external magnetic field was applied along the electron beam direction by partially exciting the objective lens. The in-plane magnetization maps of magnetic structures were obtained by the LTEM Fresnel images with a phase-retrieval QPt software on the basis of the transport of intensity equation[@ishizuka_allman_2005].
Results and discussion
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The uniaxial and biaxial stress effects on a nonlinear MEL chiral magnet
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Due to the lack of space-inversion symmetry, DMI is induced in MnSi and makes it a material of choice for study of topological spin textures [@Muhlbauer09]. In a thin film form, MnSi exhibits nonlinear MEL behavior with the first- and second MEL coefficients of $B_1$ = $1.0\times10^{6}$ J/m$^3$ and $D_{11} = -7.8\times10^{7}$ J/m$^3$, respectively[@Karhu12].
Fig. \[MnSifig\] shows the FEA results for a 18-nm-thick MnSi thin film under an external magnetic field of 0.2 T. In the strain free condition, the system shows short stripes and skyrmions of various shapes with an average diameter of about 58 nm (Fig. \[MnSifig\]a).
For a uniaxial stress along the $x$ axis, the relationships between strains along the $x$, $y$, and $z$ axes are $e_{xx}$ = $\varepsilon_u$ and $e_{yy}$ = $e_{zz}$ = $-\nu\varepsilon_u$, as inferred from equations -. Under the uniaxial *compressive* stress ($\varepsilon_u$ = $-2.0$%), these skyrmions elongate and merge to form a helical phase with the spin-spiral propagation (or $q$-) vector perpendicular to the stress direction (Fig. \[MnSifig\]b). In contrast, the uniaxial *tensile* stress does not induce a helical phase, but only the elongation and alignment perpendicular to the stress axis (Fig. \[MnSifig\]c).
Underlying mechanism of the stress-induced formation of the helical phase can be explained on the basis of strain-induced MA. For a uniaxial stress along the $x$ axis, the effective MEL field is equivalent to the following field up to addition by a field parallel to **m**
$$\label{eq:melfielduni}
\begin{split}
\textbf{H}_{mel}^u[\textbf{M}]
= & -\frac{1}{\mu_0M_s^2}\left [ 2 (1+\nu) B_1\varepsilon_u \right. \\
& \left. + (1-\nu^2) D_{11}\varepsilon_u^2 \right ] m_x \hat{\textbf{x}}
\end{split}$$
This effective MEL field has the same nature as a MA field along the $x$ axis. We note that the field can be made parallel to the stress direction ($x$) only by combination of the effective MEL fields induced not only in the direction along the stress axis, but also perpendicular to it. Under the compressive stress, due to $B_1\varepsilon_u<0$ and $D_{11}<0$ the field is parallel to $\hat{\textbf{x}}$ and thus induces an increase in the magnetization component $|m_x|$. As a result, the skyrmions and short stripes are elongated along the stress direction. These elongated skyrmions and stripes combine to form longer ones and eventually helices (Fig. \[MnSifig\]b). Due to the finite size along the thin film normal a helical phase with $q$-vector oriented along the $z$ axis is prohibited. The reason is that it will result in a lamellar structure of ferromagnetic layers parallel to the $xy$-plane, which is a relatively unstable configuration. As a result, the $q$-vector can be oriented only along the $y$ axis (Fig. \[MnSifig\]d).
![**Finite element analysis of the MnSi thin film under in-plane uniaxial stress.** Equilibrium magnetization in the $xy$-plane viewed from the top under the compressive uniaxial stress ($\varepsilon_u=$ $-2.0$%). Sample area is 500$\times$500 nm$^2$ and thickness is 18 nm. External magnetic field is 200 mT along the film normal. Uniaxial stress axis (**u**) is along the **x** axis, as indicated by the double headed arrow.[]{data-label="MnSifigD11zero"}](MnSifigD11zero.pdf)
We found that including the second order term (due to the finite $D_{11}$) is necessary for the formation of purely helical phase, which otherwise contains discontinuous helices and an isolated skyrmion (Fig. \[MnSifigD11zero\]). This second order effect can be understood in term of the additional field $\left ( -\frac{(1-\nu^2)D_{11}\varepsilon_u^2}{\mu_0M_s^2} \right )m_x\hat{\textbf{x}}$ in , which, for $D_{11}<0$, is parallel to $\hat{\textbf{x}}$ and thus enhances the in-plane MA. Therefore, it further promotes the elongation and merging of skyrmions and hence the formation of helices.
On the other hand, for tensile stress ($B_1\varepsilon_u>0$) the field points opposite to $\hat{\textbf{x}}$. Therefore, it drives magnetization equally likely in the $y$ and $z$ directions. Due to the above mentioned asymmetry between the $y$ and $z$ axes, skyrmion elongation occurs along the $y$ axis. In this case, a helical phase could, however, not be formed (Fig. \[MnSifig\]c). The reason is that the field now induces an enhancement in the perpendicular MA (PMA), which tends to stabilize magnetization along the $z$ direction. This is shown by the noticeable increase in the $m_z$ components, as indicated by the enhanced color contrast in Fig. \[MnSifig\]c, compared with that in Fig. \[MnSifig\]b. Therefore, the enhanced PMA limits the elongation and suppresses the merging, and thus stabilizes the (elongated) skyrmions .
For thin films, it is often more relevant to consider biaxial stress, which is caused by the lattice mismatch between a thin film and the underlying substrate or between the component layers in a heterostructure. Under the biaxial stress, the $x$ and $y$ directions are equivalent and, therefore, skyrmion elongation effect will not occur. The effective MEL field is now equivalent to
$$\label{eq:melfieldbiaxial}
\begin{split}
\textbf{H}_{mel}^b[\textbf{M}]
= & \frac{1}{\mu_0M_s^2}\left [ 2 \left ( \frac{1+\nu}{1-\nu} \right ) B_1\varepsilon_b \right. \\
& \left. + \frac{ ( 1 + \nu ) ( 1 - 3 \nu ) } { (1-\nu)^2} D_{11}\varepsilon_b^2 \right ] m_z \hat{\textbf{z}}
\end{split}$$
where $\varepsilon_{b}$ is strain along the $x$ and $y$ directions (see equations -). Given that for most metals the Poisson’s ratio $\nu$ is about $\approx1/3$, the equation implies that the second order term is negligibly small, owing to the factor $( 1 - 3 \nu )$ and also the second order in strain ($\varepsilon_b^2$). For the compressive stress ($B_1\varepsilon_b<0$), this field reduces PMA of the MnSi film. As a result, magnetization is driven to the $xy$ plane in most of the perimeter regions of the vortices, while still pointing downward in the core regions (Fig. \[MnSifig\]e). This magnetic configuration resembles that of merons, which are vortex-like spin textures with winding number $n=-1/2$[@Yu2018], except that there are still finite out-of-plane magnetization components in some finite parts of the perimeters. Nevertheless, the result gives rise to a possibility of creating merons and their crystal with a biaxial stress if magnetic field and temperature can be fine-tuned into a proper condition.
Under the biaxial tensile stress ($B_1\varepsilon_b>0$), PMA is enhanced, leading to a transformation of the short stripes into skyrmions and a reduction of skyrmion diameter. After a simulation time of 10.0 ns, the system was found to become all skyrmions with a diameter of about 50 nm. These skyrmions were found to keep moving and reorganizing themselves in a relatively slow process. Fig. \[MnSifig\]f shows the skyrmion system after 35.3 ns. It is expected that a hexagonal SkX will form after a sufficiently long simulation time.
As discussed above, the out-of-plane (along the $z$ axis) MEL field plays a key role in stabilization, creation, or annihilation of skyrmions. Comparing the first order terms in the equations and for the uniaxial and biaxial stress, respectively, reveals a difference by a factor of $\left ( \frac{1}{1-\nu} \right )$. Most of materials have $\nu$ ranging between 0.0 and 0.5. For the typical case of $\nu = 1/3$, this translates into a strain-induced PMA either larger or smaller by 50% for the biaxial stress, compared with that induced by the corresponding uniaxial one. Based on the above discussed results, we proposed the following criteria for efficient creation and control of skyrmions with mechanical stress: *i*) the applied stress should be biaxial; *ii*) the MEL coefficient $B_1$ and strain have to be in the same sign. Moreover, due to the factor $\left ( \frac{1}{1-\nu} \right )$ the relative efficiency of the biaxial stress depends on the Poisson’s ratio $\nu$. A higher $\nu$ value leads to a higher efficiency of the mechanical control with biaxial stress, as opposed to the uniaxial one.
The magnetoelastic sign effect on skyrmion elongation and helical phase formation
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As can be inferred from the effective MEL fields and , the strain $\varepsilon_{u(b)}$ always appears in the forms of products $B_1\varepsilon_{u(b)}$ and $D_{11}\varepsilon_{u(b)}^2$. From the above mentioned mechanism of stress effect, it is clear that formation and orientation of the elongated skyrmions and helical phase relative to the stress direction depend on sign of the MEL coefficient $B_1$. Therefore, they can vary from one material to another.
To further demonstrate the deterministic control of magnetic texture, we apply our micromagnetic framework to a $\beta$-Mn-type Co$_x$Zn$_y$Mn$_z$ alloy with $x+y+z=20$. This class of chiral magnets can host skyrmions beyond room temperature [@Tokunaga15]. As shown above, our DFT calculations predict that the $\beta$-Mn-type Co-Zn-Mn alloy has linear MEL behavior with the negative MEL coefficient $B_1$, *i.e.*, opposite to that for MnSi, and thus makes it an ideal choice for a further test of the framework.
To obtain the magnetic texture in the zero magnetic field and strain free condition, we carried out simulation at high temperature and then slowly decreased it to zero, *i.e.*, employed simulated annealing simulation to accelerate the convergence process toward the ground state. The magnetic texture exhibits discontinuous helical stripes with different orientations and an average stripe period of about 156 nm (Fig. \[CoZnMnfig\]a).
Under the external magnetic field $B_{ext}=$ 0.15 T along the thin plate normal, polydomain SkX is formed with an average skyrmion diameter of about 150 nm (Fig. \[CoZnMnfig\]b). A uniaxial tensile stress with $\varepsilon_u=$ 0.53% is then applied along the $x$ direction. The stress effect is twofold: first, it induces elongation of skyrmions; and second, it renders skyrmions aligned along the stress directions, and thus induces a polydomain-to-monodomain phase transition (Fig. \[CoZnMnfig\]c). The monodomain SkX has edge dislocations (dashed lines) and a skyrmion vacancy (dotted circle). It is worth mentioning that the local distortion induced by the vacancy is especially strong for the SkX, in sharp contrast to that for a solid crystal.
A topological magnetic texture is characterized by a winding number or topological charge defined as
$$n=\frac{1}{4\pi}\int \mathbf{m}.(\partial_x \mathbf{m}
\times \partial_y \mathbf{m})dxdy$$
which is a negative integer for the Co-Zn-Mn thin plate under the external magnetic field parallel to the $z$ axis. Fig. \[CoZnMnfig\]d shows absolute value of total winding number $|N|$ of the system as a function of strain and magnetic field below 0.3 T. Three variation regimes of $|N|$ can be identified: first, it is slowly decreased or unaltered with increasing strain; then at a certain strain value depending on the applied field, it is rapidly reduced; after that, it slowly decreases toward zero. In the first regime, the main processes are elongation and alignment of skyrmions and short stripes, which do not involve change in topology of the magnetic texture. In the second regime, merging of elongated skyrmions and short helical stripes becomes active. Moreover, in this regime strain is also sufficiently large to align magnetization along the stress direction in most of the thin plate area, leading to local transitions to ferromagnetic (FM) states, which are topologically trivial. These effects together cause the rapid decrease in $|N|$. For strong magnetic fields on the verge of a FM transition ($\sim$0.30 T), the stress-induced elongation and merging of skyrmions are suppressed for strain below a critical value. Above the critical strain, the system changes directly from a skyrmion phase to a FM phase. Therefore, for the magnetic field of 0.3 T the total winding number $|N|$ falls sharply at the strain value $\varepsilon_u=1.38$%.
For the Co-Zn-Mn thin plate, the elongation and alignment of skyrmions (Fig. \[CoZnMnfig\]c) is at 90 degree with respect to those for the MnSi thin film under the same type of strain (tensile, Fig. \[MnSifig\]c). This is due to the opposite signs of the first order MEL coefficients for the two materials. Based on the results for both of the systems, the following rule can be drawn: for a thin plate or thin film under uniaxial stress, if the product $B_1\varepsilon_u$ is negative, skyrmion elongation and alignment will be parallel to the stress direction. Moreover, the stress diminishes PMA of the system and thus tends to induce a helical phase with $q$-vector perpendicular to the stress direction. On the other hand, if $B_1\varepsilon_u$ is positive, the elongation and alignment will be perpendicular to the stress direction. Moreover, the elongation will be limited due to the stress-induced enhancement of PMA.
Experimental confirmation
-------------------------
Based on LTEM observation at room temperature, a phase diagram was established for the observed rectangular area under increasing external magnetic field, $B_{ext}$. Transitions from helical to skyrmion phase and from skyrmion to FM phase occurs at $B_{ext}\sim$ 20 and 130 mT, respectively (Fig. \[exptfig\]a). Fig. \[exptfig\]b shows helical texture in the zero field condition. These helical stripes have different orientations and average period of about 160 nm, resembling the simulation result for the zero field and zero strain condition (Fig. \[CoZnMnfig\]a). Under $B_{ext}=$ 70 mT, a SkX can be observed with average skyrmion diameter about 140 nm (Fig. \[exptfig\]d), compared to the simulation result of 150 nm.
Elliptic skyrmions aligned along the stress direction can be observed at strain $\varepsilon_u =$ 0.53% (Fig. \[exptfig\]f). The corresponding fast Fourier transform of the LTEM image has an elliptic shape with the major axis perpendicular to the stress direction (Fig. \[exptfig\]g), confirming the skyrmions are elongated along the stress direction. This agreement between the experimental and simulation results also confirms the DFT prediction of the negative MEL coefficient $B_1$ for this alloy.
Stress control of skyrmion motion on MnSi racetrack
---------------------------------------------------
To demonstrate a practical application of our findings, we consider a 900-nm long, 60-nm wide, and 18-nm thick MnSi racetrack under a perpendicular magnetic field of 0.15 T. An in-plane electron-current $j_e=$5.0$\times10^{11}$ A/m$^2$ is passed through the track along the long ($x$) axis. The current is found to induce elongation of a skyrmion. The skyrmion can grow into a long stripe of up to 260 nm in 10.0 ns (Fig.\[MnSiRacetrack\]b-f). Shorter stripes are also induced around the middle of the racetrack (Fig.\[MnSiRacetrack\]e and f). As a result, the current-induced motion of skyrmions is suppressed. As the stability of skyrmions and their steady flow are essential prerequisites for operation of skyrmion racetrack devices, such skyrmion elongation and magnetic stripes would have a detrimental effect on the devices.
Under the biaxial compressive stress ($\varepsilon_{b} = -2\%$), these stripes are transformed into skyrmions (indicated by 2 and 3 in Fig.\[MnSiRacetrack\]g-l). As a result, the skyrmions are unblocked and start to flow following the electron current. The skyrmions 1 and 2 are in free motion during time between $t=13.0$ to 14.5 ns (Fig.\[MnSiRacetrack\]i and j) and $t=12.0$ to 13.0 ns (Fig.\[MnSiRacetrack\]h and i), respectively. We note that their motion also has finite transverse components (indicated by the double headed arrows for the skyrmion 1 in Fig.\[MnSiRacetrack\]g-j). This is the observed skyrmion Hall effect caused by the magnetic Magnus force, which is induced by the skyrmion topological charge and is perpendicular to the current and the magnetic field directions[@Jonietz1648; @Jiang2016Magnus]. The transverse motion is limited due to the narrow width and reflection at the racetrack boundary. Average horizontal velocity is estimated to be 80.0 m/s, which is comparable to the experimental and predicted values for // and Pt/Co thin films, respectively[@Woo16; @Sampaio13].
Stability of skyrmions is a major challenge for their applications at room-temperature. Our results demonstrate that the biaxial stress can be employed to efficiently create, stabilize, and control skyrmions. This can be achieved, for example, by growth of the racetrack on a piezoelectric substrate, which will facilitates an energy-efficient control of strain via an applied voltage or electric field. This strain-mediated approach has already been proved to assist magnetization switching in magnetoelectric memories [@Nan19; @Hu15; @Buzzi13]. Such a device structure can be employed to control and enhance stability of skyrmion-based racetrack memories and sensors. Another advantage of such a multiferroic heterostructure is that skyrmions can also be controlled by manipulating the ferroelectric polarization of the substrate.
To summarize, we have demonstrated theoretically and experimentally the mechanical control of topological magnetic textures and their dynamics in thin films and thin plates of chiral magnets. Based on the magnetoelastic coupling, our theoretical framework elucidates the relationship between magnetoelastic properties of materials, types of stress (tensile or compressive), and their effects, and thus enables a deterministic control of the topological spin textures. It was found that the biaxial stress, rather than the uniaxial one, is more efficient to annihilate, create, and stabilize skyrmions. Moreover, creation or annihilation occurs when the magnetoelastic coefficient and strain have the same or opposite signs, respectively. Stress can induce a rich variety of topological spin textures, possibly including merons. Biaxial stress was also demonstrated to be a viable way to stabilize skyrmions and to control their current-induced motion in racetrack memory. We hope our findings stimulate further research and open prospects for deployment of mechanical stress in control and optimization of skyrmion-based devices.
This work is supported in part by Laboratory Directed Research and Development (LDRD) funds through Ames Laboratory (P.-V.O., T.K., H.Z., L.Z.) and by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Science and Engineering Division. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. DE-AC02-07CH11358. L.K. was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, Early Career Research Program following conception and initial work supported by LDRD. All TEM and related work were performed using instruments in the Sensitive Instrument Facility in Ames Lab.
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---
abstract: 'We study detailed classical-quantum correspondence for a cluster system of three spins with single-axis anisotropic exchange coupling. With autoregressive spectral estimation, we find oscillating terms in the quantum density of states caused by classical periodic orbits: in the slowly varying part of the density of states we see signs of nontrivial topology changes happening to the energy surface as the energy is varied. Also, we can explain the hierarchy of quantum energy levels near the ferromagnetic and antiferromagnetic states with EKB quantization to explain large structures and tunneling to explain small structures.'
address: 'Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, 14853-2501'
author:
- 'P. A. Houle[@houlenote], N. G. Zhang, and C. L. Henley[@clhnote]'
title: 'Semiclassical mechanics of a non-integrable spin cluster'
---
epsf
Introduction
============
When $S$ is large, spin systems can be modeled by classical and semiclassical techniques. Here we reserve “semiclassical” to mean not only that the technique works in the limit of large $S$ (as the term is sometimes used) but that it implements the quantum-classical correspondence (relating classical trajectories to quantum-mechanical behavior).
Spin systems (in particular $S=1/2$) are often thought as the antithesis of the classical limit. Notwithstanding that, classical-quantum correspondence has been studied at large values of $S$ in systems such as an autonomous single spin [@Shankar80], kicked single spin [@Haake87], and autonomous two [@Srivastava90c] and three [@Nakamura93] spin systems.
When the classical motion has a chaotic regime, for example, the dependence of level statistics on the regularity of classical motion has been studied [@Srivastava90c; @Nakamura93]. In regimes where the motion is predominantly regular, the pattern of quantum levels of a spin cluster can be understood with a combination of EBK (Einstein-Brillouin-Keller, also called Bohr-Sommerfeld) quantization and tunnel splitting (Sec. \[sec-clustering\] is a such a study for the current system.) The latter sort of calculation has potential applications to some problems of current numerical or experimental interest. Numerical diagonalizations for extended spin systems (in ordered phases) on lattices of modest size ($10$ to $36$ spins) may be analyzed by treating the net spin of each sublattice as a single large spin and thereby reducing the system to an autonomous cluster of a few spins; the clustering of low-lying eigenvalues can probe symmetry breakings that are obscured in a system of such size if only ground-state correlations are examined. [@Henley98] Nonlinear self-localized modes in spin lattices [@Lai97a], which typically span several sites, have to date been modeled classically, but seem well suited to semiclassical techniques. Another topic of recent experiments is the molecular magnets [@Gatteschi94] such as $\rm Mn_{12}Ac$ and $\rm Fe_8$, which are more precisely modeled as clusters of several interacting spins rather than a single large spin; semiclassical analysis may provide an alternative to exact diagonalization techniques [@Katsnelson99] for theoretical studies of such models.
In this paper, we will study three aspects of the classical-correspondence of an autonomous cluster of three spins coupled by easy-plane exchange anisotropy, with the Hamiltonian $${H} = \left[ \sum_{i=1}^3 {{\mathbf S}}_i \cdot {{\mathbf S}}_{i+1} - \sigma {{\mathbf S}}_i^z {{\mathbf S}}_{i+1}^z \right],
\label{eq:ham3a}$$ This model was introduced in Ref. , a study of level repulsion in regions of $(E,\sigma)$ space where the classical dynamics is predominantly chaotic [@Nakamura93]. [Eq. (\[eq:ham3a\])]{} has only two nontrivial degrees of freedom, since it conserves total angular momentum around the z-axis. As did Ref. we consider only the case of $\sum_i {{\mathbf S}}_i^z=0.$ While studying classical mechanics we set $|{{\mathbf S}}|=1;$ to compare quantum energy levels at different $S$, we we divide energies by by $S(S+1)$ to normalize them. The classical maximum energy, $E=3$, occurs at the ferromagnetic (FM) state – all three spins are coaligned in the equatorial (easy) plane. The classical ground state energy is $E=-1.5$, in the antiferromagnetic (AFM) state, in which the spins lie $120^\circ$ apart in the easy plane; there are two such states, differing by a reflection of the spins in a plane containing the $z$ axis. Both the FM and AFM states, as well as all other states of the system, are continuously degenerate with respect to rotations around the $z$-axis. The classical dynamics follows from the fact that $\cos\theta_i$ and $\phi_i$, are conjugate, where $\theta_i$ and $\phi_i$ are the polar angles of the unit vector ${{\mathbf S}}_i$; then Hamilton’s equations of motion say $$\begin{aligned}
d\cos \theta_i/dt &=& \hbar^{-1} \partial {H} / \partial \phi_i;
\nonumber \\
d\phi_i/dt &=& - \hbar^{-1} \partial {H} / \partial \phi_i.
\end{aligned}$$
In the rest of this paper, we will first introduce the classical dynamics by surveying the fundamental periodic orbits of the three-spin cluster, determined by numerical integration of the equations of motion (Sec \[sec-classical\]). The heart of the paper is Sec. \[sec-orbitspectrum\]: starting from the quantum density of states (DOS) obtained from numerical diagonalization, we apply nonlinear spectral analysis to detect the oscillations in the quantum DOS caused by classical periodic orbits; to our knowledge, this is the first time the DOS has been related to specific orbits in a multi-spin system. Also, in Sec. \[sec-flatspot\] we smooth the DOS and compare it to a lowest-order Thomas-Fermi approximation counted by Monte Carlo integration of the classical energy surface; a flat interval is visible in the quantum DOS between two critical energies where the topology of the classical energy surface changes. Finally, in Sec. \[sec-clustering\], we use a combination of EBK quantization and tunneling analysis to explain the clustering patterns of the quantum levels in our system.
Classical periodic orbits {#sec-classical}
=========================
Our subsequent semiclassical analysis will depend on identification of all the fundamental orbits and their qualitative changes as parameters are varied. Examining Poincaré sections and searching along symmetry lines of the system, we found four families of fundamental periodic orbits for the three-spin cluster. Figure \[fig:tracks\] is an illustration of their motion, and Figure \[fig:classicalEtau\] gives classical energy-time curves. Orbits of types (a)-(c) are always at least threefold degenerate, since one spin is different from the other two; orbits of types (a)-(c) are also time-reversal invariant. Orbit (a), the [*counterbalanced*]{} orbit, exists when $E>-1$ (including the FM limit) and, in the range $0<\sigma<1$ which we’ve studied, is always stable. Orbit (b), the [*unbalanced*]{} orbit, is unstable and exists when $E<E_p,$ where $$E_p={3 \over 4} \sigma - {3 \over 2}. \label{eq:polar}$$ Orbits of type (c), or [*stationary spin*]{} exist at all energies. Type (c) orbits are are unstable in the range, $-1>E>3.$ Below $E=-1$ the stationary spin orbit bifurcates into two branches without breaking the symmetry of the ferromagnetic ground state. At $$E_c(\sigma)={3 - 3 \sigma + \sigma^2 \over \sigma - 2},
\label{eq:afmcat}$$ one branch vanishes and the other branch bifurcates into two orbits that are distorted spin waves of the two AFM ground states. (Below, in Sections \[sec-flatspot\] and \[sec-clustering\], we will discuss topology changes of the entire energy surface.) Although they are not related by symmetry, all orbits of type (c) at a particular energy have the same period.
Orbits of type (d), or [*three-phase*]{} orbits are named in analogy to three-phase AC electricity, as spin vectors move along distorted circles, $120^\circ$ out of phase. The type (d) orbits break time-reversal symmetry and are hence at least twofold degenerate. A symmetry-breaking pitchfork bifurcation of the (d) family occurs (for $\sigma=0.5$ around $E=-0.75$) at which a single stable orbit, approaching from high energy, bifurcates into an unstable and two stable [*precessing three-phase*]{} orbits without period doubling. [@DeAguiar87] (Strictly speaking, the precessing three-phase orbits are not periodic orbits of the three-spin system, since after one “period” the spin configuration is not the same as before, but rather, all three spins are rotated by the same angle around the $z$-axis). The unstable three-phase orbit disappears quickly as we lower energy, but the precessing three-phase orbits persist until $E=-1.5,$ and become intermittently stable and unstable in a heavily chaotic regime near $E_A,$ but regain stability before $E \rightarrow -1.5$: thus in the AFM limit, orbits (c) are stable while orbits (b) are unstable. [@Henon64] More information on the classical mechanics of this system appears in Refs. and .
Orbit spectrum analysis {#sec-orbitspectrum}
=======================
Gutzwiller’s trace formula, the central result of periodic orbit theory, [@DeAlmedia88] $$\rho(E)={\mathrm{Re}} \sum_p
A_p(E) \exp[i S_p(E) / \hbar]
+ \rho_{tf}(E)
, \label{eq:gtf}$$ decomposes the quantum DOS $\rho(E)$ into a sum of oscillating terms contributed by classical orbits indexed by $p,$ where $S_p(E)$ is the classical action, and $A_p(E)$ is a slowly varying function of the period, stability and geometric [@FN-maslov] properties of the orbit $p$), plus the zeroth-order [*Thomas-Fermi*]{} term, $$\rho_{tf}(E)=\int { d^{2N}{{\tilde z}}\over (2 \pi \hbar)^N } \delta \left(E-H({{\tilde z}}) \right),
\label{eq:thomasFermi}$$ This integral over phase space ${{\tilde z}}$ is simply proportional to the area of the energy surface. We do not know of any mathematical derivation of (\[eq:gtf\]) in the case of a spin system.
At a fixed H, the [*orbit spectrum*]{} is, as function of $\tau,$ the power spectrum of $\rho(E)$ inside the energy window, $H-\Delta H/2<E<H+\Delta H/2.$ (Figure \[fig:c3os\], explained below, is an example of an orbit spectrum.) Since the classical period $\tau_p(E)=\partial S_p(E) / \partial E,$ [Eq. (\[eq:gtf\])]{} implies that ${O}(H,\tau)$ is large if there exists a periodic orbit with energy $H$ and period $\tau.$ The orbit spectrum can be estimated by Fourier transform, [@FN-FTnote] $${O}(H,\tau)=
\left|
\int_{H-\Delta H/2}^{H+\Delta H/2}
\rho(E) e^{-i \hbar^{-1} E \tau} dE \right|^2. \label{eq:ftospec}
\label{eq-fourier}$$ Variants of [Eq. (\[eq:ftospec\])]{} have been used to extract information about classical periodic orbits from quantum spectra. [@Baranger95; @Ezra96] Unfortunately, the resolution of the Fourier transform is limited by the uncertainty principle, $\delta E \delta t = \hbar/2.$
Nonlinear spectral estimation techniques, however, can surpass the resolution of the Fourier transform. [@Marple87] One such technique, harmonic inversion, has been successfully applied to scaling systems [@Main97] – i.e., systems like billiards or Kepler systems in which the (classical and quantum) dynamics at one energy are identical to those at any other energy, after a rescaling of time and coordinate scales. In a scaling system, windowing is unnecessary because there are no bifurcations and the scaled periods of orbits are constant. In this section, we will apply nonlinear spectral estimation to our system (\[eq:ham3a\]), which is [*nonscaling*]{}. [@FN-scaling]
Diagonalization {#subsec-diagonalize}
---------------
To get the quantum level spectrum, we wrote software to diagonalize arbitrary spin Hamiltonians polynomial in (${{\mathbf S}}_i^x,{{\mathbf S}}_i^y,{{\mathbf S}}_i^z$), where $i$ is an index running over arbitrary $N$ spins of arbitrary (and often large) spin $S$. The program, written in [Java]{}, takes advantage of discrete translational and parity symmetries by constructing a basis set in which the Hamiltonian is block diagonal, letting us diagonalize the blocks independently with an optimized version of LAPACK. Picturing the spins in a ring, the Hamiltonian [Eq. (\[eq:ham3a\])]{} is invariant to cyclic permutations of the spins, so the eigenstates are states of definite wavenumber [@Nakamura93] $k=0,\pm { 2\pi \over 3}$ (matrix blocks for $k= \pm { 2 \pi \over 3}$ are identical by symmetry). In the largest system we diagonalized (three-spin cluster with $S=65$) , the largest blocks contained $N=4620$ states.
Autoregressive approach to construct spectrum
---------------------------------------------
The input to an orbit spectrum calculation is the list of discrete eigenenergies with total $S_z=0$; no other information on the eigenstates (e.g. the wavenumber quantum number) is necessary. This level spectrum is smoothed by convolving with a Gaussian (width $10^{-3}$ for Figure \[fig:c3os\]) and discretely sampling over energy (with sample spacing $\delta=4.5 \times 10^{-4}$.)
We estimate the power spectrum by the autoregressive (AR) method. AR models a discretely sampled input signal, $y_i$ (in our case the density of states) with a process that attempts to predict $y_i$ from its previous values, $$y_i=\sum_{j=1}^N a_i y_{i-j} + x_i. \label{eq:filter}$$ Here $N$ is a free parameter which determines how many spectral peaks that model can fit; Refs. and discuss guidelines for choosing $N$. Fast algorithms exist to implement least-squares, i.e. to choose $N$ coefficients $a_i$ to minimize (within constraints) $\sum x_i^2$; of these we used the Burg algorithm [@Marple87].
To estimate the power spectrum, we discard the original $x_i$ and model $x_i$ with uncorrelated white noise. Thinking of [Eq. (\[eq:filter\])]{} as a filter acting on $x_i,$ the power spectrum of $y_i$ is computed from the transfer function of [Eq. (\[eq:filter\])]{} and is $$P(\nu)={<x_i^2> \over 1 - \sum_{j=1}^N a_j e^{i \nu \delta}}.$$ Unlike the discrete Fourier transform, $P(\nu)$ can be evaluated at any value of $\nu.$ In our application, of course, $\delta$ has units of energy, so $\nu$ (more exactly $\nu/\hbar$) actually has units of time and is to be identified with $\tau$ in (\[eq-fourier\]).
Orbit spectrum results and discussion
-------------------------------------
Figure \[fig:c3os\] shows the orbit spectrum of our system with $S=65$ and $\sigma=0.5.$; it is displayed as a $500 \times 390$ array of pixels, colored light where ${O}(H,\tau)$ is large. Each horizontal row is the power spectrum in an energy window centered at $H;$ we stack rows of varying $H$ vertically. With a window width 250 energy samples long ($\delta H = 0.1125,$) we fit $N=150$ coefficients in [Eq. (\[eq:filter\])]{}. To improve visual resolution, we let windows overlap and spaced the centers of successive windows 25 samples apart.
Comparing Figure \[fig:c3os\] and Figure \[fig:classicalEtau\] we see that our orbit spectrum detects the fundamental periodic orbits as well as multiple transversals of the orbits. Interestingly, we produced Figure \[fig:c3os\] before we had identified most of the fundamental orbits; Figure \[fig:c3os\] correctly predicted three out of four families of orbits.
We believe that, given the same data, the AR method normally produces a far sharper spectrum. This is not surprising, since the Fourier analysis allows the possibility of orbit-spectrum density at all $\tau$ values, whereas AR takes advantage of our [*a priori*]{} knowledge that there are only a few fundamental periodic orbits and hence only a few peaks. We have compared the Fourier and AR versions of the spectrum in a few cases, but have not systematically tested them against each other.
Unfortunately, the artifacts and limitations of the AR method are less understood than those of the Fourier transform. At high energies, the classical periods are nearly degenerate, so we expect closely spaced spectral peaks in the orbit spectrum. In this situation, the Burg algorithm vacillates between fitting one or two peaks causing the braiding between the (a) and (c) orbits (labeled in Figure \[fig:classicalEtau\]) in Figure \[fig:c3os\]. Also, in the range $-1<E<-1.3,$ where classical chaos is widespread, bifurcations increase the number of contributing orbits so that we cannot interpret the orbit spectrum for $\tau>10.$
Averaged density of states {#sec-flatspot}
==========================
The lowest-order Thomas Fermi approximation, [Eq. (\[eq:thomasFermi\])]{} predicts that the area of the classical energy surface is proportional to the DOS. We verify this in Figure \[fig:flatspot\], a comparison of the heavily smoothed quantum DOS to the area of the energy surface computed by Monte Carlo integration.
An energy interval is visible in which the quantum DOS appears to be constant; we then verified that the classical DOS (which is more precise) is constant to our numerical precision; a similar interval was observed for all values of $\sigma$. We identified this interval as $(E_p, -1)$, where the endpoints are associated with changes in the topology of the energy surface as the energy varies.
At energies below $E_c$ (see [Eq. (\[eq:afmcat\])]{}), the energy surface consists of two disconnected pieces, one surrounding each AFM ground state. The two parts coalesce as the energy surface becomes multiply connected at $E_c.$ For $E< E_p,$ (see [Eq. (\[eq:polar\])]{}) the anisotropic interaction confines the spins to a limited band of latitude away from the poles. At $E_p$ it becomes possible for spins to pass over the poles. At $E=-1,$ the holes that appeared in the energy surface at $E_c$ close up. A discontinuity in the slope of the area of the energy surface occurs at energy $E_c$ (not visible in Figure \[fig:flatspot\]); in the range $E_p<E<-1$ the area of the energy surface (and hence the slowly varying part of the DOS) seems to be constant as a function of energy.
In the special isotropic ($\sigma=0$) case, the flat interval is $(-1.5,-1)$ and it can be analytically derived that the DOS is constant there. This is simplest for the smoothed quantum DOS, since for $n=1,2,\ldots$ there are clusters of $n$ energy levels with level spacing proportional to $n$. (A derivation also exists for the classical case, but is less direct.) We have no analytic results for general $\sigma$.
This flat interval is specific to our three-spin cluster, but we expect that the compactness of spin phase space will, generally, cause changes in the energy surface topology of spin systems that do not occur in traditionally studied particle systems.
Level clustering {#sec-clustering}
================
The quantum levels with total ${{\mathbf S}}_z=0$ show rich patterns of clustering, some of which are visible on Figure \[fig:c3low\]. The levels that form clusters correspond to three different regimes of the classical dynamics in which the motion becomes nearly regular: (1) the FM limit (not visible in Figure \[fig:c3low\]; (2) the AFM limit (bottom edge of Figure \[fig:c3low\]) and (3) the isotropic limit $\sigma=0$ (left edge of Figure \[fig:c3low\]). Indeed, the levels form a hierarchy in as the clusters break up into subclusters. In this section, we first approximately map the phase space from four coordinates to two coordinates – with the topology of a [*sphere*]{}. (Two of the original six coordinates are trivial, or decoupled, due to symmetry, as noted in Sec. \[sec-classical\]. Then, using Einstein-Brillouin-Kramers (EBK) quantization some consideration of quantum tunneling, many features of the level hierarchy will be understood.
Generic behavior: the polyad phase sphere
-----------------------------------------
\[subsec:polyads\]
In all three limiting regimes, the classical dynamics becomes trivial. For small deviations from the limit, the equations of motion can be linearized and one finds that the trajectory decomposes into a linear combination of two harmonic oscillators with degenerate frequency $\omega$, i.e., in a 1:1 resonance; the oscillators are coupled only by higher-order (=nonlinear) terms.
There is a general prescription for understanding the classical dynamics in this situation [@Kellman]. Near the limit, the low-excited levels have approximate quantum numbers $n_{1,2}$ such that the excitation energy $\Delta E_i$ in oscillator $i$ is $\hbar \omega (n_{i}+1/2)$. (In the FM limit, regime (1), this difference is actually measured [*downwards*]{} from the energy maximum.) Clearly, the levels with a given total quantum number $P \equiv (n_1+n_2+1)$ must have nearly degenerate energies, and thus form a cluster of levels, which are split only by the effects (to be considered shortly) of the anharmonic perturbation. A level cluster arising in this fashion is called a [*polyad*]{} [@Kellman].
To reduce the classical dynamics, make a canonical transformation to the variables $\Phi$ and ${\bf P} \equiv (P_x, P_y, P_z)$, where $\Phi$ is the mean of the oscillators’ phases and $\Psi_x$ is their phase difference, and $$\begin{aligned}
P_x & \equiv & {1\over 2} (n_1-n_2), \nonumber \\
(P_y, P_z) & \equiv &
2 \sqrt{(n_1+1/2)(n_2+1/2)}(\cos\Psi_x, \sin \Psi_x) ,
\label{eq:polyad}
\end{aligned}$$ Here $\Phi$ is the fast coordinate, with trivial dynamics $d\Phi/dt = \omega$ in the harmonic limit. The slow coordinates $\bf P$ follow a trajectory confined to the “polyad phase sphere” $|{\bf P}| = P$, since $\Delta E = \hbar \omega P $ is conserved by the harmonic-order dynamics. The reduced dynamics on this sphere is properly a map ${\bf P}_i \to {\bf P}_{i+1}$, defined by (say) the Poincaré section at $\Psi_x=0 ({\rm mod}\; 2\pi )$. But $d {\bf P}/dt$ contains only higher powers of the components of $\bf P$, so near the harmonic (small $P$) limit, $|{\bf P}_{i+1}-{\bf P}_i|$ vanishes and the reduced dynamics becomes a flow. [@FN-poincare] At the limit in which it is a flow, an effective Hamiltonian $I$ can be defined so that the dynamics becomes integrable. [@FN-I] Applying EBK quantization to the reduced dynamics on the polyad phase sphere gives the splitting of levels within a polyad cluster. (Near the harmonic limit, the energy scale of $I$ is small compared to the splitting between polyads.)
In all three of our regimes, we believe this flow has the topology shown schematically in Figure \[fig:flows\]. [@FN-fourspin] Besides reflection symmetry about the “equator”, it also has a threefold rotation symmetry around the $P_z$ axis, which corresponds to the cyclic permutation of the three spins. [@FN-Paxes] (Figure \[fig:flows\] is natural for the three-spin system because it is the simplest generic topology of the phase sphere with that threefold symmetry.) The reduced dynamics has two symmetry-related fixed points at the “poles” $P_z=\pm P$, which always correspond to motions of the three-phase sort like (d) on Figure \[fig:tracks\]. There are also three stable and three unstable fixed points around the “equator”.
The KAM tori of the full dynamics correspond to orbits of the reduced dynamics. These orbits follow contours of the effective Hamiltonian $I$ of the reduced dynamics (as in Figure \[fig:flows\]). In view of the symmetries mentioned, $$I = \alpha P_z^2 + \beta (P_x^3 - 3 P_x P_y^2) + {\rm const}
\label{eq:I}$$ to leading order, where $\alpha$,$\beta$, and the constant may depend on $\sigma$, $S$, and $P$.
The KAM tori surrounding the three-phase orbits represented by the “poles” are twofold degenerate, while the tori in the stable resonant islands represented on the “equator” are threefold degenerate. Hence, the EBK construction produces degenerate subclusters containing two or three levels depending on the energy range within the polyad cluster.
The fraction of levels in one or the other kind of subcluster is proportional to the spherical areas on the corresponding side of the separatrix, which passes through the unstable points in Figure \[fig:flows\]. These areas in turn depend on the ratio of the first to the second term in Eq. (\[eq:I\]), i.e. $\alpha P^2/ \beta P^3$. Evidently, as one moves away from the harmonic limit to higher values of $P$, one universally expects to have a larger and larger fraction of threefold subclusters.
Given the numerical values of energy levels in a polyad, we can estimate the terms of Eq. (\[eq:I\]) in the following fashion: (i) the energy difference between the highest and lowest 3-fold subcluster is the difference between the stable and unstable orbits on the equator, which is $ 2 \beta P^3$ according to (\[eq:I\]); (ii) the mean of the highest and lowest 3-fold subcluster would be the energy all around the equator if $\beta$ were to vanish; the difference between this energy and that of the farthest 2-fold subcluster in the polyad is $\alpha P^2$ according to (\[eq:I\]).
Furthermore, tunneling between nearby tori creates [*fine structure*]{} splitting inside the sub-clusters. The slow part of the dynamics on the polyad phase sphere, is identical to that of a single semiclassical spin with (\[eq:I\]) as its effective Hamiltonian, so the effective Lagrangian is essentially the same, too. Then different tunneling paths connecting the same two quantized orbits must differ in phase by a topological term, with a familiar form proportional to the (real part of the) spherical area between the two paths. [@Topological92]
Results {#subsec:clusterresults}
-------
Here we summarize some observations made by examination of polyads in the three regimes, for a few combinations of $S$ and $\sigma$.
### Ferromagnetic limit
This regime is the best-behaved in that regular behavior persists for a wide range of energies. The ferromagnetic state, an energy maximum, is a fixed point of the dynamics; around it are “spin-wave” excitations (viewing our system as the 3-site case of a one dimensional ferromagnet). These are the two oscillators from which the polyad is constructed. Thus, the “pole” points in Figure \[fig:flows\] correspond to “spin waves” propagating clockwise or counterclockwise around the ring of three spins, an example of the “three-phase” type of orbit. The stable and unstable points on the “equator” are identified respectively with the orbits (a) and (c) of Figure \[fig:tracks\]. Classically, in this regime, the three-phase orbit is the fundamental orbit with lowest frequency $\omega_{3-phase}$; thus the corresponding levels in successive polyads have a somewhat smaller spacing $\hbar \omega_{3-phase}$ than other levels, and they end up at the top of each polyad. (Remember excitation energy is measured [*downwards*]{} from the FM limit.) Indeed, we observe that the high-energy end of each polyad consists of twofold subclusters and the low-energy end consists of threefold subclusters.
We see a pattern of fine structure (presumably tunnel splittings) which is just like the pattern in the four-spin problem.[@Henley98; @FN-fourspinclusters] Namely, throughout each polyad the degeneracies of successive levels follow the pattern (2,1,1,2) and repeat. (Here – as also for regime 3 – every “2” level has $k=\pm 1$ and every “1” level has $k=0$, where wavenumber $k$ was defined in Sec. \[subsec-diagonalize\].) Numerical data show that (independent of $S$) the pattern (starting from the lowest energy) begins (2112...) for even $P$, but for odd $P$ it begins (1221...). In the energy range of twofold subclusters, the levels are grouped as (2)(11)(2), i.e. one tunnel-split subcluster between two unsplit subclusters(and repeat); in the threefold subcluster regime, the grouping is (21)(12), so that each subcluster gets tunnel-split into a pair and a single level, but the sense of the splitting alternates from one subcluster to the next.
An analysis of $\sigma=0.4$, $S=30$ showed that the fraction of threefold subclusters indeed grows from around 0.3 for small $P$ to nearly $0.5$ at $P\approx 40$. Furthermore, when $\alpha P^2$ and $\beta P^3$ were estimated by the method described near the end of Subsec. \[subsec:polyads\], they indeed scaled as $P^2$ and $P^3$ respectively.
### Antiferromagnetic limit
This regime occurs at $E<E_c(\sigma)$, where $E_c(\sigma)$ is given by (\[eq:afmcat\]). That means the classical energy surface is divided into two disconnected pieces, related by a mirror reflection of all three spins in any plane normal to the easy plane. Analogous to regime one, two degenerate antiferromagnetic “spin waves” exist around [*either*]{} energy minimum, and the polyad states are built from the levels of these two oscillators. Thus the clustering hierarchy outlined in Sec. \[subsec:polyads\] – polyads clusters, EBK-quantization of $I$, and tunneling over barriers of $I$ on the polyad phase sphere – is repeated within each disconnected piece, leading to a prediction that [*all*]{} levels should be twofold degenerate.
Consequently, on the level diagram (Figure \[fig:c3low\]), there should be half the apparent level density below the line $E=E_c(\sigma)$ as above it. Indeed, a striking qualitative change in the apparent level crossing behavior is visible at that line (shown dashed in the figure). Actually, [*tunneling*]{} is possible between the disconnected pieces of the energy surface and may split these degenerate pairs. In fact this [*hyperfine*]{} splitting happens to 1/3 of the pairs, again following the (2112) pattern within a given polyad. This (2112) pattern starts to break up as the energy moves away from the AFM limit; even for large $S$ ($30$ or $65$), this breakup happens already around the polyad with $P=10$, so it is much harder than in the FM case to ascertain the asymptotic pattern of subclustering. We conjecture that the breakup may happen near the energies where, classically, the stable periodic orbits bifurcate and a small bit of phase space goes chaotic.
The barrier for tunneling between the disconnected energy surfaces has the energy scale of the bare Hamiltonian, which is much larger (at least, for small $P$) than the scale of effective Hamiltonian $I$ which provides the barrier for tunneling among the states in a subcluster. Hence, the hyperfine splittings are tiny compared to the fine splittings discussed at the end of Subsection \[subsec:polyads\]. To analyze numerical results, we replace a degenerate level pair by one level and a hyperfine-split pair by the mean level, and treat the result as the levels from [*one*]{} of the two disconnected polyad phase spheres, neglecting tunneling to the other one.
Then in the AFM limit, the “pole” points in Figure \[fig:flows\] again correspond to spin waves propagating around the ring, while the stable and unstable points on the equator are (c) and (b) on Figure \[fig:tracks\]. The three-phase orbit is the [*highest*]{} frequency orbit in the AFM limit, [@Houle98c] so again the twofold and threefold subclusters should occur at the high and low energy ends of each polyad cluster. What we observe, however, is that [*all*]{} the subclusters are twofold, except the lowest one is often threefold.
### Isotropic limit
This regime will includes only $S_{\rm tot} \leq S$ i.e. $E<-1$ – the same regime in which the flat DOS was observed (Sec. \[sec-flatspot\]). Above the critical value $E=-1$, the levels behave as in the “FM limit” described above.
At $\sigma = 0$, it is well-known that the quantum Hamiltonian reduces to ${\scriptstyle {1\over 2}}[S_{\rm tot}^2 - 3S(S+1)]$. Thus each level has degeneracy $P \equiv 2S_{\rm tot}+1$. (That is the number of ways three spins $S$ may be added to make total spin $S_{\rm tot}$, and each such multiplet has one state with $S^z_{\rm tot}=0$.) When $\sigma$ is small, these levels split and will be called a polyad. [@FN-isotropicP]
Classically, at $\sigma=0$ the spins simply precess rigidly around the total spin vector. These are harmonic motions of four coordinates; hence the polyad phase sphere can be constructed by (\[eq:polyad\]). From the threefold symmetry, there should again be three orbit types as represented generically by Figure \[fig:flows\] and Eq. (\[eq:I\]). For example, an umbrella-like configuration in which the three spin directions are equally tilted out of their plane corresponds to a three-phase type orbit, with two cases depending on the handedness of the arrangement. A configuration where one spin is parallel/antiparallel to the net moment, and the other two spins offset symmetrically from it), follows one of the threefold degenerate orbits.
Numerically, the level behavior in the near-isotropic limit is similar to the near-FM limit. The fine structure degeneracies are a repeat of the (2112) pattern as in the other regimes; the lowest levels of any polyad always begin with (1221). The fraction of threefold subclusters is large here and, as expected, grows with $S$, (from 0.5 to 0.7 in the case $S=15$). However, the energy scales of $\alpha P^2$ and $\beta P^3$ behave numerically as $\sigma P^0$ and $\sigma P^1$. What is different about the isotropic limit is that the precession frequency – hence the oscillator frequency $\omega$ – is not a constant, but is proportional to $S_{\rm tot}$. Since perturbation techniques [@FN-I] give formulas for $I$ with inverse powers of $\omega$, it is plausible that $\alpha$ and $\beta$ in (\[eq:I\]) include factors of $P^{-2}$ here, which were absent in the other two regimes.
Conclusion and summary
======================
To summarize, by using detailed knowledge of the classical mechanics of a three spin cluster [@Houle98c], we have studied the semiclassical limit of spin in three ways. First, using autoregressive spectral analysis, we identified the oscillating contributions that the fundamental orbits of the cluster make to the density of states, in fact, we detected the quantum signature of the orbits before discovering them. Secondly, we verified that the quantum DOS is proportional to the area of the energy surface; we also observed kinks in the smoothed quantum DOS, which are the quantum manifestation of topology changes of the classical energy surface; such topology changes, we expect, are more common in spin systems than particle phase space, since even a single spin has a nontrivial topology. Finally, we have identified three regimes of near-regular behavior in which the levels are clustered according to a four-level hierarchy, and we explained many features qualitatively in terms of a reduced, one degree-of-freedom system. This system appears promising for two extensions analgous to Ref. : tunnel amplitudes (and their topological phases) could be computed more explicitly; also, the low-energy levels from exact diagonalization of a finite piece of the anisotropic-exchange antiferromagnet on the triangular lattice could probably be mapped to three large spins and analyzed in the fashion sketched above in Sec. \[sec-clustering\].
This work was funded by NSF Grant DMR-9612304, using computer facilities of the Cornell Center for Materials Research supported by NSF grant DMR-9632275. We thank Masa Tsuchiya, Greg Ezra, Dimitri Garanin, Klaus Richter and Martin Sieber for useful discussions.
[10]{}
Author to whom correspondence should be addressed.
R. Shankar, Phys. Rev. Lett. [**45**]{}, 1088 (1980).
F. Haake, M. Kus, and R. Scharf, Z. Phys. B [**65**]{}, 381 (1987). The classical-quantum correspondence has been studied extensively in this system, but the Hamiltonian is time-varying and unlike that of interacting spins,
N. Srivastava and G. M[ü]{}ller, Z. Phys. B [**81**]{}, 137 (1990).
K. Nakamura, , Cambridge University Press, 1993.
C. L. Henley and N. G. Zhang, Phys. Rev. Lett. [**81**]{}, 5221 (1998).
R. Lai and A. J. Sievers, Phys. Rev. Lett. [**81**]{}, 1937 (1998). U. T. Schwartz, L. Q. English, and A. J. Sievers, Phys. Rev. Lett. [**83**]{}, 223 (1999).
D. Gatteschi, A. Caneschi, L. Pardi, and R. Sessoli, Science [**265**]{} (1994).
M. I. Katsnelson, V. V.Dobrovitski, and B. N. Harmon, Phys. Rev. B [**59**]{}, 6919 (1999).
K. Nakamura, Y. Nakahara, and A. R. Bishop, Phys. Rev. Lett. [**54**]{}, 861 (1985).
This is an example of the general phenomenon of symmetry-breaking bifurcations of orbits. For a systematic discussion, see M. A. M. de Aguiar and C. P. Malta, Annals of Physics [**180**]{}, 167 (1987); our case corresponds to (b) in their Table I.
P. A. Houle, , PhD thesis, Cornell University, 1998.
P. A. Houle and C. L. Henley, The classical mechanics of a three-spin cluster, in preparation, 1999.
A. M. O. de Almeida, , Cambridge, New York, 1988; see also M. C. Gutzwiller, , Springer, New York, 1990.
As Ref. notes, the third-order expansion of the Hamiltonian near a ground state is the same for our model (in the right coordinates) as for the famous Hénon-Heiles model (M. Hénon and C. Heiles, Astronomical J. [**69**]{}, 73 (1964)). Hence, we see similar behaviors near the limit (e.g. the dependence on excitation energy of the frequency splitting between orbits (b) and (c)).
$A_p(E)$ contains a phase factor, $e^{i\mu/4}$, where $\mu$ is the Maslov index and depends on the topology of the linearized dynamics near the orbit; see J. M. Robbins, Nonlinearity [**4**]{}, 343 (1991). As we do not consider amplitude or phase, the Maslov index is irrelevant for this paper.
Eq. (\[eq:ftospec\]) is for illustration. The square window aggravates artifacts of the Fourier transform which could be reduced by using a different window function (See Ref. ).
M. Baranger, M. R. Haggerty, B. Lauritzen, D. C. Meredith, and D. Provost, Chaos [**5**]{}, 261 (1995).
G. S. Ezra, J. Chem. Phys. [**104**]{}, 26 (1996).
S. L. [Marple Jr.]{}, , Prentice-Hall, 1987.
J. Main, V. A. Mandelshtam, and H. S. Taylor, Phys. Rev. Lett. [**78**]{}, 4351 (1997).
In fact, the [*classical dynamics*]{} of our system do scale if one rescales the spin length $S$ simultaneously. However, in contrast to the mentioned scaling systems systems, $S$ in a spin system is not just a numerical parameter, but is a discrete quantum number. In effect, constructing an orbit spectrum by varying $S$ is a mixture of scaled-energy spectroscopy and inverse-$\hbar$ spectroscopy (see J. Main, C. Jung, and H. S. Taylor, J. Chem. Phys. [**107**]{}, 6577 (1997).) Such an approach would give poor energy resolution in our system, since we can perform diagonalizations only for discrete values of $S$ in a limited range.
N. Wu, , (Springer, 1997).
L. Xiao and M. E. Kellman, J. Chem. Phys. [**90**]{}, 6086 (1989), Due to this slowness, [*any*]{} convention to define a Poincaré section gives a topologically equivalent picture. Such pictures for the AFM or FM limits are presented in Ref. , Figures 4.17 – 4.19, and in Ref. .
We expect $I$ could be calculated explicitly by some perturbation theory; it is closely related to the approximate invariants provided by the Gustavson normal form construction, or by averaging methods: see [*Perturbation methods, bifurcation theory, and computer algebra*]{} by R. H. Rand and D. Armbruster (Springer, New York, 1987).
Intriguingly, Figure \[fig:flows\] is also equivalent to the phase sphere in Figure 1 of Ref. , for an antiferromagnetically coupled cluster of [*four*]{} spins (or four sublattices of spins). The figures look different until one remembers that in Ref. , all points related by twofold rotations around the $x$, $y$, or $z$ axes are to be identified; really the sphere of Ref. has just two distinct octants, equivalent to the “northern” and “southern” hemispheres of Fig. \[fig:flows\] in this paper. The special points labeled $T_{\pm}$, $C_{x,y,z}$, and $S_{a,b,c}$ in Ref. , correspond respectively to the two “poles”, plus the three stable and three unstable points on the “equator”, in Fig. \[fig:flows\].
Observe that the “north pole” symmetry axis in Figure \[fig:flows\] is the $z$ axis. On the other hand, the axis of the polar coordinates $(P_x,\Psi_x)$ in Eq. (\[eq:polyad\]) is the $x$ axis and passes through two stationary points of types (b) and (c) in the figure. The equator in Figure \[fig:flows\] includes all motions with $\Psi_x=0$ or $\pi$, i.e. the oscillations are in phase.
D. Loss, D. P. DiVincenzo, and G. Grinstein, Phys. Rev. Lett. [**69**]{}, 3232 (1992); J. von Delft and C. L. Henley, Phys. Rev. Lett. [**69**]{}, 3236 (1992).
This is plausible in view of the topological equivalence of the reduced dynamics, suggested by footnote . However, this remains a speculation since we have [*not*]{} performed a microscopic calculation of the topological phase for the three-spin problem, analogous to the calculation in Ref. . Note that, in comparing to that work, [*all*]{} singlet levels for a given spin length in Ref. correspond to just [*one*]{} polyad cluster of levels in the present problem.
In terms of the two oscillators, $P=2S+1$ implies that polyads are allowed only with a net even number of quanta; we do not yet understand this constraint.
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---
author:
- |
S. V. Borodachov\
Department of Mathematics, Towson University,\
Towson, MD, 21252, USA\
e-mail: [sborodachov@towson.edu]{}
- |
N. Bosuwan\*\
Department of Mathematics, Vanderbilt University,\
Nashville, TN, 37240, USA\
e-mail: [nattapong.bosuwan@vanderbilt.edu]{}
title: 'Asymptotics of discrete Riesz $d$-polarization on subsets of $d$-dimensional manifolds'
---
Abstract {#abstract .unnumbered}
========
We prove a conjecture of T. Erdélyi and E.B. Saff, concerning the form of the dominant term (as $N\to \infty$) of the $N$-point Riesz $d$-polarization constant for an infinite compact subset $A$ of a $d$-dimensional $C^{1}$-manifold embedded in $\mathbb{R}^{m}$ ($d\leq m$). Moreover, if we assume further that the $d$-dimensional Hausdorff measure of $A$ is positive, we show that any asymptotically optimal sequence of $N$-point configurations for the $N$-point $d$-polarization problem on $A$ is asymptotically uniformly distributed with respect to $\mathcal H_d|_A$.
These results also hold for finite unions of such sets $A$ provided that their pairwise intersections have $\mathcal H_d$-measure zero.
Introduction {#h}
============
Let $\omega_N=\{x_1,\ldots, x_N\}$ denote a configuration of $N$ (not necessarily distinct) points in the $m$-dimensional Euclidean space $\mathbb{R}^m$ (such configurations are known as multisets, however, we will still use the word *configurations*). For an infinite compact set $A\subset \mathbb{R}^m$ and $s>0,$ we define the following quantities: $$M^{s}(\omega_N;A):=\min_{y\in A} \sum_{i=1}^N\frac{1}{|y-x_i|^{s}}$$ and $$\label {e1}
M^s_N(A):=\max_{\substack{\omega_N \subset A \\ \#\omega_N=N}} M^{s}(\omega_N;A),$$ where $\#\omega_N$ stands for the cardinality of the multiset $\omega_N$. Following [@Saff], we will call $M^{s}_N(A)$ the *$N$-point Riesz $s$-polarization constant of $A.$* The quantity $M^{s}_N(A)$ is also known as the *$N^{\textup{th}}$ $L_s$ Chebyshev constant* of the set $A$ (cf. e.g. [@AmbrusBallErdelyi]). We will call an $N$-point configuration $\omega_N\subset A$ [*optimal for $M^s_N(A)$*]{} if it attains the maximum on the right-hand side of (\[e1\]).
It is not difficult to verify that for a fixed vector $\textbf{x}_N:=(x_1,\ldots,x_N)$ in $A^{N}$ (the $N$-th Cartesian power of $A$), the potential function $f(y):=\sum_{i=1}^N |y-x_i|^{-s}, s>0,$ is lower semi-continuous in $y$ on the set $A$ and the function $g(\textbf{x}_N):=M^{s}(\textbf{x}_N;A), s>0,$ is upper semi-continuous in $\textbf{x}_N$ on $A^N.$ So, the function $f(y)$ attains its minimum on $A$ and the function $g(\textbf{x}_N)$ attains its maximum on $A^N$; i.e. an optimal configuration in (\[e1\]) exists when $A$ is an infinite compact set.
The $N$-point Riesz $s$-polarization constant was earlier considered by M. Ohtsuka in [@Ohtsuka]. In particular, he showed that for any infinite compact set $A\subset \mathbb{R}^{m}$, the following limit, called the *Chebyshev constant* of $A$, exists as an extended real number: $$\label {e2}
\mathcal{M}^{s}(A):=\lim_{N \rightarrow \infty} \frac{M^{s}_N(A)}{N}.$$ Moreover, he showed that $\mathcal{M}^{s}(A)\geq W^s(A)$, where $W^s(A)$ is the Wiener constant of $A$ corresponding to the same value of $s$. Later, Chebyshev constants were studied in [@FarkasNagy] and [@Revesz1] and used to study the so-called *rendezvous* or *average numbers* in [@Revesz2] and [@Revesz1]. In particular, it follows from [@FarkasNagy Theorem 11] that $\mathcal M^s(A)=W^s(A)$ whenever the maximum principle is satisfied on $A$ for the Riesz $s$-potential. More information on the properties of the Wiener constant can be found, for example, in the book [@Lan1972].
The optimality of $N$ distinct equally spaced points on the circle for the Riesz $s$-polarization problem was proved by G. Ambrus in [@Ambrus] and by G. Ambrus, K. Ball, and T. Erd[é]{}lyi in [@AmbrusBallErdelyi] for $s=2$. T. Erd[é]{}lyi and E.B. Saff [@Saff] established this for $s=4$. For arbitrary $s>0$, this result was proved by D.P. Hardin, A.P. Kendall, and E.B. Saff [@HardinKendallSaff] (paper [@Nikolov] earlier established this result for $N=3$). Some problems closely related to polarization were considered in [@Nikolov1].
Let $\mathcal{H}_d$ be the $d$-dimensional Hausdorff measure in $\mathbb{R}^{m}$ normalized so that the copy of the $d$-dimensional unit cube embedded in $\R^m$ has measure $1$. The inequality $\mathcal{M}^{s}(A)\geq W^s(A)$ implies that on any infinite compact set $A$ of zero $s$-capacity (i.e., when $W^s(A)=\infty$), the limit $\mathcal{M}^s(A)$ is infinite. This means that the $N$-point Riesz $s$-polarization constant $M_N^{s}(A)$ grows at a rate faster than $N.$ In particular, it was proved by T. Erdélyi and E.B. Saff [@Saff Theorem 2.4] that for a compact set $A$ in $\mathbb{R}^m$ of positive $d$-dimensional Hausdorff measure, one has $M^{d}_N(A)=O(N \ln N)$, $N\to\infty$, and $M^s_N(A)=O(N^{s/d})$, $N\to\infty$, for every $s>d$. The order estimate for $s=d$ is sharp when $A$ is contained in a $d$-dimensional $C^1$-manifold and the order estimate for $s>d$ is sharp when $A$ is $d$-rectifiable (see [@Saff Theorem 2.3]). We remark that the case $d=1$ of these order estimates when $A$ is a circle was obtained in [@AmbrusBallErdelyi].
Furthermore, when $A$ is the unit ball $\mathbb{B}^{d}$ in $\mathbb{R}^{d}$ or the unit sphere $\mathbb{S}^{d}$ in $\mathbb{R}^{d+1}$, paper [@Saff] proves that $$\label{eq2}
\lim_{N \rightarrow \infty} \frac{M^{d}_N(\mathbb{B}^{d})}{N \ln N}=1, \quad d\geq 1,$$ and $$\label{eq1}
\lim_{N \rightarrow \infty} \frac{M^{d}_N(\mathbb{S}^d)}{N \ln N}=\frac {\beta_d}{\mathcal H_d(\mathbb{S}^d)}, \quad d\geq 2,$$ where $\beta_d$ denotes the volume of the $d$-dimensional unit ball $\mathbb {B}^d$.
When $A$ is an infinite compact subset of a $d$-dimensional $C^1$-manifold, T. Erdélyi and E.B. Saff [@Saff] also show that $$\label {lower}
\liminf\limits_{N\to\infty}{\frac {M^d_N(A)}{N\ln N}}\geq \frac {\beta_d}{\mathcal H_d(A)}$$ and conjecture that the limit of the sequence on the left-hand side of exists and equals the right-hand side.
Another interesting fact established in [@Saff] is that $M^s_N(\mathbb B^d)=N$ for every $N\geq 1$ and $0<s\leq d-2$ (the maximum principle does not hold for the Riesz $s$-potential in the case $0<s<d-2$).
A more detailed review of results on polarization can be found, for example, in the papers [@AmbrusBallErdelyi], [@Saff], [@FarkasNagy], and [@Revesz1].
The polarization problem is related to the discrete minimal Riesz energy problem described below. For a set $X_N=\{x_1,\ldots,x_N\}$ of $N\geq 2$ pairwise distinct points in $\mathbb{R}^{m},$ we define its Riesz $s$-energy by $$E_s(X_N):=\sum_{1 \leq j \not=k \leq N}\frac{1}{|x_j-x_k|^s},$$ and the *minimum $N$-point Riesz $s$-energy* of a compact set $A\subset \mathbb{R}^{m}$ is defined as $$\mathcal{E}_s(A,N):=\min_{\substack{X_N \subset A \\ \# X_N=N}}E_s(X_N).$$
D.P. Hardin and E.B. Saff proved in [@HardinSaff2005] (see also [@HarSaf2004]) that if $A$ is an infinite compact subset of a $d$-dimensional $C^{1}$-manifold embedded in $\mathbb{R}^{m}$ (see Definition \[D1\]), then[^1] $$\label{eq3}
\lim_{N \rightarrow \infty} \frac{\mathcal{E}_d(A,N)}{N^{2} \ln N}=\frac{\beta_d}{\mathcal{H}_d(A)}.$$ Furthermore, if $A$ is as in above condition and $\mathcal H_d(A)>0$, then for any sequence $X_N=\{x_{k,N}\}_{k=1}^N$, $N\in \NN$, of asymptotically $d$-energy minimizing $N$-point configurations in $A$ in the sense that $$\lim_{ N \rightarrow \infty} \frac{E_d(X_N)}{\mathcal E_d(A,N)}=1,$$ we have $$\label {eqq}
\frac{1}{N}\sum_{i=1}^N \delta_{x_{i,N}} {\stackrel {*}{\longrightarrow}}\frac{\mathcal{H}_d(\cdot)|_{A}}{\mathcal{H}_d(A)}, \ \ \ N \rightarrow \infty,$$ in the weak$^\ast$ topology of measures (see Section \[n\] for the definition). Here $\delta_x$ denotes the unit point mass at the point $x.$
The dominant term of the minimum $s$-energy on $d$-rectifiable closed sets in $\RR^m$ ($s>d$) as well as relation for asymptotically optimal sequences of $N$-point configurations were obtained in [@HardinSaff2005] and [@BorHarSaf2008]. (In the case $d=1$ these results were earlier established for curves in [@MMRS]).
Relations (\[eq3\]) and (\[eqq\]) have recently been extended by D.P. Hardin, E.B. Saff, and J.T. Whitehouse to the case of $A$ being a finite union of compact subsets of $\mathbb{R}^m$ where each compact set is contained in some $d$-dimensional $C^1$-manifold in $\mathbb{R}^m$($d \leq m$) and the pairwise intersections of such compact sets have $\mathcal{H}_d$-measure zero. These authors observed that the methods of [@MMRS] could be applied (see [@BHS]).
A detailed review of known results on discrete minimum energy problems can be found, for example, in the book [@BHS].
Notation and definitions {#n}
========================
In this section we will mention the main definitions used in the paper. For a subset $K\subset A$, we will denote by $\partial_ A K$ the boundary of $K$ relative to $A$.
We say that a sequence $\{\mu_n\}_{n=1}^{\infty}$ of Borel probability measures in $\RR^m$ converges to a Borel probability measure $\mu$ in the weak$^\ast$ topology of measures (and write $\mu_n {\stackrel {*}{\longrightarrow}}\mu$, $n\to\infty$) if for every continuous function $f:\RR^m\to \RR$, $$\label {eqeq}
\int{f\ \! d\mu_n}\to \int f\ \! d\mu,\ \ \ n\to\infty.$$
\[R2.1\] [ It is well known that to prove when $\mu$ and all the measures $\mu_n$ are supported on a compact set $A\subset \RR^m$, it is sufficient to show that $$\mu_n(K)\to \mu (K),\ \ \ n\to \infty,$$ for every closed subset $K$ of $A$ with $\mu(\partial _A K)=0$. ]{}
It will be convenient to use throughout this paper the following definition of a $d$-dimensional $C^{1}$-manifold in $\mathbb{R}^m$ (see, for example, [@Spivak Chapter 5]).
\[D1\] [A set $W \subset \mathbb{R}^m$ is called a *$d$-dimensional $C^1$-manifold embedded in $\mathbb{R}^m$*, $d\leq m$, if every point $y\in W$ has an open neighborhood $V$ relative to $W$ such that $V$ is homeomorphic to an open set $U\subset \mathbb{R}^d$ with the homeomorphism $f:U\to V$ being a $C^1$-continuous mapping and the Jacobian matrix $$J^{f}_{x}:=\begin{bmatrix} ~\nabla f_1(x) ~ \\ \ldots \\ \nabla f_m(x)\end{bmatrix}$$ of the function $f$ having rank $d$ at any point $x\in U$ (here $f_1,\ldots,f_m$ denote the coordinate mappings of $f$). ]{}
Finally, we call a sequence $\{\omega_N\}_{N=1}^{\infty}$ of $N$-point configurations on $A$ [*asymptotically optimal for the $N$-point $d$-polarization problem on $A$*]{} if $$\lim\limits_{N\to\infty}{\frac {M^d(\omega_N;A)}{M^d_N(A)}}=1.$$
Main results
============
In this paper we extend relation (\[eq2\]) to the case of an arbitrary infinite compact set in $\R^d$ and relation (\[eq1\]) to the case when $A$ is an infinite compact subset of a $d$-dimensional $C^{1}$-manifold embedded in $\mathbb{R}^m$ where $m>d$ or a finite union of such sets provided that their pairwise intersections have $\mathcal H_d$-measure zero. Under additional assumption that $\mathcal{H}_{d}(A)>0,$ we also determine the weak$^\ast$ limiting distribution of asymptotically optimal $N$-point configurations for the $N$-point $d$-polarization problem on these classes of sets. Relation below proves the conjecture made by T. Erdélyi and E.B. Saff in [@Saff Conjecture 2].
\[bavv\] Let $A=\cup_{i=1}^l A_i$ be an infinite subset of $\mathbb{R}^m$, where each set $A_i$ is a compact subset contained in some $d$-dimensional $C^1$-manifold in $\mathbb{R}^m,$ $d\leq m,$ and $\mathcal H_d(A_i\cap A_j)=0$, $1\leq i<j\leq l$. Then $$\label {add_17}
\lim_{N \rightarrow \infty} \frac{M^{d}_N(A)}{N \ln N} =\frac{\beta_d}{\mathcal{H}_d{(A)}}.$$ Furthermore, under an additional assumption that $\mathcal{H}_{d}(A)>0,$ if $\omega_N=\{x_{i,N}\}_{i=1}^{N}$, $N\in \NN$, is a sequence of asymptotically optimal configurations for the $N$-point $d$-polarization problem on $A$, then in the weak$^\ast$ topology of measures we have $$\label{eq7}
\frac{1}{N} \sum_{i=1}^N\delta_{x_{i,N}} {\stackrel {*}{\longrightarrow}}\frac{\mathcal{H}_{d}(\cdot)|_{A}}{\mathcal{H}_{d}(A)}, \ \ \ N \rightarrow \infty.$$
\[ujkiol\]
To establish Theorem \[bavv\] we will use the result proved in Section \[4\], Lemma \[add\_7\], and Proposition \[add\_log\_energy\].
Upper estimate {#4}
==============
For a compact set $A\subset \RR^m$, define the quantity $$\label{eq5}
\overline{\alpha}_d(A;\varepsilon):=\sup_{0<r\leq \varepsilon} \sup_{x\in A} \frac{\mathcal{H}_d(B(x,r) \cap A)}{\beta_d r^d}.$$ Let also $$\UL h_d(A):=\liminf\limits_{N\to\infty}{\frac {M^d_N(A)}{N\ln N}} \ \ \ {\rm and}\ \ \ \OL h_d(A):=\limsup\limits_{N\to\infty}{\frac {M^d_N(A)}{N\ln N}}.$$ The main result of this section is given below.
\[upper\] Let $d,m\in \NN$, $d\leq m$, and $A\subset \RR^m$ be a compact set with $0<\mathcal H_d(A)<\infty$, containing a closed subset $B$ of zero $\mathcal H_d$-measure such that every compact subset $K\subset A\setminus B$ satisfies $$\label {3q}
\lim\limits_{\epsilon\to 0^+}\OL \alpha_d(K;\epsilon)\leq 1.$$ Then $$\label {add_eq3}
\OL h_d(A)\leq \frac {\beta_d}{\mathcal H_d(A)}.$$ If an equality holds in , then any infinite sequence $\omega_N=\{x_{k,N}\}_{k=1}^N$, $N\in \mathcal N\subset \NN$, of configurations on $A$ such that $$\label{eq6}
\lim_{N \rightarrow \infty\atop N\in \mathcal N} \frac{M^{d}(\omega_N;A)}{N \ln N} =\frac{\beta_d}{\mathcal{H}_d{(A)}}$$ satisfies $$\label {11a}
\frac{1}{N}\sum_{i=1}^N \delta_{x_{i,N}} {\stackrel {*}{\longrightarrow}}\frac{\mathcal{H}_d(\cdot)|_{A}}{\mathcal{H}_d(A)}, \ \ \ \mathcal N\ni N \rightarrow \infty.$$
We precede the proof of Theorem \[upper\] with the following auxiliary statements.
\[add\_up\] Let $0<R\leq r$, $D\subset \RR^m$ be a compact set with $\mathcal H_d(D)<\infty$, $d\in \NN$, $d\leq m$, and $y\in D$. Then $$\int\limits_{D\setminus B(y,R)}{\frac {d\mathcal H_d(x)}{\left|x-y\right|^d}}\leq r^{-d} \mathcal{H}_d(D)+\beta_d \overline{\alpha}_d(D;r) \ln \(\frac {r}{R}\)^d.$$
We have $$\begin{aligned}
\label{eq11}
\int\limits_{D\setminus B(y,R)}{\frac {d\mathcal H_d(x)}{\left|x-y\right|^d}} &= \int_{0}^{\infty} \mathcal{H}_d\{x \in D\setminus B(y,R) : |x-y|^{-d}>t\} dt\notag\\
&= \int_{0}^{\infty} \mathcal{H}_d\{x\in D\setminus B(y,R) : {t^{-1/d}} > |x-y|\} dt \notag\\
&\leq \int_{0}^{R^{-d}} \mathcal{H}_d (B(y, t^{-1/d})\cap D) dt \notag\\
&\leq r^{-d} \mathcal{H}_d(D)+ \int_{r^{-d}}^{R^{-d}} \mathcal{H}_d ( B(y,t^{-1/d}) \cap D) dt \notag\\
&\leq r^{-d} \mathcal{H}_d(D)+ \beta_d\int_{r^{-d}}^{R^{-d}} \overline{\alpha}_d(D;r) t^{-1} dt \notag\\
&=r^{-d} \mathcal{H}_d(D)+\beta_d \overline{\alpha}_d(D;r) \ln \(\frac {r}{R}\)^d,\nonumber\end{aligned}$$ which completes the proof.
\[add\_10\] Let $d,m\in \NN$, $d\leq m$, and $A\subset \RR^m$ be a compact set with $0<\mathcal H_d(A)<\infty$, containing a closed subset $B$ of zero $\mathcal H_d$-measure such that every compact subset of the set $A\setminus B$ satisfies . Then for any infinite sequence $\{\omega_N\}_{N\in \mathcal N}$, $\mathcal N\subset \NN$, of $N$-point configurations on the set $A$, the inequality $$\label {add_e1}
\frac {\mathcal H_d(K)}{\beta_d}\cdot\liminf\limits_{N\to\infty\atop N\in \mathcal N}\frac {M^d(\omega_N;A)}{N\ln N}\leq \liminf\limits_{N\to\infty\atop N\in \mathcal N}\frac {\# (\omega_N \cap K)}{N}$$ holds for any compact subset $K\subset A$ with $\mathcal H_d(K)>0$ and $\mathcal H_d(\partial _A K)=0$.
Without loss of generality, we can assume that $B\neq \emptyset$ since in the case $B=\emptyset$ we can also use as $B$ any non-empty compact subset of $A$ with $\mathcal H_d(B)=0$.
Let $x_{1,N},\ldots,x_{N,N}$ be the points in the configuration $\omega_N$, $N\in \mathcal N$, and let $K\subset A$ be any compact subset of positive $\mathcal H_d$-measure such that $\mathcal H_d(\partial _A K)=0$. Denote $$K_\rho:=\{x\in K : {\rm dist}(x,B\cup \partial _A K)\geq \rho\},\ \ \ \rho>0.$$ Choose an arbitrary number $\rho>0$ such that $\mathcal H_d(K_{2\rho})>0$. Let $r>0$ be any number such that $2\beta_d r^d<\mathcal H_{d}(K_{2\rho})$. For each $j=1,\ldots,N$, define the set $$\mathcal D_{j,N}:=K_{2\rho}\setminus B(x_{j,N},rN^{-1/d}) \ \ \ {\rm and \ let}\ \ \ \mathcal D_N:=\bigcap \limits_{j=1}^{N}{\mathcal D_{j,N}}.$$ Notice that ${\rm dist}(K_{2\rho},K\setminus K_\rho)\geq \rho>0$. Furthermore, ${\rm dist}(K_{2\rho},A\setminus K)>0$. Indeed, if there were sequences $\{x_n\}$ in $K_{2\rho}$ and $\{y_n\}$ in $A\setminus K$ such that $\left|x_n-y_n\right|\to 0$, $n\to\infty$, then by compactness of $K_{2\rho}$ and $A$ there would exist subsequences $\{x_{n_k}\}$ and $\{y_{n_k}\}$ having the same limit $z\in K_{2\rho}$. Since $\{y_{n_k}\}\subset A\setminus K$ the point $z$ must belong to $\partial _A K$, which contradicts to the definition of the set $K_{2\rho}$. Thus, we have $$h:={\rm dist}(K_{2\rho},A\setminus K_\rho)=\min\{{\rm dist}(K_{2\rho},K\setminus K_\rho),{\rm dist}(K_{2\rho},A\setminus K)\}>0.$$ Choose $N\in \mathcal N$ to be such that $rN^{-1/d}<h$ and $\OL \alpha_d(K_\rho;rN^{-1/d})\leq 2$ (such $N$ exists since $K_\rho$ is a compact subset of $A\setminus B$, and by assumption, satisfies $\lim _{N\to\infty} \OL \alpha_d(K_\rho;r N^{-1/d})\leq 1$). Then $$\mathcal H_d(\mathcal D_N)=\mathcal H_d\(K_{2\rho}\setminus \bigcup_{j=1}^{N}{B(x_{j,N},rN^{-1/d})}\)$$ $$=\mathcal H_d\(K_{2\rho}\setminus \bigcup_{x_{j,N}\in K_\rho}{B(x_{j,N},rN^{-1/d})}\)$$ $$\geq \mathcal H_d(K_{2\rho})-\sum\limits_{x_{j,N}\in K_\rho}{\mathcal H_d\(K_\rho\cap B(x_{j,N},rN^{-1/d})\)}$$ $$\geq \mathcal H_d(K_{2\rho})-\beta_d r^d \frac {\# (\omega_N \cap K_\rho)}{N}\cdot \OL \alpha_d(K_\rho;rN^{-1/d})$$ $$\geq \mathcal H_d(K_{2\rho})-\beta_d r^d \OL \alpha_d(K_\rho;rN^{-1/d})\geq \mathcal H_d(K_{2\rho})-2\beta_d r^d=:\gamma _{r,\rho}>0.$$ Let $\widetilde {\mathcal D}_{j,N}:=K_\rho\setminus B(x_{j,N},rN^{-1/d})$. Then $$M^d(\omega_N;A)=\min\limits_{x\in A}\sum\limits_{j=1}^{N}{\frac {1}{\left|x-x_{j,N}\right|^d}}$$ $$\leq \frac {1}{\mathcal H_d(\mathcal D_N)}\sum\limits_{j=1}^{N}{\ \int\limits_{\mathcal D_N}{\frac {d\mathcal H_d(x)}{\left|x-x_{j,N}\right|^d}}}
\leq\frac {1}{\gamma_{r,\rho}}\sum\limits_{j=1}^{N}\ \int\limits_{\mathcal D_{j,N}}{{\!\!\frac {d\mathcal H_d(x)}{\left|x-x_{j,N}\right|^d}}}$$ $$\leq\frac {1}{\gamma_{r,\rho}}\(\sum\limits_{x_{j,N}\in K_\rho}\ \int\limits_{\widetilde {\mathcal D}_{j,N}}{\!\!\frac {d\mathcal H_d(x)}{\left|x-x_{j,N}\right|^d} }+\!\!\sum\limits_{x_{j,N}\in A\setminus K_\rho}{\ \int\limits_{\mathcal D_{j,N}}{\!\!\frac {d\mathcal H_d(x)}{\left|x-x_{j,N}\right|^d}}}\).$$ Taking into account Lemma \[add\_up\] with $R=rN^{-1/d}$ and $D=K_\rho$ and the fact that ${\rm dist}(\mathcal D_{j,N},A\setminus K_\rho)\geq {\rm dist}(K_{2\rho},A\setminus K_\rho)= h>0$, we will have $$M^d(\omega_N;A)\leq \frac {1}{\gamma_{r,\rho}} \Bigg( \# (\omega_N\cap K_\rho)\(\frac {\mathcal H_d(K_\rho)}{r^d}+\beta_d\OL \alpha_d(K_\rho;r)\ln N\)$$ $$+\sum\limits_{x_{j,N}\in A\setminus K_\rho}\frac {\mathcal H_d(\mathcal D_{j,N})}{h^d} \Bigg).$$ Consequently, $$\label {add_eq2}
\frac {M^d(\omega_N;A)}{N\ln N}\leq \frac {1}{\gamma_{r,\rho}}\(\frac {\# (\omega_N\cap K_\rho)}{N}\(\frac {\mathcal H_d(K_\rho)}{r^d\ln N}+\beta_d\OL \alpha_d(K_\rho;r)\)+\frac {\mathcal H_d(A)}{h^d\ln N} \).$$ Passing to the lower limit in we will have $$\tau:=\liminf\limits_{N\to\infty\atop N\in \mathcal N}{\frac {M^d(\omega_N;A)}{N\ln N}}
\leq \frac {\beta_d \OL \alpha_d(K_\rho;r)}{\mathcal H_d(K_{2\rho})-2\beta_d r^d}\liminf\limits_{N\to\infty\atop N\in \mathcal N}{\frac {\# (\omega_N\cap K_\rho)}{N}}.$$ Letting $r\to 0$ and taking into account and the fact that $K_\rho \subset K$, we will have $$\tau\leq \frac {\beta_d}{\mathcal H_d(K_{2\rho})}\liminf\limits_{N\to\infty\atop N\in \mathcal N}{\frac {\# (\omega_N\cap K_\rho)}{N}}\leq \frac {\beta_d}{\mathcal H_d(K_{2\rho})}\liminf\limits_{N\to\infty\atop N\in \mathcal N}{\frac {\# (\omega_N\cap K)}{N}}.$$ Since $\lim\limits_{\rho \to 0^+}\mathcal H_{d}(K_{2\rho})=\mathcal H_d(K\setminus (B\cup \partial_ A K))=\mathcal H_d(K)$, we finally have $$\tau\leq \frac {\beta_d}{\mathcal H_d(K)}\liminf\limits_{N\to\infty\atop N\in \mathcal N}{\frac {\# (\omega_N\cap K)}{N}},$$ which implies .
[**Proof of Theorem \[upper\].**]{} Let $\mathcal N_0\subset \NN$ be an infinite subset such that $$\OL h_d(A)=\lim\limits_{N\to\infty\atop N\in \mathcal N_0}\frac {M^d_N(A)}{N\ln N}.$$ Let $\{\OL\omega_N\}_{N\in \mathcal N_0}$ be a sequence of $N$-point configurations on $A$ such that $M^d_N(A)=M^d(\OL\omega_N;A)$, $N\in \mathcal N_0$. Then applying Lemma \[add\_10\] with $K=A$, we will have $$\OL h_d(A)=\lim\limits_{N\to\infty\atop N\in \mathcal N_0}\frac {M^d(\OL\omega_N;A)}{N\ln N}\leq \frac {\beta_d}{\mathcal H_d(A)}\liminf\limits_{N\to\infty\atop N\in \mathcal N_0}{\frac {\# (\overline{\omega}_N\cap A)}{N}}=\frac {\beta_d}{\mathcal H_d(A)}$$ and inequality follows.
Assume now that $\OL h_d(A)=\beta_d \mathcal H_d(A)^{-1}$ and let $\{\omega_N\}_{N\in \mathcal N}$, $\mathcal N\subset \NN$, be any infinite sequence of $N$-point configurations on $A$ satisfying . For any closed subset $D\subset A$ with $\mathcal H_d(D)>0$ and $\mathcal H_d(\partial _A D)=0$, by Lemma \[add\_10\] we have $$\label {add_q}
\liminf\limits_{N\to\infty\atop N\in \mathcal N}\frac {\# (\omega_N \cap D)}{N}\geq \frac {\mathcal H_d(D)}{\beta_d}\lim\limits_{N\to\infty\atop N\in \mathcal N}{\frac {M^d(\omega_N;A)}{N\ln N}}= \frac {\mathcal H_d(D)}{\mathcal H_d(A)}.$$
Let now $P\subset A$ be any closed subset of zero $\mathcal H_d$-measure. Show that $$\label {add_p}
\lim\limits_{N\to \infty\atop N\in \mathcal N}{\frac {\# (\omega_N \cap P)}{N}}=0.$$ If $P=\emptyset$, then holds trivially. Let $P\neq \emptyset$. Since $\mathcal H_d(A)<\infty$, for every $\epsilon>0$, there are at most finitely many numbers $\delta>0$ such that the set $P[\delta]:=\{x\in A : {\rm dist}(x,P)=\delta\}$ has $\mathcal H_d$-measure at least $\epsilon$. This implies that there are at most countably many numbers $\delta>0$ such that $\mathcal H_d(P[\delta])>0$. Denote also $P_\delta=\{x\in A : {\rm dist}(x,P)\geq \delta\}$, $\delta>0$. Then there exists a positive sequence $\{\delta_n\}$ monotonically decreasing to $0$ such that every set $\partial _A P_{\delta_n}\subset P[\delta_n]$ has $\mathcal H_d$-measure zero. Since $P_{\delta_n}$ is closed and $\mathcal H_d(P_{\delta_n})>0$ for every $n$ greater than some $n_1$, in view of , we have $$\liminf\limits_{N\to\infty\atop N\in \mathcal N}\frac {\# (\omega_N \cap (A\setminus P))}{N}\geq \liminf\limits_{N\to\infty\atop N\in \mathcal N}\frac {\# (\omega_N \cap P_{\delta_n})}{N}\geq \frac {\mathcal H_d(P_{\delta_n})}{\mathcal H_d(A)},\ \ \ n>n_1.$$ Since $\mathcal H_d(P_{\delta_n})\to \mathcal H_{d}(A\setminus P)=\mathcal H_d(A)$, $n\to\infty$, we have $$\lim\limits_{N\to \infty\atop N\in \mathcal N}{\frac {\# (\omega_N \cap (A\setminus P))}{N}}=1,$$ which implies .
Since the set $\OL {A\setminus D}$ is also a closed subset of $A$ and $\mathcal H_d(\partial _A (A\setminus D))=\mathcal H_d(\partial _A D)=0$, by and (with $P=\partial_A D$) we have $$\limsup\limits_{N\to\infty\atop N\in\mathcal N}{\frac {\# (\omega_N \cap D)}{N}}=1-\liminf\limits_{N\to\infty\atop N\in\mathcal N}{\frac {\# (\omega_N \cap (A\setminus D))}{N}}$$ $$= 1-\liminf\limits_{N\to\infty\atop N\in\mathcal N}{\frac {\# (\omega_N \cap \OL {A\setminus D})}{N}}\leq 1-\frac {\mathcal H_d(\OL {A\setminus D})}{\mathcal H_d(A)}=\frac {\mathcal H_d(D)}{\mathcal H_d(A)}.$$ Thus, $$\label {15}
\lim\limits_{N\to\infty\atop N\in \mathcal N}{\frac {\# (\omega_N \cap D)}{N}}=\frac {\mathcal H_d(D)}{\mathcal H_d(A)}$$ for any closed subset $D\subset A$ with $\mathcal H_d(D)>0$ and $\mathcal H_d(\partial _A D)=0$. In view of relation also holds when $D\subset A$ is closed and $\mathcal H_d(D)=0$. Then in view of Remark \[R2.1\] we have .$\square$
Auxiliary statements
====================
We will show in this section that for every set $A$ satisfying the assumptions of Theorem \[bavv\], the assumptions of Theorem \[upper\] necessarily hold.
\[Pr1\] Let $A$ be a compact subset of a $d$-dimensional $C^1$-manifold embedded in $\R^m$, $d\leq m$. Then for such a set $A,$ $$\label{eq4}
\lim_{\varepsilon \rightarrow 0^{+}} \overline{\alpha}_d(A;\varepsilon) \leq1.$$
The proof of this statement is given in the Appendix.
\[add\_7\] Let $A=\cup_{i=1}^{l}A_i$, where each set $A_i$ is a compact set contained in some $d$-dimensional $C^1$-manifold in $\RR^m$, $d\leq m$, and $\mathcal H_d(A_i\cap A_j)=0$, $1\leq i< j\leq l$. Then there is a compact subset $B\subset A$ with $\mathcal H_d(B)=0$ such that every compact subset $K\subset A\setminus B$ satisfies $\lim\limits_{\epsilon\to 0^+}{\OL \alpha_d(K;\epsilon)}\leq 1$.
Denote $B:=\bigcup\limits_{1\leq i<j\leq l}{A_i\cap A_j}$. Let $K\subset A\setminus B$ be a compact subset. Then $$\delta_0:=\min\limits_{1\leq i<j\leq l}{{\rm dist}(A_i\cap K,A_j\cap K)}>0.$$ Choose any $\epsilon\in (0,\delta_0)$. Choose also arbitrary $r\in (0,\epsilon]$ and $x\in K$. We have $x\in A_i$ for some $1\leq i\leq l$ and $x\notin A_j$ for every $j\neq i$. Since $r<\delta_0$, we have $B(x,r)\cap K\subset B(x,r)\cap A_i$ and consequently, $$\frac {\mathcal H_d(B(x,r)\cap K)}{\beta_d r^d}\leq \frac {\mathcal H_d(B(x,r)\cap A_i)}{\beta_d r^d}$$ $$\leq \sup\limits_{t\in (0,\epsilon]}{\sup\limits_{y\in A_i}}\frac {\mathcal H_d(B(y,t)\cap A_i)}{\beta_d t^d}=\OL \alpha_d(A_i;\epsilon)\leq \max\limits_{1\leq j\leq l}{\OL \alpha_d(A_j;\epsilon)}.$$ Consequently, $$\label {add_5}
\OL \alpha_d(K;\epsilon)=\sup\limits_{r\in (0,\epsilon]}{\sup\limits_{x\in K}}\frac {\mathcal H_d(B(x,r)\cap K)}{\beta_d r^d}\leq \max\limits_{1\leq j\leq l}{\OL \alpha_d(A_j;\epsilon)}.$$ Since each $A_i$ is a compact subset of a $d$-dimensional $C^1$-manifold, by Proposition \[Pr1\], we have $\lim\limits_{\epsilon\to 0^+}\OL \alpha_d(A_i;\epsilon)\leq 1$, $i=1,\ldots,l$. Then in view of we have $\lim\limits_{\epsilon\to 0^+}\OL \alpha_d(K;\epsilon)\leq 1$.
The following proposition is a part of the result by D.P. Hardin, E.B. Saff, and J.T. Whitehouse mentioned at the end of Section \[h\]. For completeness, we will reproduce its proof.
\[add\_log\_energy\] Let $A=\cup_{i=1}^{l}A_i$, where each $A_i$ is a compact set contained in some $d$-dimensional $C^1$-manifold in $\RR^m$ and $\mathcal H_d(A_i\cap A_j)=0$, $1\leq i< j\leq l$. Then $$\UL g_d(A):=\liminf\limits_{N\to\infty}{\frac {\mathcal E_d(A,N)}{N^2\ln N}}\geq \frac {\beta_d}{\mathcal H_d(A)}.$$
Since every set $A_i$ is a compact subset of a $d$-dimensional $C^1$-manifold, in view of Theorem 2.4 in [@HardinSaff2005], there holds $
\UL g_d(A_i)\geq \beta_d \mathcal H_d(A_i)^{-1},
$ $i=1,\ldots,l$. In view of inequality (34) from Lemma 3.2 in [@HardinSaff2005], we then have $$\UL g_d(A)=\UL g_d\(\bigcup\limits_{i=1}^{l}{A_i}\)\geq \(\sum\limits_{i=1}^{l}\UL g_d(A_i)^{-1}\)^{-1}\!\!\geq \(\frac {1}{\beta_d}\sum\limits_{i=1}^{l}\mathcal H_d(A_i)\)^{-1}\!\!=\frac {\beta_d}{\mathcal H_d(A)},$$ which yields the desired inequality.
Proof of Theorem \[bavv\]
=========================
The proof of the lower estimate in will repeat the proof of inequality (2.9) in [@Saff]. It is known that (see [@Saff], [@FarkasNagy], or [@Revesz1]) for any infinite compact set $A\subset \mathbb{R}^m,$ $$\label{eq15}
M^{s}_N(A) \geq \frac{1}{N-1} \mathcal{E}_s(A,N), \quad N \geq 2,\ \ \ s>0.$$
Then Proposition \[add\_log\_energy\] and inequality (\[eq15\]) give the lower estimate for $M^{d}_N(A)$: $$\liminf_{N \rightarrow \infty} \frac{M_N^d(A)}{N \ln N} \geq\liminf_{N \rightarrow \infty} \frac{\mathcal{E}_d(A,N)}{(N-1)N \ln N} \geq\frac{\beta_d}{\mathcal{H}_d(A)}.$$ Note that if $\mathcal{H}_d(A)=0,$ then $\lim_{N \rightarrow \infty} {M_N^d(A)}/{(N \ln N)}=\infty.$
Now, assume that $\mathcal{H}_d(A)>0.$ In view of Lemma \[add\_7\] and Remark \[ujkiol\], the set $A$ satisfies the assumptions of Theorem \[upper\]. Consequently $$\limsup_{N \rightarrow \infty} \frac{M_N^d(A)}{N \ln N} \leq \frac{\beta_d}{\mathcal{H}_d(A)}.$$ This implies .
Every sequence $\{\omega_N\}_{N=1}^{\infty}$ of $N$-point configurations, which is asymptotically optimal for the $N$-point $d$-polarization problem on $A$ must satisfy with $\mathcal N=\NN$. Since $\OL h_d(A)=\beta_d\mathcal H_d(A)^{-1}$, by Theorem \[upper\] we obtain .
Appendix
========
In this part of the paper we prove Proposition \[Pr1\].
We say that a set $B$ in $\mathbb{R}^m$ is [*bi-Lipschitz homeomorphic to a set $D\subset \mathbb{R}^n$ with a constant $M\geq 1$*]{}, if there is a mapping $\varphi:B\to D$ such that $\varphi(B)=D$ and $$M^{-1}\left|x-y\right|\leq \left|\varphi(x)-\varphi(y)\right|\leq M\left|x-y\right|,\ \ \ x,y\in B.$$
\[lemma1\] Let $U\subset \mathbb{R}^d$ be a non-empty open set and $f:U\to \mathbb{R}^m$, $m\geq d$, be an injective $C^1$-continuous mapping such that its inverse $f^{-1}:f(U)\to U$ is continuous and the Jacobian matrix $$\label{tgfr}
J^{f}_{x}:=\begin{bmatrix} ~\nabla f_1(x) ~ \\ \ldots \\ \nabla f_m(x)\end{bmatrix}$$ of $f$ has rank $d$ at any point $x\in U$. Then for every $\epsilon>0$ and every point $y_0\in f(U)$, there is a closed ball $B$ centered at $y_0$ such that the set $B\cap f(U)$ is bi-Lipschitz homeomorphic to some compact set in $\mathbb{R}^d$ with a constant $1+\epsilon$.
Let $x_0\in U$ be the point such that $f(x_0)=y_0$. Choose any $\epsilon>0$ and let $\delta=\delta (x_0,\epsilon)>0$ be such that $B[x_0,\delta]\subset U$ and $$\left|\nabla f_i(x)-\nabla f_i(x_0)\right|<\epsilon,\ \ \ x\in B[x_0,\delta],\ i=1,\ldots,m.$$ Let $x,y\in B[x_0,\delta]$ be two arbitrary points. Define the function $g_i(t):=f_i(x+t(y-x))$, $t\in [0,1]$. Then there exists $\xi_i\in (0,1)$ such that $$f_i(y)-f_i(x)=g_i(1)-g_i(0)=g'_i(\xi_i)=\nabla f_i(z_i)\cdot (y-x)$$ $$=\nabla f_i(x_0)\cdot (y-x)+\(\nabla f_i(z_i)-\nabla f_i(x_0)\)\cdot (y-x),$$ where $z_i=x+\xi_i(y-x)$, $i=1,\ldots,m$. Since $z_i\in B[x_0,\delta]$, we have $$\left|f_i(y)-f_i(x)-\nabla f_i(x_0)\cdot (y-x)\right|$$ $$=\left|\(\nabla f_i(z_i)-\nabla f_i(x_0)\)\cdot (y-x)\right|\leq \epsilon \left|y-x\right|,\ \ \ \ i=1,\ldots,m,$$ and hence (we treat $x$ and $y$ as vector-columns below), $$\label{eq8}
\left|f(y)-f(x)-J^{f}_{x_0}(y-x)\right|\leq \epsilon\sqrt m \left|y-x\right|, \ \ x,y\in B[x_0,\delta].$$ Since the matrix $J^{f}_{x_0}$ has rank $d$, for every standard basis vector $e_i$ from $\R^d$, there is a vector $v_i\in R^m$ such that $(J^{f}_{x_0})^T v_i=e_i$, $i=1,\ldots,d$, where $(J^{f}_{x_0})^T$ denotes the transpose of the matrix $J^{f}_{x_0}$. Then the $d\times m$ matrix $Z:=\[v_1,\ldots,v_d\]^T$ satisfies $ZJ^{f}_{x_0}=I_d$, where $I_d$ is the $d\times d$ identity matrix. Taking into account we have $$\left|f(y)-f(x)-J^{f}_{x_0}(y-x)\right|\leq \epsilon\sqrt m \left|ZJ^{f}_{x_0}(y-x)\right|$$ $$\leq \epsilon\sqrt m \|Z\| \left|J^{f}_{x_0}(y-x)\right|,\ \ \ x,y\in B[x_0,\delta],$$ where $
\|Z\|:=\max \{\left|Zu\right| : u\in \R^m,\ \left|u\right|=1\}.
$ Consequently, $$\(1-\epsilon \sqrt{m} \|Z\|\)\left|J^{f}_{x_0}(y-x)\right|\leq \left|f(y)-f(x)\right|$$ $$\leq \(1+\epsilon \sqrt{m} \|Z\|\)\left|J^{f}_{x_0}(y-x)\right|,\ \ \
x,y\in B[x_0,\delta].$$
Let $u_1,\ldots,u_d$ be an orthonormal basis in the subspace $H$ of $R^m$ spanned by the columns of the matrix $J^{f}_{x_0}$ and let $D:=\[u_1,\ldots,u_d\]$ be the $m\times d$ matrix with columns $u_1,\ldots, u_d$. Since the columns of $J^{f}_{x_0}$ also form a basis in $H$, there exists an invertible $d\times d$ matrix $Q$ such that $D=J^{f}_{x_0}Q$.
Let $V\subset \R^d$ be the open set such that $\Phi(V)=B(x_0,\delta)$, where $\Phi:\R^d\to \R^d$ is the linear mapping given by $\Phi(v)=Qv$. Since the columns of the matrix $D$ are orthonormal, for every $u,v\in \OL V$, we will have $$\left|f\circ \Phi(u)-f\circ \Phi(v)\right|=\left|f(Qu)-f(Qv)\right|$$ $$\leq \(1+\epsilon \sqrt{m} \|Z\|\)\left|J^{f}_{x_0}Q(u-v)\right|=\(1+\epsilon \sqrt{m} \|Z\|\)\left|D(u-v)\right|$$ $$= \(1+\epsilon \sqrt{m} \|Z\|\)\left|u-v\right|.$$ Similarly, $$\left|f\circ \Phi(u)-f\circ \Phi(v)\right|\geq \(1-\epsilon \sqrt{m} \|Z\|\)\left|u-v\right|,\ \ \ u,v\in \OL V,$$ which implies that for $0<\epsilon <(\sqrt {m}\|Z\|)^{-1}$, the restriction of the mapping $\psi:=f\circ \Phi$ to the set $\OL V$ is a bi-Lipschitz mapping onto the set $f(\Phi(\OL V))=f(B[x_0,\delta])$ with constant $M_\epsilon:=\max\{ 1+\epsilon \sqrt{m} \|Z\|,(1-\epsilon \sqrt {m}\|Z\|)^{-1}$}.
Since $f$ is a homeomorphism of $U$ onto $f(U)$, the set $f(B(x_0,\delta))$ is open relative to $f(U)$. Then there is a closed ball $B$ in $\R^m$ centered at $y_0=f(x_0)$ such that $B\cap f(U)\subset f(B(x_0,\delta))$. Then the set $B\cap f(U)=B\cap f(B[x_0,\delta])$ is bi-Lipschitz homeomorphic (with constant $M_\epsilon$) to the set $$V_1:=\psi^{-1}(B\cap f(U))=\psi^{-1}(B\cap f(B[x_0,\delta])),$$ which is compact in $\R^d$. Since $M_\epsilon\to 1$ as $\epsilon\to 0^+$, the assertion of the lemma follows.
Let $W$ denote the $d$-dimensional $C^1$-manifold that contains $A$ and let $\epsilon>0$ be arbitrary. In view of Definition \[D1\], for every point $x\in W$, there is an open neighborhood $V_x$ of $x$ relative to $W$ which is homeomorphic to an open set $U_x\subset \R^d$ such that the homeomorphism $f:U_x\to V_x$ is a $C^1$-continuous mapping and the Jacobian matrix $J^f_u$ (see the definition $J^{f}_u$ in (\[tgfr\])) has rank $d$ for every $u\in U_x$. There is also a number $\epsilon_x>0$ such that $B(x,\epsilon_x)\cap W\subset V_x$. By Lemma \[lemma1\], there is a number $0<\delta (x)<\epsilon_x/2$ such that the set $B[x,2\delta (x)]\cap W=B[x,2\delta (x)]\cap f(U_x)$ is bi-Lipschitz homeomorphic to a compact set $D_x$ from $\R^d$ with constant $1+\epsilon$. Since $A$ is compact, the open cover $\{B(x,\delta (x))\}_{x\in A}$ has a finite subcover $\{B(x_i,\delta (x_i))\}_{i=1}^{p}$.
Denote $\delta_\epsilon:=\min\limits_{j= {1, \ldots,p}}\delta (x_j)$. Let $x$ be any point in $A$ and $r \in (0,\delta_\epsilon]$. There is an index $i$ such that $x\in B(x_i,\delta (x_i))$. Since $B(x,r)\cap A\subset B[x_i,2\delta (x_i)]\cap W$, the set $B(x,r)\cap A$ is bi-Lipschitz homeomorphic to a set $D_i\subset D_{x_i}$ with constant $1+\epsilon$. If $\varphi:B(x,r)\cap A\to D_i$ denotes the corresponding bi-Lipschitz mapping, we have $D_i\subset B(\varphi(x),(1+\epsilon)r)$. Then $$\mathcal H_d(B(x,r)\cap A)\leq (1+\epsilon)^d \mathcal L_d(D_i)\leq \beta_d r^d(1+\epsilon)^{2d}.$$ Consequently, $$\OL \alpha_d (A;\delta_\epsilon)=\sup\limits_{r\in (0,\delta_\epsilon]}\sup\limits_{x\in A}\frac {\mathcal H_d(B(x,r)\cap A)}{\beta_d r^d}\leq (1+\epsilon)^{2d},$$ which implies that $\lim\limits_{\delta\to 0^+}{\OL \alpha_d (A;\delta)}\leq 1$.
Acknowledgements
================
The authors would like to thank Professor E.B. Saff who introduced this question to the authors and provided useful references and comments on this paper.
[9]{}
G. Ambrus, *Analytic and Probabilistic Problems in Discrete Geometry*; Ph.D. Thesis, University College London, 2009.
G. Ambrus, K. Ball, and T. Erdélyi, *Chebyshev constants for the unit circle*, Bull. Lond. Math. Soc., **45**(2) (2013), 236–248.
S.V. Borodachov, D.P. Hardin, and E.B. Saff, *Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets*, Trans. Amer. Math. Soc., **360** (2008), 1559–1580.
S.V. Borodachov, D.P. Hardin, and E.B. Saff, Minimal Discrete Energy on the Sphere and other Manifolds. Springer (to appear).
T. Erdélyi and E.B. Saff, *Riesz polarization inequalities in higher dimensions*, J. Approx. Theory, **171** (2013), 128–147.
B. Farkas and B. Nagy, *Transfinite diameter, Chebyshev constant and energy on locally compact spaces*, Potential Anal., **28** (2008), 241–260.
B. Farkas and Sz.Gy. Révész, *Rendezvous numbers in normed spaces*, Bull. Aust. Math. Soc., **72**(3) (2005), 423–440.
B. Farkas and Sz.Gy. Révész, *Potential theoretic approach to rendezvous numbers*, Monatsh. Math., **148** (2006), 309–331.
D.P. Hardin, A.P. Kendall, and E.B. Saff, *Polarization optimality of equally spaced points on the circle for discrete potentials*, Discrete Comput. Geom., **50**(1) (2013), 236–243.
D.P. Hardin and E.B. Saff, [*Discretizing manifolds via minimum energy points*]{}, Notices Amer. Math. Soc., **51(10)** (2004), 1186–1194.
D.P. Hardin and E. B. Saff, *Minimal Riesz energy point configurations for rectifiable d-dimensional manifolds*, Adv. Math., **193(1)** (2005), 174–204.
N.S. Landkof, Foundations of Modern Potential Theory. Springer, New York–Heidelberg, 1972.
A. Martinez-Finkelshtein, V. Maymeskul, E.A. Rakhmanov, and E.B. Saff, [*Asymptotics for minimal discrete Riesz energy on curves in $\RR^d$*]{}, Canad. J. Math., **56** (2004), 529–552.
N. Nikolov and R. Rafailov, *On the sum of powered distances to certain sets of points on the circle*, Pacific J. Math., **253(1)** (2011), 157–168.
N. Nikolov and R. Rafailov, *On extremums of sums of powered distances to a finite set of points*, Geom. Dedicata, 1–21, doi: 10.1007/s10711-012-9804-3.
M. Ohtsuka, *On various definitions of capacity and related notions*, Nagoya Math. J., **30** (1967), 121–127.
M. Spivak, Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. Perseus Books Publishing L.L.C., 1965.
[^1]: The results and techniques of [@HardinSaff2005], in fact, yield relations and under a more general assumption that $A$ is a compact set in $\RR^m$ which for every $\epsilon>0$ can be partitioned into finitely many subsets bi-Lipschitz homeomorphic to some sets from $\RR^d$ with constant $1+\epsilon$ and having boundaries relative to $A$ of $\mathcal H_d$-measure zero (see [@BHS]).
|
[to]{}
**Exotic Lepton Searches via Bound State Production at the LHC**
**Neil D. Barrie$^{1,2}$, Archil Kobakhidze$^{1}$, Shelley Liang$^{1}$,\
Matthew Talia$^{1}$ and Lei Wu$^{3}$**\
****
[ *$^{1}$ ARC Centre of Excellence for Particle Physics at the Terascale,*\
*School of Physics, The University of Sydney, NSW 2006, Australia*\
*$^{2}$ Kavli IPMU (WPI), UTIAS, University of Tokyo, Kashiwa, Chiba 277-8583, Japan*\
*$^{3}$ Department of Physics and Institute of Theoretical Physics, Nanjing Normal University,*\
*Nanjing, Jiangsu 210023, China*\
*E-mails: neil.barrie, archil.kobakhidze, shelley.liang, matthew.talia, lei.wu1@sydney.edu.au*\
**]{}
**Abstract**
Heavy long-lived multi-charged leptons (MCLs) are predicted by various new physics models. These hypothetical MCLs can form bound states, due to their high electric charges and long life times. In this work, we propose a novel strategy of searching for MCLs through their bound state productions and decays. By utilizing LHC-8 TeV data in searching for resonances in the diphoton channel, we exclude the masses of isospin singlet heavy leptons with electric charge $|q|\geq 6$ (in units of electron charge) lower than $\sim$1.2 TeV, which are much stronger than the corresponding 8 TeV LHC bounds from analysing the high ionisation and the long time-of-flight of MCLs. By utilising the current 13 TeV LHC diphoton channel measurements the bound can further exclude MCL masses up to $\sim$1.6 TeV for $|q|\geq 6$. Also, we demonstrate that the conventional LHC limits from searching for MCLs produced via Drell-Yan processes can be enhanced by including the contribution of photon fusion processes.
Introduction
============
Heavy long-lived multi-charged leptons (MCLs) are predicted by various extensions of the Standard Model (SM) (for a review, see [@Fairbairn20071]). The charge conservation of such MCLs typically implies that they are long-lived. Theoretically, the observed charge quantisation of known quarks and leptons lacks a fundamental explanation within the SM due to the Abelian nature of the hypercharge gauge symmetry. Various theoretical frameworks have been proposed to accommodate charge quantisation, such as quantum-mechanical monopoles [@Dirac:1931kp], grand unified theories [@Georgi:1974sy] and gauge anomaly cancellation [@Foot:1992ui]. Experimental observation of non-integer and multiply charged particles may thus have important implications for the charge quantisation problem and beyond the Standard Model physics in general.
In LHC experiments, MCLs with lifetimes greater than ${\cal O}(1)$ ns can be observed with the ATLAS and CMS detectors as high-momentum tracks with anomalously large rates of energy loss through ionization. MCLs could also be highly penetrating so that the fraction reaching the muon system of the detectors would be sizeable. Therefore, the muon system could be used to help in identification and in the measurement of the time-of-flight (TOF) of the particles. So far, the ATLAS and CMS collaborations have extensively searched for long-lived MCLs by analysing the anomalously high ionisation and the long TOF to the outer muon system at the LHC [@cms-multicharged; @Chatrchyan:2013oca; @atlas-multicharged]. Based on the 8 TeV dataset, considering Drell-Yan (DY) like signals, the ATLAS collaboration has excluded the MCLs mass range from 50 GeV up to 660, 740, 780, 785, and 760 GeV for integer charges $|q|=2-6$, respectively [@atlas-multicharged]. Similar bounds are obtained by the CMS collaboration. In this case, integer charges $|q|=1-8$ are excluded for masses below 574, 685, 752, 793, 796, 781, 757, and 715 GeV/c$^{2}$, respectively [@Chatrchyan:2013oca]. The non-integer charge analysis for certain charge ranges and masses are also given by the CMS collaboration in [@cms-multicharged].
However, the signal efficiency in searches for the MCLs drops significantly with the increase of the mass and charge of leptons. For example, when the charge is higher than 6, the signal efficiency is expected to be less than 5% [@atlas-multicharged]. Such low efficiencies mean that a different approach is required.
Due to their high charges and long life times, the MCLs are expected to form bound states via the electromagnetic interaction, which will subsequently decay to SM particles. We call these bound states ‘leptonium’. In this paper we propose a new search for the MCLs through their bound state productions and decays at the LHC. To demonstrate our method, we consider vector-like, weak isospin singlet leptons $L$, charged only under $U(1)_{Y}$ with hypercharge (equal to the electric charge) $Y_{L}$. Our proposal can also be applied to other representations of MCLs with modifications in the production and decay calculations.
Constraints on MCLs
===================
In current LHC experiment of searches for MCLs [@cms-multicharged; @Chatrchyan:2013oca; @atlas-multicharged], only pair production of MCLs via DY processes is included. However, these MCLs can also be produced in pairs via the photon fusion process. In Fig. \[pair\], we compare the LHC-8 TeV exclusion limits of MCLs from DY processes (left panel) with those from DY+photon fusion processes (right panel). We compute these processes using `Madgraph 5` [@Alwall:2014hca] with the `NNPDF2.3QED` parton distribution function (PDF) [@Ball2013]. We can see that the DY process alone can exclude the masses of the heavy charged lepton up to 640 and 800 GeV for $|q|=2$ and $3$ respectively [^1]. While the DY+photon fusion process can exclude masses up to 810 and 1000 GeV for $|q|=2$ and $3$ respectively. Therefore, we can conclude that the contribution of the photon fusion process to LHC searches for MCLs is not negligible and should be included in LHC analyses.
Since the multi-charged lepton can contribute to the electroweak observables, we consider the dependence of the $\Delta S,\Delta T,\Delta U$ parameters [@Peskin:1991sw] on the hypercharge and mass of the new lepton. $$\Delta S=\frac{4}{3\pi}\frac{Y_{L}^{2}\sin^{4}\theta_{w}}{1-4m_{L}^{2}/m_{Z}^{2}},~%{equation}
%
\Delta T=\frac{1}{3\pi}\frac{Y_{L}^{2}\tan^{2}\theta_{w}}{1-4m_{L}^{2}/m_{Z}^{2}},~%{equation}
%
\Delta U=\frac{1}{3\pi}\sin^{4}\theta_{w}\frac{m_{Z}^{2}}{m_{L}^{2}}\left(Y_{L}^{2}+\frac{2}{3\sin^{2}\theta_{w}}\right)$$ comparing with the experimental values ($\pm1\sigma$) are [@Baak:2014ora]: $$\Delta S=0.05\pm0.11,~~\Delta T=0.09\pm0.13,~~\Delta U=0.01\pm0.11$$ we find that the $T$ parameter is the most constraining, and require:
$$|Y_{L}|<9\left(\frac{m_{L}}{300~\textrm{GeV}}\right),$$
in the limit $m_{L}\gg m_{Z}$. Hence for MCLs masses above $\sim300$ GeV with $ Y_{L}\leq 9 $, the oblique parameter bounds can be easily satisfied.
The Millikan-type experiments require the abundance of fractionally charged particles be less than $\sim10^{-22}$ per ordinary matter nucleon [@Perl:2004qc]. However, this bound can be easily satisfied by assuming a low reheating temperature after inflation, $T_{rh}<m_{L}$ [@Giudice:2000ex].
The introduction of these leptons would also be expected to have implications for the running of the hypercharge coupling. It is found that the large hypercharge $Y_{L}$ accelerates the running of the effective hypercharge coupling constant such that it hits the Landau pole at the following energy scale: $$\Lambda^{2}=m_{Z}^{2}\exp\left(\frac{1}{8+Y_{L}^{2}}\frac{3\pi}{\alpha(m_{Z}^{2})}\right)$$ which gives $\sim10^{5}-10^{17}$ GeV for hypercharges $Y_{L}=8-3$ at the TeV mass scale, respectively. Although this shows that our perturbative calculations in the energy domain well below the Landau pole are valid, the hypercharge $U(1)$ should probably be embedded into a larger non-Abelian group in order to avoid theoretical inconsistency. Alternatively, the hypercharge $U(1)$ may avoid the Landau pole by developing an ultraviolet non-Gaussian fixed point [@Holdom:2010qs].
MCL Bound State Production at the LHC
=====================================
Besides the production of free MCLs, these hypothetical MCLs can form bound states via the electromagnetic interaction, due to their high electric charges and long life times. Such heavy leptoniums can be copiously produced at the LHC and serve as a smoking gun to probe the MCLs.
In the non-relativistic approximation, the heavy lepton $L$ is described by the Schrödinger equation $$\left(-\frac{\nabla^{2}}{m_{L}}+V(r)\right)\psi=E\psi~,\label{2.1}$$ with the binding Coulomb potential $$V(r)=-\frac{Y_{L}^{2}\alpha}{r}~,\label{2.2}$$ where $\alpha\approx1/128$ is the fine structure constant evaluated at $m_{Z}$ [^2]. In this approximation, the ground state ($n=1,~l=0$) energy is given by: $$E=-\frac{1}{4}m_{L}\left(Y_{L}^{2}\alpha\right)^{2}~,\label{2.3}$$ We also include the leading $\left[\sim\mathcal{O}\left(Y_{L}^{8}\alpha^{4}\right)\right]$ relativistic Breit correction to the binding energy given in Eq. (\[2.3\]): $$\delta E_{{\rm Breit}}=-\frac{1}{2m_{L}}\left(E^{2}-2E\left\langle V\right\rangle +\left\langle V^{2}\right\rangle \right)=-\frac{5}{16}m_{L}\left(Y_{L}^{2}\alpha\right)^{4}~.\label{2.4}$$ The mass of the para-leptonium $\psi_{para}$ ($J^{PC}=0^{-+}$) is then given by: $$m_{\psi_{para}}=2m_{L}+E+\delta E_{{\rm Breit}},\label{2.5}$$ and from this we can derive the mass of the constituent leptons from a resonance observed in the diphoton channel which can be identified as the para-leptonium state, $$m_{L}=m_{\psi_{para}}\left(2-\frac{1}{4}(\alpha Y_{L}^{2})^{2}-\frac{5}{16}(\alpha Y_{L}^{2})^{4}\right)^{-1}~.\label{lepton_mass}$$
The wave function $\psi_{para}(0)$ is given by, $$\left|\psi_{para}(0)\right|^{2}=\left(\frac{1}{\sqrt{4\pi}}\right)^{2}\left|R_{para}(0)\right|^{2},\label{2.8}$$ with the radial part evaluated as: $$\left|R_{para}(0)\right|^{2}=\frac{\left(Y_{L}^{2}\alpha m_{L}\right)^{3}}{2}.\label{2.8b}$$ In addition to para-leptonium, our model predicts a heavier spin-1 ortho-leptonium bound state $\psi_{ortho}$ ($J^{PC}=1^{--}$) with mass [@Efimov:2010ih], $$m_{\psi_{ortho}}\simeq m_{\psi_{para}}\left(1+\frac{7}{12}\frac{\left(Y_{L}^{2}\alpha\right)^{4}}{\left(2-\frac{1}{4}\left(Y_{L}^{2}\alpha\right)^{2}-\frac{5}{16}\left(Y_{L}^{2}\alpha\right)^{4}\right)}\right),\label{ortho}$$ The wave function $\psi_{ortho}(0)$ is equal to $\psi_{para}(0)$ at leading order, since they satisfy the same Schr'’oinger equation.
![[The Feynman diagrams for inelastic (left), semi-elastic (middle) and elastic (right) scattering photoproduction subprocesses.]{}[]{data-label="feyn"}](combined){width="100.00000%"}
At the LHC the spin-0 para-leptonium is predominantly produced via photon-fusion processes in three distinct ways, shown in Fig. \[feyn\]. Photoproduction is dominated by inelastic scattering, which is followed by the semi-elastic and elastic processes in the ratio 63:33:4 [@csaki2016]. Here it should be noted that this relation changes only slightly with the heavy lepton mass since its dependence on the mass is cancelled in the ratio [@Luszczak2015]. So we use this ratio as a good approximation in the numerical calculations that follow. The parton level cross section for photoproduction of the bound state $\psi_{para}$ can be written in terms of the decay width of $\psi_{para}\to\gamma\gamma$, since both the production and decay processes share the same matrix elements. This is given by [@Bertulani:2001zk; @Kats:2012ym]: $$\hat{\sigma}_{\gamma\gamma\rightarrow\psi_{para}}(\hat{s})=8\pi^{2}\frac{\Gamma_{\psi_{para}\rightarrow\gamma\gamma}}{m_{\psi_{para}}}\delta(\hat{s}-m_{\psi_{para}}^{2})\label{2.7}$$ The decay width $\Gamma_{\psi_{para}\rightarrow\gamma\gamma}$ in turn is given in terms of the annihilation cross section of a free lepton-antilepton pair into two photons and the wave function for the leptonium bound state evaluated at the origin: $$\Gamma_{\psi_{para}\rightarrow\gamma\gamma}=\frac{16\pi\alpha^{2}Y_{L}^{4}\left|\psi_{para}(0)\right|^{2}}{m_{\psi_{para}}^{2}}~.$$ Then we can calculate the two photon production and decay cross section by convolution with the parton distribution function (PDF) for the photon in the proton, $f_{\gamma}(x)$: $$\sigma_{\gamma\gamma\rightarrow\psi_{para}\rightarrow\gamma\gamma}=\frac{8\pi^{2}}{sm_{\psi_{para}}}\frac{\Gamma_{\psi_{para}\rightarrow\gamma\gamma}^{2}}{\Gamma_{\psi_{para}}}\int\delta(x_{1}x_{2}-m_{\psi_{para}}^{2}/s)f_{\gamma}(x_{1})f_{\gamma}(x_{2})dx_{1}dx_{2}\label{2.9}$$ where $\sqrt{s}$ is the centre-of-mass energy, and $\Gamma_{\psi_{para}}$ denotes the leptonium total decay width. It should be noted that the para-leptonium state would also decay into $\gamma Z$ and $ZZ$ , the ratio of these signals would be $1:2\tan^{2}\theta_{W}:\tan^{4}\theta_{W}$, for $\gamma\gamma:\gamma Z:ZZ$ respectively ($\theta_{W}$ is the weak mixing angle, $\tan\theta_{W}\approx0.55$). Given the sizeable branching ratio and clean signatures in the diphoton channel, we use the LHC data of diphoton resonance searches to obtain bounds on MCLs. Other searches in the $Z\gamma$ and $ZZ$ channels, are suppressed by the subsequent decay branching ratios of the $ Z $. In the numerical calculations, we use the `NNPDF2.3QED` photon PDF to calculate the hadronic cross section at the LHC.
In Fig. \[plot\_diphoton\], we plot the dependence of the production rate of para-leptonium diphoton decay on the masses of $m_{\psi_{para}}$ and $m_{L}$ for integer hypercharges $Y_{L}=4-8$. We can see that 8 TeV LHC data searching for resonances in the diphoton channel can exclude the mass of a heavy lepton with hypercharge $Y_{L}=6$ and $ 7 $ up to 1.2 and 1.5 TeV respectively, which is much stronger than the corresponding ATLAS and CMS bound from analysing the high ionisation and the long time-of-flight in DY production. Also we checked and found that even with the inclusion of the photon fusion process given in Fig. \[pair\], the bounds derived from the diphoton channel are still stronger. The LHC-13 TeV data for the diphoton channel further excludes the mass of an MCL with $Y_{L}\geq6$ below $\sim$ 1.7 TeV at $95\%$ C.L..
While we have concentrated on the lowest energy bound state, a tower of multiple higher energy excited states is also present. The energy difference between these resonances $\Delta E$ are small enough to be resolved at the LHC. Therefore, in a more accurate treatment, interference between multi-resonance amplitudes must be taken into account. However, we have verified that the width of the lightest para-leptonium resonance is much smaller than $\Delta E$, $\Gamma_{\psi_{para}}/\Delta E \propto (\alpha Y^2_L)^3 \ll 1$ for the $Y_L$ range considered, meaning that the interference effects with heavier resonances can be safely ignored for the purpose of obtaining the conservative bounds.
Besides, a spin-1 ortho-leptonium bound state $\psi_{ortho}$ ($J^{PC}=1^{--}$) can be produced via quark-antiquark annihilation at the LHC and may decay into $W^{+}W^{-}$, $f\bar{f}$ ($f=e,\mu,\tau,u,d,c,s,b,t$) or $3\gamma$ final states. The interference effects between higher resonance states for large $Y_L$ may also affect the observation of $\psi_{ortho}$ at the LHC, which requires a more detailed analysis and is beyond the scope of this work.
Conclusion
==========
Due to their high electric charges and long lifetimes, the heavy exotic leptons can form bound states, which can be copiously produced at the LHC. In this paper, we have proposed a new method using the LHC data of searches for heavy resonances to probe these heavy long-lived multi-charged leptons. The bounds derived from the 8 TeV LHC diphoton results are much stronger than the currently available 8 TeV LHC limits from analysing the high ionisation and long time-of-flight of freely produced exotic leptons. Furthermore, the mass of an isospin singlet heavy lepton with electric charge $Y_{L}\geq6$ can be excluded below $\sim$1.7 TeV from diphoton channel 13 TeV data. Therefore, our proposal will prove to be an invaluable new tool in the search for MCLs in the future LHC experiment.
#### Acknowledgement.
This work is partially supported by the Australian Research Council. NDB is supported in part by World Premier International Research Center Initiative (WPI), MEXT, Japan. LW is also supported in part by the National Natural Science Foundation of China (NNSFC) under grant No. 11705093.
[10]{} M. Fairbairn et. al., Phys. Rep. 438, 63 (2007)
P. A. M. Dirac, Proc. Roy. Soc. Lond. A **133**, 60 (1931). doi:10.1098/rspa.1931.0130
H. Georgi and S. L. Glashow, Phys. Rev. Lett. **32**, 438 (1974). doi:10.1103/PhysRevLett.32.438
R. Foot, H. Lew and R. R. Volkas, J. Phys. G **19**, 361 (1993) Erratum: [\[]{}J. Phys. G **19**, 1067 (1993)[\]]{} [\[]{}hep-ph/9209259[\]]{}. CMS Collaboration, CMS-PAS-EXO-11-090.
S. Chatrchyan *et al.* [\[]{}CMS Collaboration[\]]{}, JHEP **1307**, 122 (2013)
G. Aad *et al.* [\[]{}ATLAS Collaboration[\]]{}, Eur. Phys. J. C **75**, 362 (2015)
J. Alwall [*et al.*]{}, JHEP [**1407**]{} (2014) 079 \[arXiv:1405.0301 \[hep-ph\]\]. R. D. Ball *et al.* [\[]{}NNPDF Collaboration[\]]{}, Nucl. Phys. B **877**, 290 (2013)
M. E. Peskin and T. Takeuchi, Phys. Rev. D **46**, 381 (1992).
M. Baak *et al.* [\[]{}Gfitter Group Collaboration[\]]{}, Eur. Phys. J. C **74** (2014) 3046
M. L. Perl, E. R. Lee and D. Loomba, Mod. Phys. Lett. A **19**, 2595 (2004). G. F. Giudice, E. W. Kolb and A. Riotto, Phys. Rev. D [**64**]{} (2001) 023508 B. Holdom, Phys. Lett. B **694**, 74 (2011)
G. V. Efimov, “QED and ortho-para- positronium mass difference,” arXiv:1010.0144 [\[]{}hep-ph[\]]{}. C. Csáki et. al., Phys. Rev. D **93**, 9 (2016)
Luszczak, M., Szczurek, A. & Royon, C. J. High Energ. Phys. **98** (2015)
C. A. Bertulani and F. Navarra, Nucl. Phys. A **703**, 861 (2002)
Y. Kats and M. J. Strassler, JHEP **1211**, 097 (2012)
Martin, A. D. and Roberts, R. G. and Stirling, W. J. and Thorne, R. S. Eur. Phys. J. **C39**, 155-161 (2005)
CMS Collaboration, CMS-PAS-EXO-12-045
V. Khachatryan *et al.* [\[]{}CMS Collaboration[\]]{}, Phys. Lett. B **767**, 147-170 (2017)
[^1]: Since the 95% C.L. observed limits on MCLs are within 1 TeV in ATLAS analysis at 8 TeV LHC, we will not explicitly state the comparison between DY and DY+photon fusion for charges $|q|=4-6$.
[^2]: Note that since the leptonium Bohr radius, $r_{L}=\frac{2}{m_{L}}\frac{1}{Y_{L}^{2}\alpha}$, is larger than the Compton wavelength of the $Z$-boson, $r_{L}\gtrsim1/m_{Z}$, in deriving the potential (\[2.2\]) we only take into account massless photon exchange.
|
---
author:
- 'Pedro Nevado[^1]'
- 'Diego Porras[^2]'
title: 'Mesoscopic mean-field theory for spin-boson chains in quantum optical systems'
---
Introduction {#intro}
============
Experimental progress in quantum optical setups has opened up new research areas where the controllability of those experimental systems meets the complexity of quantum many body physics. For example, trapped ions and ultracold atoms can now be controlled to the extent in which they emulate the physics of Condensed Matter systems in a way that may lead to the implementation of quantum simulators of many-body models [@Cirac12natphys; @Schneider12rpp]. The latter are devices in which quantum states may be prepared and measured and interactions may be tuned, so as to mimic complex dynamics with a practical and scientific interest in material science. The research field of Analogical Quantum Simulation has emerged to exploit this idea.
Remarkably the physical elements of many of those systems, particularly trapped ions and atoms interacting with photons, can be understood by means of spin-boson models. Effective spins are implemented by two-level systems which can be either atomic transitions of trapped ions [@Leibfried03rmp; @Porras04aprl] or isolated energy levels in solid-state qubits [@Hartmann06natphys; @Schoelkopf08nat]. The bosonic degree of freedom is provided by vibrations in ion Coulomb crystals [@Porras04bprl] or photons in optical or microwave cavities. Bosonic modes may be collective excitations either because of the vibrational couplings in the case of ions, or because of the collective nature of photonic bands in cavities. Finally, a number of different spin-boson couplings may be implemented in those systems, either by laser-induced forces in the case ions, or by the light-matter coupling between qubits and cavities. To focus our discussion, let us define $\sigma_{j}^z = |1\rangle_j \langle 1|-|0\rangle_j \langle 0|$,with $|0\rangle_j, |1\rangle_j$ internal energy levels of an atom (qubit), and annihilation, creation operators for a bosonic mode at site $j$. A natural coupling in quantum optical systems is $$H_{\rm I} = g \sum_j \sigma^z_j \left( a_j + a^\dagger_j \right).
\label{intro}$$ If an additional process couples different bosonic modes, qubit-boson couplings provide us with a mechanism to couple different qubits. This idea is the basis for quantum gate designs [@Cirac00nat; @Schoelkopf08nat; @Peropadre10prl]. In the field of Analogical Quantum Simulation, the use of couplings like (\[intro\]) has been mainly an instrument to get effective models, like quantum magnetic Hamiltonians, in which tracing out bosons results in effective interactions of the form $\sum_{j,k} J_{j,k} \sigma^z_j \sigma^z_k$ [@Porras04aprl; @Deng05pra; @Friedenauer08natphys; @Bermudez09pra; @Islam11natcomm; @Britton12nat]. Also, strongly coupled spin-boson chains have been considered in a few previous works, for example in the context of disordered systems [@Bermudez10njp] and quantum dissipation involving a single spin interacting with a bosonic bath [@Porras08pra].
Quite remarkably, however, interaction (\[intro\]) poses an outstanding many-body model defined on a quantum lattice, with a quantum phase diagram which is intrinsically interesting. It belongs to a family of many-body models previously considered in the context of solid-state physics, in particular, in the description of the cooperative Jahn-Teller E$\otimes$$\beta$ distortion [@Englman.book]. The latter is a many-body effect which appears in solids, where two-level systems correspond to orbital electronic levels localized in atoms, which interact with the lattice vibrations. This effect has received attention for its connections with high-Tc superconductivity and colossal magnetoresistance [@Millis96prb; @Millis96prl; @Tokura00sci]. Very recently it was shown that a generalization of interaction (\[intro\]) can be implemented in a quite natural way in a linear ion crystal placed in a magnetic field gradient [@Porras12bprl; @Ivanov12arX]. The prospects for experimental investigation of the cooperative Jahn-Teller effect would be very interesting, since they would allow us to observe novel quantum structural phase transitions in clean controllable systems. Solid-state systems pose many limitations to observe quantum effects in cooperative Jahn-Teller models, since experiments are typically performed at high temperatures, and also because Jahn-Teller couplings are masked by magnetic or electronic dynamics. A variety of related models are currently under theoretical investigation by several groups including the single particle case [@Casanova10prl], Jaynes-Cummings-Hubbard models [@Leib10njp; @Tureci12prl], structural phase transitions in quantum potentials [@Cormick12prl] and spin-Peierls transitions [@Bermudez12prl].
This paper is dedicated to the spin-wave theory of cooperative (E$\otimes$$\beta$) Jahn-Teller models arising from longitudinal $\sigma^z$ couplings of the form (\[intro\]). A natural starting point is mean-field theory together with a spin-wave description of Gaussian fluctuations [@Auerbach]. The main goal of this paper is to test the validity of mean-field theory by calculating the intensity of quantum fluctuations. The outline of the paper is the following. In section 2 we introduce the Hamiltonian that we are considering for cooperative Jahn-Teller systems, and present some qualitative considerations arising from the comparison with the Dicke model. In section 3 we review the mean-field in this model. In section 4 we present a calculation of the Gaussian fluctuations by means of the definition of Holstein-Primakoff bosons. In section 5 we study in detail the case of Periodic Boundary Conditions and check the validity of the mean-field approximation in a region that depends both on the magnetic order of the system, as well as on mesoscopic cooperative effects as a function of the number of particles. Finally in section 6 we present our conclusions.
Cooperative Jahn-Teller systems {#sec:cJT_model}
===============================
We consider a chain of $N$ spins coupled to $N$ bosonic modes. In addition, there are couplings between different bosonic modes as well as a magnetic field acting on the spins. This leads to a cooperative Jahn-Teller Hamiltonian of the form, $$\begin{aligned}
H_{\rm JT} &=& H_{\mathrm{s}}+H_{\mathrm{b}}+H_{\mathrm{sb}},
\nonumber \\
H_{\mathrm{s}} &=& \sum_{j}\frac{\Omega }{2}\sigma _{j}^{x}, \ \ \ \
H_{\mathrm{b}} =\sum_{j}\omega _{j}a_{j}^{\dagger
}a_{j}+\sum_{j,l}t_{j,l}a_{j}^{\dagger }a_{l}, \ \ \ \
H_{\mathrm{sb}} = g \sum_{j}\sigma _{j}^{z}(a_{j}+a_{j}^{\dagger }),
\label{Hamiltonian}\end{aligned}$$ where $a_j$, $a_j^\dagger$ are annihilation/creation operators of bosons that are localized near the effective spin. The physical meaning of those bosonic modes can be either vibrations of a trapped ion, or photons confined in a cavity. $\Omega$ is a transverse magnetic field, and $\omega_j$ and $t_{j,l}$ are boson local energies and hopping amplitudes; the latter determine the boson dispersion. There are several experimentally relevant cases. However, in this work we focus on nearest-neighbours boson tunneling, $$t_{j,l} = -t(\delta _{j,l+1}+\delta _{j,l-1}),\,t > 0 .
\label{tunn}$$ The latter describes coupled cavities, and to a fair approximation also the radial phonons of an ion Coulomb chain [@Deng08pra]. We diagonalize the bosonic Hamiltonian by transforming to a collective mode basis, $a_j = \sum_n M_{j,n} \bar{a}_n$, where $M_{j,n}$ is the collective mode amplitude at site $j$. The relation $$\bar{\omega}_{n} \delta_{n,m} =
\sum_{j,l}
M_{j,n}^*(t_{j,l} + \omega _{j} \delta _{j,l}) M_{l,m}$$ determines the collective mode energies, $\bar{\omega}_n$, such that $H_{\rm b} = \sum_n \bar{\omega}_n \bar{a}^{\dagger}_n \bar{a}_n$. Finally, the spin-boson coupling in the collective modes basis reads $$H_{\mathrm{sb}}
=
g\sum_{j,n} \sigma _{j}^{z}
\left( M_{j,n} \bar{a}_{n} + M_{j,n}^* \bar{a}_{n}^{\dagger} \right) .$$ The problem posed by $H_{\rm JT}$ resembles the Ising quantum magnet in a transverse field. For large $\Omega$ the ground state is a product state of the boson vacuum and spins pointing in $x$. In the case of large $g$ magnetic order is in the $z$-direction. To show this result, let us see how to get effective spin-spin Ising interactions [@Porras04aprl; @Deng05pra]. Consider the canonical transformation $U=e^{-S}$ with $$S = \sum_{j,n} (g/\bar{\omega} _{n}) \sigma _{j}^{z}
\left(M_{j,n} \bar{a}_n - M_{j,n}^*\bar{ a}_{n}^{\dagger} \right),$$ representing a spin-dependent boson displacement. We apply $U$ to the $\Omega = 0$ case, $$e^{-S} (H_{\rm b} + H_{\rm sb}) e^{S} =
\bar{H}_{\rm b}+\bar{H}_{\rm sb}=\sum_{n} \bar{\omega}_n \bar{a}^{\dagger}_n \bar{a}_n -\sum_{j,l,n} \frac{g^2}{\bar{\omega}_{n}}M_{j,n}^* \sigma^{z}_{j} \sigma^{z}_{l}M_{l,n},$$ which shows that the $\Omega = 0$ Jahn-Teller model is equivalent to an Ising model with interaction strength, $$J_{j,l}=-g^{2}\sum_{n}M_{j,n}^*\frac{1}{\bar{\omega} _{n}}M_{l,n}.
\label{interaction}$$ The former argument is equivalent to the adiabatic elimination of bosons [@Deng05pra]. Adding the transverse field term, we find $$e^{-S} H_{\rm s} e^{S} =
\bar{H}_{\mathrm{s}}=\frac{\Omega }{2}\sum_{j}\left( \sigma
_{j}^{+}e^{-2\sum_{n}M_{j,n}\frac{g}{\bar{\omega} _{n}}\left( \bar{a}_{n}-\bar{a}_{n}^{\dagger
}\right) }+\mathrm{H.c.}\right) ,$$ which shows that as long as $g \ll \bar{\omega}_n$, $H_{\rm s} \approx \bar{H}_{\rm s}$. Previous works exploited this idea for the quantum simulation of Ising and Heisenberg models with trapped ions [@Porras04aprl]. Here we are interested on the whole phase diagram of the model, where bosons suffer a displacement which cannot be neglected.
We aim to give a mean-field description and to test its validity. Let us first qualitatively discuss the problem. Mean-field theory may be well justified by two lines of reasoning that arise in two different fields:
- [**Quantum magnetism.-**]{} The limits of large $\Omega$ and large $g$ should present a well-defined magnetic order in the $x$ and $z$ directions, respectively. Mean-field theory is a good approximation there.
- [**Cooperativity effect.-**]{} In the limit of large $t$, we expect $\bar{\omega}_n$ to form a set of energies with a large energy separation $\Delta \bar{\omega} \propto t$. In the case that $\Delta \bar{\omega} \gg g$, we may simply ignore high energy modes and keep $n=0$ only. We recover the celebrated Dicke model with infinite range interactions, for which mean-field theory is exact in the thermodynamic limit [@Hepp73]. However, as $N \to \infty$ we expect $\Delta \bar{\omega} \to 0$ so that this argument is only valid in a mesoscopic regime of spin-boson chains.
[*The interplay between magnetic order and finite-size effects is a unique feature of the spin-boson model described by $H_{\rm JT}$*]{}.
Mean-field theory {#sec:mf_theory}
=================
Let us present our mean-field variational ansatz, which consists of a product state of spins in the $x$-$z$ plane, and bosons displaced in the collective mode basis [@Porras12bprl], $$| \Psi_{\rm MF} \rangle = \bigotimes_j
| \theta_j \rangle \otimes
e^{-\sum_n (\bar{\alpha}^*_n \bar{a}_n-\bar{\alpha}_n \bar{a}^\dagger_n )} | 0 \rangle_{\rm b},$$ where $| \theta_j \rangle =
\cos (\theta_j /2) |0 \rangle_j + \sin (\theta_j /2) | 1 \rangle_{j}$. The mean-field energy is $$\langle \Psi_{\rm MF} | H_{\rm JT} | \Psi_{\rm MF} \rangle =
\sum_n \bar{\omega}_n \bar{\alpha}_{n}^* \bar{\alpha}_n -
g \sum_{j,n} \cos \theta_j (M_{j,n}^* \bar{\alpha}_{n}^{*} + M_{j,n} \bar{\alpha}_n)
+\frac{\Omega}{2} \sum_j \sin \theta_j.$$ We minimize the energy and get a set of coupled equations for the variational parameters $\theta_j$ and $\bar{\alpha}_n$,
&{
[l]{} |\_n= \_j M\_[j,n]{}\^\* \_j,\
\
\_l J\_[l,j]{} \_l=- \_j, J\_[l,j]{}=2 \_n ( M\_[j,n]{}\^\*M\_[l,n]{}).
. \[min\_lig\]
Let us consider Periodic Boundary Conditions (PBC) to simplify the discussion, together with the nearest-neighbours coupling (\[tunn\]). The problem is diagonalized by plane-wave modes such that $$\bar{\omega}_n = \bar{\omega}_0 + 2t (1 - \cos(2 \pi n / N)),$$ with $\bar{\omega}_0$ the lowest mode energy; we have assumed constant local energies in (\[Hamiltonian\]), $\omega_j = \bar{\omega}_0 + 2 t$. The latter is a convenient parametrization so that the lowest collective energy mode is $\bar{\omega}_0$, independent on the value of $t$. Collective mode amplitudes become simply $$M_{j,n} = e^{-i \frac{2 \pi n}{ N} j}/ \sqrt{N} ,
\label{pbc}$$ with $n = 0, \dots, N-1$. It is worth noting that the eigenfunctions are normalized and that they follow the closure relation $\sum_{j} e^{i\frac{2 \pi j}{N} (m-n)}=N \delta_{m,n}$.
By virtue of the PBC we can drop the index in $\theta_j,\theta_l$ in the equations (\[min\_lig\]), turning out that
|\_n&= \_[j]{} M\_[j,n]{}\^\* = \_j e\^[i j]{}\
&= \_[n0]{}= {
[l]{} |\_[n 0]{} =0,\
|\_0= ,
. \[alpha\]
and that
\_l J\_[l,j]{}&=\_[l]{} 2g\^2 ( \_n M\_[j,n]{}\^\* M\_[l,n]{} ) = 2g\^2 (\_[n,l]{} e\^[ij]{} e\^[-il]{} )\
&= 2g\^2 ( \_n e\^[ij]{} N \_[n0]{} )=:=J. \[J\]
Inserting (\[J\]) in the mean-field Eqs. (\[min\_lig\]) we find the minimum at $$\begin{aligned}
\sin(\theta) &=& -\Omega / (2 J), \ \ |\Omega|/(2 J) \leq 1,
\nonumber \\
\sin(\theta) &=& -1, \hspace{1.2cm} |\Omega|/(2 J) > 1.
\label{critical}\end{aligned}$$ Thus, mean-field theory predicts a phase transition at a critical value $|\Omega_{\rm c}| = 2J$. The behaviour of spins on both sides of the critical point can be defined by taking the two different asymptotic limits,
&{
[l]{} J =0 | = (|0|1),\
\
J =0 | = { |0 , |1}.
.
The meaning of the two phases is obvious and corresponds to the discussion in the previous section. Note that the mean-field theory yields the same result that would be obtained by an effective Ising spin model. However, fluctuations around the mean-field solution are different in the original spin-boson lattice and thus, the validity of the mean-field description cannot be addressed with an effective Ising Hamiltonian.
Gaussian fluctuations {#sec:gaussian_fluctuations}
=====================
Quantum fluctuations are expected to diverge at the critical point. Near the latter our mean-field description is no longer valid. To quantify fluctuations we use a Gaussian approximation around the mean-field solution. As we are going to consider a particular case of the mean-field solution later on, which presents homogeneity in the spin states, we drop the index in $\theta_j$.
Let us define first fluctuations operators with respect to the bosonic degrees of freedom, $\delta \bar{a}_n=\bar{a}_n-\bar{\alpha}_n$. Spin fluctuations are more involved and we quantify them by using Holstein-Primakoff bosons [@Auerbach]. We have to work in a spin rotated frame such that the mean-field state is the reference state for the Holstein-Primakoff transformation. To this aim we define the rotated operators $\tilde{\sigma}_{j}^{x,y,z}$,
&{
[l]{} \_[j]{}\^y=\_[j]{}\^y,\
\
\_[j]{}\^z=\_[j]{}\^z+\_[j]{}\^x,\
\
\_[j]{}\^x=\_[j]{}\^x-\_[j]{}\^z.
.
The rotated Hamiltonian (from the original Hamiltonian (\[Hamiltonian\]) in the collective mode basis) becomes
H\_[JT]{} &= \_[j]{} (\_[j]{}\^x-\_[j]{}\^z)+\_[n]{}| \_[n]{}|[a]{}\_[n]{}\^|[a]{}\_[n]{}\
&+g\_[j,n]{}(\_[j]{}\^z+\_[j]{}\^x) (M\_[j,n]{} |[a]{}\_[n]{}+M\_[j,n]{}\^\*|[a]{}\_[n]{}\^).
After the rotation, we can use the usual Holstein-Primakoff transformation [@Auerbach] where the reference state is taken in the $\tilde{\sigma}_{j}^z$ basis. The spin fluctuations are expressed by means of bosons $b_j$, $b_{j}^\dagger$,
&{
[l]{} \_[j]{}\^+ = b\_[j]{}\^ ,\
\
\_[j]{}\^-= b\_j,\
\
\_[j]{}\^z = 2 n\_[s,j]{} - 1,
. \[HP\]
where $n_{s,j} = b_{j}^{\dagger} b_j$. It is straightforward to show that those operators indeed obey the commutation relations corresponding to the Pauli matrices. In the limit of small fluctuations we approximate
&{
[l]{} \_[j]{}\^+b\_[j]{}\^,\
\
\_[j]{}\^- b\_[j]{}.\
.
The former approximations are only valid in the limit $\langle b_{j}^\dagger b_j \rangle \ll 1$ which we shall verify later for self-consistency. Near the critical point, we expect that large spin fluctuations such that nonlinear terms in Eq. (\[HP\]) become relevant.
Substituting in the rotated Hamiltonian we arrive at the Gaussian fluctuations Hamiltonian,
&H\_[G]{}=\_n |\_n |[a]{}\_[n]{}\^|[a]{}\_n + \_j \_j b\_[j]{}\^b\_[j]{}\
&+ \_[j,n]{} g (M\_[j,n]{}\^\*|[a]{}\_[n]{}\^+ M\_[j,n]{}|[a]{}\_n)(b\_[j]{}\^+b\_[j]{}), \[H.g\]
where $\Delta_j:=-\Omega \sin \theta + 2 g \cos \theta \sum_n \left(M_{j,n} \bar{\alpha}_n+{\rm c.c.}\right)$. $H_\GG$ is diagonalized by means of a Bogoliubov transformation to spin-phonon fluctuation operators $c_{m}$, $$H_{\rm G}=\sum_{m = 1, \dots, 2 N} E_m c_{m}^{\dagger} c_m.$$ The new $2 N$ bosons are related to the $N$ boson and $N$ spin fluctuation operators by a relation of the form $$\begin{aligned}
\delta \bar{a}_n &=&
\sum_{m = 1,\ldots, 2N}
\left(
W_{n,m}^{(a)} c_m + V_{n,m}^{(a)} c_{m}^{\dagger} \right),
\notag \\
b_{j} &=&
\sum_{m = 1,\ldots, 2N}
\left(
W_{j,m}^{(b)}c_{m} + V_{j,m}^{(b)} c_m^{\dagger}
\right),
\label{bogoliubov}\end{aligned}$$ where the matrices $W_{n,m}^{(a)}, W_{j,m}^{(b)}, V_{n,m}^{(a)}, V_{j,m}^{(b)}$ define a generalized Bogoliubov transformation, which has to satisfy the condition that bosonic commutation relations are left invariant. We will find below a very convenient way to get an explicit result for that transformation. For the moment, we just show how to compute quantum fluctuations. Define first the vacuum of the eigenmodes of $H_{\GG}$, $|\Omega \rangle$ by the condition $c_m |\Omega \rangle = 0$. Then we define the variance per atom for a set of the original spin-phonon fluctuation modes and get the result $$\begin{aligned}
F_{\{ \delta a \}} &=& \frac{1}{N} \sum_n \langle \Omega | \delta \bar{a}_{n}^\dagger \delta \bar{a}_n |\Omega \rangle=\sum_{n,m} |V_{n,m}^{(a)}|^2,\notag\\
F_{\{ b\}} &=& \frac{1}{N} \sum_j \langle \Omega | b_{j}^\dagger b_{j} |\Omega \rangle=\sum_{j,m} |V_{j,m}^{(b)}|^2.
\label{fluctuations}\end{aligned}$$ In the next section we use these expressions to calculate quantum fluctuations in the case of translational invariant systems with PBC.
Analysis of the problem under Periodic Boundary Conditions {#pbc_analysis}
==========================================================
To impose PBC we substitute the eigenfunctions $M_{j,n}$ defined in Eq. (\[pbc\]). We transform fluctuation operators to the plane-wave basis by the relations $\delta a_j = \sum_n M_{j,n} \delta \bar{a}_n$, $b_j = \sum_n M_{j,n} \bar{b}_n$, and use the relations (\[alpha\], \[J\], \[critical\]) in the Hamiltonian (\[H.g\]), to get $$\begin{aligned}
H_{\rm G} =
\sum_n \bar{\omega}_n \delta \bar{a}_n^{\dagger} \delta \bar{a}_n +
\sum_n g \sin \theta (\delta \bar{a}_{-n}^{\dagger} + \delta \bar{a}_n)( \bar{b}_{n}^{\dagger}+ \bar{b}_{-n}) -\sum_n \frac{\Omega}{\sin \theta} \bar{b}_{n}^{\dagger} \bar{b}_{n}.\end{aligned}$$ Note that the closure relation in the plane-wave basis yields couplings between modes with opposite linear momentum, $n$ and $-n$, in terms of the form $\delta \bar{a}_n \bar{b}_{-n}$, for example. To diagonalize $H_\GG$ we work in the $X,P$ representation. We define $$\left\{
\begin{array}{l}
X_{a,n}=\frac{1}{\sqrt{2\bar{\omega}_n }}(\delta \bar{a}_{-n}^{\dagger }+\delta \bar{a}_{n}), \\
P_{a,n}=\frac{1}{i}\sqrt{\frac{\bar{\omega}_n}{2}}(\delta \bar{a}_{-n}-\delta \bar{a}_{n}^{\dagger }),
\end{array}
~~~\left\{
\begin{array}{l}
X_{b,n}=\sqrt{\frac{|\sin\theta|}{2\Omega }}( \bar{b}_{-n}^{\dagger }+ \bar{b}_{n}), \\
P_{b,n}=\frac{1}{i}\sqrt{\frac{\Omega}{2|\sin\theta|}}( \bar{b}_{-n}- \bar{b}_{n}^{\dagger }).
\end{array}
\right. \right.$$ Note the relations $X_{\mu,n}^\dagger = X_{\mu,-n}$, $P_{\mu,n}^\dagger = P_{\mu,-n}$ $(\mu = a,b)$. Up to an additive constant corresponding to the vacuum energy, the problem can be stated as a sum of separable Hamiltonians for each mode $n$, $$\begin{aligned}
%
H_{\GG} &=& \sum_n H_{\GG,n},
\nonumber \\
H_{\GG,n} &=&
\frac{1}{2} P_{a,n}P_{a,-n}+\frac{1}{2} \bar{\omega}_{n}^2 X_{a,n}X_{a,-n} +\frac{1}{2} P_{b,n}P_{b,-n}+\frac{1}{2} \left(\frac{\Omega}{|\sin\theta|}\right)^2 X_{b,n}X_{b,-n}
\nonumber \\
&-& g\sqrt{\Omega \bar{\omega}_n |\sin\theta|} \left( X_{a,n} X_{b,-n} + X_{b,n} X_{a,-n} \right).\end{aligned}$$ In matrix notation each of the Hamiltonians $H_{\GG,n}$ can be written as
H\_[, n]{} &=\_[,]{} K\_[,]{}\^[(n)]{} X\_[,n]{} X\_[,-n]{} + \_P\_[,n]{} P\_[,-n]{},\
&K\^[(n)]{}=
| \_[n]{}\^[2]{} & -2g\
-2 g & ()\^[2]{}
,
where $\mu$ and $\nu$ run over labels $a,b$.
Now we have to find a transformation defined by a matrix $U_{\mu,\nu}^{(n)}$, $\hat{X}_{\mu, n} = \sum_\nu U_{\mu,\nu}^{(n)} X_{\nu, n}$, $\hat{X}_{\mu,-n} = \sum_\nu U_{\mu,\nu}^{(n)} X_{\nu,-n}$, $\hat{P}_{\mu, n} = \sum_\nu U_{\mu,\nu}^{(n)} P_{\nu,n}$, $\hat{P}_{\mu,-n} = \sum_\nu U_{\mu,\nu}^{(n)} P_{\nu,-n}$ that diagonalizes $K^{(n)}$, and conserves the canonical commutation relations. The latter requirement is fulfilled automatically since the transformation is orthogonal due to the fact that $K^{(n)}$ is real and symmetric.
The eigenvalues of $K^{(n)}$ can be split up into two branches, $$E_{\pm,n}^2=\frac{\left(\frac{\Omega}{|\sin\theta|}\right)^2+\bar{\omega}_{n}^2 \pm \sqrt{16g^2\Omega \bar{\omega}_n|\sin\theta| + \left(\left(\frac{\Omega}{|\sin\theta|}\right)^2-\bar{\omega}_{n}^2\right)^2}}{2}.$$ The transformed eigenvectors $\hat{X}_{\pm,n}$ and $\hat{P}_{\pm,n}$ are the columns of the $U^{(n)}$ matrix $$U^{(n)}=\begin{pmatrix}
-\frac{2g\sqrt{\Omega \bar{\omega}_n|\sin\theta|}}{v_n} & \frac{E_{-,n}^2-\left(\frac{\Omega}{|\sin\theta|}\right)^2}{v_n} \\
\frac{E_{+,n}^2-\bar{\omega}_{n}^2}{v_n} & -\frac{2g\sqrt{\Omega \bar{\omega}_n|\sin\theta|}}{v_n}
\end{pmatrix};$$ there the normalization factor is $$v_n^2 =
\left(E_{-,n}^2-\left(\frac{\Omega}{|\sin\theta|}\right)^2\right)^2+4g^2 \Omega \bar{\omega}_{n}|\sin\theta|.$$ We write the new operators $\hat{X}_{\pm,n}$,$\hat{P}_{\pm,n}$ in second quantized form, $$\left\{
\begin{array}{l}
\hat{X}_{\pm,n}=\frac{1}{\sqrt{2 E_{\pm,n} }}(c_{\pm,-n}^{\dagger}+c_{\pm,n}), \\
\hat{P}_{\pm,n}=\frac{1}{i}\sqrt{\frac{E_{\pm,n}}{2}}(c_{\pm,-n}-c_{\pm,n}^{\dagger}).
\end{array}
\right.$$ From here we can easily get a transformation to relate $\delta \bar{a}_{s, n}$, $\delta \bar{a}_n$ to normal modes $c_{\pm,n}$ like in Eq. (\[bogoliubov\]) and use those to get a final expression for the fluctuations defined in Eq. (\[fluctuations\]), $$\begin{aligned}
F_{\{\delta a\}} &=& \sum_n F_{\{\delta \bar{a}_n\}},
\notag \\
F_{\{ \delta \bar{a}_n \}} &=&
\frac{1}{N} \frac{1}{v_{n}^2}
\left(
g^2\Omega |\sin\theta| E_{+,n} \left(\frac{\bar{\omega}_n}{E_{+,n}}-1\right)^2
\right.
\notag \\
&& \hspace{2cm}
+
\left.
\frac{\left(E_{-,n}^2-\frac{\Omega^2}{|\sin\theta|^2}\right)^2}{4} \frac{E_{-,n}}{\bar{\omega}_n} \left(1-\frac{\bar{\omega}_n}{E_{-,n}}\right)^2
\right),
\notag \\
F_{\{b\}} &=& \sum_n F_{\{\bar{b}_{n}\}},
\notag \\
F_{\{ \bar{b}_{n}\}} &=& \frac{1}{N} \frac{1}{v_{n}^2}
\left( g^2\bar{\omega}_n |\sin\theta|^2 E_{-,n} \left(\frac{\Omega}{|\sin\theta|}\frac{1}{E_{-,n}}-1\right)^2 \right.
\notag \\
&& \hspace{2cm}
+
\left. \frac{(E_{+,n}^2-\bar{\omega}_{n}^2)^2}{4} \frac{1}{E_{+,n}}\frac{\Omega}{|\sin\theta|} \left(1-\frac{E_{+,n}}{\Omega}|\sin\theta|\right)^2 \right)
\label{fluctuations.pbc}\end{aligned}$$ In the following we are going to analyze the results obtained by means of Eq. (\[fluctuations.pbc\]). In Fig. \[fluc1\] we plot the quantum fluctuations as a function of $g$ and check that they diverge at the critical point. Although this divergence is a straightforward effect it will shed light on the scaling of fluctuations with $N$. We have chosen units such that the spin and boson fluctuations are of the same order.
The mean-field critical point is defined by the condition $\Omega = 2 J = 4 g^2/\bar{\omega}_0$.
First we notice that $E_{-,n} = 0$ if $n=0$ since , with $c$ a constant determining the speed of acoustic fluctuations at the critical point. Therefore from (\[fluctuations.pbc\]) it can be shown that the the fluctuations are dominated by the term $1/E_{-,n}$. This fact, indeed, makes the fluctuations ill-defined for $n=0$. On the other hand the contributions $n\neq 0$ to $1/E_{-,n}$ in (\[fluctuations.pbc\]) are the source of a logarithmic divergence expected as $N\rightarrow \infty$, which can be studied separately from the $n=0$ divergence, by looking at the well-defined modes with $n \neq 0$, $$\frac{1}{N} \sum_{n=1}^N \frac{1}{E_{-,n}}=\frac{1}{N} \sum_{n=1}^{N} \frac{1}{E \left(\frac{n}{N} \right)} \sim \frac{1}{N} \int_1^N \frac{dn}{E \left(\frac{n}{N} \right)}= \int_{1/N}^1 \frac{dx}{E(x)} \sim \log N.
\label{log.div}$$ We conclude that the amplitude of quantum fluctuations has to be understood by studying separately two different contributions:
- Modes $n=0$ show the same behaviour as the one expected for the Dicke model: fluctuations diverge at the critical point, however, they are suppressed for $g \neq g_{\rm c}$. As $N \to \infty$, the critical point becomes singular and any $g \neq g_c$ shows no fluctuations.
- Modes $n \neq 0$ contribute with a logarithmic divergence at the critical point due to the infrared divergence in the fluctuations. Out of the critical point, we expect an infrared cut-off in the fluctuation spectrum, and thus there is a well-defined $N \to \infty$ limit.
Out of this discussion we get the following picture: Exactly at the critical point ($g = g_{\rm c}$) fluctuations diverge and they are governed by the $n=0$ mode. Out of the critical point both $n = 0$ and $n \neq 0$ modes contribute but the $n = 0$ contribution will be suppressed in the thermodynamic $N \to \infty$ limit.
Following those arguments, we calculate separately the total fluctuations ($F_{b}$, $F_{\delta a}$), the contribution from $n = 0$, ($F_{\bar{b}_{0}}$, $F_{\delta \bar{a}_0}$), and the fluctuations from modes $n \neq 0$, ($\sum_{n \neq 0} F_{\bar{b}_{n}}$, $\sum_{n \neq 0} F_{\delta \bar{a}_n}$). In Fig. \[fluc2\] (left), we plot the total fluctuations as function of $g$ for different values of $N$. Note that, for concreteness, we consider the case $\Omega = \omega_0$, which leads to similar spin and boson quantum fluctuations. We find convergence in the large $N$ limit. In Fig. \[fluc2\] (right) we plot the $n=0$ contribution and check that it contributes to the divergence at the critical point, but it can be neglected in the noncritical region for large $N$, as expected from the single-mode Dicke model.
In Fig. \[FvsN\] we plot the $n\neq0$ contribution and the total fluctuations as a function of the number of particles. We observe that fluctuations converge to the $n \neq 0$ contribution for large $N$, however, for the values used in this calculations there is an intermediate [mesoscopic]{} regime of $N \approx 10$ particles, where fluctuations decrease with $N$. This effect is induced by the suppression of the $n=0$ contribution as $N$ increases. Finally, Fig. \[FvsT\] shows the scaling of the $n \neq 0$ contribution for different values of $t$. At the critical point (right), we check the logarithmic divergence estimated in Eq. (\[log.div\]). Out of the critical phase, fluctuations converge to a steady value. A crucial observation is that for increasing $t$, the contribution from $n \neq 0$ decreases, due to the energy cost of $n \neq 0$ fluctuations.
Conclusions and Outlook
=======================
We have presented a mean-field theory for cooperative Jahn-Teller models that appear in a natural way in a variety of quantum optical setups. Our investigation has relied on the use of mean-field and spin-wave theory. The latter is limited in what concerns the description of the critical phase. However, we have shown that there is a regime of validity for mean-field theory that can be checked self-consistently by calculating the amplitude of quantum fluctuations. Our results show that the mean-field phase is determined by the interplay between the $n=0$ fluctuations typical from long-range Dicke models and the $n \neq 0$ fluctuations that are well-defined in the thermodynamic limit. Our calculations show that quantum fluctuations decrease with $N$ due to the suppression of the $n=0$ contribution and arrive at a steady value for large values of $N$.
Our work is relevant to several experimental setups in quantum optics. In particular, in the case of trapped ions, it was recently shown that a generalization of the cooperative Jahn-Teller model (\[Hamiltonian\]) could be implemented. This can be achieved either inducing interactions with lasers [@Porras04aprl] or with magnetic field gradients [@Porras12bprl]. Typical energy scales are $\bar{\omega}_0, \Omega, g, t \approx 100$ kHz, and ion chains of $N = 2, \dots 50$ ions. Another promising system is circuit QED, where qubit-field couplings in the ultrastrong coupling correspond to values $g \approx \bar{\omega}_0, \Omega$ in the GHz regime [@Schoelkopf08nat; @Houck12natphys]. Recent theoretical proposal could allow one to induce sidebands in the qubit-field coupling to achieve a high degree of controllability of the parameters of the model [@Porras12aprl]. In addition, our work could be complemented by the study of Jaynes-Cummings-like qubit-field interactions, which also arise in a variety of quantum optical systems. An interesting issue that could be addressed by means of spin-wave theory is the effect of finite temperature. Experimental quantum optical systems like trapped ions and circuit QED systems are typically out-of-equilibrium systems. Ground states are created by adiabatic evolution, so that the correct physical description would correspond to an initial finite temperature state which evolves by adiabatically turning on some Hamiltonian parameter (for example, the spin-boson coupling, g). The theoretical description would thus rely on a non-equilibrium spin-wave theory to account for temporal evolution. Finite temperature effects would then lead to a finite number of gaussian excitations in the final state after the adiabatic evolution. We acknowledge QUITEMAD S2009-ESP-1594, FIS2009-10061, CAM-UCM/910758 and RyC Contract Y200200074, and COST action “IOTA”.
J.I. Cirac and P. Zoller, Nature Phys. [**8**]{}, 264 (2012). Ch. Schneider, D. Porras and T. Schaetz, Reports on Progress in Physics [**75**]{}, 024401 (2012). D. Leibfried, R. Blatt, C. Monroe and D. Wineland Rev. Mod. Phys. [**75**]{}, 281 (2003). D. Porras and J. I. Cirac, Phys. Rev. Lett. [**92**]{}, 207901 (2004). X.-L. Deng, D. Porras, and J. I. Cirac, Phys. Rev. A [**72**]{}, 063407 (2005). M. J. Hartmann, F. G. S. L. Brandao and M. B. Plenio, Nature Phys. [**2**]{}, 849 (2006). R. J. Schoelkopf and S. Girvin, Nature (London) [**451**]{}, 664 (2008). D. Porras and J. I. Cirac, Phys. Rev. Lett. [**93**]{}, 263602 (2004). J.I. Cirac and P. Zoller, Nature (London) [**404**]{}, 579 (2000). B. Peropadre, P. Forn-Díaz, E. Solano and J. J. García-Ripoll, Phys. Rev. Lett. [**105**]{}, 023601 (2010). A. Friedenauer, H. Schmitz, J.T. Glueckert, D. Porras, T. Schaetz, Nature Physics [**4**]{}, 757 (2008). A. Bermúdez, D. Porras, M.A. Martín-Delgado, Phys. Rev. A [**79**]{}, 060303 (2009). R. Islam, E.E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J.K. Freericks and C. Monroe, Nature Communications [**2**]{}, 377 (2011). J.W. Britton, B.C. Sawyer, A.C. Keith, C.-C. Joseph Wang, J.K. Freericks, H. Uys, M.J. Biercuk and J.J. Bollinger, Nature (London) [**484**]{}, 489 (2012). A. Bermúdez, M.A. Martín-Delgado and D. Porras, New Jour. of Phys. [**12**]{}, 123016 (2010). D. Porras, F. Marquardt, J. von Delft, and J. I. Cirac, Phys. Rev. A [**78**]{}, 010101 (2008). R. Englman, [*The Jahn-Teller Effect in Molecules and Crystals*]{} (Wiley, New York 1972). A. J. Millis, Phys. Rev. B [**53**]{}, 8434 (1996). A. J. Millis, B. I. Shraiman, and R. Mueller, Phys. Rev. Lett. [**77**]{}, 175 (1996). Y. Tokura and N. Nagaosa, Science [**288**]{}, 462 (2000). D. Porras, P.A. Ivanov and F. Schmidt-Kaler, Phys. Rev. Lett. [**108**]{}, 235701 (2012). P.A. Ivanov, D. Porras, S.S. Ivanov, F. Schmidt-Kaler, eprint arXiv:1207.0452. J. Casanova, G. Romero, I. Lizuain, J. J. Garc�a-Ripoll, and E. Solano Phys. Rev. Lett. [**105**]{}, 263603 (2010). M. Leib and M.J. Hartmann, New J. Phys. [**12**]{}, 093031 (2010). M. Schiró, M. Bordyuh, B. Öztop, and H. E. Türeci, Phys. Rev. Lett. [**109**]{}, 053601 (2012). C. Cormick and G. Morigi, Phys. Rev. Lett. [**109**]{}, 053003 (2012). A. Bermúdez and M. B. Plenio, Phys. Rev. Lett. [**109**]{}, 010501 (2012). A. Auerbach, *Interacting electrons and quantum magnetism* (Graduate texts in contemporary physics, Springer-Verlag, 1994) X.-L. Deng, D. Porras and J.I. Cirac, Phys. Rev. A [**77**]{}, 033403 (2008). K. Hepp and E. H. Lieb, Ann. Phys. (N.Y.) [**76**]{}, 360 (1973). A.A. Houck, H.E. T�reci and Jens Koch, Nature Physics [**8**]{}, 292 (2012). D. Porras and J.J. García-Ripoll, Phys. Rev. Lett. [**108**]{}, 043602 (2012).
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author:
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Eric <span style="font-variant:small-caps;">Brunet</span> and Bernard <span style="font-variant:small-caps;">Derrida</span>\
Laboratoire de Physique Statistique, ENS, 24 rue Lhomond, 75005 Paris, France
date: 'Physical Review E 1997, **56** (3), 2597–2604'
title: 'Shift in the velocity of a front due to a cut-off'
---
Introduction
============
Equations describing the propagation of a front between a stable and an unstable state appear in a large variety of situations in physics, chemistry and biology. One of the simplest equations of this kind is the Fisher-Kolmogorov [@Fisher.Genes.37; @KPP.Diffusion.37] equation $${\partial h \over \partial t} = {\partial^2h\over\partial x^2} +h-h^3,
\label{eqn:FK}$$ which describes the evolution of a space and time dependent concentration $h(x,t)$ in a reaction-diffusion system. This equation, originally introduced to study the spread of advantageous genes in a population [@Fisher.Genes.37], has been widely used in other contexts, in particular to describe the time dependence of the concentration of some species in a chemical reaction .
For such an equation, the uniform solutions $h=1$ and $h=0$ are respectively stable and unstable and it is known that for initial conditions such that $h(x,0) \to 1 $ as $ x \to -\infty$ and $h(x,0) \to 0 $ as $ x \to +
\infty$ there exists a one parameter family $F_v$ of traveling wave solutions (indexed by their velocity $v$) of the form $$h(x,t)=F_v(x-vt),
\label{eqn:sol}$$ with $F_v$ decreasing, $F_v(z) \to 1$ as $z \to - \infty$ and $F_v(z)
\to 0$ as $z \to \infty$. The analytic expression of the shape $F_v$ is in general not known but one can determine the range of velocities $v$ for which solutions of type (\[eqn:sol\]) exist. If one assumes an exponential decay $$F_v(z) \simeq e^{-\gamma z}\qquad\text{for large $z$},
\label{eqn:exp}$$ it is easy to see by replacing (\[eqn:sol\]) and (\[eqn:exp\]) into (\[eqn:FK\]) that the velocity $v$ is given by $$v(\gamma)=\gamma+{1\over\gamma}.
\label{eqn:v(g)1}$$
As $\gamma$ is arbitrary, this shows the well known fact that the range of possible velocities is $v \geq 2$. The minimal velocity $v_0=2$ is reached for $\gamma_0=1$ and for steep enough initial conditions $h(x,0)$ (which decay faster than $e^{-\gamma_0 x}$), the solution selected for large $t$ is the one corresponding to this minimal velocity $v_0$.
Equations of type (\[eqn:FK\]) are obtained either as the large scale limit or as the mean field limit [@DerridaSpohn.Polymers.88] of physical situations which are discrete at the microscopic level (particles, lattice models, etc.) As the number of particles is an integer, the concentration $h(x,t)$ could be thought as being larger than some $\epsilon$, which would correspond to the value of $h(x,t)$ when a single particle is present. Equations of type (\[eqn:FK\]) appear then as the limit of the discrete model when $\epsilon\rightarrow0$. Several authors have already noticed in their numerical works that the speed $v_\epsilon$ of the discrete model converges slowly, as $\epsilon$ tends to 0, towards the minimal velocity $v_0$. We believe that the main effect of having $\epsilon\neq0$ is to introduce a cut-off in the tail of the front, and that this changes noticeably the speed.
The speed of the front is in general governed by its tail. In the present work, we consider equations similar to (\[eqn:FK\]), which we modify in such a way that whenever $h(x,t)$ is much smaller than a cut-off $\epsilon$, it is replaced by 0. The cut-off $\epsilon$ can be introduced by replacing (\[eqn:FK\]) by $${\partial h \over \partial t}={\partial^2h\over\partial
x^2}+(h-h^3)a(h),
\label{eqn:cutoff}$$ with $$\begin{aligned}
\label{eqn:a(h)}
a(h)&=1 &\qquad\text{if $ h > \epsilon$},\\
a(h)&\ll1 &\qquad\text{if $h \ll \epsilon$}.\nonumber\end{aligned}$$ For example, one could choose $a(h)=1$ for $h\ge\epsilon$ and $a(h)=h/\epsilon$ for $h\le\epsilon$. Another choice that we will use in section \[sec:calcul\] is simply $a(h)=1$ if $h>\epsilon$ and $a(h)=0$ if $h\le\epsilon$.
The question we address here is the effect of the cut-off $\epsilon$ on the velocity $v_\epsilon$ of the front. We will show that the velocity $v_\epsilon$ converges, as $\epsilon \to 0$, to the minimal velocity $v_0$ of the original problem (without cut-off) and that the main correction to the velocity of the front is $$v_\epsilon \simeq v_0-{\pi^2\gamma_0^2\over2} \ v''(\gamma_0) \
{1\over\left(\log\epsilon\right)^2}
\label{eqn:resu}$$ for an equation of type (\[eqn:FK\]) for which the velocity is related to the exponential decay $\gamma$ of the shape (\[eqn:sol\]) by some relation $v(\gamma)$. (Everywhere we note by $v_0$ the minimal velocity and $\gamma_0$ the corresponding value of the decay $\gamma$.) In the particular case of equation (\[eqn:FK\]), where $v(\gamma)$ is given by (\[eqn:v(g)1\]), this becomes $$v_\epsilon \simeq 2-{\pi^2\over\left(\log\epsilon\right)^2}.
\label{eqn:resuFK}$$
In section \[sec:model\] we describe an equation of type (\[eqn:FK\]) where both space and time are discrete, so that simulations are much easier to perform. The results of the numerical simulations of this equation are described in section \[sec:nume\]: as $\epsilon\to0$, the velocity is seen to converge like $\left(\log\epsilon\right)^{-2}$ to the minimal velocity $v_0$, and the shape of the front appears to take a scaling form.
In section \[sec:calcul\] we show that for equations of type (\[eqn:FK\]) in presence of a small cut-off $\epsilon$ as in (\[eqn:cutoff\]), one can calculate both the shape of the front and the shift in velocity. The results are in excellent agreement with the numerical data of section \[sec:nume\].
In section \[sec:stoch\] we consider a model defined, for a finite number $N$ of particles, by some microscopic stochastic dynamics which reduces to the front equation of sections \[sec:nume\] and \[sec:calcul\] in the limit $N\to\infty$. Despite the presence of noise, our simulations indicate that in this case too, the velocity dependence of the front decays slowly (as $\left(\log N\right)^{-2}$) to the minimal velocity $v_0$ of the front.
A discrete front equation {#sec:model}
=========================
To perform numerical simulations, it is much easier to study a case where both time and space are discrete variables. We consider here the equation
\[eqn:model\] $$h(x,t+1)= g(x,t)\ \Theta[g(x,t) - \epsilon],$$ where $$g(x,t)= 1 - \bigl[1 - p h(x-1,t) - (1-p)h(x,t) \bigr]^2.$$
Time is a discrete variable and if initially the concentration $h(x,0)$ is only defined when $x$ is an integer, $h(x,t)$ remains so at any later time. Because $t$ and $x$ are both integers, the cut-off $\epsilon$ can be introduced as in (\[eqn:model\]) in the crudest way using a Heaviside $\Theta$ function. (We have checked however that other ways of introducing the cut-off $\epsilon$ as in (\[eqn:cutoff\], \[eqn:a(h)\]) do not change the results.)
Equation (\[eqn:model\]) appears naturally (in the limit $\epsilon=0$) in the problem of directed polymers on disordered trees [@DerridaSpohn.Polymers.88; @Derrida.Norvege.91] (where the energy of the bonds is either $1$ with probability $p$ or $0$ with probability $1-p$). At this stage we will not give a justification for introducing the cut-off $\epsilon$. This will be discussed in section \[sec:stoch\].
We consider for the initial condition a step function $$\begin{aligned}
\label{eqn:init}
h(x,0)& =0 &\qquad\text{if $x\ge0$,}\\
h(x,0)& =1 &\qquad\text{if $x<0$.}\nonumber\end{aligned}$$ Clearly for such an initial condition, $h(x,t)=1$ for $x < 0$ at all times. As $h(x,t)\simeq1$ behind the front and $h(x,t)\simeq0 $ ahead of the front, we define the position $X_t$ of the front at time $t$ by $$X_t=\sum_{x=0}^{+\infty} h(x,t).
\label{eqn:pos}$$ The velocity of the front $v_\epsilon$ can then be calculated by $$v_\epsilon = \lim_{t \to \infty}{ X_t \over t}
=\left\langle X_{t+1}-X_t\right\rangle,
\label{eqn:speed}$$ where the average is taken over time. (Note that as $h(x,t)$ is only defined on integers, the difference $X_{t+1}-X_t$ is time dependent and has to be averaged as in (\[eqn:speed\]).)
When $\epsilon=0$, the evolution equation (\[eqn:model\]) becomes $$h(x,t+1)=
1 - \bigl[1 - p h(x-1,t) - (1-p)h(x,t) \bigr]^2.
\label{eqn:mf}$$ As for (\[eqn:FK\]), there is a one parameter family of solutions $F_v$ of the form (\[eqn:sol\]) indexed by the velocity $v$ which is related (\[eqn:exp\]) to the exponential decay $\gamma$ of the shape by $$v(\gamma)={1\over\gamma}\log\bigl(2pe^{\gamma}+2(1-p)\bigr).
\label{eqn:v(g)2}$$ (This relation is obtained as (\[eqn:v(g)1\]) by considering the tail of the front where $h(x,t)$ is small and where therefore (\[eqn:mf\]) can be linearized.)
One can show that for $p<{1\over2}$, $v(\gamma)$ reaches a minimal value $v_0$ smaller than 1 for some $\gamma_0$, whereas for $p\ge{1\over2}$, $v(\gamma)$ is a strictly decreasing function of $\gamma$, implying that the minimal velocity is $v_0=
\lim\limits_{\gamma\to\infty}v(\gamma)=1$.
We will not discuss here this phase transition and we assume from now on that $p<{1\over2}$. Table \[tab:limspeed\] gives some values of $v_0$ and $\gamma_0$ obtained from (\[eqn:v(g)2\]).
$p$ 0.05 0.25 0.45
------------ ----------- ----------- -----------
$\gamma_0$ 2.751111… 2.553244… 4.051851…
$v_0$ 0.451818… 0.810710… 0.979187…
: Values of $\gamma_0$ and $v_0$ for some $p$ when $\epsilon =0$.[]{data-label="tab:limspeed"}
It is important to notice that for $p<{1\over2}$, the function $v(\gamma)$ has a single minimum at $\gamma_0$. Therefore, there are in general two choices $\gamma_1$ and $\gamma_2$ of $\gamma$ for each velocity $v$. For $v\neq v_0$, the exponential decay of $F_v(z)$ is dominated by min$(\gamma_1,\gamma_2)$. As $v \to v_0$, the two roots $\gamma_1$ and $\gamma_2$ become equal, and the effect of this degeneracy gives (in a well chosen frame) $$F_{v_0}(z) \simeq A\, z\,e^{-\gamma_0z} \qquad \text{for large $z$,}
\label{eqn:general}$$ where $A$ is a constant. This large $z$ behavior can be recovered by looking at the general solution of the linearized form of equation (\[eqn:mf\]) $$h(x,t+1)=
2 p h(x-1,t) +2 (1-p)h(x,t).
\label{eqn:linear}$$
Numerical determination of the velocity {#sec:nume}
=======================================
We iterated numerically (\[eqn:model\]) with the initial condition (\[eqn:init\]) for several choices of $p < {1\over2}$ and for $\epsilon$ varying between $0.03$ and $10^{-17}$. We observed that the speed is usually very easy to measure because, after a short transient time, the system reaches a periodic regime for which $$h(x,t+T)=h(x-Y,t)
\label{eqn:cycle}$$ for some constants $T$ and $Y$. The speed $v_\epsilon$ of the front is then simply given by $$v_\epsilon={Y\over T}.$$ For example, for $p=0.25$ and $\epsilon=10^{-5}$, we find $T=431$ and $Y=343$ so that $v_\epsilon=343/431$. The emergence of this periodic behavior is due to the locking of the dynamical system of the $h(x,t)$ on a limit cycle. Because $Y$ and $T$ are integers, our numerical simulations give the speed with an *infinite accuracy.*
For each choice of $p$ and $\epsilon$, we measured the speed of the front, as defined by (\[eqn:speed\]) and its shape. Figure \[fig:vit\] is a log-log plot of the difference $v_0-v_\epsilon$ versus $\epsilon$ (varying between $0.03$ to $10^{-17}$) for three choices of the parameter $p$. The solid lines on the plot indicate the value predicted by the calculations of section \[sec:calcul\].
We see on this figure that the velocity $v_\epsilon$ converges slowly towards the minimal velocity $v_0$ as $\epsilon\rightarrow0$. Our simulations, done over several orders of magnitude (here, fifteen), reveal that the convergence is logarithmic: $v_0 - v_\epsilon \sim
\left(\log\epsilon\right)^{-2}$.
As the front is moving, to measure its shape, we need to locate its position. Here we use expression (\[eqn:pos\]) and we measure the shape $s_\epsilon(z)$ of the front at a given time $t$ relative to its position $X_t$ by $$s_\epsilon(z)=h(z+X_t,t)$$
When the system reaches the limit cycle (\[eqn:cycle\]), the shape $s_\epsilon(z)$ becomes roughly independent of the time chosen. (In fact it becomes periodic of period $T$, but the shape $s_\epsilon$ has a smooth envelope.) We have measured this shape at some arbitrary large enough time to avoid transient effects. As we expect $s_\epsilon(z)$ to look more and more like $F_{v_0}(z)$ as $\epsilon$ tends to 0, we normalize this shape by dividing it by $e^{-\gamma_0z}$. The result $s_\epsilon(z) e^{\gamma_0z} $ is plotted versus $z$ for $p=0.25$ and $\epsilon=10^{-9}$, $10^{-11}$, $10^{-13}$, $10^{-15}$ and $10^{-17}$ in figure \[fig:shape\].
On the left part of the graph, our data coincide over an increasing range as $\epsilon$ decreases, indicating that far from the cut-off, the shape converges to expression (\[eqn:general\]) of $F_{v_0}(z)$. On the right part, the curves increase up to a maximum before falling down to some small value which seems to be independent of $\epsilon$. When $\epsilon$ is multiplied by a constant factor (here $10^{-2}$), the maximum as well as the right part of the curves are translated by a constant amount. This indicates that for $\epsilon$ small enough, the shape $s_\epsilon(z)$ in the tail (that is for $z$ large) takes the scaling form $$s_\epsilon(z) \simeq \left|\log\epsilon\right|\
G\left({z\over\left|\log \epsilon\right|}\right)\
e^{-\gamma_0z}.
\label{eqn:scaling}$$
We will see that our analysis of section \[sec:calcul\] does predict this scaling form. As one expects this shape to coincide with the asymptotic form (\[eqn:general\]) of $F_{v_0}(z)$ for $ 1 \ll z \ll
|\log \epsilon|$, the scaling function $G(y)$ should be linear for small $y$.
Calculation of the velocity for a small cut-off {#sec:calcul}
===============================================
The first remark we make is that as soon as we introduce a cut-off through a function $a(h)$ which is everywhere smaller than 1, the velocity $v_\epsilon$ of the front is lowered compared to the velocity obtained in the absence of a cut-off. This is easy to check by comparing a solution $h_\epsilon(x,t)$ of (\[eqn:cutoff\]) where $a(h)$ is present and a solution $h_0(x,t)$ of (\[eqn:FK\]). If initially $h_\epsilon(x,0) < h_0(x,0)$, the solution $h_\epsilon$ will never be able to take over the solution $h_0$. Indeed, would the two functions $h_\epsilon(x,0)$ and $h_0(x,0)$ coincide for the first time at some point $x$, we would have at that point $\partial^2 h_\epsilon / \partial
x^2 \leq \partial^2 h_0 / \partial x^2 $ and together with the effect of $a(h)$ this would bring back the system in the situation where $h_\epsilon(x,t) < h_0(x,t)$ . This shows that $v_\epsilon\le v_0$.
For the calculation of the velocity $v_\epsilon$, we will consider first the modified Fisher-Kolmogorov equation (\[eqn:cutoff\]) when the cut-off function $a(h)$ is simply given by $$a(h)= \Theta(h- \epsilon).
\label{eqn:a(h)1}$$
In this section we will calculate the leading correction to the velocity when $\epsilon $ is small and we will obtain the scaling function $G$ which appears in (\[eqn:scaling\]). Then we will discuss briefly how our analysis could be extended to more general forms of the cut-off function $a(h)$ or to other traveling wave equations such as (\[eqn:model\]).
As $v_\epsilon$ is the velocity of the front, its shape $s_\epsilon(z)=h(z+v_\epsilon t,t)$ in the asymptotic regime satisfies $$v_\epsilon s_\epsilon' + s_\epsilon'' + (s_\epsilon-s_\epsilon^2)
a(s_\epsilon) =0.$$ When $\epsilon$ is small, with the choice (\[eqn:a(h)1\]) for $a(h)$, we can decompose the range of values of $z$ into three regions:
> Region I
>
> : where $s_\epsilon(z)$ is not small compared to 1.
>
> Region II
>
> : where $\epsilon < s_\epsilon(z) \ll 1$.
>
> Region III
>
> : where $s_\epsilon(z) <\epsilon$.
>
In region I, the shape of the front $s_\epsilon$ looks like $F_{v_0}$ whereas in regions II and III, as $s_\epsilon$ is small, it satisfies the linear equations $$\begin{aligned}
v_\epsilon s_\epsilon'+s_\epsilon''+s_\epsilon=0 &\qquad\text{in region~II,}
\label{eqn:linearII} \\
v_\epsilon s_\epsilon'+s_\epsilon''=0 &\qquad\text{in region~III.}
\label{eqn:linearIII} \end{aligned}$$ These linear equations (\[eqn:linearII\],\[eqn:linearIII\]) can be solved easily. The only problem is to make sure that the solution in region II and its derivative coincides with $F_{v_0}$ at the boundary between I and II and with the solution valid in region III at the boundary between II and III. If we call $\Delta$ the shift in the velocity $$\Delta=v_0-v_\epsilon,$$ and if we note $\gamma_r \pm i \gamma_i$ the two roots of the equation $v(\gamma) = v_\epsilon$, the shape $s_\epsilon$ is given in the three regions by $$\begin{aligned}
\label{eqn:3ways}
s_\epsilon(z) \simeq & F_{v_0}(z)
&\text{in region I,} \nonumber\\
s_\epsilon(z) \simeq & C e^{-\gamma_r z} \sin (\gamma_i z +D)
&\text{in region II,} \\
s_\epsilon(z) \simeq & \epsilon e^{-v_\epsilon (z-z_0)}
&\text{in region III,} \nonumber\end{aligned}$$ and we can determine the unknown quantities $C$, $D$, $z_0$ and $v_\epsilon$ by using the boundary conditions.
For large $z$ we know from (\[eqn:general\]) that $F_{v_0}(z) \simeq A z
e^{-\gamma_0 z}$ for some $A$. Therefore, as $\gamma_0- \gamma_r \sim
\Delta$ and $\gamma_i \sim \Delta^{1/2}$, the boundary conditions between regions I and II impose, to leading order in $\Delta^{1/2}$, that $C= A/
\gamma_i$ and $D=0$.
At the boundary between regions II and III, we have $s_\epsilon(z)=\epsilon$ and $z=z_0$. If we impose the continuity of $s_\epsilon$ and of its first derivative at this point, we get
\[eqn:system\] $$A e^{-\gamma_r z_0} \sin (\gamma_i z_0) = \epsilon\gamma_i,$$ and $$A e^{-\gamma_r z_0} [ -\gamma_r \sin (\gamma_i z_0)
+\gamma_i \cos (\gamma_i z_0)] = - v_\epsilon \epsilon\gamma_i.$$
Taking the ratio between these two relations leads to $$\gamma_r - {\gamma_i \over \tan(\gamma_i z_0)} = v_\epsilon.
\label{eqn:eqq}$$
When $\Delta$ is small, $\gamma_r \simeq \gamma_0 = 1$, $v_\epsilon \simeq
v_0 = 2$ and $\gamma_i \sim \Delta^{1/2}$. Thus the only way to satisfy (\[eqn:eqq\]) is to set $ \gamma_i z_0 \simeq
\pi$ and $\pi - \gamma_i z_0 \simeq \gamma_i \sim
\Delta^{1/2}$. Therefore, (\[eqn:system\]) implies to leading order that $z_0 \simeq -(\log \epsilon)/ \gamma_0$ and the condition $\gamma_i
z_0 \simeq \pi$ gives $$\gamma_i \simeq {\pi \over z_0} \simeq {\pi \gamma_0 \over \left|\log
\epsilon\right|}
\label{eqn:gammai}$$ Then, as $\gamma_i$ is small, the difference $\Delta=v_0-v_\epsilon$ is given by $$v_0 - v_\epsilon \simeq {1 \over 2} v''(\gamma_0) \gamma_i^2
\simeq { v''(\gamma_0) \pi^2 \gamma_0^2 \over
2 \left(\log \epsilon\right)^2}
\label{eqn:prediction}$$ which is the result announced in (\[eqn:resu\]) and (\[eqn:resuFK\]).
A different cut-off function $a(h)$ should not affect the shape of $s_\epsilon$ in the region II or the size $z_0$ of region II. Only the precise matching between regions II and III might be modified and we do not think that this would change the leading dependency of $z_0$ in $\epsilon$ which controls everything. In fact there are other choices of the cut-off function $a(h)$ (piecewise constant) for which we could find the explicit solution in region III, confirming that the precise form of $a(h)$ does not change (\[eqn:gammai\]). The generalization of the above argument to equations other than (\[eqn:FK\]) (and in particular to the case studied in sections \[sec:model\] and \[sec:nume\]) is straightforward. Only the form of the linear equation is changed and the only effect on the final result (\[eqn:resu\]) is that one has to use a different function $v(\gamma)$.
When expression (\[eqn:resu\]) is compared in figure \[fig:vit\] to the results of the simulations, the agreement is excellent. Moreover, in region II, one sees from (\[eqn:3ways\]) and (\[eqn:gammai\]) that $$s_\epsilon(z) \simeq
{A\over\pi\gamma_0}\left|\log\epsilon\right| \sin \left( \pi \gamma_0 z \over
\left|\log \epsilon \right|\right) e^{-\gamma_0 z},
\label{eqn:scaling2}$$ which also agrees with the scaling form (\[eqn:scaling\]).
Recently, for a simple model of evolution governed by a linear equation, the velocity was found to be the logarithm of the cut-off to the power $1\over3$. This result was obtained by an analysis which has some similarities to the one presented in this section.
A stochastic model {#sec:stoch}
==================
Many models described by traveling wave equations originate from a large scale limit of microscopic stochastic models involving a finite number $N$ of particles . Here we study such a microscopic model, the limit of which reduces to (\[eqn:mf\]) when $N\to\infty$. Our numerical results, presented below, indicate a large $N$ correction to the velocity of the form $v_N\simeq v_0 - a \left(\log N\right)^{-2}$ with a coefficient $a$ consistent with the one calculated in section \[sec:calcul\] for $\epsilon={1\over N}$.
The model we consider in this section appears in the study of directed polymers [@CookDerrida.Lyapunov.90] and is, up to minor changes, equivalent to a model describing the dynamics of hard spheres [@vanBeijeren.Lyapunov.97]. It is a stochastic process discrete both in time and space with two parameters: $N$, the number of particles, and $p$, a real number between 0 and 1. At time $t$ ($t$ is an integer), we have $N$ particles on a line at integer positions $x_1(t)$, $x_2(t)$, …, $x_N(t)$. Several particles may occupy the same site. At each time-step, the $N$ positions evolve in the following way: for each $i$, we choose two particles $j_i$ and $j'_i$ at random among the $N$ particles. (These two particles do not need to be different.) Then we update $x_i(t)$ by $$x_i(t+1)= \max\bigl(x_{j_i}(t) + \alpha_i,\,
x_{j'_i}(t) + \alpha'_i\bigr),
\label{eqn:stochmodel}$$ where $\alpha_i$ and $\alpha'_i$ are two independent random numbers taking the value 1 with probability $p$ or 0 with probability $1-p$. The numbers $\alpha_i$, $\alpha'_i$, $j_i$ and $j'_i$ change at each time-step. Initially ($t=0$), all particles are at the origin so that we have $x_i(0)=0$ for all $i$.
At time $t$, the distribution of the $x_i(t)$ on the line can be represented by a function $h(x,t)$ which counts the fraction of particles strictly at the right of $x$. $$h(x,t)={1\over N}\sum_{x_i(t)>x} 1.$$ Obviously $h(x,t)$ is always an integral multiple of $1\over N$. At $t=0$, we have $h(x,0)=1$ if $x<0$ and $h(x,0)=0$ if $x\ge0$. One can notice that the definition of the position $X_t$ of the front used in (\[eqn:pos\]) coincides with the average position of the $N$ particles $$X_t=\sum_{x=0}^{+\infty} h(x,t) = {1\over N}\sum_{i=1}^N x_i(t).$$
Given the positions $x_i(t)$ of all the particles (or, equivalently¸ given the function $h(x,t)$), the $x_i(t+1)$ become independent random variables. Therefore, given $h(x,t)$, the probability for each particle to have at time $t+1$ a position strictly larger than $x$ is given by $$\begin{aligned}
\label{eqn:stoch}
&\bigl\langle h(x,t+1)\,|\,h(x,t)\bigr\rangle\\
&\qquad = 1-\bigl[1-ph(x-1,t)- (1-p)h(x,t) \bigr]^2.\nonumber\end{aligned}$$
The difficulty of the problem comes from the fact that one can only average $h(x,t+1)$ over a single time-step. On the right hand side of (\[eqn:stoch\]) we see terms like $h^2(x,t)$ or $h(x-1,t)h(x,t)$ and one has to calculate all the correlations of the $h(x,t)$ in order to find $\bigl\langle h(x,t+1)\bigr\rangle$. This makes the problem very difficult for finite $N$. However, given $h(x,t)$, the $x_i(t+1)$ are independent and in the limit $N\to\infty$, the fluctuations of $h(x,t+1)$ are negligible. Therefore, when $N\to\infty$, $h(x,t)$ evolves according to the deterministic equation (\[eqn:mf\]). As the initial condition is a step function, we expect the front to move, in the limit $N\to\infty$, with the minimal velocity $v_0$ of (\[eqn:v(g)2\]).
For large but finite $N$, we expect the correction to the velocity to have two main origins. First, $h(x,t)$ takes only values which are integral multiples of $1\over N$, so that $1\over N$ plays a role similar to the cut-off $\epsilon$ of section \[sec:model\]. Second, $h(x,t)$ fluctuates around its average and this has the effect of adding noise to the evolution equation (\[eqn:mf\]). In the rest of this section we present the results of simulations done for large but finite $N$ and we will see that the shift in the velocity seems to be very close to the expression of section \[sec:calcul\] when $\epsilon={1\over N}$.
With the most direct way of simulating the model for $N$ finite, it is difficult to study systems of size much larger than $10^6$. Here we use a more sophisticated method allowing $N$ to become huge. Our method, which handles many particles at the same time, consists in iterating directly $h(x,t)$.
Knowing the function $h(x,t)$ at time $t$, we want to calculate $h(x,t+1)$. We call respectively $\xmin$ and $\xmax$ the positions of the leftmost and rightmost particles at time $t$ and $l=\xmax-\xmin+1$. In terms of the function $h(x,t)$, one has $0<h(x,t)<1$ if and only if $\xmin\le x < \xmax$. Obviously, all the positions $x_i(t+1)$ will lie between $\xmin$ and $\xmax+1$. The probability $p_k$ that a given particle $i$ will be located at position $\xmin+k$ at time $t+1$ is $$p_k=\bigl\langle h(\xmin+k-1,t+1)\bigr\rangle-\bigl\langle
h(\xmin+k,t+1)\bigr\rangle,$$ with $\bigl\langle h(x,t+1)\bigr\rangle$ given by (\[eqn:stoch\]). Obviously, $p_k\neq0$ only for $0 \le k \le l$.
The probability to have, for every $k$, $n_k$ particles at location $\xmin+k$ at time $t+1$ is given by $$\begin{aligned}
P(n_0,n_1,\ldots,n_l)=&{N!\over n_0!\,n_1!\,\ldots\,n_l!}\
p_0^{n_0}\,p_1^{n_1}\,\ldots\,p_l^{n_l}\nonumber\\
&\times\delta(N-n_0-n_1-\cdots-n_l).
\label{eqn:proba}\end{aligned}$$
Using a random number generator for a binomial distribution, expression (\[eqn:proba\]) allows to generate random $n_k$. This is done by calculating $n_0$ according to the distribution $$P(n_0)={N!\over n_0!\,(N-n_0)!}\ p_0^{n_0}(1-p_0)^{N-n_0},$$ then $n_1$ with $$\begin{aligned}
P(n_1 \,|\, n_0)
&=& {(N-n_0)!\over n_1!\,(N-n_0-n_1)!}\
\left(p_1\over 1-p_0\right)^{n_1}\nonumber\\
& &\times
\left(1-{p_1\over 1-p_0}\right)^{N-n_0-n_1},\end{aligned}$$ and so on. This method can be iterated to produce the $l+1$ numbers $n_0$, $n_1$, …, $n_l$ distributed according to (\[eqn:proba\]). Then we construct $h(x,t+1)$ by $$\begin{aligned}
h(x,t+1)=& 1 && \text{if $x<\xmin$,}\nonumber\\
h(x,t+1)=& \displaystyle{1\over N} \sum_{i=k+1}^l n_i
&& \vcenter{\hbox{if $\xmin\le x\le\xmax+1$}
\hbox{and $x=\xmin+k$,}}\nonumber\\
h(x,t+1)=& 0 && \text{if $x>\xmax+1$.}\end{aligned}$$ As the width $l$ of the front is roughly of order $\log N$, this method allows $N$ to be very large.
Using this method with the generator of random binomial numbers given in [@NumericalRecipes], we have measured the velocity $v_N$ of the front for several choices of $p$ (0.05, 0.25 and 0.45) and for $N$ ranging from 100 to $10^{16}$. We measured the velocities with the expression $$v_N={X_{10^6}-X_{10^5} \over 9\,10^5}.$$ Figure \[fig:vitR\] is a log-log plot of the difference $v_0-v_N$ versus $1\over N$ compared to the prediction (\[eqn:resu\]) for $\epsilon={1\over N}$. The variation of $v_N$ when using longer times or different random numbers were not larger than the size of the symbols.
We see on figure \[fig:vitR\] that the speed $v_N$ of the front seems to be given for large $N$ by $$v_N\simeq v_0 - {K\over \left(\log N\right)^2},$$ where the coefficient $K$ is not too different from the prediction (\[eqn:resu\]).
The agreement is however not perfect. The shift $v_0-v_N$ seems to be proportional to $\left(\log N\right)^{-2}$, but the constant looks on figure \[fig:vitR\] slightly different from the one predicted by (\[eqn:resu\]). A possible reason for this difference could have been the discretization of the front: instead of only cutting off the tail as in sections \[sec:nume\] and \[sec:calcul\], here the whole front $h(x,t)$ is constrained to take values multiple of $1\over N$. One might think that this could explain this discrepancy. However, we have checked numerically (the results are not presented in this paper) that equation (\[eqn:mf\]) with $h(x,t)$ constrained to be a multiple of a cut-off $\epsilon$ does not give results significantly different from the simpler model of sections \[sec:nume\] and \[sec:calcul\] with only a single cut-off. So we think that the full discretization of the front can not be responsible for a different constant $K$. The discrepancy observed in figure \[fig:vitR\] is more likely due to the effect of the randomness of the process. It is however not clear whether this mismatch would decrease for even larger $N$. It would be interesting to push further the numerical simulations and check the $N$-dependence of the front velocity for very large $N$.
Conclusion
==========
We have shown in the present work that a small cut-off $\epsilon$ in the tail of solutions of traveling wave equations has the effect of selecting a single velocity $v_\epsilon$ for the front. This velocity $v_\epsilon$ converges to the minimal velocity $v_0$ when $\epsilon\to0$ and the shift $v_0-v_\epsilon$ is surprisingly large (\[eqn:resu\], \[eqn:resuFK\]).
Very slow convergences to the minimal velocity have been observed in a number of cases as well as the example of section \[sec:stoch\]. As the effect of the cut-off on the velocity is large, it is reasonable to think that it would not be much affected by the presence of noise. The example of section \[sec:stoch\] shows that the cut-off alone gives at least the right order of magnitude for the shift and it would certainly be interesting to push further the simulations for this particular model to see whether the analysis of section \[sec:calcul\] should be modified by the noise. The numerical method used in section \[sec:stoch\] to study a very large ($N\sim
10^{16}$) system was very helpful to observe a logarithmic behavior. We did not succeed to check in earlier works whether the correction was logarithmic, mostly because the published data were usually too noisy or obtained on a too small range of the parameters. Still even if the cut-off was giving the main contribution to the shift of the velocity, other properties would remain very specific to the presence of noise like the diffusion of the position of the front [@Breuer.MacroscopicLimit.95].
Our approach of section \[sec:calcul\] shows that the effect of a small cut-off is the existence of a scaling form (\[eqn:scaling\], \[eqn:scaling2\]) which describes the change in the shape of the front in its steady state. The effect of initial conditions for usual traveling wave equations (with no cut-off) leads to a very similar scaling form for the change in the shape of the front in the transient regime. This is explained in the appendix where we show how the logarithmic shift of the position of a front due to initial conditions [@Bramson.Convergence.83; @Bramson.Displacement.78] can be recovered.
Effect of initial conditions on the position and on the shape of the front {#app:init}
==========================================================================
In this appendix we show that ideas very similar to those developed in section \[sec:calcul\] allow one to calculate the position and the shape at time $t$ of a front evolving according to (\[eqn:FK\]), or to a similar equation, given its initial shape. The main idea is that in the long time limit, there is a region of size $\sqrt{t}$ ahead of the front which keeps the memory of the initial condition. We will recover in particular the logarithmic shift in the position of the front due to the initial condition [@Bramson.Convergence.83; @Bramson.Displacement.78], namely that if the initial shape is a step function $$\begin{aligned}
h(x,0)=& 0 & \qquad\text{if $x>0$,}\\
h(x,0)=& 1 & \qquad\text{if $x<0$,}\nonumber\end{aligned}$$ then the position $X_t$ of the front at time $t$ increases like $$X_t\simeq 2t-{3\over2} \log t.
\label{eqn:pos3/2}$$ More generally, if initially $$\begin{aligned}
h(x,0)=& x^\nu e^{-\gamma_0x} & \qquad\text{if $x>0$,}\\
h(x,0)=& 1 & \qquad\text{if $x<0$,}\nonumber\end{aligned}$$ we will show that for $\nu>-2$ $$X_t\simeq 2t+ {\nu-1\over2}\log t,
\label{eqn:pos(nu-1)/2}$$ whereas the shift is given by (\[eqn:pos3/2\]) for $\nu<-2$. Here, there is no cut-off but the transient behavior in the long time limit gives rise to a scaling function very similar to the one discussed in section \[sec:calcul\].
If we write the position of the front at time $t$ as $$X_t=v_0 t - c(t),
\label{eqn:posb}$$ we observed numerically (as in figure \[fig:shape\] of section \[sec:nume\]) and we are going to see in the following that the shape of the front takes for large $t$ the scaling form $$h(x,t)=t^\alpha G\left({x-X_t \over t^\alpha}\right)
e^{-\gamma_0(x-X_t)},
\label{eqn:scalingF}$$ very similar to (\[eqn:scaling\], \[eqn:scaling2\]).
If we use (\[eqn:posb\]) and (\[eqn:scalingF\]) into the linearized form of equation (\[eqn:FK\]), we get using, the fact that $v_0=2$ and $\gamma_0=1$, $${1\over t^\alpha}G''+{1\over t^{1-\alpha}}\left(\alpha z G'-\alpha
G\right)+t^\alpha \dot{c}\, G= \dot{c}\, G',
\label{eqn:diff1}$$ where $z=(x-X_t)t^{-\alpha}$. By writing that the leading orders of the different terms of (\[eqn:diff1\]) are comparable, we see that we must have $$\begin{aligned}
\alpha&=&{1\over2}, \\
\dot{c} &\simeq&{\beta\over t}, \end{aligned}$$ for some $\beta$, and that the right hand side of (\[eqn:diff1\]) is negligible. Therefore, the equation satisfied by $G$ is $${d^2\over dz^2}G+{z\over2} {d\over dz}G +\left(\beta-{1\over2}\right) G =0,
\label{eqn:diff2}$$ and the position of the front is given by $$X_t\simeq v_0t-\beta \log t.
\label{eqn:pos1}$$
As in section \[sec:calcul\], we expect that as $t\to\infty$, the front will approach its limiting form and therefore that for $z$ small, the shape will look like (\[eqn:general\]). Therefore we choose the solution $G_\beta(z)$ of (\[eqn:diff2\]) which is linear at $z=0$. This solution can be written as an infinite sum $$\begin{aligned}
G_\beta(z) &= A\sum_{n=0}^\infty
{(-1)^n\over(2n+1)!}z^{2n+1}\prod_{i=0}^{n-1}(\beta+i) ,\nonumber\\
&=A\sum_{n=0}^\infty
{(-1)^n\over(2n+1)!}\,{\Gamma(n+\beta)\over\Gamma(\beta)}\,z^{2n+1}.
\label{eqn:sum}\end{aligned}$$ (The second expression is not valid when $\beta$ is a non-positive integer.)
To determine $\beta$, one can notice that the scaling form (\[eqn:scalingF\]) has to match the initial condition when $x$ is large and $t$ of order 1. We thus need to calculate the asymptotic behavior of $G(z)$ when $z$ is large.
For certain values of $\beta$, there exist closed expressions of the sum (\[eqn:sum\]). For instance, $$\begin{aligned}
\label{eqn:Gbeta}
G_{-2}(z) &=& A\left(z+{z^3\over3}+{z^5\over60}\right),\nonumber\\
G_{7\over2}(z) &=&
A\left(z-{z^3\over3}+{z^5\over60}\right)e^{-{z^2\over4}},\nonumber \\
G_{-1}(z) &=& A\left(z+{z^3\over6}\right), \nonumber\\
G_{5\over2}(z)&=& A\left(z-{z^3\over6}\right)e^{-{z^2\over4}}, \\
G_{0}(z)&=& Az, \nonumber\\
G_{3\over2}(z)&=& Aze^{-{z^2\over4}},\nonumber\\
G_1(z) &=& Ae^{-{z^2\over4}}\int_0^z e^{t^2\over4}dt,\nonumber\\
G_{1\over2}(z)&=&A\int_0^ze^{-{t^2\over4}}dt,\nonumber\end{aligned}$$ One can check directly on (\[eqn:diff2\]) that $G_\beta$ has a symmetry $$G_\beta(z)=-i e^{-z^2/4}\,G_{{3\over2}-\beta}(iz).$$
For any $\beta$, one can obtain the large $z$ behavior of $G(z)$. To do so, we note that for $\beta>0$, one can rewrite (\[eqn:sum\]) as $$\begin{aligned}
G_\beta(z)&=&{A\over\Gamma(\beta)}\int_0^\infty dt\; t^{\beta-{3\over2}}
\sin(\sqrt{t} z) e^{-t},\nonumber\\
&=&{2A\over\Gamma(\beta)}z^{1-2\beta}\int_0^\infty
dt\;t^{2\beta-2} \sin(t) e^{-{t^2\over z^2}}.
\label{eqn:integ}\end{aligned}$$ For $0<\beta<1$, the second integral in (\[eqn:integ\]) has a non zero limit and this gives the asymptotic behavior of $G_\beta(z)$ $$G_\beta(z)\simeq-{2A\over\Gamma(\beta)}\cos(\pi\beta)\,
\Gamma(2\beta-1) z^{1-2\beta}.
\label{eqn:asymptot}$$ From (\[eqn:sum\]), one can also show that $$G_\beta''=-{\Gamma(\beta+1)\over\Gamma(\beta)}G_{\beta+1},$$ implying that (\[eqn:asymptot\]) remains valid for all $\beta$ except for $\beta={3\over2}$, $5\over2$, $7\over2$, etc., where the amplitude in (\[eqn:asymptot\]) vanishes. For these values of $\beta$, $G_\beta(z)$ decreases faster than a power law (see (\[eqn:Gbeta\])).
The functions $G_\beta$ calculated so far are acceptable scaling functions for the shape of the front only for $\beta\le{3\over2}$. Indeed, one can see in (\[eqn:asymptot\]) that for ${3\over2}<\beta<{5\over2}$ the function $G_\beta(z)$ is negative for large $z$. In fact, for all $\beta>{3\over2}$, this function changes its sign at least once, so that the scaling form (\[eqn:scalingF\]) is not reachable for an initial $h(x,0)$ which is always positive. It is only for $\beta\le{3\over2}$ that $G_\beta$ remains positive for all $z>0$.
Looking at the asymptotic form (\[eqn:asymptot\]), we see that if initially $h(x,0)=x^\nu e^{-\gamma_0 x}$, the only function $G_\beta(z)$ which has the right large $z$ behavior is such that $1-2\beta=\nu$, and this gives, together with (\[eqn:pos1\]), the expression (\[eqn:pos(nu-1)/2\]) for the shift of the position. As the cases $\beta>{3\over2}$ are not reachable, all initial conditions corresponding to $\nu<-2$ or steeper (such as step functions) give rise to $G_{3\over2}$ and the shift in position given by (\[eqn:pos3/2\]).
All the analysis of this appendix can be extended to other traveling wave equations such as (\[eqn:mf\]), with more general functions $v(\gamma)$ (having a non-degenerate minimum at $\gamma_0$) as in (\[eqn:v(g)2\]). Then the expressions (\[eqn:pos3/2\], \[eqn:pos(nu-1)/2\]) of the shift become $$X_t\simeq v_0 t -{3\over2\gamma_0}\log t$$ and $$X_t\simeq v_0 t -{1-\nu\over2\gamma_0}\log t.$$
We thank C. Appert, V. Hakim and J.L. Lebowitz for useful discussions.
[10]{}
R. A. Fisher, “The Wave of Advance of Advantageous Genes,” Annals of Eugenics [**7,**]{} 355–369 (1937).
A. Kolmogorov, I. Petrovsky, and N. Piscounov, “Étude de l’Équation de la Diffusion avec Croissance de la Quantité de Matière et son Application à un Problème Biologique,” Moscou Univ. Bull. Math. A [**1,**]{} 1 (1937).
D. G. Aronson and H. F. Weinberger, “Multidimensional Nonlinear Diffusion Arising in Population Genetics,” Advances in Mathematics [**30,**]{} 33–76 (1978).
G. Dee and J. S. Langer, “Propagating Pattern Selection,” Physical Review Letters [**50,**]{} 383–386 (1983).
M. Bramson, P. Calderoni, A. D. Masi, P. Ferrari, J. L. Lebowitz, and R. H. Schonmann, “Microscopic Selection Principle for a Diffusion-Reaction Equation,” Journal of Statistical Physics [**45,**]{} 905–920 (1986).
W. van Saarloos, “Front Propagation into Unstable States. [L]{}inear Versus Nonlinear Marginal Stability and Rate of Convergence,” Physical Review A [**39,**]{} 6367–6390 (1989).
P. Collet and J.-P. Eckmann, [*Instabilities and Fronts in Extended Systems*]{} (Princeton University Press, 1990).
A. R. Kerstein, “Computational Study of Propagating Fronts in a Lattice-Gas Model,” Journal of Statistical Physics [**45,**]{} 921–931 (1986).
D. G. Aronson and H. F. Weinberger, “Nonlinear Diffusion in Population Genetics, Combustion, and Nerve Propagation,” Lecture Notes in Mathematics [**446,**]{} 5–49 (1975).
M. Bramson, [*Convergence of Solutions of the Kolmogorov Equation to Traveling Waves*]{}, No. 285 in [*Memoirs of the American Mathematical Society*]{} (AMS, 1983).
W. van Saarloos, “Dynamical Velocity Selection: Marginal Stability,” Physical Review Letters [**58,**]{} 2571–2574 (1987).
W. van Saarloos, “Front Propagation into Unstable States: Marginal Stability as a Dynamical Mechanism for Velocity Selection,” Physical Review A [ **37,**]{} 211–229 (1988).
H. P. Breuer, W. Huber, and F. Petruccione, “Fluctuation Effects on Wave Propagation in a Reaction-Diffusion Process,” Physica D [**73,**]{} 259–273 (1994).
J. Cook and B. Derrida, “Lyapunov Exponents of Large, Sparse Random Matrices and the Problem of Directed Polymers with Complex Random Weights,” Journal of Statistical Physics [**61,**]{} 961–986 (1990).
R. van Zon, H. van Beijeren, and C. Dellago, “Largest Lyapunov Exponent for Many Particle Systems at Low Densities,” Physical Review Letter [**80,**]{} 2035–2038 (1998).
H. P. Breuer, W. Huber, and F. Petruccione, “The Macroscopic Limit in a Stochastic Reaction-Diffusion Process,” Europhysics Letters [**30,**]{} 69–74 (1995).
B. Derrida and H. Spohn, “Polymers on Disordered Trees, Spin Glasses, and Traveling Waves,” Journal of Statistical Physics [**51,**]{} 817–840 (1988).
B. Derrida, “Mean field theory of directed polymers in a random medium,” Physica Scripta [**38,**]{} 6–12 (1991).
D. A. Kessler, H. Levine, D. Ridgway, and L. Tsimring, “Evolution on a Smooth Landscape,” Journal of Statistical Physics [**87,**]{} 519–544 (1997).
L. Tsimring, H. Levine, and D. A. Kessler, “[RNA]{} Virus Evolution via a Fitness-Space Model,” Physical Review Letters [**76,**]{} 4440–4443 (1996).
W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, [ *Numerical Recipes in C*]{} (Cambridge University Press, 1994).
A. R. Kerstein, “A Two-Particle Representation of Front Propagation in Diffusion-Reaction Systems,” Journal of Statistical Physics [**53,**]{} 703–712 (1988).
M. D. Bramson, “Maximal Displacement of Branching [B]{}rownian motion,” Communications In Pure and Applied Mathematics [**31,**]{} 531–581 (1978).
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abstract: 'Previously, we proposed that the polarization and capacitive charge in screens the external electric field that hinders charge transport. We argue here that this screening effect is in significant part responsible for the power conversion characteristics and hysteresis in photovoltaic cells. In this paper, we implement capacitive charge and polarization charge into the numerical model that we have developed for perovskite solar cells. Fields induced by these two charges screen the applied hindering field, promote charge transport, and improve solar cell’s performance, especially in solar cells with short diffusion lengths. This is the reason why perovskite solar cells made from simple fabrication methods can achieve high performance. More importantly, with relaxations of capacitive charge and polarization charge, we quantitatively reproduce experimental “anomalous” hysteresis J-V curves. This reveals that both polarization relaxation and ions relaxation could contribute to anomalous hysteresis in perovskite solar cells.'
author:
- Yecheng Zhou
- Fuzhi Huang
- 'Yi-Bing Cheng'
- 'Angus Gray-Weale'
bibliography:
- 'numerical\_hysteresis.bib'
nocite:
- '[@Zhou2016]'
- '[@Zhou2016]'
- '[@Zhao2016]'
- '[@Hou2016; @Kim2015; @Miyano2016a; @Heo2015]'
- '[@Zhou2014]'
title: Numerical analysis of a hysteresis model in perovskite solar cells
---
Introduction
============
Perovskite solar cells have achieved power conversion efficiencies (PCEs) up to 22% in just five years.[@Kojima2009; @Record] They attract great attention due to their high performance and anomalous hysteresis. It is believed that the large charge carrier diffusion lengths and the compensated field are two key factors for high performances of hybrid perovskite solar cells. In experiment it was observed that the slowly built compensated field contributes to the anomalous hysteresis.[@Tress2015; @Unger2014] It is believed that the compensated field is induced by ion migration and electronic charge traps.[@Tress2015; @Unger2014; @VanReenen2015] The compensated field works as a screening effect resulting in a high dielectric constant, which has been observed up to 1000 for .[@Juarez-Perez2014; @Lin2014]
At high frequencies only electronic orbitals are able to respond and become polarized, whereas at low frequencies all of the electronic orbitals, defect charges and ions are able to respond. As a result, at high frequencies low dielectric constants of 6-7 are observed,[@Zhou2015; @Juarez-Perez2014] compared to dielectric constants approximately 100 times higher at low frequencies. [@Juarez-Perez2014] This frequency dependent dielectric constant behaviour is consistent with the compensated field and the extremely slow photo-conductivity response in solar cells.[@Gottesman2014] We argue that both compensated field and slow photo-conductivity come from certain slow relaxations. This relaxation screens external fields and increases the dielectric constant. It is widely believed that ion migration is one possible reason for this slow relaxation.[@Unger2014; @Beilsten-Edmands2015a; @Yang2015; @Snaith2014; @Tress2015; @VanReenen2015] But the slow response experiment carried out by Gottesman *et al.* shows two opposite behaviours of decreased/increased photo-conductivity in identically constructed devices. This cannot be explained by ion migration.[@Gottesman2014] It needs to be revised that ion migration is not the only origin of hysteresis. VanReenen *et al.* modeled this hysteresis and found the combination of ion migration and electronic traps brings about hysteresis.[@VanReenen2015]
Besides ion migration, polarization is another possible reason for hysteresis. Beilsten-Edmands *et al.* claimed that there is no ferroelectric nature contribution to hysteresis due to the intrinsic polarization being too small.[@Beilsten-Edmands2015a] They treated the polarization at very high frequency ($f\rightarrow \infty$) as its intrinsic polarization, however this is incorrect. For any ferroelectric polarizations under a very high frequency it should be zero as the electronic orbitals and ions are unable to respond. There is no direct evidence to deny ferroelectric polarization in perovskite solar cells. Additionally, Kutes *et al.* showed a direct observation of ferroelectric polarizations.[@Kutes2014] Debate of polarization in hysteresis continues.
From a theoretical standpoint most research supports the existence of polarization. First-principles studies have shown that the energy barriers for defect migrations are from 0.08 eV to 0.40 eV depending on ion types[@Haruyama2015; @Azpiroz2015] This energy barrier is low enough to be crossable at room temperature. We also reported the energy barrier for methylammonium ions () reorientation is about 0.01 eV to 0.098 eV, which depends on the initial and final orientations and neighbor orientations.[@Zhou2015] From the base of their energy landscapes, polarization is easier to respond and screen external fields. ions are able to be rotated collectively under an applied external field, which then polarises the crystal or thin film. This collective reorientation and polarization combined with capacitive charges screen external hindering field and promote power conversion efficiency. Hence, another slow relaxation should be polarization relaxation. Our argument is in good agreement with Sanchez’s experiment that the slow dynamic process depends strongly on the organic cation, or .[@Sanchez2014] The rotation of ions and migration of ions are systemically discussed by Frost *et al.*.[@Frost2014; @Frost2016; @Weller2015b] They argued that the internal electrical fields associated with polarization contribute to hysteresis in J-V curves.[@Frost2014] They also observed that a single cation rotation and anion migration take several picoseconds.[@Frost2016; @Weller2015b] Hence, we argue here that polarization and ion migration are both possible to induce screening fields and contribute to hysteresis.
We propose that the screening charge contains two components: one is the polarization charge resulted from reorientation and inorganic frame, which we name the polarization charge; another is capacitive charges from defects and trapped charges including ions. For normal perovskite solar cells without polarization, photon generated charge carriers accumulate in defects near interfaces, shown as blue charge in Figure \[fig:mechanism\]. These accumulated charges induce a field that counteracts the applied hindering field. We name the applied field the hindering field because it is opposite to the work current vector. For perovskite solar cells, polarization charges can further counteract the hindering field and promote charge carrier transport. Due to these capacitive and polarization screening effects, high PCEs are expected for hybrid perovskite solar cells. This is the first hypothesis we are going to test.
![Our proposed mechanism of perovskite solar cells[]{data-label="fig:mechanism"}](mechanism){width="8.5cm"}
The second hypothesis we are going to test is that hysteresis in J-V curves come from these two kinds of slow relaxations. As their relaxations are slow, screening fields fall behind the applied hindering field if the measurement scanning is fast enough. This delay induces hysteresis. We apply numerical simulations to reproduce and explain the anomalous hysteresis effect in perovskite solar cells. Our results show both capacitive charge and polarization charge could contribute to hysteresis effects. Relaxation times of these two charges determine the overall behaviour of scan rate dependent hysteresis.
Methods
=======
Our model is based on the continuity equations and Poisson’s equation in one dimension, adapted for perovskite solar cells:[@Zhou2016] $$\label{eq:sc}
\begin{split}
J_n= eD_n\frac{\partial n}{\partial x}+n \mu_n F \\
J_p= -eD_p\frac{\partial p}{\partial x}+p \mu_p F\\
\frac{\partial J_n}{\partial x}=-eG+eR \\
\frac{\partial J_p}{\partial x}=eG-eR \\
\frac{\partial F}{\partial x}= \frac{p-n}{\varepsilon \varepsilon_0 }
\end{split}$$ where, $J_n$ and $J_p$ are electron current and hole current respectively; $n$ is electron density and $p$ is hole density; $\mu$ and $D$ are charge carriers mobility and diffusion coefficient respectively; footnote symbols $_n$ and $_p$ mean they belong to electron and hole respectively; $G$ and $R$ are generation rate and recombination rate; $F$ is the external applied electric field. Boundary conditions and parameters are shown in reference [ ]{}.
![Integral and fitting incident photon density of AM 1.5 Standard Spectrum. The insert shows the integral and fitting result of between band gaps of 1.45 eV and 1.62 eV. Error in this region is smaller than 2%.[]{data-label="fig:AM1_5"}](AM1_5){width="8.5cm"}
Light harvest and charge generation are expressed as $G=IPCE\times N$, where $IPCE$ is Incident Photon-to-Current Efficiency. $N$ is the incident photon density calculated by $\int \frac{I(\lambda)}{hc/\lambda} d\lambda$, where $I(\lambda)$ is incident light density, $h$ is the Plank constant, $c$ is the speed of light and $\lambda$ is the photon’s wavelength. According to the Beer-Lambert law, light intensity inside a material decays exponentially from the surface as: $I(\lambda,x)=I(\lambda,0)e^{-\alpha(\lambda) x}$, where $x$ is the incident depth from the surface and $\alpha(\lambda)$ is the absorption coefficient. Therefore, the charge generation rate becomes: $$\begin{aligned}
\label{eq:ag}
G(x)=\int_0^{\lambda_0}G(\lambda,x) d\lambda =\int_0^{\lambda_0} IPCE(\lambda)\times \frac{I(\lambda,0)\times \alpha(\lambda) \times e^{-\alpha(\lambda) x}}{hc/\lambda} d\lambda\end{aligned}$$ $\lambda_0$ is the absorption edge, corresponding to the bandgap. After a photon is absorbed, an exciton formed by a separated hole and electron pair is generated. The hole and electron attract each other and try to combine. We assume all excitons separate to pairs of free holes and electrons, which means $IPCE(\lambda)=100\%$.[@Zhou2016] Then the charge generation rate becomes: $$\label{eq:gr}
G(x)=\int_0^{\lambda_0} IPCE(\lambda)\times \frac{I(\lambda,0)\times \alpha(\lambda) \times e^{-\alpha(\lambda) x}}{hc/\lambda} d\lambda=\alpha N_0 e^{-\alpha x}$$ where, $N_0 = \int_0^{\lambda_0} \frac{I(\lambda,0)}{hc/\lambda} d\lambda$; $I(\lambda,0)$ is the AM1.5 Standard Solar Spectra. The experiment band gap of is in the region from 1.45 eV to 1.70 eV.[@Yin2014; @Kim2012b; @Stoumpos2013; @Zhou2014; @Yamada2014; @Schulz2014] In our simulations, we use a linear fitting to estimate incident photon density near 1.55 eV. $N_0 =-2.20\times 10 ^{17} \times E_{bgap}$ (eV)+5.12$\times 10 ^{17}$ (cm$^{-2}$), where $E_{bgap}$ is the band gap of the perovskite thin film. As shown in Figure \[fig:AM1\_5\], for the band gap in the range of 1.45 eV to 1.62 eV, the accurate integral density and fitted density are almost the same. The incident photon density is calculated to be $1.59\times 10^{17}$ cm$^{-2}$, if the band gap is 1.60 eV.
A planar perovskite solar cell has a sandwich structure. Two electrodes clip a compact () layer, a perovskite layer and a hole transport layer (Spiro-OMeTAD layer). Electrodes are conductors and the potential in a conductor is constant, hence we neglect their potential drop in this discussion. The most important parts are the clipped compact layer, perovskite layer and hole transport layer. Under light irradiation, charge carriers are generated and flow. Here we describe the resistances of the compact layer, perovskite layer and hole transport layer as $R_t$, $R_p$ and $R_s$ respectively. Based on Ohm’s law, the voltage drop in the perovskite layer is $V_p=\frac{V_0 \times R_p}{R_t+R_p+R_s}$, where $V_0$ is the applied voltage. Two facts refute this assumption. The first is that Ohm’s law only refers to drift current, in which case the current is in the same direction as the field, while current in solar cells is opposite to the applied field. The second is that the diffusion current, which is beyond the Ohm’s law and the Drude model description, is larger than the drift current. Hence, it is much more reasonable to consider them as dielectric materials. Then the voltage drop across the perovskite layer is: $$\label{eq:vp}
V_p=\frac{V_0\times \epsilon_t \times \epsilon_s \times d_p}{(\epsilon_t\times \epsilon_s \times d_p+\epsilon_t\times \epsilon_p \times d_s+\epsilon_s\times \epsilon_p\times d_t )}= A\times V_0$$ where $\epsilon$ and $d$ are dielectric constant and thickness of corresponding layers, footnote $_t$ for , $_p$ for perovskite and $_s$ for Spiro-OMeTAD. $A$ is the percentage of applied voltage drop across the perovskite layer, which equals $\frac{ \epsilon_t \times \epsilon_s \times d_p}{(\epsilon_t\times \epsilon_s \times d_p+\epsilon_t\times \epsilon_p \times d_s+\epsilon_s\times \epsilon_p\times d_t )}$. When $\epsilon_t$, $\epsilon_p$ and $\epsilon_s$ are 100,[@tio] 1000,[@Juarez-Perez2014] and 3,[@Snaith2006] and their corresponding layers thickness are 50, 380 and 200 nm,[@Zhou2014] then the voltage drop across the perovskite layer is A=0.56%. For the following simulations, the boundary field at the ends of the perovskite layer are the same, which are $F(x=0)=F(x=d)=\frac{V_p}{d_p}$.
Defects and traps in semiconductor interfaces are able to charge and discharge as capacitors do. Hence, we name these charges the capacitive charge. In our previous work,[@Zhou2015] we also showed that can be polarized by external fields through rotating ions and tilting inorganic frame. Due to energy barriers, ion needs some time to respond. We name this response the polarization relaxation. We assume that the polarization field relaxes (increases/decreases) exponentially with delay time ($\delta t$): $\delta \textbf{F}[\delta t]= \delta \textbf{F}[\infty] \times (1-e^{- \delta t /\tau_c})$, where $\delta \textbf{F}[\infty]$ is the field difference between the initial field and final field with infinite relaxation time.
The field in bulk materials within a static external field ($\textbf{F}_0$) is $\textbf{F}=\textbf{F}_0+\textbf{F}_{c}$, where $\textbf{F}_{c}=-S_{c}\textbf{F}_0$ and $S_{c}$ is the screening coefficient for capacitive charges. If there is a polarization field, then the total field becomes $\textbf{F}=\textbf{F}_0+\textbf{F}_{c}+\textbf{F}_{p}$, where $\textbf{F}_{p}=-S_{p}\textbf{F}_0$ and $S_{p}$ is the screening coefficient due to polarization. If the measurement voltage is applied step-by-step, then the field at time $t$ with applied voltage $V$ can be expressed: $$\textbf{F}[V,t]=\textbf{F}_0[V,t]+\textbf{F}_{c}[V,t]+\textbf{F}_{p}[V,t],$$ where: $$\textbf{F}_c[V,t]=(\textbf{F}_{c}[V,\infty]-\textbf{F}_c[V-\delta V,t- \delta t])(1-e^{- \delta t /\tau_c})+\textbf{F}_c[V-\delta V,t- \delta t]$$ and $$\textbf{F}_p[V,t]=(\textbf{F}_{p}[V,\infty]-\textbf{F}_p[V-\delta V,t- \delta t])(1-e^{- \delta t /\tau_p})+\textbf{F}_p[V -\delta V,t- \delta t]$$ in which, $\textbf{F}_{c}[V,\infty]$ and $\textbf{F}_{p}[V,\infty]$ are screening fields under applied voltage $V$ with infinite delay time for capacitive charge and polarization charge, respectively; $\delta t$ is the measurement delay time and $\tau_c$ and $\tau_p$ are relaxation times of capacitive charge and polarization charge respectively; $\textbf{F}[V,t]$ and $\textbf{F}_0[V,t]$ are the total field and the hindering field at time $t$ with applied voltage $V$; $\textbf{F}_c[V-\delta V,t- \delta t]$ and $\textbf{F}_p[V-\delta V,t- \delta t]$ are the field of capacitive charge and the field of polarization charge of last step, respectively.
Results and discussion
======================
According to Equation (\[eq:vp\]), the voltage drop across the perovskite layer is a function of dielectric constant and thickness of each layer. At very slow scan rates (such as 25 mV/s), low frequency dielectric constants are exhibited. The low frequency dielectric constants of [@tio] and perovskite[@Juarez-Perez2014] have values of up to 173 and 1000 respectively. In this case $A$ is 0.5%. For quick scans, dielectric constants at high frequency are exhibited. The high frequency dielectric constants of , perovskite, Spiro-OMeTAD are 86,[@tio] 6,[@Juarez-Perez2014] and 3,[@Snaith2006] respectively. In this case the potential drop across the perovskite layer is 43% of the applied voltage. With a long delay time, screening fields of capacitive charge and polarization charge contribute to the large dielectric constant, and this leads to a small $A$–the percentage of applied voltage drop across the perovskite layer. We propose that the increased dielectric constant of perovskites come from the slow polarization relaxation and ion migration, which has been shown to be possible via DFT calculations.[@Zhou2015; @Haruyama2015; @Azpiroz2015; @Weller2015b]
A $V_{oc}$ (mV) $J_{sc}$ (mA) PCEs (%) FF
------------ --------------- --------------- ---------- --------
0 1176 22.77 22.44 0.8380
25 1154 22.77 21.36 0.8131
43 1142 22.77 20.93 0.8044
100 1117 22.77 19.90 0.7821
Experiment 1130 22.75 19.30 0.7507
: Performance of perovskite solar cells with different values of $A$.[]{data-label="tab:Comparison"}
Based on experimental parameters, the percentage of voltage drop across the perovskite layer is between 0.5% to 43%. Here we estimate the solar cell’s performance with respect to $A$ with values from 0 to 100%. The thicknesses of simulated solar cells are 350 nm.[@Zhou2014] Diffusion coefficients of the perovskite were set to be (for electrons) and (for holes).[@Stranks2013a] The band gap is 1.55 eV.[@Zhou2014] Carrier lifetimes in bulk materials without interfaces are assumed to be 736 ns.[@Zhou2014] The presence of interfaces decreases charge carrier lifetime which indicates a high recombination rate at the interface. More details have been discussed in reference [ ]{}. Thus, we assume the interface recombination region is 2 nm and its charge carriers lifetime is 7 ns, which is determined according to experimental $J_{sc}$. In our simulation, for all values of $A$, $J_{sc}$ is , which is close to the experimental value of . Figure \[fig:screen\] and Table \[tab:Comparison\] show experiment and simulation current density–voltage (J-V) curves of solar cells with different $A$. The orange square line is the performance of a solar cell fabricated by Zhou.[@Zhou2014] Our simulated FFs are larger than the experiment value of 75%. $V_{oc}$ in these simulations are around the experiment value of 1130 mV. When $A$ is 100%, no screening effect is exhibited, and the J-V curve is closest to experiment. However, as shown in the insert of Figure \[fig:screen\], the experiment current decreases between theoretical lines with $A=25\%$ and $A=43\%$ between 0 - and 22 - . Below 300 mV, the blue diamonds line simulated with $A=25\%$ is the closest to Zhou’s experiment. The simulated $V_{oc}$, $J_{sc}$, FF and PCE are 1154 mV, 22.77 mA, 81.3% and 21.36%, respectively.
As we proposed that polarization charges can further reduce the hindering field at very low scan rate, then the dielectric constant can be up to 1000.[@Juarez-Perez2014] In this case, the voltage drop across the perovskite layer is almost zero, which implies the hindering field has disappeared. With these parameters, our model gives a PCE of 22.4%. Therefore. we conclude that by measuring with slow scan rate a high dielectric constant of the perovskite layer results, which reduces the voltage drop across the perovskite layer. Therefore, charge carriers are easier to transfer out, and a higher PCE is achieved. This conclusion is consistent with Sherkar’s result that devices having polarization in the plane of devices show high $J_{sc}$ and FF.[@Sherkar2015] The difference is that the screening effect in our simulation comes from both polarization and ion migration. It is worth noticing that the polarization comes from both ions and the inorganic frame.[@Zhou2015]
![Performance of an unoptimized perovskite solar cell and simulated perovskite solar cells with different screening coefficients, $S$=1-$A$. The thickness of the simulated solar cell is 350 nm. The diffusion coefficients of the perovskite layer were assumed to be and for electrons and holes respectively. The band gap is 1.45 eV. The lifetime in the cell is 57 ns. The working conditions were set to be at 300 K and 1 sun (1.5AM). The charge carrier lifetime in the perovskite layer is 57 ns. The interface region is set as 12 nm thick with an interface charge carrier lifetime of 0.37 ns.[]{data-label="fig:yibing"}](fityibing1){width="8.5cm"}
The screening improvement is small for solar cells with very high mobilities and long lifetimes. But PCEs of solar cells with poor electron transport ability can be significantly improved through screening effects. A simply made, unoptimized perovskite solar cell shows poor performance as shown by the green triangle in Figure \[fig:yibing\]. Its performance can be reproduced by our model with short charge carriers lifetime and a thick interface recombination. The simulated $J_{sc}$ is 10.5 , in agreement with experiment. For an ideally screened solar cell its current is almost constant before 700 mV. Whereas, for a solar cell without screening effect its current decreases drastically near 0 mV then linearly with voltage until 700 mV. The simulated curve with 70% screen coefficient is in good agreement with experimental curve when the output voltage is lower than 600 mV. The variation at high voltage is due to higher fill factor of our idealized model. This implies the remaining percentage of the applied voltage in the experimental real solar cell is about 30%, which is in good agreement with the range of 0.5% - 43% calculated from experimental parameters. This is evidence that screening effects are present in perovskite solar cells. The PCE of the solar cell without screening is 4.3%, which is improved to 5.9% if a 70% screening effect is present. With an ideal screening the PCE would reach 7.5%. This results in a more than 70% improvement compared to the unscreened solar cell. The screening effect in solar cells improves PCE, especially in solar cells with poor charge carrier conductivity. Therefore, we draw a conclusion that the screening effect is the reason why simply made perovskite solar cells could achieve high performance.
In a normal solar cell without polarization, screening is also present. But all of screening charges are capacitive charges due to defects and trapped charge carriers. These trapped charges will take part in charge recombination, which in turn reduces its current and PCE. For , polarized charges cannot be combined unless polarizations become totally disordered. This also gives the benefit of a large current in the device. It is worth noticing that the polarization is not only from ions but also from the inorganic frame. For perovskite materials, even though there are no cations with dipoles, the materials can be ferroelectrically polarized as in .[@batio3] This means perovskite materials, such as and also have the potential to make high performance solar cells.
In the above simulations, diffusion coefficients were assumed constant. Actually, screening fields also make charge carrier transport easier increasing the diffusion coefficients. Therefore, PCEs of perovskite solar cells can be further improved. Capacitive charge is believed to be the main factor for hysteresis in silicon solar cells and DSCs.[@Koide2004; @Koide2005; @Herman2012] When the measurement scan rate is too fast, capacitive charges are unable to catch up with the changing of scanning field and hysteresis is observed. Usually, more obvious hysteresis is observed with faster scanning. As charges are able to be trapped and de-trapped in silicon solar cells quickly, an extremely fast scan rate (short delay time) is required to observe hysteresis in silicon solar cells. The measurement delay time is estimated to be around 1 ms.[@Herman2012] Charge in DSCs moves slower and the trap and de-trap process takes longer. The relaxation time in DSCs can be up to 100 ms. The required scanning speed to observe hysteresis can be down to .[@Koide2004; @Koide2005]
What makes perovskite solar cell hysteresis mysterious is not only its large relaxation time, but also the changing of hysteresis with scan rate. For silicon solar cells and DSCs, a shorter delay time (higher scan rate) brings about more obvious hysteresis.[@Koide2004; @Koide2005; @Herman2012] In contrast, in perovskite solar cells, a shorter delay time can either induce a more or less obvious hysteresis.[@Unger2014; @Tress2015; @Snaith2014] As shown in Figure \[fig:Ehys\], even in the same time region, some experiments increase and others decrease. The hysteresis constant is defined as the difference between the maximum PCEs of forward and backward measurement. The presence of two peaks may reflect two different mechanisms, polarization and ion migration. A single ion takes several ps to rotate,[@Frost2016; @Weller2015b; @Leguy2015] while, the slow charge relaxation time in perovskite solar cells is in the order of 1-30 s.[@Unger2014; @Snaith2014; @Gottesman2014; @Tress2015; @Wei2014e] This is because the collective relaxation of millions of ions should take much longer. The timescale for a domain wall to traverse a typical device is about 0.1-1 ms.[@Leguy2015] For ion migration, it should take longer due to its larger energy barrier than rotation. If we select a typical 0.2 eV [@Haruyama2015; @Azpiroz2015]energy barrier for ion migration and 0.05 eV[@Zhou2015] for rotation, and assume the relaxation time is proportional to $e^{-E_b/kT}$, where $E_b$ is energy barrier and kT is thermal energy, the relaxation time of ions is about 320 times higher than the relaxation time of ions polarization. Considering longer relaxation time for the experiment, we set capacitive charge relaxation time as 2.5 s and 250 ms for polarization in below simulations.
Lots of experiments have been designed to study hysteresis, the most systematic experiment is the work done by Tress *et al*.[@Tress2015] They measured one hybrid solar cell forward and backward with different scan rates from 10 to 100,000 mVs$^{-1}$. Every measurement begins with certain polarization states which are pre-polarized with the same condition. As all the measurements were taken from a single solar cell with a certain initial state we could model their J-V curves under different scan rates with one set of basic parameters. The parameters used to model the work by Tress *et al.* are shown in Table \[tab:param\]. In Figure \[fig:fit-tress\], experimental J-V curves are drawn with dashed lines, while simulated J-V curves are plotted with solid lines. The PCE is higher during backward scanning. The voltage falls during the backward scan, and the screening field lags behind. It follows that the screening field is higher during the backward scan than during the forward scan. That means that the hindering field is more screened during the backward scan leading to a higher efficiency.
Our simulated $V_{oc}$ are about 915 mV, which are in good agreement with Tress’s experiment. As we expect, simulated performance of the backward scan is higher than that of the forward scan. The current at voltage of -1 V with scan rate of is , which is close to the experimental value of . At -1 V, currents modeled with scan rates of , , and are , , and , respectively. All of these current densities are close to the experiment , , and with errors smaller than . Not only do these typical values agree, but the J-V curve also have a similar shape. By applying one set of parameters for a certain solar cell, we reproduce its J-V curves under different conditions. The agreement between Tress’ experiment and our simulation suggests that our model and the proposed mechanism are correct.
Symbol Meaning Value
------------- -------------------------------------- ------------------------------------------
$E_{bgap}$ Band gap 1.52 eV
T Temperature 300 K
$I_l$ Light intensity 1.5 AM
$IPCE$ IPCE 100%
$d$ Perovskite thickness 350 nm
$N_c$,$N_v$ Density of States $3.97 \times 10^{18}cm^{-3}$.[@Zhou2015]
$\alpha$ Absorption coefficient $5.7 \times10^4cm^{-1}$.[@Xing2013]
A Voltage drop percentage 8.4%
$D_n$ Electron diffusion coefficient 0.030 $cm^2s^{-1}$
$D_p$ Hole diffusion coefficient 0.063 $cm^2s^{-1}$
$\tau_p$ Polarization relaxation time 0.25 s
$\tau_c$ Capacitive charge relaxation time 2.5 s
$S_p$ Polarization screen coefficient 50%
$S_c$ Capacitive charge screen coefficient 49%
$\tau$ Charge carriers lifetime 10.2 ns
: Assumed and experimental parameters used to simulate Tress’ solar cells.[]{data-label="tab:param"}
For normal hysteresis from typical capacitive charges, as seen in Dye Sensitized Solar Cells made by Koide *et al.*[@Koide2004; @Koide2005] it becomes more extreme when the scan rate increases, as shown in Figure \[fig:Ehys\]. In contrast, Snaith observed anomalous hysteresis, which becomes more extreme as the scan rate is reduced. Even at extremely slow scan rates, it is still significant.[@Snaith2014] We interpret this phenomenon as being a result of slower ion migration. As observed in experiment, the relaxation time can be as short as 1 s,[@Sanchez2014] or alternatively as long as several tens of seconds.[@Unger2014; @Snaith2014; @Gottesman2014; @Tress2015; @Wei2014e] The performance is similar in Tress’ and Snaith’s experiments which suggests their solar cells have similar electronic parameters. If we change the capacitive relaxation time to 50 s without changing other parameters, our simulations give hysteresis in good agreement with Snaith’s experiment.[@Snaith2014] Jeon’s solar cells show higher PCEs, hence, their solar cells should have better electronic properties.[@Jeon2014] To model Jeon’s solar cells, we increased the charge carrier lifetime to 80 ns and decreased $S_{c}$ to 10%. Hysteresis seen in Jeon’s experiment is also repeated with a capacitive charge relaxation time of 0.9 s. All of the simulation and experimental hysteresis are shown in Figure \[fig:Ehys\]. It is found that for very low or high scan rate measurements, the hysteresis decreases. This is due to the relaxation field cannot follow the changing of external fields for an extremely high scan rate. The screening field is constant during a forward-backward measurement. No hysteresis is expected as there is no difference between forward and backward scans. While, for the case of very slow scan, there is enough time to relax, the screening field is always proportional to the external applied field. Screening fields are the same under a certain applied voltage no mater it is a forward or a backward scan. Hence, hysteresis will not happen at very slow scan rate.
![Scanning rate dependent hysteresis constants. The hysteresis constant is defined as the difference between the maximum PCEs of forward and reverse measurement. Experimental data is depicted by solid lines with filled points. Numerical simulation result are plotted as dashed lines with open points. T$_c$ is the polarization relaxation time used in the model.[]{data-label="fig:Ehys"}](Ehys2){width="8.5cm"}
Hysteresis-free inverted perovskite solar cells have been made from interface engineering or using electron transfer layer.[@Kim2015; @Miyano2016a; @Hou2016] We argue these hysteresis-free devices are due to the improvement of interface states rather than the inverted structure, because normal structure perovskite solar cells have been made without large hysteresis by implementing [@Zhao2016] or PCBM[@Heo2015]. As illustrated in Figure \[fig:mechanism\], both the polarization charge and the ionic charge are accumulated near interfaces. These charges could be compensated or neutralized by contact layers. For an ideal crystal without any defects, ions cannot migrate because there is no vacancy to go. Whereas, defects exist at interfaces, such as dangling bonds. Therefore, it is possible to reduce and even to eliminate ion migration and it induced hysteresis through interface engineering or decreasing defects in thin films. For uniform polarizations, there is no net charge overall and also no net charge in the bulk. All of the polarization charges are near surfaces or interface. In perovskite solar cells, if polarizations present, net charges should locate at interfaces between perovskite and contact layers: one end of perovskite is positive and another is negative. If we impose a contact layer with negative charges at the surface on the end with positive charges, and a contact layer with positive charges at the surface on the end with negative charges, these polarization charges are neutralized. Therefore, this part of the hysteresis also could be reduced or eliminated through interface engineering. The slow relaxed polarization of perovskite is also possible to be compensated by the polarization of contact layers. Fullerene and its derivatives, such as [@Chaban2015] and PCBM[@Tada2011; @Ryno2016] are polarisable. Hence they are good candidates to compensate the polarization of perovskite and eliminate hysteresis of PSCs. Some of these materials have been successfully applied to reduce or eliminate hysteresis, such as in Ref.[ ]{} and PCBM in Ref.[ ]{}.
Conclusion
==========
Using numerical simulations we have confirmed that the screening effect improves perovskite solar cell performance. This improvement is more obvious in solar cells made from simple methods. On the basis of our previous DFT calculations, we argue that the screening field comes from both ion migration and polarization charge. This field weakens the hindering field, promotes charge transport and improve PCEs.
Due to slow polarization and ion relaxation, the screening field is delayed, which leads to hysteresis. As the relaxation time of capacitive and polarization charges are in different scales, rate dependent hysteresis behaviors become more complicated. With the relaxation of capacitive and polarization charges we reproduced various measured hysteresis curves.[@Unger2014; @Tress2015; @Jeon2014] Using similar parameters and different scan rates, we also reproduced the hysteresis effects observed by Snaith *et al.*. These results suggest that hysteresis is caused by two kinds of very slow relaxations. This agreement with measured hysteresis, in turn, supports our proposed mechanism that polarization and capacitive charge take part in the screening of the hindering field and improves its PCE. We show that not only can ion migration cause hysteresis, but also polarization can as well. Although, both ion migration and polarization are bulk properties, they could be affected by interface states and contact layers. Polarizable contact materials, such as fullerene and its derivatives, may be good candidates to compensate screening fields from polarization or ionic charges and eliminate hysteresis of PSCs.
In this paper we assumed that one of the relaxations is polarization based on Gottesman’s experiment,[@Gottesman2014] but this is not necessary. Any slow relaxation response with the ability to screen external hindering field could bring about hysteresis. This slow relaxation could also be different kinds of ion migration. As polarization relaxation and ion migration usually exhibit similar behavior, such as thickness dependence.[@Kim2007; @Bazant2004] To determine whether these slow responses are polarization or ion migration, more experiments and theoretical works should be carried out further.
Acknowledgment
==============
We greatly appreciate the contribution of Professor Yang Yang of UCLA and Professor Qi Chen for providing their initial data in reference [ ]{}. Their initial data which is used to plot the data “Exp-Zhou" in Figure \[fig:screen\] allowed us to compare the experiment and theory in detail.
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abstract: |
We present X-ray observations of the interacting pair of galaxies NGC 4410a/b with the ROSAT HRI and PSPC. The ROSAT HRI images reveal a point-like source corresponding to NGC 4410a and an X-ray halo, extending 10 from the nucleus toward the southeast. The halo emission accounts for $\sim$1/3 ($L_{\mathrm X}$ = 1.3 erg s$^{-1}$ in the 0.1 - 2.4 keV ROSAT band) of the total X-ray emission detected from NGC 4410a. The spectrum of the total X-ray emission from NGC 4410a can be fitted at best with a two-component emission model, combining a Raymond-Smith spectrum with $T$ = 10$^7$ K and a power-law spectrum ($\Gamma$ = 2.2). The fraction of the thermal component to the total flux within this model is 35%, supporting the results of the HRI observation. The total unresolved X-ray luminosity detected with the PSPC amounts to 4 erg s$^{-1}$ in the ROSAT PSPC band (for $D$ = 97 Mpc).
A preferable explanation might be the interaction between the two galaxies causing a circumnuclear starburst around the active nucleus in the peculiar late type Sab galaxy NGC 4410a. As a cumulative effect of exploding stars a superbubble forms a bipolar outflow from the galactic disk with an expansion time of $\sim$8 Myr. The central source injects mechanical energy at a constant rate of a few times 10$^{42}$ erg s$^{-1}$ into the superbubble.
A HST WFPC2 image decovers that NGC 4410a is seen almost face-on so that only the approaching outflow is visible. The HRI contours reveal an elongation of this outflow indicative for a faint X-ray ridge below the detection limit ($L_{\mathrm X} \le$ 1.4 erg s$^{-1}$) toward the neighbouring galaxy NGC 4410b caused by its tidal interaction. NGC 4410b houses only a faint X-ray source ($L_{\mathrm X} \le$ 3.8 erg s$^{-1}$).
author:
- 'D. Tschöke'
- 'G. Hensler'
- 'N. Junkes'
date: 'Received 1998 July 20 / Accepted 1998 November 6'
title: |
ROSAT X-ray observations of the interacting pair of galaxies\
NGC 4410: evidence for a central starburst
---
Introduction
============
The pair of interacting galaxies NGC 4410a/b belongs to a group of 11 members (Hummel et al. 1986, hereafter HKG86) which are located behind the Virgo cluster. It consists of a peculiar Sab (NGC 4410a) (Thuan & Sauvage 1992) and an E galaxy (NGC 4410b) (HKG86) located in east-west direction and separated by 187 (8.8 kpc at a distance of 97 Mpc) (Mazzarella & Boroson 1993, hereafter MB93). NGC 4410a is located at RA = 12$^\mathrm{h}$ 26$^\mathrm{m}$ 279, Dec = +09 01 18 (J2000). Spectral analysis of the nuclei of both components led to the classification as two LINERs (Mazzarella et al. 1991; Bicay et al. 1995; Thuan & Sauvage 1992). In the eastern component NGC 4410b a supernova type [i]{} has been detected in 1965 (SN 1965 A) from which Turatto et al. (1989) derived a distance of 139 Mpc. The system has a radial velocity of about 7300 km s$^{-1}$ (Mazzarella et al. 1991; Batuski et al. 1992; Thuan & Sauvage 1992; MB93). With a Hubble constant of 75 km s$^{-1}$ Mpc$^{-1}$ this leads to a distance of 97 Mpc. In this paper we apply the latter value to all distance-dependent parameters, like e.g. luminosities etc. A distance of 97 Mpc for NGC 4410 results in an absolute length scale of 470 pc arcsec$^{-1}$.
HKG86 found a spatial coincidence of the western optical component NGC 4410a with a radio point source of luminosity $L_{\mathrm R} \approx$10$^{39}$ erg s$^{-1}$ embedded in a strong, extended radio source around NGC 4410a with a total radio luminosity $L_{\mathrm R} \approx$7 erg s$^{-1}$.
The phenomenon of interaction between galaxies is closely related to the occurrence of starbursts (SBs): A large fraction of interacting galaxies ($\sim$70%) exhibits typical characteristica of SBs (Bushouse 1986) and vice versa. It is also striking that the majority of infrared-bright galaxies shows evidence for recent interactions as indicated by the presence of close neighbours, or by their disturbed morphology and tidal tails (Joseph et al. 1984; Lonsdale et al. 1984; Telesco 1988). The fraction of mergers increases drastically in the infrared luminosity range $L_{IR}$ from 10$^{10} L_{\sun}$(12%) to 10$^{12} L_{\sun}$(95%) (Sanders & Mirabel 1996). Although a large IR luminosity is not necessarily a tracer for enhanced star formation, it is one of the typical features for SB galaxies.
Norman & Scoville (1988) demonstrated that gas inflow from the galactic disk toward the central region, as required for building a central SB, will be triggered by a non-axialsymmetric gravitational potential like in interacting galaxies. They argued that perturbed orbits of gas clouds caused by an encounter lead to enhanced cloud collisions within the galactic central region and trigger the formation of massive stars. An alternative model for triggering a central SB was given by Jog & Das (1992) and Jog & Solomon (1992). The infall of giant molecular clouds (GMCs) into an intercloud medium with a higher mean pressure in the central region drives a radiative shock into the GMCs and ignites the SB.
The inflow of gas from the galactic disk is not only required for a central SB but also for fuelling an AGN. It is still under debate whether SBs are progenitors of AGN (Norman & Scoville 1988) or whether both are different physical processes. Weedman (1983) proposed that if a large number of massive stars form fast in a small central volume, the compact stellar remnants from these could act as accretors. Also the stellar dynamical merging of a dense cluster of massive stellar remnants would plausibly form a blackhole nucleus (Rees 1984).
The nuclei of several nearby galaxies, like NGC 1068, NGC 1097, and NGC 7469, can be resolved into a central AGN and a circumnuclear SB ring (Keel 1985; Pérez-Olea & Colina 1996, hereafter PC96). Some host galaxies of quasars also exhibit evidence for SBs, such as 3C48 (Stockton 1990).
Assuming an evolution from starbursts to AGNs during the interaction of galaxies one would expect more AGNs with the age of the merger process. Recent observations of ultraluminous IR galaxies with the infrared satellite ISO however do not show any obvious tendency of an increasing fraction of AGNs within interacting galaxies with advanced merging process (Lutz et al. 1998; Genzel et al. 1998). Half of the observed galaxies reveal both an AGN and starburst activity. It seems more likely that more local and shorter term conditions like time-dependent compression of the circumnuclear interstellar gas, the accretion rate onto the central black hole, and the associate radiation efficiency determine AGN or starburst dominated luminosities.
In this paper we present the spectral and imaging X-ray properties of NGC 4410 observed by the ROSAT Position Sensitive Proportional Counter (PSPC) and High Resolution Imager (HRI), respectively. We are able to resolve the two optical components in the ROSAT HRI image. The paper is structured as follows: In the next section we are considering the X-ray observations in general, detecting and trying to identify the emergent X-ray sources in the field and discussing the data reduction. In Sect. 3 we concentrate on NGC 4410 presenting the HRI results at first before we study the spectral flux distribution of the PSPC data with particular concern of comparison with a sequence of X-ray radiation models. The results are then discussed in Sect. 4 making use of a HST WFPC2 image of NGC 4410a for a geometrical consideration of its inclination.
Observations and data reduction
===============================
The data presented in this paper were taken with the HRI and the PSPC detectors on board of the X-ray satellite ROSAT. This X-ray telescope is operating in the energy range of 0.1 to 2.4 keV. The spatial resolutions of the HRI is 17. The point spread function (PSF) of this detector at the optical axis in combination with the telescope is 3. The PSPC has a PSF of 25. The two detectors have a field of view of 38 and 2, respectively. For details concerning ROSAT and its instruments see the ROSAT User’s Handbook (Briel et al. 1996).
NGC 4410 was observed on June 28-30, 1993 with the ROSAT PSPC detector for a total effective exposure time of 23.4 ksec. The total number of background-subtracted counts from the central source associated with NGC 4410 is 870$\pm$31. Figure 1 shows the central 42 of the PSPC field of view. For the spectral analysis of the source we used the software package IDL. The source photons were extracted from a circular area of 92 around the central source and corrected for telescope vignetting and detector dead-time. The spectrum of the source was binned according to a signal-to-noise ratio of 8. For the background correction we selected three uncontaminated circular areas close to NGC 4410 with radii of 86, 101 and 141. The background contributes an X-ray flux of (1.36$\pm$0.27) cts arcsec$^{-2}$ in the ROSAT bandpass. In order to fit a model spectrum to the PSPC data we used the X-ray spectral-fitting software package XSPEC (Arnaud 1996).
Our ROSAT HRI observations have been taken between June 27 and July 1, 1995 with a total effective exposure time of 36.7 ksec which led to 152$\pm$24 counts for the central source associated with NGC 4410. The background X-ray flux amouts to (1.6$\pm$1.1) cts arcsec$^{-2}$.
Source detection
----------------
Ten X-ray sources above a limit of 5$\sigma$ are found within the field of view of the HRI detector using a standard source-detection algorithm in the EXSAS X-ray analysis software package (Zimmermann et al. 1997) (Fig. 2). In Table 1 the detected sources are listed with position, background subtracted count rates and optical identifications.
----- ------------------------------------- ------------ ---------------------------- ------------------------------ -----------------------------------
No. RA (2000) Dec (2000) countrate (HRI)$^\dag$ angular distance from identification
\[10$^{-3}$ cts s$^{-1}$\] NGC 4410a$^\ddag$ \[arcmin\]
1 12$^\mathrm{h}$ 27$^\mathrm{m}$ 165 +914118 3.90$\pm$0.50 17.70 QSO 1224+095
2 12$^\mathrm{h}$ 26$^\mathrm{m}$ 038 +912594 2.38$\pm$0.33 13.27
3 12$^\mathrm{h}$ 25$^\mathrm{m}$ 494 +906568 1.17$\pm$0.25 11.30
4 12$^\mathrm{h}$ 27$^\mathrm{m}$ 115 +906533 1.97$\pm$0.31 12.20 unidentified optical point source
5 12$^\mathrm{h}$ 26$^\mathrm{m}$ 457 +904059 0.57$\pm$0.15 5.21
6 12$^\mathrm{h}$ 26$^\mathrm{m}$ 024 +902164 0.90$\pm$0.18 6.58
7 12$^\mathrm{h}$ 26$^\mathrm{m}$ 160 +901184 0.54$\pm$0.14 3.09
8 12$^\mathrm{h}$ 26$^\mathrm{m}$ 282 +901130 4.13$\pm$0.36 0.00 NGC 4410a
9 12$^\mathrm{h}$ 26$^\mathrm{m}$ 191 +900369 1.32$\pm$0.20 2.39
10 12$^\mathrm{h}$ 27$^\mathrm{m}$ 260 +850067 5.30$\pm$0.78 18.20 Abell 1541
----- ------------------------------------- ------------ ---------------------------- ------------------------------ -----------------------------------
\
$^\dag$ background subtracted\
$^\ddag$ NGC 4410a coorinates see no. 8\
\
From these detected X-ray sources in the HRI image, four coincidences with optical objects can be found (see Table 1 and Fig. 2). The X-ray source no. 8 can be clearly identified with NGC 4410a, while an extended X-ray source (no. 10, 44 FWHM) belongs to the galaxy cluster Abell 1541. In addition, the source no. 1 coincides with the QSO 1224+095 and no. 4 is still unidentified in the optical. In order to investigate further identifications of X-ray sources with optical objects we decrease the detection limit to 3$\sigma$. This leads to 32 sources in the HRI image with 10 counterparts in the optical. Three of them are identified as the Abell galaxy cluster 1541 (extended), the QSO 1224+095 (1 offset) and the K7-star G60-2 (8 offset). The offsets between optical and X-ray sources do not indicate any systematical shift or rotation, requiring a position correction of the superposition of the two images at this point.
Results
=======
X-ray imaging
-------------
Using the coordinates of the ROSAT HRI pointing the overlay of X-ray and optical image reveals a shift between the optical (NGC 4410a) and X-ray maximum of 4. Strikingly, an additional X-ray source, that is visible at the 3$\sigma$ contour level at a distance of $\sim$18 from the central X-ray maximum to the east, has the same displacement from the maximum of the eastern optical component NGC 4410b. It contains about 1% of the count rate of the central source in the HRI image. From the detection limit of 3$\sigma$ = 9.0 cts s$^{-1}$ arcsec$^{-2}$ in the HRI image and the energy conversion factor (ECF) of 6.7 cts cm$^2$ erg$^{-1}$ for the HRI and a power-law spectrum with $\Gamma$ = 2.4 from the ROSAT User’s Handbook (Briel et al. 1996) we get an upper limit of 3.8 erg s$^{-1}$ for the X-ray luminosity of NGC 4410b.
Furthermore, radio observations of NGC 4410 present a similar emission feature as the X-ray contours. The maximum radio contours correspond spatially to the optical maximum of NGC 4410a and the lower emission levels are elongated toward the southeast (HKG86). To check whether the eastern X-ray maximum has to coincide with the optical structure of NGC 4410, we compare its X-ray luminosity with the blue luminosity $L_{\mathrm B}$ of NGC 4410b. With m$_{\mathrm B}$= 15.28 (MB93) we derive a blue luminosity $L_{\mathrm B}$ = 1.2 $L_{\sun}$ for NGC 4410b. Brown & Bregman (1998) analysed a sample of 34 early-type galaxies observed with ROSAT PSPC and HRI and studied the X-ray-to-optical distribution. The estimated X-ray luminosity of $\sim$ 10$^{39}$ erg s$^{-1}$ for NGC 4410b fits well into the observed distribution of the sample. This obtrudes to superpose the X-ray maxima of both components NGC 4410a and b to the optical image of the pair. This fine tuning requirement is not detectable and therefore not necessary on the scale of the full HRI image (Fig. 2) as pointed out in Sect. 2.1.
Figure 3 shows the position corrected contour plot of the central 2 of the HRI image with the X-ray source overlaid onto an optical image of NGC 4410 of the Digitized Palomar sky survey plates. The HRI image was smoothed with a Gaussian filter FWHM of 3.
The contours of the HRI image indicate a radial extension from the maximum of NGC 4410a toward the southeast by 3 to 10. Since this elongation is close to the resolution limit it has to be checked whether it is real or artificial. Morse (1994) found out that in some ROSAT HRI observations the contours of point-like X-ray sources are elongated over a scale of $\sim$5 to 10 because of errors in the attitude correction as the satellite was wobbled during the observation. On the other hand, it is striking that the extended radio emission of NGC 4410a (HKG86) agrees in its direction with the X-ray contours. Moreover, the elongation of the X-ray contours is not symmetrical with respect to the elongation axis, as one would expect if it is caused by the wobbling satellite, similar to the point-like sources in the paper by Morse (1994). Another possible error namely that the elongation is caused by an incorrect superposition of the different used observation intervalls (OBIs) was also checked by us but didn’t explain the effect.
We fit a Gaussian distribution with FWHM 35 (spatial resolution of HRI = 17, image smoothed with Gaussian filter FWHM 3 corresponding to the HRI PSF) to the point-like source and subtract it from the original image. The result is plotted in the Figs. 4 and 5. The residual reveals a $\sim$10$\times$5 emission feature 7 to the southeast of the X-ray maximum visible up to 13$\sigma$ above the background. As expected, the wings of the Gaussian distribution in the Figs. 5b-f fit very well to the X-ray source except in the direction of deformation of the X-ray contours.
As a working hypothesis which will be supported by the finding of a thermal emission (see Sect. 3.2) we assume that the extended X-ray contours represent a superbubble originating from type [ii]{} SNe in the starburst region, and expanding into the halo. Since its elongation is too small to resemble the edge-on view of an outflow we further assume that the residual emission in Fig. 5 represents the front part of a bipolar outflow from the central region of the galactic disk. To verify this assumption, we estimate the ratio of the integrated flux from the residual image with the total flux of the original image. For that we integrate the flux within a radius of 13 from the maximum in both the residual image and the original one. We derive a ratio of 0.34$\pm$0.05 (52$\pm$8 counts). The error is estimated by varying the integration radius around the source by 2 and from the mean fluctuation of the background flux. This result is supported by the spectral analysis of PSPC data where a flux of $L_{\mathrm X}$ = 1.3 erg s$^{-1}$ for the thermal emission component is derived as part of the combined thermal and power-law spectrum (see Sect. 3.2). However, since the fraction of the superbubble within the point-like source is neglected, this estimate is only a lower limit.
An X-ray ridge toward the optical position of NGC4410b is discernable at the 3$\sigma$ level, which corresponds to an upper count level of 3.0 cts s$^{-1}$ arcsec$^{-2}$. Using the HRI ECF of 1.8 cts cm$^2$ erg$^{-1}$ for a 1 keV Raymond-Smith plasma with log $N_{\mathrm H}$ = 20.0 from the ROSAT User’s Handbook (Briel et al. 1996) one gets an upper limit for the X-ray luminosity of 1.4 erg s$^{-1}$. Two regions of enhanced X-ray emission are visible 15 and 22 south of the maximum X-ray source that were not contained in our source list with a 3$\sigma$ confidence level. Depending on the spectral model and therefore on the ECF the upper limit for the X-ray flux of these sources is 2.25/ECF erg s$^{-1}$ cm$^{-2}$.
Spectral analysis
-----------------
-------------------------- -------------------------- ---------------------- ------------------------ -------------------- --------------- -------- ---------------- ----------------
model $N_\mathrm{H}$ $T$ $\Gamma$ norm red. $\chi^2$ d.o.f. $F_\mathrm{X}$ $L_\mathrm{X}$
(1) (2) (3) (4) (5) (6) (7) (8) (9)
\[1.5mm\]BS 2.06$^{+0.30}_{-0.27}$ 11.0$^{+1.6}_{-1.4}$ – 8.0 1.6 10 3.62 4.08
\[2mm\] RS (sol) 1.4$^{+1.2}_{-1.0}$ 12.7$^{+0.5}_{-0.6}$ – 162 4.7 10 2.96 3.33
\[2mm\] RS (sol) 1.73 $^\dag$ 13.0$^{+0.5}_{-0.3}$ – 180 10.7 11 2.66 2.99
\[2mm\] RS (ww) (0.06$^{+2000}_{-0.06}$) 11.7$^{+0.6}_{-0.8}$ – 17 7.0 10 3.28 3.69
\[2mm\] RS (ww) 1.73 $^\dag$ 11.4$\pm$0.3 – 17 12.7 11 2.81 3.16
\[2mm\] RS+RS (sol) 0.45$^{+0.24}_{-0.21}$ 4.4$^{+1.6}_{-0.8}$ – 35 1.0 8 3.33 3.75
32$^{+68}_{-13}$ – 224
\[2mm\] RS+RS (sol) 1.73 $^\dag$ 1.7$^{+0.2}_{-1.0}$ – 100 6.6 9 2.73 3.07
8.8$^{+3.4}_{-0.5}$ – 65
\[2mm\] RS+RS (ww) 0.35$^{+0.58}_{-0.24}$ 2.3$^{+0.1}_{-0.2}$ – 2.4 0.9 8 3.65 4.10
12.0$^{+0.8}_{-1.0}$ – 15
\[2mm\] RS+RS (ww) 1.73 $^\dag$ 0.9$^{+0.8}_{-0.9}$ – 2.6 3.1 9 2.78 3.13
9.3$^{+2.0}_{-0.1}$ – 11
\[2mm\] RS+PO (sol) 1.16$^{+0.51}_{-0.33}$ 7.2$^{+2.3}_{-2.6}$ 1.85$\pm$0.14 76 0.8 8 3.46 (0.88) 3.90 (0.99)
\[2mm\] RS+PO (sol) 1.73 $^\dag$ 7.7$^{+1.6}_{-2.2}$ 2.08$^{+0.10}_{-0.11}$ 25 (RS); 8.0 (PO) 0.7 9 3.44 (0.78) 3.87 (0.87)
\[2mm\] RS+PO (ww) 1.41$^{+0.52}_{-0.25}$ 9.1$^{+1.4}_{-2.6}$ 2.03$^{+0.21}_{-0.19}$ 64 (RS); 4.8 (PO) 0.7 8 3.41 (1.15) 3.84 (1.30)
\[2mm\] RS+PO (ww) 1.73 $^\dag$ 9.3$^{+0.9}_{-1.0}$ 2.17$\pm$0.10 5.4 (RS); 6.5 (PO) 0.6 9 3.47 (1.20) 3.91 (1.35)
\[2mm\] RS+PO (ww) $^\#$ 1.73 $^\dag$ 1.0$^{+0.3}_{-1.0}$ 2.6 (RS) 1.3 8 3.20 (0.38) 3.60 (0.43)
30.6$^{+5.8}_{-4.4}$ 4.24$^{+0.27}_{-0.28}$ 275 (PO)
\[2mm\] PO 3.42$^{+0.30}_{-0.35}$ – 2.39$\pm$0.9 12 2.3 10 3.83 4.31
\[1mm\]
-------------------------- -------------------------- ---------------------- ------------------------ -------------------- --------------- -------- ---------------- ----------------
\
$^\dag$ fixed to Galactic foreground value\
$^\#$ separate absorption for each component\
\
Col.(1)— Emission models abbreviated as: BS = thermal Bremsstrahlung, RS = Raymond-Smith, PO = power-law. The specification in brackets indicates the used element abundances for the hot thermal plasma, solar (sol) or abundances of a supernova calculated by Woosley and Weaver (ww) and integrated over all stars between 10 and 100 M$_{\sun}$ for a Salpeter IMF.\
Col. (2)— column density in units of 10$^{20}$ cm$^{-2}$.\
Col. (3)— plasma temperature in units of 10$^6$ K.\
Col. (4)— photon index.\
Col. (5)— scaling factor: for BS in units of (10$^{-19}$/(4$\pi D^2$))$\int n_\mathrm{e}n_\mathrm{I}$d$V$, $n_\mathrm{e}, n_\mathrm{H}$ = electron and ion densities (cm$^{-3}$); for RS in units of (10$^{-20}$/(4$\pi D^2$))$\int n_\mathrm{e}n_\mathrm{H}$d$V$, $n_\mathrm{e}, n_\mathrm{H}$ = electron and H densities (cm$^{-3}$); for PO in units of 10$^{-5}$ photons keV$^{-1}$ cm$^{-2}$ s$^{-1}$ at 1 keV.\
Col. (6)— reduced $\chi^2$.\
Col. (7)— degrees of freedom.\
Col. (8)— X-ray flux in units of 10$^{-13}$ erg cm$^{-2}$ s$^{-1}$, corrected for the fitted absorption. Values in brackets give the contribution of the thermal component.\
Col. (9)— X-ray luminosity in units of 10$^{41}$ erg s$^{-1}$. Values in brackets give the contribution of the thermal component.
In addition to the spatial information of emission components we got from the HRI data, we now want to decompose these components by using the spectral information obtained from the PSPC data. With the PSF of 25 of the PSPC we get no satisfactory spatial resolution.
The 870 background corrected counts of the central source detected with the PSPC in the energy range of 0.1-2.4 keV were binned in order to achieve a constant signal-to-noise ratio of 8. To fit the observed PSPC X-ray spectrum we apply different models of absorbed emission spectra including a thermal Bremsstrahlung spectrum (BS), the spectrum of an optically thin thermal plasma in collisional equilibrium (RS) as calculated by Raymond & Smith (1977) and a power-law spectrum (PO) with a photon energy distribution $\propto E^{-\Gamma}$, as well as several combinations of those. The absorption of the spectra depends on the hydrogen column density $N_\mathrm{H}$ between the source and the observer. As a first-order estimate and as a lower limit of $N_\mathrm{H}$ we use the Galactic foreground value in the direction of NGC 4410. This value for $N_\mathrm{H}$ amounts to 1.73 cm$^{-2}$ (Dickey & Lockman 1990).
The model fits for the X-ray emission to the observation are shown in Table 2. Usually the RS model is applied under the assumption of solar abundances (Anders & Grevesse 1989). Reasonably, this is not an ideal approach because expanding SN ejecta should contain the nucleosynthesis products of their massive stellar progenitors. For this reason we use two different sets of chemical abundances. Beside the classical model with solar abundances we fit one with enhanced abundances of SNe according to the yields by Woosley & Weaver (1985), but integrated over all massive stars between 10 and 100 M$_{\sun}$ and with a Salpeter initial mass function (IMF).
The main uncertainty to fit the data comes in by the value of the hydrogen column density $N_\mathrm{H}$. In each model, except the combinations of RS+PO and the single component models BS and PO, the free parameter $N_\mathrm{H}$ turns out to become much too low for the best fit. For the single thermal model RS it is not possible to fit both, the soft and the hard part of the spectrum reasonably without an unrealistically low value of $N_\mathrm{H}$.
This is also true for the combined two temperature plasma model RS+RS(ww). The best fit results in an unrealistically low $N_\mathrm{H}$ value of 0.35 cm$^{-2}$. For the same model with solar abundances RS+RS(sol), the best fit yields $N_\mathrm{H}$ = 0.45 cm$^{-2}$. In addition, the plasma temperature of more than 3 K for the second component is abnormally high for halo gas. This temperature may only be reached for gas in the central regions of starburst galaxies, for example NGC 253 (Ptak et al. 1997) and M82 (Mathews & Doane 1994). In the case of fixing the column density to the Galactic value, the resulting plasma temperatures of 1.7 K and 8.8 K for RS+RS(sol) and 0.9 K and 9.3 K for RS+RS(ww) are much more reasonable. But both fits to the observed spectra are much worse (see Table 2).
In the case of fitting the observation with a single PO the column density of 3.4 cm$^{-2}$ becomes plausible because of an expected additional absorption within NGC 4410. The photon index of 2.39$\pm$0.09 is in agreement with the results from Turner et al. (1993) and Mulchaey et al. (1993), who found a mean value for Seyfert1 and Seyfert2 galaxies of $\Gamma$ = 2.4. Comparing all these single-component models the model PO and BS provide the best fits to the observation. The fits are plotted in Fig. 6.
The best fit by far for the whole observed spectrum can be reached by applying a two-component model which combines a thermal and a power-law spectrum (RS+PO) with a photon index of 2.17$\pm$0.10 and a plasma temperature of (9.3$\pm$1.0) K (see Fig. 9). Nevertheless, the fit becomes worse for applying a higher value for $N_\mathrm{H}$ than the Galactic one. The combined PO+RS fit yields the best result with an absorption corrected luminosity of (3.91$\pm$0.55) erg s$^{-1}$ for the assumed distance of 97 Mpc.
Under the assumption, that the nonthermal emission originates from a central active nucleus heavily obscured by dust and gas, and that hot plasma has expanded into the outer parts of NGC 4410a, we fit the RS+PO model with different column density values for each component. One would expect a column density $N_\mathrm{H,RS} \approx N_\mathrm{H,Gal}$ for the less obscured plasma outflow and a much higher $N_\mathrm{H,PO}$ for the AGN because of the intrinsic absorption within the NGC 4410 nucleus. While we apply $N_\mathrm{H,RS}$ = $N_\mathrm{H,Gal}$ = 1.73 cm$^{-2}$ for the RS component, $N_\mathrm{H,PO}$ for the PO component is set to 3.1 cm$^{-2}$. This yields the best fit for a lower plasma temperature of 1.0 K and a very steep power-law for the central AGN with $\Gamma$ = 4.24 (see Fig. 11). In the very soft range (0.1-0.3 keV) the spectrum is determined by the intrisically unobscured RS component. Raising the absorption for the PO component prohibits its contribution to the soft spectral range, which means that the total X-ray flux there has to originate from the hot ionized gas. As a consequence the temperature of the plasma is almost one order of magnitude lower than in the RS+PO model with only one common absorption. The hard spectrum $>$0.5 keV originates completely from the nonthermal component. Nevertheless the photon index is unplausibly steep.
Discussion
==========
The X-ray halo
--------------
The HRI image strongly suggests a twofold spatial correlation between the X-ray and the radio emission: we have a compact central emission peak apparently coinciding with NGC 4410a and a diffuse halo slightly elongated about 10 toward the southeast. Subtraction of the central point-like source from the total X-ray emission enables us to unveil a clear extention in this direction. Because of the asymmetrical elongation and the spatial correlation in the X-ray and radio flux we are convinced not to be mistaken by an artefact. Furthermore, the flux ratio of 0.34$\pm$0.05 from this feature compared to the total flux corresponds to the best fitting model for the observed spectrum, a two-component model, combining a thermal Raymond-Smith spectrum and a power-law spectrum. As a hypothesis we assume that the thermal emission originates from an expanding X-ray halo gas with a plasma temperature of a few 10$^{6}$ K whose expulsion is driven by a central engine (SB + AGN).
The SE elongation of the emission contours could be caused by two effects:\
(1) If the pec Sab galaxy NGC 4410a is seen at moderate inclination, a circumnuclear superbubble or a bipolar outflow is visible in soft X-ray which expands vertically to both sides out of the galactic disk only the part in front of the H[i]{} disk. X-ray emission in the ROSAT band originating from gas outflowing into the reverse halo hemisphere is absorbed by cold gas in the disk of NGC 4410. This fact also accounts for the low N$_{\mathrm H}$ that is required from the best model fit and is lacking of any additional contribution to the Galactic foreground value.\
(2) The X-ray halo gas is exposed to the tidal force of the companion galaxy NGC 4410b.
In order to decide which of the two effects causes the deformation of the X-ray halo, information about the inclination angle of NGC 4410a is necessary. Fig. 12 shows the [*Hubble Space Telescope*]{} (HST) WFPC2 image of the central part of NGC 4410a in the V (5843 Å) broad band filter taken from the Hubble Space Telescope archive. One can discern the nucleus, a bright emission region at the lower left, and a line of additional smaller bright knots at the left. At the right-hand side of the nucleus a long tidal arm probably caused by the interaction is clearly visible. Since the morphology in this image leads to a very low inclination, we prefer the second explanation of an interaction-induced deformation of the X-ray halo. However, the HST image suggests an additional possibility, namely, that the bright emission region at the lower left is a starburst region, and that the extended part of the X-ray contours originates from this region.
The X-ray contours reveal a faint ridge toward NGC 4410b ($L_{\mathrm X} \le$ 1.4 erg s$^{-1}$) which has only a very faint corresponding X-ray counterpart centrally of NGC 4410b ($L_{\mathrm X} \le$ 3.8 erg s$^{-1}$) as indication of any activity in or around the nucleus of NGC 4410b. The non-detection of a similar radio emission admits at best a radioquiet AGN.
Physical conditions of the X-ray emission
-----------------------------------------
As one can see in Table 2 the derived values for the X-ray luminosity are all of the same order of 3–4 erg s$^{-1}$ and independent of the applied model. From this point of view it is not possible to distinguish between the different models. The two most acceptable models with reasonable values for N$_{\mathrm H}$ are the two-component combination RS+PO and the single-component PO model, respectively.
Although the photon index of $\Gamma$ = 2.39$\pm$0.09 of the single PO model is in good agreement with observed values for low luminousity AGNs (Turner et al. 1993; Mulchaey 1993), it is obvious that the observed X-ray spectrum of NGC 4410 cannot be fitted properly by a single PO model but requires an additional component as can be seen in Table 2 and by comparing Fig. 6 with Fig. 9. The RS+PO model for the X-ray source leads to a luminosity of the thermal component $L_\mathrm{X}$ = (1.35$\pm$0.19) erg s$^{-1}$ (35% of the total X-ray luminosity). Nevertheless, the X-ray luminosity can vary by a factor of 4 because of the uncertainty of the distance to the source. To compare the derived properties of NGC 4410 with theoretical calculations, we apply the following model: We assume a central SB to drive an expanding superbubble. The central source injects mechanical energy since a time $t_\mathrm{exp}$ at a constant rate $L_\mathrm{mech}$ by sequential supernova explosions into the bubble. The supersonic expansion leads to a shock front and is heating the ambient gas. Assuming a homogeneous density of this ambient gas would lead to a spherically expanding bubble, as calculated by MacLow & McCray (1988).
This model, however, does not correspond to the realistic conditions of a SB in a disk galaxy. Because of the vertically decreasing gas density the superbubble forms a “chimney” and blows out of the galactic disk into the halo by a bipolar outflow (Norman & Ikeuchi 1989). The dynamics, X-ray emission and disk-halo interaction of these outflows are modelled under different conditions of ambient disk gas density, halo gas density, energetic input into the superbubble, and expansion time of the superbubble in a number of papers (Tomisaka & Ikeuchi 1986; Tomisaka & Ikeuchi 1988; MacLow et al. 1989; Suchkov et al. 1994, hereafter S94; Michaelis et al. 1996). S94 found that the soft X-ray emission of the suberbubble in the range of (0.1–2.2) keV primarily originates from the shock-heated halo and disk gas with temperatures of 10$^6$ – 10$^7$ K, rather than from the supernova material itself. Depending on the density of the disk and halo gas, the scale-height of the disk, and the energy deposition rate from the SB, the morphology of the bipolar outflow varies in structural parameters: opening angle within the disk, radius of the chimney, vertical extension of the superbubble, and ratio of the two last properties. Independent of these different conditions, the models always achieve a bipolar morphology. S94 derived an analytic expression for the shell X-ray luminosity in the (0.1–2.2) keV band:
$$L_{\mathrm{X}} = 9.7\mtento{40} (L_{42})^{3/5} (n_{0.01})^{7/5} (t_{7})^{9/5} \mathrm{erg\ s^{-1}} ,$$
where $L_{42}$ is the mechanical input energy in units of 10$^{42}$ erg s$^{-1}$, $n_{0.01}$ is the particle density of the halo gas in units of 0.01 cm$^{-3}$, and $t_{7}$ is the expansion time of the superbubble in units of 10$^{7}$ yr.
From the HRI observation we can estimate a maximum expansion of the outflowing gas into the halo of 10, scaling to 4.7 kpc at a distance of 97 Mpc. The models of Tomisaka & Ikeuchi (1986), Tomisaka & Ikeuchi (1988), MacLow et al. (1989) and S94 reveal chimney radii within the galactic disk between 200 pc and 600 pc. The much more extended X-ray emission observed in NGC 4410a can therefore only originate from the outflow into the less dense galactic halo where it can expand in each direction. This involves that the SB has to be old enough in order to account for the escape of the expanding superbubble from the disk into the halo. That means a lower limit of $t_\mathrm{exp} >$ 4 Myr for the expansion time (Tomisaka & Ikeuchi 1988).
We estimate the mechanical energy input by the starburst into the superbubble from the emitted H$\alpha$ radiation using the shock model by Binette et al. (1985). They calculated the radiative cooling mechanism of shock-heated gas, emitting optical line radiation, and found that $L_\mathrm{H\alpha} \approx$ 10$^{-2} L_\mathrm{mech}$. With $L_\mathrm{H\alpha}$ = 6.6 erg s$^{-1}$ for NGC 4410 (MB93) this leads to $L_\mathrm{mech} \approx$ 6 erg s$^{-1}$. Applying 10$^{51}$ erg for the energy release per and taking into account that roughly only 20% is converted into mechanical luminosity we derive a rate of $\sim$1.0 yr$^{-1}$.
Under the simplified consideration of a spherically expanding gas one can estimate its density. With the scaling factor of the RS model and setting the electron density equal to the hydrogen density we get the following expression for the electron density of the hot gas:
$$n^2_e = \frac{3D^2 N_{RS}}{fr^3}\times10^{14}cm^{-5},$$
where $N_{RS}$ is the scaling factor, $D$ is the distance to the source, $r$ is the radius of the superbubble and $f$ is a filling factor taking into account that the hot gas is not distributed homogeneously but broken up into separate bubbles. With the parameters of $D$ = 97 Mpc, $r$ = 4 kpc, $N_{RS}$ = 5 cm$^5$ and an assumed filling factor of 0.1 one obtains an electron density of 0.03 cm$^{-3}$. Changing the filling factor to $f$ = 0.9 leads to $n_e$ = 0.01 cm$^{-3}$.
An expansion time of 10$^7$ yr with a halo gas density of 0.01 cm$^{-3}$ leads to an X-ray luminosity of the shell of 2.8 erg s$^{-1}$. A lower rate of 0.5 yr$^{-1}$ and a slightly smaller expansion time of 8 yr reduce the obtained X-ray luminosity to 1.2 erg s$^{-1}$ in the ROSAT band. The derived plasma temperature of 10$^{7}$ K lies at the upper bound of the range with log $T$ = 6.0 – 6.9 found by S94 for the (0.1–2.2) keV band.
For a Salpeter IMF, a activity between 10 and 100 M$_{\sun}$ and a SN rate of 0.5 yr$^{-1}$ the star formation rate results to $\sim$95 M$_{\sun}$ yr$^{-1}$. Depending on the fraction of the mechanical energy release of a this value can increase up to a factor of 5.
Comparison with other galaxies of similar properties
----------------------------------------------------
Each galaxy and, in particular, mergers, galaxy pairs or SB galaxies are unique systems. In order to get an insight on whether NGC 4410 and its derived structures are somehow typical for close encounters, we compare the derived X-ray luminosity in the ROSAT band with other disturbed and isolated SB galaxies. The peculiar galaxy NGC 2782 e.g. is thought to be a merger of two disk galaxies of unequal mass and has $L_\mathrm{X}$ = 4 erg s$^{-1}$ (Schulz et al. 1998). Another galaxy with disturbed morphology and comparable X-ray luminosity ($L_\mathrm{X}$ = 1.4 erg s$^{-1}$) is NGC 1808. In contrast the PSPC data of this object do not show any X-ray outflow out of the central source into the halo. But one has to mention that NGC 1808 has a SFR of only 10 M$_{\sun}$ yr$^{-1}$ (Junkes et al. 1995). Relatively isolated systems without any companion, like e.g. NGC 253 (Fabbiano et al. 1992), NGC 2903 and NGC 4569 (Junkes et al. in preparation), contain X-ray luminosities of a few 10$^{40}$ erg s$^{-1}$, emphasizing the importance of interaction for star-forming activity.
PC96 found a significant difference in the $L_\mathrm{X}/L_\mathrm{H\alpha}$ ratio between pure AGN, pure SBs and galaxies with circumnuclear star-forming rings with an active nucleus. The pure active nuclei show log($L_\mathrm{X}/L_\mathrm{H\alpha}$) between 0.00 and +1.68, while the pure SBs in the sample lie between -1.46 and -0.36. The three galaxies with combined X-ray emission from AGN and SB have values of -0.26 (NGC 1097), +0.16 (NGC 1068) and +0.63 (NGC 7469), indicating a continuous decrease from AGN to SB. From this tendency one would expect a log($L_\mathrm{X}/L_\mathrm{H\alpha}$)$<$0 for the RS+PO model for NGC 4410. Our results, however yields +0.77 thereby, approximately the same as for a single PO model with log($L_\mathrm{X}/L_\mathrm{H\alpha}$) = +0.81.
The fraction of $L_\mathrm{H\alpha}$ from the SB relative to the total H$\alpha$ luminosity amounts to 98%, 80% and 40% for NGC 1097, NGC 1068 and NGC 7469, respectively. Comparing only the contributions from the SB to the H$\alpha$ and X-ray luminosity, PC96 found log($L_\mathrm{X}/L_\mathrm{H\alpha}$) = -0.99, -0.70 and -0.36 for NGC 1097, NGC 1068 and NGC 7469, respectively. If we concider the fraction of H$\alpha$ luminosity from the SB in these galaxies, and assume that 90% of the total H$\alpha$ luminosity originates from the SB within NGC 4410a, $L_\mathrm{H\alpha}$ would result in 5.9 erg s$^{-1}$ and log($L_\mathrm{X}/L_\mathrm{H\alpha}$) = +0.36, which is quite high compared to the sample analysed by PC96.
Conclusions
===========
We observed the interacting pair of galaxies NGC 4410 with the ROSAT HRI and PSPC. Spectral investigations of NGC 4410 suggest that the integral X-ray emission ($L_\mathrm{X}$ = 3.9 erg s$^{-1}$) can be decomposed into a thermal component (described by a RS spectrum) and a component from the AGN (described by a power-law spectrum). The HRI image reveals an extended X-ray halo related to NGC 4410a with an extension of 10 from the nucleus of NGC 4410a to the southeast. Combining spatial and spectral informations reveals an X-ray luminosity of the halo gas of $L_\mathrm{X}$ = 1.3 erg s$^{-1}$ (1/3 of the total X-ray emission). The companion galaxy NGC 4410b houses only a very faint central point-like source below the 3$\sigma$ level, corresponding to an upper limit of 3.8 erg s$^{-1}$ for the X-ray luminosity.
As a reasonable model we can assume that the tidal interaction in the pair of galaxies NGC 4410 has two effects on the one partner, the face-on pec Sab galaxy NGC 4410a:\
(1) A central monster is either formed due to this interaction or has already existed before and is now fed by infalling gas during the merging event, producing AGN signatures. Evidence for an existing AGN comes from the ROSAT X-ray spectrum supported by the spatially correlated radio emission.\
(2) The merging process has ignited a nuclear SB ejecting an X-ray gas into the halo. Due to the tidal forces of NGC 4410b, the X-ray halo of NGC 4410a is deformed toward its neighbouring galaxy, either directly by gravitational forces or indirectly, originating from a decentred starburst region as a consequence of the merging process.
This system of merging galaxies represents a highly interesting object because it is a close interacting pair where the effects of tidal forces on nuclear activity and SB can be studied in detail with large telescopes. The HST WFPC2 image of NGC 4410a suggests a possible relation between the deformed X-ray halo and a bright emission region near the nucleus. With ROSAT imaging it is not possible to distinguish between the nucleus and the bright emission region resolved by HST. Further spectroscopic observations of this region and additional infrared imaging could help answering the question of the exact origin of the X-ray emission.
The authors are grateful to an unknown referee for his substantial and constructive report. The ROSAT project is supported by the German Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie (BMBF) and the Max-Planck-Society. The optical image shown is based on photographic data of the National Geografic Society - Palomar Observatory Sky Survey (NGS-POSS) obtained using the Oschin Telescope on Palomar Mountain. The NGS-POSS was funded by a grant from the National Geographic Society to the California Institute of Technology. The plates were processed into the present compressed digital form with their permission. The Digitized Sky Survey was produced at the Space Telescope Science Institute (STScI) under US Government grant NAG W-2166. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, Caltech, under contract with the NASA. Observations made with the NASA/ESA Hubble Space Telescope were used, obtained from data archive at STScI. STScI is operated by the Association of Universities for Research in Astronomy, Inc. (AURA) under the NASA contract NAS 5-26555.
Anders E., Grevesse N. 1989, Geochimica et Cosmochimica Acta 53, 197 Arnaud K.A. 1996, Astronomical Data Analysis Software and Systems V, ASP Conf. Ser. vol. 101, eds. Jacoby G. and Barnes J., p. 17 Batuski D.J., Hanisch R.J., Burns J.O. 1992, AJ 103, 1077 Bicay M.D., Kojoian R.J., Seal J., et al. 1995, ApJS 98, 369 Binette L., Dopita M.A., Tuohy I.R. 1985, ApJ 297, 476 Briel U., Aschenbach B., Hasinger G., et al. 1996, ROSAT User’s Handbook (MPE, Garching) Brown B.A., Bregman J.N. 1998, ApJ 495, L75 Bushouse H.A. 1986 AJ 91, 255 Dickey J.M., Lockman F.J. 1990, ARAA 28, 215 Fabbiano G., Kim D.-W., Trinchieri G. 1992, ApJS 80, 531 Genzel R., Lutz D., Sturm E., et al. 1998, ApJ 498, 579 Hummel E., Kotanyi C.G., van Gorkom J.H. 1986, A&A 155, 161 (HKG86) Jog C.J., Das M. 1992, ApJ 400, 476 Jog C.J., Solomon P.M. 1992, ApJ 387, 152 Joseph R.D., Meikle W.P.S., Robertson N.A., Wright G.S. 1984, MNRAS 209, 111 Junkes N., Zinnecker H., Hensler G., et al. 1995, A&A 294, 8 Keel W.C. 1985, in “Astrophysics of Active Galaxies and Quasi-Stellar Objects”, ed. Miller J.S., University Science Books, Mill Valley, p.1 Lonsdale C.J., Persson S.E., Matthews K. 1984, ApJ 287,95 Lutz. D., Spoon H.W.W., Rigopoulou D., et al. 1998, ApJ 505 L103 MacLow M.-M., McCray R. 1988, ApJ 324, 776 MacLow M.-M., McCray R., Norman M.L. 1989, ApJ 337, 141 Mathew W.G., Doane J. 1994, in “Panchromatic View of Galaxies – Their Evolutionary Puzzle”, eds. Hensler G., Theis C., Gallagher J.S., Éditions Frontières, p. 221 Mazzarella J.M., Boroson T.A. 1993, ApJS 85, 27 (MB93) Mazzarella J.M., Bothun G.D., Boroson T.A. 1991 AJ 101, 2034 Michaelis O., Hensler G., Samland M. 1996, in IAP Proc. “The Interplay between Massive Star Formation, the ISM and Galaxy Evolution”, eds. Kunth D. et al.,Editions Frontiers, Gif-sur-Yvettes, p. 525 Morse J.A. 1994, PASP 106, 675 Mulchaey J.S., Colbert E., Wilson A.S., et al. 1993, ApJ 414, 144 Norman C.A., Ikeuchi S. 1989, ApJ 345, 372 Norman C.A., Scoville N.Z. 1988, ApJ 332, 124 Pérez-Olea D.E., Colina L. 1996, ApJ 468, 191 (PC96) Ptak A., Serlemitsos P., Yaqoob T., Mushotzky R. 1997, AJ 113, 1286 Raymond J.C., Smith B.W. 1977, AJS 35, 419 Rees M.J. 1984, ARAA 22, 471 Sanders D.B., Mirabel I.F. 1996, ARAA 34, 749 Schulz H., Komossa S., Berghöfer T.W., Boer B. 1998, A&A 330, 823 Stockton A. 1990, in “Dynamics and Interaction s of Galaxies”, ed. Wielen R., Springer Verlag, Berlin, p.440 Suchkov A.A., Balsara D.S., Heckman T.M., Leitherer C. 1994, ApJ 430, 511 (S94) Telesco C.M. 1988, ARAA 26, 343 Thuan T.X., Sauvage M. 1992 A&AS 92, 749 Tomisaka K., Ikeuchi S. 1986, PASJ 38, 697 Tomisaka K., Ikeuchi S. 1988, ApJ 330, 695 Turatto M., Capellaro E., Petrosian A.R. 1989, A&A 217, 79 Turner T.J., George I.M., Mushotzky R.F. 1993, ApJ 412, 72 Weedman D.W. 1983, ApJ 266, 479 Woosley S.E., Weaver T.A. 1985, in “Radiation Hydrodynamics in Stars and Compact Objects”, Proc. of the IAU Colloquium No. 89, eds. Mihalas D., Winkler K.H., p.91 Zimmermann H.U., Becker W., Belloni T., et al. 1997, EXSAS User’s Guide, Edition 5, MPE Report, Garching
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abstract: |
The Cygnus X region is known as the richest star-forming region within a few kpc and is home to many particle accelerators such as supernova remnants, pulsar wind nebulae or massive star clusters. The abundance of accelerators and the ambient conditions make Cygnus X a natural laboratory for studying the life cycle of cosmic-rays (CRs). This naturally makes the Cygnus X complex a highly interesting source in neutrino astronomy, in particular concerning a possible detection with the IceCube Neutrino Observatory, which has a good view of the northern hemisphere.\
In this paper, we model the multiwavelength spectrum of the Cygnus Cocoon, for the first time using a broad data set from radio, MeV (COMPTEL), GeV (Fermi), TeV (Argo) and 10s of TeV (Milagro) energies. The modeling is performed assuming a leptohadronic model. We solve the steady-state transport equation for leptons and hadrons injected homogeneously in the region and test the role of diffusive transport and energy loss by radiation and interaction.\
The result shows that diffusion loss plays a significant role in Cygnus X and always exceeds the advection loss as well as almost all other loss processes. The best-fit parameters we find are a magnetic field of $B=8.9\times10^{-6}$ G, a target density of $N_t=19.4$ cm$^{-3}$, a cosmic ray spectral index of $\alpha=2.37$ and neutral gas distribution over a depth of 116 pc. We find that the fit describes the data up to TeV energies well, while the Milagro data are underestimated. This transport model with a broad multiwavelength fit provides a neutrino flux which approaches the sensitivity of IceCube at very high energies ($>$ 50 TeV). In the future, the flux sensitivity of IceCube will be improved. With this rather pessimistic model, leaving out the influence of possible strong, high-energy point sources, we already expect the flux in the Cygnus X region to suffice for IceCube to measure a significant neutrino flux in the next decade.
address:
- 'Fakultät für Physik und Astronomie, RAPP Center, TP IV, Ruhr-Universität Bochum, D-44801, Germany'
- 'Department of Physics, IceCube Collaboration, University of Wisconsin, Madison, WI 53706,USA'
author:
- Mehmet Guenduez
- Julia Becker Tjus
- Björn Eichmann
- Francis Halzen
bibliography:
- 'literatur.bib'
title: ' Identification of Gamma-Rays and Neutrinos from the Cygnus-X Complex Considering Radio Gamma Correlation '
---
Cygnus X,Cygnus Cocoon , Cosmic Rays ,Transport equation ,Hadronic , Leptonic ,Neutrinos ,Multi Wavelength ,Gamma Rays ,Radio Emission
Introduction {#Introduction}
============
One of the main unsolved problems in astroparticle physics is based upon the origin of high energetic cosmic rays (CRs). Although more than 100 years have passed since Victor Hess discovered cosmic rays ([@VictorHess]), the search for a reliable answer still continues. An insight into the acceleration mechanism which causes the energy gain plays an important role in that regard. Active Galactic Nuclei (AGN), Supernova Remnants (SNRs), Gamma-Ray Bursts (GRBs) and pulsars wind nebulae (PWN) are showing great promise for being accelerators of CRs (i.e. [@HighEnergyAstrophysics]).\
CRs are deflected by magnetic fields and interact with the ambient medium ([@Ginzburg]). The deflection makes it harder to locate the source of CR, whereas for high energy photons and neutrinos this problem does not occur. In fact, high energy (TeV) photons, but especially astrophysical neutrinos, can point to the direction of the source ([@Julia1]). These neutrinos also allow us to draw conclusions about the hadronic particles themselves since their generation process is based on the interaction between hadronic particles and the ambient medium by producing pions.\
Galactic CR accelerators are called PeVatrons as they accelerate up to the knee, i.e. PeV energies. The associated particles should be observable by multiple observatories such as Fermi, Argo or IceCube. Even if these detectors provide accurate data, it is necessary to identify the relevant radiation processes to give a realistic interpretation of the experimental data. In particular, the origin of $\gamma$-rays can be explained by several processes.\
The brightest diffuse $\gamma$-ray emission in the northern hemisphere is detected from the Cygnus X complex. It reeveals that this astrophysical region is rich on cosmic-ray accelerators (see e.g. [@ARGOCyg]). Cygnus X can expose the secret behind the acceleration mechanism of the CRs because of the short distance between the Earth and Cygnus X and the content of well-studied sources (such as J2032.2+4126, J2021.0 + 3651, J2021.5+4026 or J2030.0+36542). In order to comprise them and potential accelerators such as SNR and PWN, our region of interest (ROI) includes these known sources within a radius of $3.14 \deg$ centered in Cygnus X and a solid angle of $32 \deg ^2$, respectively, as this part of Cygnus includes the dominant part of the high-energy emission.\
In previous works [@Francis] and [@IdentifyingPevatrons], the neutrino flux from Cygnus X was calculated by simple approximations and assuming parameters which are averaged over the Galaxy, e.g. the magnetic field strength ($B$=1 $\mu$G ). Today, it is known that these parameters could deviate significantly from the used values, indicated by different astrophysical observations. In our model, we use parameters like the magnetic field and the column depth as free parameters and determine them via a best-fit scenario.\
Recently, [@TovaPaper] investigated Cygnus X more extensively by assuming that the CR spectrum observed at Earth is also a representative for the Cygnus X and by adding emission from the Cocoon as well as from point sources separately to the diffuse emission modeled in their work. Moreover, continuous momentum loss and losses due to advection and diffusion were considered. All calculations were carried out for 5 deg $\times$ 5 deg region which is subdivided in 0.25 deg $\times$ 0.25 deg, and considered observation data from 150 MeV (Fermi) up to 16 TeV (Milagro).\
In this paper, we add information from radio wavelength and also take into account the COMPTEL-detected 10-MeV signal. This broad energy range gives strong constraints on the possible leptonic (synchrotron, non-thermal Bremsstrahlung, inverse Compton) and hadronic (pion production) processes in Cygnus X. In doing so, the transport and the loss mechanisms in Cygnus X can be investigated, such that the resulting neutrino spectrum is derived.\
The primary requirement to generate high energetic CRs is an appropriate accelerator, which in Cygnus X is thought to be PWN or SNRs.\
In the same regard, different losses will be considered:
1. Continuous momentum losses by: Synchrotron, Inverse Compton, non-thermal Bremsstrahlung, ionization and hadronic pion production.
2. Catastrophic losses by advection and diffusion.
Per definition, continuous momentum losses conserve the total number of the particles in Cygnus X, whereas catastrophic losses do not. This means that particles escape the region of interest due to diffusion or advection. In the same vein, both the flattening of the hadronic pion production for energies greater than 200 GeV as well as the different cooling behaviors of electrons and protons will be considered ([@Kelner]).\
Since the exact particle accelerator is not known, the acceleration mechanism of CRs from that region will remain unspecified. Therefore, a CR emission from a non-thermal electron-proton plasma with a power-law in momentum will be used. This work will rely on the mathematically convenient description by assuming a spatially homogeneous and spherically symmetric CR density distribution in Cygnus X since the region is very complex and small inhomogeneities vanish at a larger scale.\
Nevertheless, the rigidity difference between electrons and protons will be considered. Also, the messenger particle from secondaries of CRs will be considered to find confirmation indirectly for the proposed model by examining experimental data.
Cygnus X
========
Cygnus X is a part of the largest star-forming region of the constellation Cygnus in the northern galactic plane, which is located in the galactic local spiral arm, more precisely at galactic longitudes between $70^{\circ}$ and $90^{\circ}$ and $4^{\circ}$ and $8^{\circ}$ below and above the plane (see figure \[fig:fermimap\]) ([@CygnusByFermi2]). It is one of the most structurally complex regions in the galactic plane. Moreover, it is formed by a massive molecular cloud complexes. This property is important for CR formation and characteristics. Nevertheless, as indicated in [@CygnusByFermi] the CR population is similar to the local interstellar space.
![Fermi’s color map from -60$^{\circ}$ - +60$^{\circ}$ in the vertical plane and from 150$^{\circ}$ - 300$^{\circ}$ in the horizontal plane, which is distributed by Skyview HEASARC - HEALPixed by CDS; the map was edited with Aladin v9.0. The red color in the map denotes photons in the energy range 0.3-1 GeV, green 1-3 GeV and blue 3-300 GeV ([@AladinFermi]).[]{data-label="fig:fermimap"}](FermiAllsky2){width="0.8\linewidth"}
There are many reasons why Cygnus X is an excellent region to investigate the origin of CRs:
- The emission is observable from radio to high-energy gamma-ray frequencies ([@GammaSourcesCygnus]), whereby in the energy range from GeV up to TeV Cygnus X has the brightest emission in the northern hemisphere ([@BrightestSourceCygnus]). Moreover, many other gamma-ray sources exist in that region.
- It contains sources which accelerate particles at least up to 100 TeV ([@Milagro]).
- Many potential accelerators such as supernova remnants[^1], pulsar wind nebulae[^2] and Wolf-Rayet (WR) stars or OB associations (Cyg OB2, Cyg OB1) can be found. Many of these constituents are pictured in figure \[fig:fermimapzoom2b\]. Here, it is important to mention that approximately 20% of CRs are produced in WR stars nearly $10^5$ years before they become accelerated ([@CygnusByFermi2]). These stars are a phase of OB stars, which appear in Cygnus X as clusters (OB associations).
- Most of the objects are at a distance of 1.4 kpc.
- It consists of H$_{\text{II}}$ regions ([@IdentificationTeVCygnusCocoon]).
All of these characteristics make Cygnus X a suitable natural laboratory for the astronomer to look beyond the usually constrained view.
![Fermi’s color map of Cygnus X (red cycle), which is distributed by Skyview HEASARC - HEALPixed by CDS; the map was edited with Aladin v9.0. The scales are given by galactic coordinates. Cygnus X is represented here by the big red cycle, pulsars abbreviated with “PSR” by green cycles, OB2 association by a blue cycle and Cygnus Cocoon by a yellow one. All sources identified by Fermi 3FGL are represented by blue quads.[]{data-label="fig:fermimapzoom2b"}](FermiMapZoom2b){width="0.8\linewidth"}
The supernova remnant (SNR), $\gamma$ Cygni, was firstly investigated using Fermi data, which provide information about the interstellar background by subtracting the radiation from $\gamma$ Cygni.\
Moreover, Cygnus X has a Cocoon where freshly accelerated CR can be found, and the emission exceeds 100 GeV. The SNR $\gamma$ Cygni, which is located in the Cocoon, could cause the acceleration of protons even up to 80-300 TeV and electrons up to 6-30 TeV. The accelerated particles could fill the whole Cocoon if it is assumed that the primary transport mechanism is diffusion. On the other hand, advection could dominate the transport mechanism, if an anisotropic emission from $\gamma$ Cygni was observed ([@CygnusByFermi2]). However, there is no proof for this scenario yet.\
The Cocoon can give hints about the transport mechanism and escape of CRs from their source. In the model built in this work, the influence of diffusion and advection in Cygnus X can be investigated. Thus, at the very least a suggestion of the role of $\gamma$ Cygni in the Cocoon can be given as our ROI includes these objects.\
In order to properly model the CR interactions, the column depth needs to be known. Following [@CygnusByFermi], we will use $7\times10^{21}$ atoms/cm$^2$ for ROI.\
Local distribution in radio and gamma range
-------------------------------------------
In our model, we assumed a spatially homogeneous injection of accelerated CRs. It is important to investigate the local distribution to see the reliability of this assumption.
![Fermi map band 1: 30-300 MeV[]{data-label="fig:fermimapzoom11"}](Fermi1){width="0.7\linewidth"}
{width="0.7\linewidth"}
{width="0.7\linewidth"}
{width="0.7\linewidth"}
{width="0.7\linewidth"}
![Fermi map band 5 with contours by reducing the background emission[]{data-label="fig:fermimapzoom12"}](Fermi5bb){width="0.7\linewidth"}
![Decomposed non-thermal emission from the total intensity in Cygnus X at 4800 MHz with an angular resolution of 9’.5 ([@thermalNonThermal1]).[]{data-label="fig:non_thermal"}](non_thermal){width="1.0\linewidth"}
Figures \[fig:fermimapzoom11\] -\[fig:fermimapzoom12\] which are extracted from [@AladinFermi] for Fermi bands represent a photon count map and serve here as visualization of the structure of Cygnus X dependent on the energy. The density of counts is anti-proportional to the brightness. However, for our calculations we will use directly data from Fermi.\
For Fermi band 1, the structure is to some extent compatible with our assumptions of a spatially homogeneous and spherical symmetric distribution of CRs in Cygnus X. This agreement deteriorates with higher energies. Therefore, a stronger agreement for lower $\gamma$-ray energies can be expected than for higher energies.
Cygnus X-1 {#cygnus-x-1 .unnumbered}
----------
Cygnus X-1 is a binary X-ray system similar to Cygnus X-3. It is also a microquasar, which could produce PeV $\gamma$-rays in Cygnus X ([@Schlicki]). A microquasar is a binary star system with a stellar black hole or a neutron star. Cygnus X-1 contains a black hole with nearly 15 $M_{\odot}$. The main star is a blue giant (HD 226868), and the constellation seems to have been in existence for almost $5\times10^6$ years ([@UniverseinGammaRays]).\
This constellation has a soft and hard state and most of the time remains in the latter. This non-thermal component is thought to be caused in an optically thin and hot corona by thermal Comptonization of disk photons ([@CygnusX1]). During the latter, the energy of the constellation is in the power-law component. Therefore, especially at 10 MeV, a purely non-thermal emission will be expected.
Introduction of the model {#Model}
=========================
The dynamics in a star-forming region can be very complicated a fortiori in the Cygnus X complex. It is, therefore, necessary to work by simplifying assumptions, which are reasonable and favorable to the relevant conditions.\
The ability to describe particle transport phenomena is indispensable for predicting processes in star-forming regions. The massive molecular clouds in Cygnus X demand a reliable transport mechanism. Beyond the common continuous momentum losses, the catastrophic losses due to advection and diffusion of particles will also be considered.\
Considering the generic state transport equation, the transport equation for a plasma with a differential particle density $n$ yields: $$\begin{split}
&\underbrace{\frac{\partial n(\textbf{x},\textbf{v},t)}{\partial t}}_{\substack{\text{Storage}}} -\underbrace{\nabla_{\textbf{v}}(F(\textbf{v})\,n(\textbf{x},\textbf{v},t)))}_{\substack{\text{Continuous Loss}}}+ \underbrace{\nabla(\textbf{v}\,n(\textbf{x},\textbf{v},t))}_{\substack{\text{Advection}}}\\&+\underbrace{\nabla(D\,\nabla n(\textbf{x},\textbf{v},t) )}_{\substack
{\text{Diffusion}}}=\underbrace{q(\textbf{x},\textbf{v},t)}_{\substack{\text{Generation}}}\, .
\end{split}
\label{generalTrEq}$$ Here, $\textbf{v}$ is the advection velocity, in consideration it represents the galactic wind velocity. The diffusion tensor is approximated by the scalar $D$.\
The first term describes the storage; the second term in eq.(\[generalTrEq\]) the continuous losses in momentum; the third, the catastrophic losses due to advection in magnetic fields; the fourth, the catastrophic diffusion losses; and the last term, the source rate which obeys a power-law.\
\
This work is developed to describe an emission from an isotropic and spatially homogeneous part of the Cygnus region in its steady state. Additionally, it assumes an isotropic diffusion of the particles within Cygnus X. This assumption is reasonable since an extended region with a diameter of 77 pc will be considered, emission outside this region is negligible, the region is very complex and small inhomogeneities vanishes at larger scales[^3]\
In our model, we follow a general leaky box approach in which the cosmic-ray density does not depend on spatial coordinates and is characterized by some average values [@Ginzburg]. This general scheme is well-established with previous analytical solutions for different scenarios given in [@Schlicki] and references therein. Here, we particularly follow the model of [@Bjoern], who meets the requirements we apply to our source region, i.e. a homogeneous steady-state CR sea with a power-law-injection, continuous momentum as well as diffusion and advection loss.\
With these assumptions, from eq.(\[generalTrEq\]) the following equation is obtained ([@Schlicki; @Bjoern]) as the steady state transport equation: $$0=\underbrace{\frac{\partial}{\partial \gamma}\left( \Gamma_{e,p}n_{e,p}(\gamma)\right) }_{\substack{Continuous \\momentum\ loss}}-\underbrace{\frac{n_{e,p}(\gamma)}{\tau^{e,p}_{diff}(\gamma)}}_{\substack{Diffusion\\ Loss}}-\underbrace{\frac{n_{e,p}(\gamma)}{\tau^{e,p}_{adv}(\gamma)}}_{\substack{Advection\\ Loss}}+\underbrace{q_{e,p}(\gamma)}_{\substack{CR\ source\\ rate}}\, .
\label{TransportEq}$$ Here, $\gamma$ represents the Lorentz factor, $\Gamma$ is the term which includes the continuous loss and $\tau_{adv}$ the advection loss timescale. Using the assumption mentioned above for catastrophic losses the diffusion coefficient for electrons or protons can be approximated by: $$D_{e,p}(\gamma)\simeq\frac{c\,\lambda_{e,p}\,\gamma^{\beta}}{3}\ .
\label{diffsuionCoeff}$$ The diffusion timescale $\tau_{diff}^{e,p}$ can then be approximated ([@Bjoern]) by: $$\tau_{diff}^{e,p}(\gamma)\simeq \frac{R^2}{3D(\gamma)}\simeq \frac{R^2}{c\,\lambda_{e,p}}\gamma^{-\beta}\ .
\label{sigmaDiff}$$ Here, $\lambda_{e,p}\, \gamma^{\beta}$ denotes the related diffusion length or the mean free path, $\beta$ the related spectral index of the diffusion coefficient, $R$ the radius of the considered region and $c$ the speed of light. The diffusion in the astrophysical context does not describe deflection by collision but interaction with local magnetic fields. As more deflections and interaction force the particles to be spread in the region, the diffusion timescale equals the particle escape timescale and is a quantity for particle conservation.\
Observation of C/O nuclei spectra shows that the scalar diffusion coefficient index (see eq.(\[diffsuionCoeff\])) in the Galaxy is given by $\beta\approx 0.5$ ([@DiffGalaxyRatio]). Though, to accord with the observation data, this work uses a Kolmogorov-spectrum referring to $\beta=1/3$.\
The rigidity difference between electrons and protons will consistently be taken into account. This has an effect on the maximum energy and ratio of momentum loss through various phenomena. In this manner, electrons can reach higher energies faster than protons because of their mass difference. Since particles at higher energies are more conducive to continuous momentum loss, it follows that the latter is more prevalent among electrons. The loss mechanisms are not the same for electrons and protons due to their characteristics as leptons and baryons.\
For protons in Cygnus X, the diffusion length factor is supposed to be $\lambda_p=2.5\times10^{17}$ cm. Due to the mass dependence of the Larmor radius the electron mean free path can be calculated as follows: $$\lambda_e=\left(\frac{m_p}{m_e}\right) ^{-\beta}\lambda_p\, .$$ The influence of the spectral index $\beta$ on the diffusion timescale can be seen in figure \[Timescale\] and \[TimescaleE\]. This index describes the energy dependence of the diffusion coefficient.
The timescale becomes smaller at higher energies, and the $\beta$ reinforces this behavior with larger value. However, the influence of diffusion for electrons and protons in Cygnus X can be seen by considering the blue dashed line in figures \[TimescaleE\] and \[Timescale\]. Since the advection timescale is constant, the influence of the diffusion length factor $\lambda$ on the diffusion timescale in Cygnus X can also be seen in figure \[fig:timescaleleelectron\]. The galactic wind speed can be used to determine the advection timescale according to $$\tau^{e}_{adv}=\tau^{p}_{adv}=\tau_{adv}\simeq\frac{R}{v_{wind}}\, .
\label{sigmaAdv}$$ For Cygnus X the advection velocity is assumed to be $v_{adv}\simeq50$ km/s as we now it from the Galactic Disk [@WindVel], which is comparable to the Alfvén speed of the CRs. Using the relation in eq.(\[sigmaDiff\]) and eq.(\[sigmaAdv\]) the diffusion and advection loss for Cygnus X can be compared by considering the ratio of both timescales.
![The ratio of diffusion and advection timescale as a function of the diffusion length coefficient $\lambda$ and Lorentz factor $\gamma$. Additionally, the dashed line for the proton (red) and electron (blue) diffusion length exhibits the ratio in Cygnus X.[]{data-label="fig:timescaleleelectron"}](Lambda.pdf){width="0.8\linewidth"}
Figure \[fig:timescaleleelectron\] shows the ratio as a function of the diffusion length coefficient $\lambda$ and the Lorentz factor. The ratio for relativistic electrons and protons in Cygnus X can be found by considering the advection timescale $\tau_{adv}=4.752\times10^{13}$ s and the diffusion timescale $\tau_{diff}^e=6.67\times10^{13}\times\gamma^{-\beta}$ s and $\tau_{diff}^p=3.26\times10^{9}\times\gamma^{-\beta}$ s. Concerning catastrophic losses, relativistic particles in Cygnus X are more subject to diffusion loss than to advection loss, as the diffusion timescale is much shorter.\
The timescale for continuous loss be approximated by $$\tau_{con}^{e,p}= \frac{\gamma}{\Gamma_{e,p}}\, .$$ In the same manner, the timescale dependency for electron and proton continuous loss can be calculated when the momentum loss rate $\Gamma_{e,p}$ is known.
Relation between electrons and protons {#RelationEP}
--------------------------------------
If a steady state is considered and a homogeneous distribution of CRs without charge imbalances is assumed, the total amount of injected electrons and protons will be the same. Here, we suppose the total acceleration time $T_a$ for electrons and protons is the same. The total number of accelerated protons or electrons can then be described by: $$N_0=\int_{0}^{T_a} \rm d t\int_{\gamma_0^i}^{\infty}\rm d\gamma\,q_i(\gamma)=T_a\int_{\gamma_0^i}^{\infty}\rm d\gamma\,q_i(\gamma)
\label{totalNumber}$$ for $i= e_1, p$, where $\gamma_0^i$ represents the minimum Lorentz factor and $e_1$ the primary electrons. Here, it is necessary to distinguish between primary and secondary electrons, as electrons resulting from injection and hadronic interaction are present.\
Primary electrons $e_1$ denote electrons which received their energy from an accelerator and in the present model obey a power-law, i.e. electrons from the injection.\
In contrast, secondary electrons $e_2$ denote electrons from the decay of muons from hadronic pion production. If an effective particle acceleration from shock waves with the velocity $v_s$ is assumed, the particles can be accelerated when their kinetic energy is at least $E_{min,kin}=4\, (\frac{m_p\,v_s}{2})$ which equals 10 keV considering a velocity of $ v_s= 700$ km/s ([@Bell]). Given this energy gain, the minimum Lorentz factor becomes: $$\gamma_0^i=1+ \frac{10 \text{ keV}}{m_ic^2}\ .$$ By assuming a power-law spectrum for the injected relativistic particles the related source rate can be expressed by: $$q_i(\gamma)=q_0^i\, \gamma(\gamma-1)^{-\frac{\alpha+1}{2}}\ .$$ Considering that the particle energy has an upper limit the source rate becomes: $$q_i(\gamma)=q_0^i\, \gamma(\gamma-1)^{-\frac{\alpha+1}{2}}\,H[\gamma_{max}-\gamma]\,H[\gamma-\gamma_{min}],
\label{SourceRate}$$ where $q_0^i$ denotes the source rate normalization factor and $\alpha$ the energy spectral index[^4].\
Since in Cygnus X many accelerators reside which may complement each other, it is of use to relate the maximal Lorentz factor to the magnetic field and adapt it to the observed spectrum. Hence, $\gamma_{max}^p=10^{13}\times(\frac{B}{G})$ and $\gamma_{max}^e=(\frac{m_p}{m_e})\times\gamma_{max}^p$ is obtained.\
With eq.(\[totalNumber\]) and eq.(\[SourceRate\]), a relation between the electron and proton source rate due to the normalization factor can simply be established. $$\eta(\alpha)=\frac{q_0^e}{q_0^p}=\frac{\left( (\gamma_{max}^p)^2-1\right) ^{\frac{1+\alpha}{2}}-\left( (\gamma_{0}^p)^2-1\right) ^{\frac{1+\alpha}{2}}}{\left( (\gamma_{max}^e)^2-1\right) ^{\frac{1+\alpha}{2}}-\left( (\gamma_{0}^e)^2-1\right) ^{\frac{1+\alpha}{2}}}\, ,
\label{qRatio}$$ which leads for very high energies and an unbroken power-law to a constant ratio of $$\frac{q_0^e}{q_0^p}\simeq (\frac{m_p}{m_e})^{\frac{\alpha-1}{2}}.$$ As a function of the Lorentz factor the electron-proton source rate ratio would lead to $$\frac{q_p(\gamma)}{q_e(\gamma)}\simeq (\frac{m_p}{m_e})^{\frac{\alpha+1}{2}}$$ or in the space of the momentum $q_i(\gamma)\rightarrow q_i(p)$ which due to the fact $p_i=\gamma\, m_i$, leads to ([@Schlicki; @Pohl1]) $$\frac{q_p(p)}{q_e(p)}\simeq (\frac{m_p}{m_e})^{\frac{\alpha-1}{2}}\, ,$$ where $p$ denotes the momentum.\
The ratio of the electron and proton source rates as a function of the spectral index and Lorentz factor $\gamma$ is shown in figure \[fig:qratiowithoutsecondary2\]. Here, $q_e(\gamma)$ denotes only the primary electron source rate.
![ The ratio of primary electrons and protons as a function of the spectral index $\alpha$ and Lorentz factor $\gamma$, whereby the dashed line represents the relation in Cygnus X.[]{data-label="fig:qratiowithoutsecondary2"}](ProtonElectron.pdf){width="0.9\linewidth"}
As it will be shown later, the total number of electrons in Cygnus X is composed predominantly of primary electrons. Since this is the case, figure \[fig:qratiowithoutsecondary2\] also exhibits the ratio of the source rate of all electrons and protons. The quotient of the normalization factor in Cygnus X leads to the consequence $q_p(\gamma)=172\cdot q_ e(\gamma)$, i.e. the total injection rate of protons is 172 times greater than the injection rate of electrons.\
Solution of the transport equation
==================================
The specification of the general solution of electrons or protons relies on the adjustment of the loss processes. The catastrophic losses are already specified and distinguished between electrons and protons. The remaining loss mechanism is the continuous loss, which likewise makes a distinction between protons and electrons since they are based on different interactions and the particles have different rigidities.
Solution for electrons
----------------------
Electrons are subject to many losses, which are individually distinctive in different energy ranges. All of the following loss processes will be considered ([@Bjoern]):
&\_[ion]{}\^e(N\_t)7.210\^[-13]{} ( ) s\^[-1]{} ,\
&\_[syn]{}(B)1.310\^[-9]{}( )\^2 s\^[-1]{} ,\
&\_[Br]{}(N\_t)10\^[-15]{}( ) \^[-1]{} ,\
&\_[IC]{}(U\_[IR]{})5.210\^[-29]{}( )( ) \^[-1]{} .
Here, $ion$ denotes the ionization loss, $syn$ synchrotron loss, $Br$ non-thermal Bremsstrahlung and $IC$ Inverse Compton loss. In this context $N_t$ represents the constant target density in the plasma, $B$ the predominant magnetic field, $R$ the radius and $U_{IR}$ the infrared energy density. Hence, the total continuous loss for electrons yields: $$\Gamma_e \simeq \left( \Lambda_{IC}(U_{IR})+\Lambda_{syn}(B)\right) \gamma^2 +\Lambda_{Br}(N_t)\,\gamma\, + \, \Lambda_{ion}^e(N_t)\ .
\label{ContLoss}$$ The associated progression can be found in section \[CCLosses\]. Using the variation of constant and skilled integration, we obtain the following expression for the differential CR particle density ([@Bjoern]): $$\begin{split}
n_e(\gamma)=\frac{\Lambda_{ion}^e\exp\left(\chi_e(\gamma)+T_{diff}^e(\gamma) \right)}{(\Lambda_{IC}+\Lambda_{syn})\gamma^2+\Lambda_{Br}\gamma+\Lambda_{ion}^e} \\
\cdot\int_{\gamma_l}^{\gamma_{max}^e} \rm d\gamma'\, \frac{q_0^e\, \gamma'(\gamma'-1)^{\frac{\alpha-1}{2}}}{(\Lambda_{IC}+\Lambda_{syn})\gamma'^2+\Lambda_{Br}\gamma'+\Lambda_{ion}^e}\\ \cdot\exp\left(-(\chi_e(\gamma')+T_{diff}^e(\gamma' )\right)\, ,
\end{split}$$ $$\chi_e(\gamma)=\int\frac{\rm d\gamma}{\Gamma_e\,\tau_{adv}}=\frac{2}{\sqrt{4 a c-b^2 }} \arctan\left( \frac{b + 2 a \gamma}{\sqrt{4 a c-b^2} }\right)
\label{Advection}$$ and $$\begin{aligned}
\begin{split}
T_{diff}^e=&\frac{c\;\lambda_e}{R^2}\int \rm d\gamma\frac{\gamma^{\beta}}{a\gamma^2+b\gamma+c}
\\=&\frac{c\;\lambda_e}{R^2}\cdot \frac{a}{2\beta\theta} \cdot \big[ (2a\gamma+b-\theta)^{\beta}\cdot\:F_1\left(-\beta,-\beta;1-\beta;\: \frac{b-\theta}{2a\gamma+b-\theta}\right)\\& -(2a\gamma+b+\theta)^{\beta} \cdot F_1\left( -\beta,-\beta;1-\beta;\: \frac{b+\theta}{2a\gamma+b+\theta}\right) \big]\, .
\end{split}
\label{Diffusion}\end{aligned}$$ Here, the function $F_1$ represents the hyper-geometric function with $$F_1(-\beta,-\beta,\, 1-\beta, z)=\sum_{k=0}^{\infty}\frac{z^k}{k!}\prod_{k=0}^{\infty}\frac{(-1)(\beta+k)^2}{\beta-1+k}%=e^z\sum_{k=0}^{\infty}\prod_{k=0}^{\infty}\frac{(-1)(\beta+k)^2}{\beta-1+k}$$ For the lower integration limit, two cases must be considered $$\begin{aligned}
\gamma_l =
\begin{cases}
\ \gamma\, , \ \ \ \ \ \ \ \ \text{for}\ \gamma_{min}^e<\gamma<\gamma_{max}^e\, ,\\
\ \gamma_{min}^e\, , \ \ \ \ \text{for}\ \gamma<\gamma_{min}^e\, .
\end{cases}
\label{gamma_l}\end{aligned}$$ In the following the minimum Lorentz factor is supposed to be $\gamma_{min}=1$ and the maximal Lorentz factor the same as supposed in Section \[RelationEP\] above.
Solution for protons
--------------------
In contrast to electrons, protons are also subject to strong interaction. Hence, after an inelastic collision, they can produce a meson by generating a quark anti-quark or change the flavor due to the weak interaction. So, protons are influenced by different losses than electrons, which are also individually distinctive in various energy ranges.\
In this vein, the ionization loss for the protons as well as the hadronic pion production will be considered (e.g. [@SchlickiKrakau2015]): $$\begin{split}
\Gamma_{p,\pi}&\simeq4.4\cdot 10^{-16}\cdot\left( \frac{N_t}{cm^{-3}}\right) \gamma^{1.28}(\gamma+187.6)^{-0.2}\: s^{-1}\\ &= \Lambda_{p,\pi}(N_t)\cdot\gamma^{1.28}(\gamma+187.6)^{-0.2}\, .
\end{split}$$ The proton ionization loss is given by (e.g. [@Bjoern]): $$\Lambda^p_{ion}(N_t)\simeq 1.9\cdot 10^{-16}\cdot\left( \frac{N_t}{cm^{-3}}\right) \: s^{-1} \ .$$ Hence, the total loss rate is obtained as the sum of the two: $$\Gamma_p\simeq \Lambda_{p,\pi}\cdot\gamma^{1.28}(\gamma+187.6)^{-0.2}+\Lambda^p_{ion}.$$ The associated progression can be found in section \[CCLosses\]. Considering the same procedure as for electrons the differential CR particle density yields ([@Bjoern]): $$\begin{split}
n_p(\gamma)=&\frac{\exp\left( T_{diff}^p(\gamma)+\chi_p(\gamma)\right)}{\Lambda_{p,\pi}\cdot\gamma^{1.28}(\gamma+187.6)^{-0.2}+\Lambda_{io}} \\
&\cdot \int_{\gamma_l}^{\gamma^p_{max}} \rm d\gamma' q_0^p\, \gamma'(\gamma'-1)^{\frac{\alpha-1}{2}}\\&\cdot\exp\left( -T_{diff}^p(\gamma)-\chi_p(\gamma)\right)\, .
\end{split}$$ $$\begin{split}
\chi_p(\gamma)=&\int \frac{\rm d\gamma}{|\Gamma|_p\,\tau_{adv}}\\
\simeq &\frac{1}{\Lambda_{p,\pi}\tau_{adv}} \frac{3.571(\gamma+187.6)^{-0.2}}{(0.00533\gamma+1)^{0.2}\gamma^{0.28}}\\
&\cdot F_1\left( -0.28,\ -0.2;\ 0.72;\ -0.00533\, \gamma\right)
\end{split}$$ and $$\begin{split}
&T_{diff}^p\simeq\frac{c\lambda_p}{R^2\,\Lambda_{p,\pi}}\frac{(\gamma+187.6)^{-0.2}\gamma^{\beta-0.28}}{(\beta-0.28)(0.00533\gamma+1)^{0.2}}\\
&\cdot F_1\left( -0.2,\ \beta-0.28;\ 1+(\beta-0.28);\ 0.00533\gamma\right)\, .
\end{split}$$ The lower integration limit will be considered in the same way as before.
Radiation processes
===================
To understand the radiation from the Cygnus region, it is essential to derive the theoretical expressions for the most critical processes in the astrophysical context. In the following, an expression for the theoretical flux of each is given. The theoretical differential flux $\Phi(\gamma)$ can be described as a function of the emissivity and source function $\varepsilon(\gamma)$, respectively, by ([@RadiationProcesses]) : $$\Phi_{i}(E_j)=\frac{V}{4\pi\,d^2}\varepsilon_i(E_j)\ .
\label{differentialFLux}$$ Here, $i= IC, \, Br,\, \pi^0$ represents an individual process equal Inverse Compton, non-thermal Bremsstrahlung or hadronic $\pi^0$ decay and $j=\gamma, \nu, e^+ ,e^-$ the radiation type. The factor $(4\pi\,d^2)^{-1}$ is a correction for a fraction of the emission, which reaches the observer. The total $\gamma$-ray differential flux is obtained by summation over all present processes. $$\Phi_{\gamma}(E_{\gamma})=\frac{V}{4\pi d^2}\left( \varepsilon_{IC}(E_{\gamma})+\varepsilon_{Br}(E_{\gamma})+\varepsilon_{\pi^0}(E_{\gamma})\right)\ .$$ In addition, the integral flux is given by: $$\phi_i(E)=\int_{E}^{\infty}\rm d E' \Phi_{i}(\gamma')\ .$$ The source function in cm$^{-3}$s$^{-1}$ eV$^{-1}$ must be found individually for each process. It has the following proportionality: $$\varepsilon_{p_1,\,p_2}\propto c\int_{\gamma_{min}}^{\infty}\rm d\gamma\,n_{p_1}(\gamma)\,n_{p_2}(\gamma)\frac{d\sigma_{p_1,\,p_2}}{d\gamma}\, .
\label{emissivity}$$ It mainly depends on the differential density $n_i(\gamma)$ of the interacting particles $i=p_1,\ p_2$ and the related differential cross section $\rm d\sigma_{p_1,\,p_2}/{\rm d\gamma}$.
Synchrotron radiation
---------------------
To use only synchrotron radiation as the vital radio emission process and to avoid free-free emission and Bremsstrahlung, respectively, only the non-thermal emission from Cygnus X will be taken into account by considering a radio spectrum $\lesssim 10\ GHz$ ([@Bjoern]). The emissivity of synchrotron radiation is given by ([@Bjoern; @RadiationProcesses]): $$\varepsilon_{syn}(\nu)=\frac{1}{4\pi}\int_{\gamma_{min}}^{\gamma_{max}}n_e(\gamma)P_{syn}(\nu,\gamma)\, d\gamma$$ $$\begin{split}
&\text{with}\ \ P_{syn}=P_0\cdot\left( \frac{\nu}{\gamma^2\nu_s}\right)^{1/3}\exp(- \frac{\nu}{\gamma^2\nu_s}) \\ &\text{and}\ \ P_0= 2.65\cdot10^{-10}\cdot\left( \frac{B}{1\, G}\right)\ \text{eV s$^{-1}$ Hz$^{-1}$}.
\end{split}$$ In contrast to electrons, protons do not emit synchrotron radiation at the same intensity level, because the emitted power of synchrotron radiation $\dot{E}_{syn}$ is proportional to $ m^{-4}$. $$\frac{\dot{E}_{syn}^e}{\dot{E}_{syn}^p}= \left( \frac{m_p}{m_e}\right)^4\simeq 1.13\cdot10^{13}$$ Since the ratio of the proton-electron mass is $\sim 1836$, the proton synchrotron radiation requires inconceivably high energies and a strong magnetic field ([@Proton_Syn]).
Inverse Compton
---------------
In astrophysical context, the Inverse Compton process is based on the interaction of a relativistic CR electron with an ambient photon. The relativistic electron transfers a part of its kinetic energy to the target photon, whereby a minimum energy of $$E_{min}= \frac{E_{\gamma,f}}{2}\left[1+\left( 1+\frac{m_e^2c^4}{E_{\gamma,i} E_{\gamma,f}}\right) \right]$$ is necessary. Here, $E_{\gamma,i}$ and $E_{\gamma,f}$ represent the initial and final photon energy respective to the scattering ([@Schlicki]).\
\
Since Cygnus X contains a large amount of dust, the primary photon field is represented by infrared emission, as the starlight is absorbed and then re-radiated in the infrared range. Thus, $\gamma E_{\gamma,i}\ll m_e\,c^2$ is valid, and the Inverse Compton loss can be considered in the Thomson limit. Only a small fraction of the injected electrons are within the condition $\gamma>m_e\,c^2/E_{\gamma,i}$, the Klein-Nishina (KN) regime. The following further considers a maximal Lorentz factor (see Section \[RelationEP\] for more details). Considering a gray body or rather a modified blackbody according to ([@Casey]) and an isotropic and uniform spatial distribution, the differential infrared photon density yields ([@Bjoern; @Casey]):
$$\begin{split}
\frac{\rm{d} \it n_{IR} (E)}{\rm{d} \it E}=&1.125\cdot 10^{19}\cdot\frac{U_{IR}}{E_0}\left( \frac{E}{h\, c}\right) ^3\\&\cdot\frac{1-\exp\left( ({E}/{E_0})^{\beta}\right) }{\exp\left( E/(k_B\,T_D)\right) -1}\ .
\end{split}$$
Here, $T_D$ denotes the dust temperature, $U_{IR}$ the infrared photon energy density, $\beta=1.5$ the emissivity index ([@Casey]) and $E_0=12.4\cdot10^{-3}$ eV the energy ([@Casey]), where the optical depth equals unity. In Cygnus X the dust temperature is supposed to be approximately 25 K ([@DustTemp]) and the infrared photon energy density 5 eV/cm$^3$ ([@TovaPaper]). The photon density is then given by $$\begin{split}
n_{IR}(E)&=1.125\cdot 10^{19}\frac{U_{IR}}{E_0(h\, c)^3}\\&\cdot\int \rm d \it E\: E^3 \frac{1-\exp\left(- ({E}/{E_0})^{\beta}\right) }{\exp(E/k_B\,T_D)-1}.
\end{split}$$ According to eq.(\[emissivity\]) the differential cross section for the Inverse Compton process is needed, which is given by the Klein-Nishina formula: $$\begin{split}
&\frac{\rm d\sigma(E_{\gamma,f},E_{\gamma,i},\gamma)}{\rm \gamma}=\frac{3}{4}\frac{\sigma_T}{E_{\gamma,i}\gamma^2}G(q,\Gamma), \ \text{with}\\
&G(q,\Gamma)= 2q\ln(q)+(1-q)(1+2q)+\frac{(\Gamma q)^2(1-q)}{2(1+\Gamma q)}\\
&q=\frac{E_{\gamma,f}}{\Gamma(\gamma m_e\,c^2-e_{\gamma})}\ \ \, \Gamma =\frac{4\gamma E_{\gamma,i}}{m_e\,c^2}\ ,
\end{split}$$ whereby $\gamma$ denotes the electron Lorentz factor. According to eq.(\[emissivity\]) the emissivity yields $$\varepsilon_{e,\,\gamma}^{IC}(E_{\gamma})=\frac{3}{4}c\,\sigma_T\,n_{IR}\int_{\frac{E_{min}}{m_ec^2}}^{\infty} \frac{\rm d\gamma}{\gamma^2} n_e(\gamma)G(q,\Gamma)\ .$$
Non-thermal Bremsstrahlung
--------------------------
In an astrophysical context, Bremsstrahlung is produced in a hot and predominantly ionized plasma, where the particles are free before and after the deflection. Bremsstrahlung is an important process in Cygnus X, since it contains H$_{\text{II}}$ regions, ionized gas clouds around hot and young stars ([@IdentificationTeVCygnusCocoon; @HighEnergyAstrophysics]).\
The general emissivity produced by Bremsstrahlung can be described by ([@Stecker]): $$\varepsilon^{Br}_{e,\,\gamma}(E_{\gamma})=\int_{\gamma}^{\infty}d\gamma\,N_t\,c\,\frac{\sigma_{Br}}{E_{\gamma}}n_e(\gamma)=\frac{N_t\,c\,\sigma_{Br}}{E_{\gamma}}\int_{E_{\gamma}/m_e}^{\infty}d\gamma\,n_e(\gamma)\ .$$ Here, $\sigma_{Br}=3.38\cdot 10^{-26}$ cm$^2$ denotes the Bremsstrahlung cross section and $N_t$ the proton target density. The flux can be calculated using eq.(\[differentialFLux\]).
Gamma-rays from hadronic pion production
-----------------------------------------
After the interaction of protons with the ambient medium pions are generated. Thus, the inelastic proton-proton cross section $\sigma_{pp, inel}$ must be considered by equation(\[ppCrossSection\]) ([@Kelner]). In succession, the $\pi^0$ decays and generates two $\gamma$-rays. This process is thought to cause most of the $\gamma$-rays in star forming regions ([@Bjoern]).\
Additionally, the energy spectrum $F_{\gamma}(x,E_p)$ of secondary particles (here $\gamma$-rays), which are produced in one interaction with a proton of the energy $E_p$, must also be considered. This includes the intermediate production and decay of $\pi^0$, whereby $x_{\gamma}=E_{\gamma}/E_{\pi}$ is the ratio of the energy of the incident proton to the produced $\pi^0$. In this regard $F_{\gamma}(x_{\gamma},E_p)\cdot dE_{\gamma}/E_p$ is the number of $\gamma$-rays from a single proton-proton interaction in the interval $[E_{\pi}, E_{\pi}+dE_{\pi}]$ ([@Kelner]). $$\begin{split}
\sigma_{pp, inel}(E_p)=&
\begin{cases}
(34.3+1.88\,L+0.25\,L^2)\left[ 1-(\frac{E'}{E_p})^4\right]\ \text{mb, } \text{for}\ E'\leq E_p\leq0.1\ \text{TeV} \\
(34.3+1.88\,L+0.25\,L^2)\ \text{mb, } \ \ \ \ \ \qquad\qquad\text{for}\ E_p>0.1\ \text{TeV}\
\end{cases}\\
&\text{with}\ E'=\left( m_p+2m_{\pi} +m_{\pi}^2/2m_p\right) c^2 \ \text{ and }\ L=\ln(E_p/1\ \text{TeV})\, . \ \ \
\end{split}
\label{ppCrossSection}$$ Using this relation, the total emissivity is obtained by: $$\varepsilon_{p,\pi^0,\gamma}^{had}(E_{\gamma})=c\, N_t\int_{E_{\gamma}}^{\infty}\frac{dE_p}{E_p}\sigma_{pp, inel}(E_p) \cdot n_p(E_p)F_{\gamma}(x_{\gamma},E_p)\ .$$
Neutrinos
---------
In the astrophysical context, neutrinos can be produced from hadronic charged pion production and after that from leptonic muon decay.\
In the same manner, as the emissivity was determined for $\gamma$-rays from the hadronic pion decay by considering its energy spectrum, the energy spectra can be replaced for another particle $i$, i.e. $F_{\gamma}(E_{\gamma}/E_{\pi^0},\ E_p)\longmapsto F_{i}(E_{i}/E_{\pi^{\pm}},\ E_p)=F_i(x_i,E,p)$ for $i=\nu_e,\nu_{\mu}, e$.\
Since $F_{i}(x_{i},E_p){dE_{\pi}}/{E_p}$ denotes the number of the particle $i$ for a single proton-proton interaction in the interval $[E_{\pi}, E_{\pi}+dE_{\pi}]$, the following approximation is valid: $F_e(x_e,\ E_p)\simeq F_{\nu_e}(x_{\nu_e},\ E_p)$. The deviation is less than 5% ([@Kelner]) when using this approximation. If the muon neutrino from the process $$\pi^{\pm}\longrightarrow \mu^{\pm}+\nu_{\mu}/\bar{\nu}_{\mu}
\label{PionDecay}$$ is denoted as $\nu_{\mu1}$ and the muon from the process $$\mu^{\pm}\longrightarrow e^{\pm}+ \nu_e/\bar{\nu}_e+\bar{\nu}_{\mu}/\nu_{\mu}
\label{MuyonDecay}$$ as $\nu_{\mu2}$, the following can be approximated:\
$F_e(x_e,E_p)\simeq F_{\nu_{\mu2}}(x_{\nu_{\mu2}},\ E_p)$. Hence, the emissivities can be described by: $$\varepsilon_{p,\pi,\nu_{\mu2}}(E_{\nu_{\mu2}})=c\, N_t \cdot\int_{E_{\nu_{\mu2}}}^{\infty}\frac{dE_p}{E_p}\sigma_{pp, inel}(E_p)\,n_p(E_p)F_{e}(x_{\nu_{\mu2}},E_p),$$ $$\varepsilon_{p,\pi,\nu_{e}}(E_{\nu_{e}})=c\, N_t \cdot\int_{E_{\nu_{e}}}^{\infty}\frac{dE_p}{E_p}\sigma_{pp, inel}(E_p)\,n_p(E_p)F_{e}(x_{\nu_{e}},E_p)$$ and $$\varepsilon_{p,\pi,\nu_{\mu1}}(E_{\nu_{\mu1}})=c\, N_t\cdot\int_{E_{\nu_{\mu1}}}^{\infty}\frac{dE_p}{E_p}\sigma_{pp, inel}(E_p)\,n_p(E_p)F_{\nu_{\mu1}}(x_{\nu_{\mu1}},E_p)\ .$$ The total neutrino emissivity is the sum of the contributions: $$\varepsilon_{p,\pi,\nu}(E_{\nu})=c\, N_t\int_{E_{\nu}}^{\infty}\frac{dE_p}{E_p}\sigma_{pp, inel}(E_p)\,n_p(E_p)\cdot\left( F_{\nu_{\mu}}(x_{\nu},\, E_p)+2F_{e}(x_{\nu},\, E_p)\right)\ ,$$ where we used the differential particle density of relativistic protons from Section \[Model\].\
Hereby, the ratio of the appearance of different type of neutrino at the source is (1:2:0) for ($\nu_e:\nu_{\mu}:\nu_{\tau}$). Due to the neutrino oscillation at the large distance the ratio changes to 1:1:1 (e.g. [@Julia1]), whereby this condition is fulfilled for Cygnus X.\
Secondary electrons {#secondary-electrons .unnumbered}
-------------------
Analogously to the case of the neutrino, the emissivity for secondary electrons which are generated from hadronic pion production is calculated with $$\varepsilon_{p,\pi,{e}}(E_{{e}})=c\, N_t\int_{E_{{e}}}^{\infty}\frac{dE_p}{E_p}\cdot\sigma_{pp, inel}(E_p)\,n_p(E_p)F_{e}(x_{{e}},E_p)\, .
\label{sFktElectron}$$
Primary and secondary electrons in Cygnus X
===========================================
Because secondary electrons result from proton interaction or rather a hadronic pion production, they depend on the proton density and on its cross section to produce charged pions. Therefore, it also depends on the Lorentz factor and the target density. Since $\gamma_{max}^i$ in our model is related to the magnetic field strength and the primary electron source rate includes the spectral index, the dependence will be plotted to present the ratio of primary to secondary electron source rate as a function of the target density $N_t$, magnetic field strength $B$ and the spectral index $\alpha$. The associated figures can be seen in figures \[fig:primarysecondariesNt\]-\[fig:primarysecondariesB\].
Here, $q_{e_1}(\gamma)$ denotes the primary and $q_{e_2}(\gamma)$ the secondary electron source rate function with $q_{e_2}(\gamma)=\varepsilon_{p,\pi,{e}}(E_{{e}})\cdot m_ec^2$. The ratio of primary to secondary electrons as a function of the Lorentz factor for certain parameters which in this work seems to describe Cygnus X, is represented by the dashed red line.\
The figures \[fig:primarysecondariesNt\] - \[fig:primarysecondariesB\] clarify that the amount of secondary electrons increases with greater $\gamma$ relative to the primary electrons. Additionally, the rise for Cygnus X seems to be quite uniform. In any event, in Cygnus X the primary electrons always dominate.
Continuous and catastrophic losses {#CCLosses}
==================================
Whether the continuous or catastrophic momentum loss adopts the main loss mechanism in Cygnus X or whether both are equally significant can be determined by regarding the related timescale. The total continuous loss timescale is given by: $$\begin{split}
\varSigma_{con}^e\equiv&\frac{1}{\tau_{con}^e}=\frac{\Gamma_e}{\gamma}=\frac{\Lambda^e_{ion}(N_t)}{\gamma}+ \Lambda_{IC}(U_{IR})\, \gamma\\&+{\Lambda_{syn}(B)\, \gamma} + \Lambda_{Br}(N_t) \\=&\frac{1}{\tau_{ion}^e}+\frac{1}{\tau_{IC}}+\frac{1}{\tau_{syn}}+\frac{1}{\tau_{Br}}
\end{split}$$ $$\begin{split}
&\varSigma_{con}^p\equiv\frac{1}{\tau_{con}^p}=\frac{\Gamma_p}{\gamma}=\frac{\Lambda^p_{ion}(N_t)}{\gamma}+\Lambda_{p,\pi}(N_t)\\&\cdot\gamma^{0.28}(\gamma+187.6)^{-0.2}=\frac{1}{\tau_{ion}^p}+\frac{1}{\tau_{p,\pi}}\, .
\end{split}$$ In figures \[fig:timescaleallE\] and \[fig:timescaleallP\] the continuous timescales are represented by solid lines and catastrophic timescales by dashed lines. Here, $\tau_{fs}=R/c$ denotes the timescale of a free streaming particle with the velocity of light c which is needed to pass through the considered region of Cygnus X within 77 pc.\
As the diffusion timescale represents a quantity for the entrapment of the particle particles in Cygnus X, it must be smaller than $\tau_{fs}$ because the particle cannot move faster than the speed of light. If $\tau_{diff}\approx\tau_{fs}$ then the diffusion is negligible as in this case the particles will move undisturbed.
For most of the time, the diffusion timescale has the smallest value. In particular, the Inverse Compton and synchrotron timescales are most important at higher energies. They decrease with a similar slope which is according to the amount the strongest. The timescales of Bremsstrahlung and advection do not differ significantly from each other and are constant.\
The importance of diffusion is recognizable for protons as it is always the dominant process. In contrast, ionization loss is even at the lowest energies almost negligible. Moreover, hadronic pion production is the second most important loss mechanism at higher energies and advection is relevant for $\gamma < 10^4$.\
Thus in total, the diffusion loss timescale for catastrophic losses is smaller than that for advection for all energy ranges. Additionally, the Inverse Compton loss becomes the main loss mechanism for electrons at very high energies.\
![The ratio of $\Sigma_{con}^e$ and $1/\tau_{diff}^e$ for electrons as a function of the magnetic field strength $B$ and the Lorentz factor $\gamma$ with $N_t=19.4$; the dashed line shows the dependency in Cygnus X.[]{data-label="fig:electronmesh"}](ElectronConB){width="1.0\linewidth"}
![The ratio of $\Sigma_{con}^e$ and $1/\tau_{diff}^e$ for electrons as a function of the target density $N_t$ and the Lorentz factor $\gamma$ with $B=8.9\times10^{-6}$ G; the dashed line shows the dependency in Cygnus X.[]{data-label="fig:electronmesh1"}](ElectronConNt){width="0.95\linewidth"}
![The ratio of $\Sigma_{con}^p$ and $1/\tau_{diff}^p$ for protons as a function of the target density $N_t$ and the Lorentz factor $\gamma$; the dashed line shows the dependency in Cygnus X.[]{data-label="fig:protonmesh"}](ProtonConNt){width="0.95\linewidth"}
It is meaningful to investigate when the continuous loss can exceed the diffusion loss. Therefore the dependency of ${\Sigma_{con}^i}/{(1/\tau_{diff}^i)}$ for $i=e,p$ on the target density and magnetic field is shown in figures \[fig:electronmesh\]-\[fig:protonmesh\].\
In all variations that are meaningful for Cygnus X, which can be seen in figure \[fig:ChiAlpha\], the diffusion loss almost always exceeds other losses. In conclusion, it is crucial to consider the diffusion.\
Retrospectively, this conclusion could give us some hints about the relation of the young supernova remnant $\gamma$-Cygni and the Cygnus Cocoon if we assume the same parameters for the Cocoon as for the whole Cygnus X. It must be mentioned that $\gamma$-Cygni may have delivered protons and electrons at TeV range approximately five kyrs ago ([@CygnusByFermi2]). So, our steady-state model is not appropriate enough for the Cygnus Cocoon but the whole Cygnus for the reason as mentioned above. However, assuming the same parameters for the Cygnus Cocoon as for the whole Cygnus X, we can give hints about the relation between $\gamma$-Cygni and Cygnus Cocoon.\
According to [@CygnusByFermi2], the condition for $\gamma$-Cygni to maintain as the only accelerator for the Cygnus Cocoon is that the dominant particle transport mechanism should be diffusion and the diffusion coefficient largely be similar to the coefficient in our Galaxy. In that case, the particles released by $\gamma$-Cygni could maintain CRs from the whole Cocoon. The diffusion-dominated scenario leads to an isotropic particle release from the young remnant. It is important to answer the question whether the particles from $\gamma$-Cygni could maintain the CRs in the Cygnus Cocoon due to diffusion transport mechanism. It can be answered by considering the mean free path and the average distance $d_{\gamma, Cyg}$ of $\gamma$-Cygni to the Cygnus Cocoon, which is pictured in figure \[fig:cyggal\].
Here, the solid blue line represents the mean free paths in Cygnus X, the red line the one in the Galaxy ([@D0diffCoeff]) and the dashed black line shows the distance of $\gamma$ Cygni from the Cocoon. Nearly all CRs from $\gamma$ Cygni in direction to the Cocoon could reach the Cocoon if the mean free path is similar or larger than the mean free path of the Galaxy according to ([@CygnusByFermi2]) or at least longer than the distance of these objects to each other.\
However, the mean free path is for electrons always smaller than the distance and than the value in the Galaxy. At energies $\gtrsim 10^5$ GeV, protons from $\gamma$-Cygni may be reasonable for the freshly accelerated CRs from the Cygnus Cocoon since the mean free path becomes larger than the distance between them. For lower energies indeed $\gamma$ Cygni is not favored to be the only injector of CRs.
Parameters for Cygnus X
-----------------------
Data from different observatories can be used to restrict the magnitude of the parameters on the one hand, and the number of free parameters on the other hand. In the following $\gamma$-ray data from Fermi, Argo and Milagro will be used. Additionally, the non-thermal radio data from the work of [@thermalNonThermal1] and [@thermalNonThermal2] will be considered. When doing so, for example fitting methods can be used to find the normalization factor $q_0^{e,p}$ of the source rate function. The non-thermal radio data will be used to determine $q_0^e$ and $\gamma$-ray data to find $q_0^p$. Additionally, by considering emission in the energy range, which requires only electrons, the amount of injected electrons can be determined and thus the appearance of a leptonic process. Especially, the contribution of non-thermal Bremsstrahlung can be estimated. This correlation will be discussed in the results in chapter \[Results\]. In order to constrain the appearance of leptonic processes, the differential fluxes for non-thermal radio emission and $\gamma$-rays from Cygnus X will be presented followed by the correlation between them. In the following, a magnetic field strength of $B$=1 $\mu$G ([@Francis]) and a target density of $N_t$=70 cm$^{-3}$ ([@CygnusByFermi]) will be used. Using the brightness temperature spectral index of non-thermal data a spectral index of $\alpha$=2.6 ([@thermalNonThermal2; @thermalNonThermal1]). The Cygnus X region can be summarized by the following parameters:
Parameters Cygnus X
--------------------------------------- --------------------------------------------
Electron diffusion length \[cm\] $l_{e}=2\times10^{16}$
Proton diffusion length \[cm\] $l_{p}=l_{e}(\frac{m_{p}}{m_{e}})^{\beta}$
Diffusion index $\beta=1/3$
Infrared photon density \[eV/cm$^3$\] $U_{IR}= 5$
Advection velocity \[km/s\] $v_{adv}=50$
Dust temperature \[K\] $T_{dust}= 25$
Distance to Cygnus X \[kpc\] $d=1.4$
Radius of Cygnus X \[pc\] $R= 77$
Radius of Cygnus X \[$\deg$\] $\Theta_{CygX}=3.15$
: Input parameters for the Cygnus X region.[]{data-label="list1"}
Results with parameters from previous calculations (PC) {#RFCM}
=======================================================
In the following, the non-thermal radio and $\gamma$-ray spectra will be presented by considering parameters from PCs (see figure \[fig:aaa\] and \[fig:aaa2\]). The consideration of the $\gamma$-ray spectrum is paramount as all relevant subatomic processes occur in it and the radio data assist in constraining the leptonic processes as it is based on one.
Here, “**IC**” denotes the contribution of the Inverse Compton process to the total differential $\gamma$-ray flux. In the same way “**BR**” denotes Bremsstrahlung and “**HADRONIC**” the hadronic pion production. The solid line represents the progression of the total differential $\gamma$-ray flux of this work. Besides, the results are presented without diffusion which is denoted as “WO-D”, and with diffusion loss as diffusion is the main loss mechanism. The diffuse fluxes observed by Fermi LAT (green cross), ARGO-YBJ (red triangle) and Milagro (turquoise circle) are pictured. The Fermi $\gamma$-ray data points are taken from [@CygnusByFermi] and adapted for our region of interest. The same procedure is done for ARGO-YBJ from [@ARGOData] and Milagro from [@MilagroData]. From these diffuse fluxes the extragalactic point sources (AGNs) J2000.1+4212 and J2018.5+3851 are subtracted. It is not necessary to subtract galactic point sources as our model considers them by fitting the source rate normalization factor $q_0$ on the observed $\gamma$-ray flux.\
Additionally, the flux from Cygnus X-1 measured by Comptel can be seen (blue circle).\
Cygnus X-1 is a very well studied black hole and a front runner microquasar candidate in the Galaxy. It contains a dominant power-law component at 10 MeV. Consequently, it can be assumed that the flux from the whole Cygnus X region should be equal or higher.\
Considering the constituents of the total flux, the progression at 10 MeV is of particular importance, since the total differential flux at 10 MeV is distinctly dominated by Bremsstrahlung which is caused by the leptonic process. The agreement with this data point and the consideration of the non-thermal radio data can point towards the real relevance of the diffusion loss mechanism. This is plausible as at the relatively low energy 10 MeV the entrapment of the electrons due to diffusion is more efficient than for higher energies (see figure \[fig:timescaleallE\]) so that more Bremsstrahlung can be produced. Moreover, the real parameters such as the magnetic field and target density and the previous assumption regarding the diffusion of the particles can be investigated.\
Comparing these results, there is no agreement between non-thermal radio and$\gamma$-ray data (see figure \[fig:aaa\] and \[fig:aaa2\]). The used parameters have been investigated with PCs which do not consider diffusion loss as this work does. The value of the magnetic field e.g. is averaged over the whole Galaxy and is therefore not accurate enough for our calculations. As the structure of Cygnus X is very complicated, and it has many constituents the target density also might be not accurate enough for our model. Moreover, for Cygnus X the non-thermal radio and $\gamma$-ray have not been correlated before. In this work, the assumed parameters do not lead to an agreement between data from these radiations.
Best-fit procedure
------------------
Since the parameters used in the previous results do not seem to describe the Cygnus X sufficiently well, they must be changed. On this basis, the total deviation in square $\chi^2$ of each data point to the theoretical flux is defined by the reduced $\chi^2$: $$\chi^2= (\chi^2_{\gamma}+\chi^2_{\text{syn}})\, ,$$ $\chi_{\gamma}$ denotes the deviation of the theoretical $\gamma$-ray flux from the $\gamma$-ray data over the degree of freedom. In the same regard, $\chi_{\text{syn}}$ describes the deviation of the theoretical synchrotron flux from the non-thermal radio data over the associated degree of freedom. They can be calculated with eq.(\[chi1\]) and eq.(\[chi2\]). The degree of freedom $F$ estimates the population standard deviation calculated from a sample. The degree of freedom is given by $F=n-1$, where $n$ is the size of the sample. In this work two samples have been used: Non-thermal radio and $\gamma$-ray data. $$\begin{split}
\chi^2_{\gamma}&=\bigr[ \Phi_{\gamma,\text{obs}}(E_{\gamma})-\left( \Phi_{\gamma,IC}(E_{\gamma})+\Phi_{\gamma,Br}(E_{\gamma})\right.\\ &\left. +\Phi_{\gamma,had}(E_{\gamma})\right)\bigr] ^2 /\Delta\Phi_{\gamma,\text{obs}}(E_{\gamma}) ^2\cdot \frac{1}{F-1}
\end{split}
\label{chi1}$$ $$\chi^2_{\text{syn}}=\left( \dfrac{\Phi_{\text{syn,obs}}(\nu)-\Phi_{syn}(\nu) }{\Delta\Phi_{\text{syn,obs}}(\nu)}\right) ^2\cdot \frac{1}{F-1}\, ,
\label{chi2}$$ Here, $\Delta\Phi_{\gamma,\text{obs}}(E_{\gamma})$ and $\Delta\Phi_{\text{syn,obs}}(\nu)$ denote the uncertainties which result from the measurements.\
As discussed above it is a reasonable step to set the magnetic field and the target density as free parameter as the correlation of radio and gamma data proved problematic and the relation was strongly affected by these parameters. The variation of $B$ and $N_t$ can also change the spectral index $\alpha$. Furthermore, the diffusion of the particles causes a steepening of the spectrum and increases the flux produced by leptonic processes which are necessary to reach especially the data point at 10 MeV, as the entrapment of the particle due to diffusion is at the relatively low energy 10 MeV more efficient. The whole variation range of the three fitting parameters can be taken from table \[variation2\]:
Physical parameters Variation range
------------------------------------ -----------------------
Magnetic field $B$ \[G\] $[10^{-7};\ 10^{-4}]$
Target density $N_t$ \[cm$^{-3}$\] \[$10^1;\ 10^{2.7}$\]
Spectral index $\alpha$ \[$2.0;\ 3.0$\]
: Variation range of magnetic field, target density and spectral index.[]{data-label="variation2"}
The deviation can be illustrated graphically, such that the influence through the whole range and the best-fit parameters may be seen easily. Figure \[fig:ChiAlpha\] shows $\chi^2$ as a function of the target density $N_t$ and the magnetic field $B$.
![The total deviation $\chi^2$ from the $\gamma$-ray and non-thermal radio data as a function of the magnetic field B and target density $N_t$. The best fit parameters are represented by “+”. For each calculation of $\chi^2$ a best-adapted spectral index $\alpha\in[2.0-3.0]$ was also used.[]{data-label="fig:ChiAlpha"}](2a=2-3_B=-6,-3,N=1+27-027e2e+16.pdf){width="1.1\linewidth"}
The dark red area shows a stronger agreement between theoretical and experimental fluxes than the other colors.\
These two parameters, in particular, appear to be markedly different than anticipated. The magnetic field is larger than the previously assumed value by a factor of nearly one order of magnitude, and the target density is smaller by a factor of 3.6. Considering figure \[fig:ChiAlpha\], the magnetic field is not supposed to be smaller than 3 $\mu$G or larger than 100 $\mu$G. The target density has an upper limit of 300 cm$^{-3}$. The best-fit point is represented by a “+” sign.\
The lowest $\chi^2$ provides a magnetic field strength of $B=8.9\times10^{-6}$ and a target density of $N_t=19.4$ cm$^{-3}$.\
In order to find a reliable spectral index, the best-adapted spectral index is presented as a function of the magnetic field and target density in figure \[fig:alpha\].
![The best adapted-spectral index $\alpha$as a function of the magnetic field $B$ and target density $N_t$.[]{data-label="fig:alpha"}](alpha60){width="1.\linewidth"}
If one considers the range from figure \[fig:ChiAlpha\], where the smallest $\chi^2$ was found, it may be asserted that $\alpha=2.32-2.4$ is the best value for Cygnus X. However, the smaller range of variation and the calculations in this work show that the best-fit is obtained for $\alpha=2.37$.\
In addition, the energy loss in erg/s can be investigated by: $$\dot{{E}}^i_j=\frac{4}{3}\pi R^3m_i c^2\int_{\gamma_{0}^i}^{\infty}d\gamma\gamma\frac{n_i(\gamma)}{\tau_j}\, ,$$ whereby the index $i$ refers to the quantity of an electron or proton, $\tau$ the loss timescale and $j$ the loss mechanism. Since diffusion is the dominant loss mechanism, $\dot{{E}}^e_{diff}$ and $\dot{{E}}^p_{diff}$ as a function of the magnetic field and target density respectively will be presented in figure \[fig:energylosselectronzoom\]. Here, for each data point, an adapted spectral index $\alpha$ and the source rate normalization factor $q_0^p$ have been used. The result is a calculation of the energy loss with different spectral indices and $q_0$, which lead to a smallest total deviation $\chi^2$.
In Cygnus X the energy loss yields $\dot{{E}}^e_{diff}=1.08\times10^{56}$ erg/s and $\dot{{E}}^p_{diff}=3.39\times10^{59}$ erg/s. This implies that the proton is more prone to be lost due to diffusion than electrons. This behavior is comprehensible in view of figure \[fig:timescaleleelectron\] and also eq.(\[sigmaDiff\]), which show that $\lambda_p>\lambda_e$ and thus $\tau^p_{diff}<\tau^e_{diff}$.
Results with best-fit parameters {#Results}
================================
The best procedure leads to the following parameters
Parameters Cygnus X
--------------------------------------------- --------------------------------
Magnetic field strength \[G\] $B=8.9\times10^{-6}$
Proton target density \[cm$^{-3}$\] $N_t=19.4\ $
Spectral index $\alpha=2.37$
Proton source rate
normalization factor \[cm$^{-3}$ s$^{-1}$\] $q_0^p=9.8\times10^{-22}$
Electron source rate
normalization factor cm$^{-3}$ s$^{-1}$\] $q_0^e\approx 172\times q_0^p$
: Best fit parameters for the Cygnus X region.[]{data-label="list2"}
The new target density provides the depth of Cygnus X in the spiral arm, as the the column density in Cygnus X is known. Again considering the work of [@CygnusByFermi] a column density of $C_{H_I}=70$ cm$^{-2}$ is obtained. Putting the target and column density in a simple relation, the neutral gas distribution over a depth of $d_t$ is obtained: $$d_t=C_{H_I}/N_t\approx3\times10^{20}\ \rm cm\approx116\ \rm pc$$ Furthermore, the relation between $q_e(\gamma)$ and $q_p(\gamma)$ leads to an injection rate of protons 172 times greater than that of electrons, i.e. $q_p(\gamma)\approx 172\times q_e(\gamma)$. Finally, the $\gamma$-ray and non-thermal radio spectrum can be presented in figure \[fig:a252b24e-5nt23q0p236e-23\] and \[fig:2a252b24e-5nt23q0p236e-23\], respectively.\
The $\gamma$-ray spectrum shows that for high energies ($>40$ MeV) the dominant constituent of the total differential $\gamma$-ray flux is caused by the hadronic pion production. The Bremsstrahlung represents the second largest component of the total flux. This process is also the most significant element of $\gamma$-ray flux at lower energies ($<40$ MeV). The Inverse Compton differential flux is for the most part in the background.
![$\gamma$-ray energy spectrum **with** and **without** consideration of diffusion loss; the source rate normalization factor $q_0$ was fitted on the observed $\gamma$-ray data. Additionally, the new and better-adapted parameters from list \[list2\] were used.[]{data-label="fig:a252b24e-5nt23q0p236e-23"}](NewParameterGamma){width="0.85\linewidth"}
![Synchrotron differential flux spectrum as a function of the frequency **with** and **without** consideration of diffusion loss; the source rate normalization factor $q_0$ was fitted on the observed gamma data. Here, the new and better-adapted parameters from list \[list2\] were used.[]{data-label="fig:2a252b24e-5nt23q0p236e-23"}](NewParameterRadio){width="0.85\linewidth"}
The new results for $\gamma$-rays show an adamant agreement between ARGO-YBJ and Fermi data. The condition for the flux at 10 MeV is also fulfilled. In contrast, the Milagro differential flux is less than circa four times greater than the differential flux calculated in this work. This leads to the suspicion that Milagro may be overestimating the flux in Cygnus X. Indeed, Daniele Gaggero et al. 2015 ([@AliPaper]) have shown for the Galactic diffuse emission that a conventional representative model with the properties found by Fermi data fails to reproduce the large flux measured by Milagro, meaning the overestimation of Milagro is still an open issue. As our diffuse flux can not explain the Milagro flux, a point source, i.e. might me responsible at these high energies. In a recent paper [@TovaPaper] have evaluated the radiation in the Cygnus region resulting from the propagation of the average cosmic ray flux in the Galaxy. The result depends on the spectrum assumed but, in any case, cannot accommodate the observed photon emission, especially at TeV energy and above as measured by Milagro and HAWC. The conclusion is that accelerators in the region, most likely in the Cocoon, must be responsible for the high-energy flux. This is consistent with our modeling of the Cygnus.\
Nevertheless, a natural explanation for the “Milagro anomaly” in our Galaxy has been found by considering the radial dependence for diffusion coefficient spectral index $\beta$ and the advective wind. However, this fact does not hold great relevance for the present work, since Cygnus X is small in comparison to the Galaxy. The prediction that the agreement between theoretically and experimentally determined fluxes deteriorates when examining high-energy particles may therefore be validated. Also, the new parameter leads to a much stronger agreement for the non-thermal radio data than before.\
In the same way, we identified the theoretical spectra without considering diffusion of the particles. The agreement between non-thermal radio and $\gamma$-ray data is worse than with considering diffusion and the conditions at 10 MeV is not fulfilled. This also shows us that diffusion is very meaningful for Cygnus X.\
Finally, the neutrino differential flux spectrum considering the new parameters is presented in figure \[fig:neutrinoNewPar\]. As a comparison, the spectrum considering parameters from PCs is pictured in figure \[fig:neutrinoNewPar2\]. The limits in these figures are normalized with an $E^{-2.6}$ spectrum of Cygnus X from [@CygLimitDis]. The first spectrum predicts a flux which coincides with the limit of IceCube at very high energies ($>$50 TeV). As IceCube has the highest sensitivity at 100 TeV and the spectral index of the predicted flux and the limit does not differ much from each other, a significance measurement by IceCube or IceCube-Gen2 may be soon possible.\
According to the model without diffusion loss (WO-D) the predicted flux is above the sensitivity of IceCube. The neutrino differential flux at 100 TeV when considering diffusion loss is here almost 2.3 times smaller than the model without diffusion loss.\
The second spectrum is worth measurable than the first one. Even the flux from WO-D which is not realistic, is not measurable. Overall, the $\gamma$-ray and non-thermal radio spectra provides a very strong agreement suggesting that the used parameters, transport mechanism, and model indeed seem to describe Cygnus X in an accurate way.
Conclusion {#Summary}
==========
In this paper, the leptonic and hadronic cosmic ray transport in the Cygnus is modeled with a leaky box model that takes into account energy loss processes via radiation and interaction as well as advective and diffusive transport, assuming the homogeneous injection of cosmic rays into Cygnus X. The solution of the transport equation is based on a semi-analytical approach, while the resulting radiation processes are fit to the broadband multiwavelength spectrum with a statistical procedure. Because of the complex structure of Cygnus X and the miscellaneous processes which take place there, this work distinguishes between them and considers all relevant scenarios as far as possible, determining the best-fit scenario to draw conclusions about the possible dominance of a diffuse cosmic ray sea for the radiation signatures. Moreover, the relation between the electron and proton source rate is based on the quasi-neutrality of the plasma and depends primarily on the spectral index $\alpha$. Therefore, one aim of this work is to find a reliable spectral index which considers all relevant transport and cooling mechanisms, whereby a consistent spectral index of $\alpha=2.37$ is found. If electrons and protons are not injected with the same spectral index, or if they are injected with different minimal energies, the value for $q_e/q_p$ may change significantly. As there is no concrete evidence for differences in the spectrum the standard number was used.\
Concerning the parameters of the interstellar medium, in our model, the radio flux in contrains a combination of the number of electrons and the magnetic field. The MeV emission can only be explained by bremsstrahlung losses, fixing the number of electrons in combination with the target densities. Highe-energy $\gamma$-rays then further constrain the electron and magnetic field strength via the inverse Compton process, while the proton number in combination with the target density are relevant for those $\gamma$-rays coming from $\pi^{0}-$ decays.\
At first, $\gamma$-ray and synchrotron spectra have been presented with those parameters that were used in early fits to the high-energy data. For this case, it could be shown that either the progression of the predicted $\gamma$-ray flux or radio flux do not satisfy the observed data. In a second step, a best-fit procedure has been performed to find stable parameters. This fit leads to the following conclusinos:
1. The fit parameters lead to an adamant agreement between predicted fluxes and the data measured by Fermi, ARGO-YBJ also to the non-thermal radio data, while the $10$ TeV data of Milagro are not well-fit.
2. It can be shown that diffusion dominates the loss processes in Cygnus X and is important to consider in the transport equation: By considering the flux of Bremsstrahlung at 10 MeV, which is the dominant radiation process at this energy, a mean free path of $2\cdot10^{16}\cdot\gamma^{1/3}$ cm is found. The energy loss due to diffusion is also investigated. The protons in Cygnus X lose nearly $3.6\times10^3$ more energy due to diffusion than electrons.\
If we transfer this information to the Cygnus Cocoon, we can assert that at energies $ \lesssim10^5$ GeV the freshly accelerated protons in the Cygnus Cocoon have unlikely their origin only in $\gamma$-Cygni.
3. The condition for the flux at 10 MeV particularly fixes the contribution from bremsstrahlung, which is the only process that is able to contribute in such a high amount at these energies. This in term implies strong constraints for the main input parameters of the bremsstrahlung process, i.e. the differential electron number and the target column depth.
4. The Milagro differential flux at $\sim 10$ TeV energies is about a factor of four times higher than the differential flux calculated in this work. These results show that a diffuse, homogeneous component could be responsible for the multiwavelength spectrum up to TeV energies, but a further component is necessary to explain the data at 10 TeV energies. Such an additional component could be a localized and/or short term accelerator within the region like $\gamma$-Cygni.
5. The new parameters provide a neutrino flux which approaches the sensitivity of IceCube at very high energies ($>$50 TeV). Considering that the difference between the spectral index of the flux and limit of IceCube for Cygnus X is less than 0.05, the coincidence is surely valid at high energies. In the future, the flux sensitivity of IceCube will be improved, so that the sensitivity for Cygnus X will suffice to measure the neutrino flux within the next decade.\
Additionally, considering the relation between column and target density, our results indicate that the depth of the neutral gas $d_t$ of Cygnus X should be close to $d_t$=116 pc.
Overall, despite its complexity, the present work investigates Cygnus X in a fundamental way, so as to reveal certain information about the transport mechanism, injection of cosmic rays and possible sources for acceleration.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank the people of the “Theoretische Physik IV” of Ruhr-Universiät Bochum and the IceCube Collaboration for stimulating discussions and Rosa-Luxemburg-Foundation for supporting financially. We would like to express our profound gratitude to Steven Young Eulig, Mike Kroll, James Doing, Fabian Bos, Sebastian Schöneberg, Lukas Merten, Donglian Xu, Tova M. Yoast-Hull, Markus Ahlers, Paolo Desiati but especially to Ali Kheirandish. Moreover, we are grateful for the comments of Dominik Bomans and Reinhard Schlickeiser.\
We further acknowledge the support from the MERCUR project St-2014-0040 (RAPP Center) and the BMBF, FZ 05A14PC1.
References {#references .unnumbered}
==========
[^1]: For example $\gamma$ Cygni J2021.0+4031e, which Milagro also detected at very high energies ([@MilagroGammaCygni]).
[^2]: List of pulsars in the region of interest: J2032.2+4126, J2021.0 + 3651, J2021.5+4026, J2030.0+36542.
[^3]: Small inhomogeneities vanish especially at larger scales than the gyro-radius.
[^4]: Notice that the source rate normalization factor in the momentum space as in the work of [@Pohl1], differs by a factor of $m_i^{-1}$.
|
---
author:
- |
\
Max-Planck-Institut für Kernphysik, Postfach 103980, 69029 Heidelberg, Germany\
E-mail:
- |
Vikram Dwarkadas\
Department of Astronomy and Astrophysics, University of Chicago, 5640 S Ellis Ave, Chicago, IL 60637, USA\
E-mail:
- |
Alexandre Marcowith\
Laboratoire Univers et Particules de Montpellier (LUPM) Université Montpellier, CNRS/IN2P3, CC72, place Eugène Bataillon, 34095, Montpellier Cedex 5, France\
E-mail:
- |
Andrea Chiavassa\
Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Lagrange, CS 34229, Nice, France\
E-mail:
title: 'Numerical Simulations of Cosmic-Ray Acceleration at Core-Collapse Supernovae'
---
Introduction {#Sec_Introduction}
============
Several studies have suggested that core-collapse supernovae (SN) exploding in dense winds could accelerate cosmic-rays (CR) to PeV energies for a few decades [@Tatischeff:2009kh; @Bell:2013kq; @Marcowith:2014; @Marcowith:2018ifh]. The situation during the first few days of the SN is somewhat more complicated. After core collapse, a radiation-mediated shock travels through the progenitor. Such a shock is not expected to accelerate CRs because its width is larger than the gyroradius of suprathermal particles. Later, at shock breakout (SB), it stalls in the outer layers of the star at an optical depth $\tau \sim c/U_{\rm s}$, where $U_{\rm s}$ denotes the shock velocity. At this point, the radiation in the immediate shock downstream escapes, producing a flash of photons [@Colgate74], which accelerates the circumstellar wind. A collisionless shock (CS) later forms [@ChevalierKlein79]: The dilution of photons as $1/R^{2}$, where $R$ is the stellar radius ($R=0$ in the centre of the star), ensures that the shocked outer layers of the star ram supersonically into the wind further out. Once the conditions are favourable for the first order Fermi mechanism to operate at the newly formed CS, CR acceleration should start.
In this work, we focus on the first day of the explosion of a red supergiant, and study the beginning of CR acceleration in such an extreme environment. In Sect. \[Sec\_Simulations\], we present our simulations of the progenitor and of its explosion, and describe our calculations of the quantities related to particle acceleration. We present our results in Sect. \[Sec\_Results\] and conclude in Sect. \[Sec\_Conclusions\].
Numerical simulations {#Sec_Simulations}
=====================
Stellar convection modelling is carried out using three-dimensional (3D) radiative hydrodynamical (RHD) code CO$^5$BOLD [@2012JCoPh.231..919F]. It solves the coupled non-linear equations of compressible hydrodynamics and non-local radiative energy transfer in the presence of a fixed external gravitational field. The computational domain is a cubic grid equidistant in all directions, and the same open boundary condition is employed for all sides of the computational box (see, e.g., Ref. ). The code employs realistic input physics either for the equation of state [@2012JCoPh.231..919F] or for the radiative transfer opacities . A detailed and precise solution of the radiative transfer is essential for a realistic treatment of convection because it is the radiative losses in the surface layers of the star that drain the convective movements and thus influence the whole simulation domain.
{width="44.00000%"} {width="54.00000%"}
The fundamental stellar parameters of the RHD simulation used in this work are reported in Table 1 of Ref. . We plot in Fig. \[Fig1\] a snapshot of the resulting density profile. In the left panel, we plot the logarithm of the density in a plane cut of the 3D simulated domain. The plane contains the centre of the star, $R=0$, in the centre of the panel. The very dense central region of the star is not represented. The dense, optically thick layers of the progenitor are in yellow/orange, while the surrounding dark regions correspond to the optically thin circumstellar material. Deviations from spherical symmetry are clearly visible. In the right panel, we plot with the thick red line the radial density profile of this star, averaged over all radial directions. The thin lines, denoted D1 to D4, represent the density profiles along four given radial directions.
The simulation of the SN is carried out using our Eulerian 1D-spherical radiation-hydro-dynamics code, presented in Ref. [@GGABPaper1]. We use the average radial density and temperature profiles of the above star as an input. CRs have been added as a pressure term in our code, see Ref. [@GGetal_In_Prep] for more details. The code is two-temperature, i.e. electron and proton temperatures are assumed to be equal. For the radiation, we use a gray frequency average, and represent it by its internal energy $E_{\rm rad}$ with characteristic temperature $T_{\rm rad} = ( cE_{\rm rad} / 4\sigma )^{1/4}$, where $\sigma$ denotes the Stefan-Boltzmann constant. The radiation transport is solved using a square-root flux-limited diffusion approximation. We take into account Compton cooling and bremsstrahlung for the transfer of energy between fluid and radiation, using the formulae from Ref. [@ChevalierKlein79]. At $t=0$ in the simulation, we trigger the SN by depositing $10^{51}$erg at the centre of the star as a spike in temperature. We follow the evolution of the physical conditions around the shock in the simulation, and determine the formation time of the CS, using the method of Ref. [@GGABPaper1]. Once the CS is formed, we calculate, at each time in the simulation, the typical Coulomb loss time of suprathermal particles at the CS, $\tau_{\rm Coul, T_{\rm e,u}}$, and their typical acceleration time to 1GeV, $\tau_{\rm acc, 1GeV}$. For the latter, we assume Bohm diffusion of the CRs, in an [*initial*]{} circumstellar magnetic field of 1mG strength. For stronger magnetic fields, particle acceleration would proceed faster. We also calculate the characteristic times for CR energy losses due to inelastic $pp$ collisions, $\tau_{\rm pp}$, and adiabatic losses, $\tau_{\rm adiab}$. For the latter, we take the conservative lower value $\tau_{\rm adiab} = R_{\rm s}/U_{\rm s}$, where $R_{\rm s}$ denotes the radius of the CS. See Refs. [@GGABPaper1; @GGetal_In_Prep] for more details. We assume that particle acceleration starts in our simulation once the acceleration time to 1GeV is shorter than all loss times.
We assume an $E^{-2}$ spectrum for the CRs accelerated at the CS. We further assume that magnetic field amplification at the shock is due to Bell’s instability [@Bell2004], which is driven by the escape of the highest energy CRs in the upstream of the CS [@Bell:2013kq]. We calculate the maximum CR energy at the CS, $E_{\max}$, using the results of Ref. [@Bell:2013kq]. At each time step in the simulation, we calculate the CR current escaping ahead of the CS, the growth rate of the instability in the upstream, and $E_{\max}$. See Ref. [@GGetal_In_Prep] for technical details.
Finally, we calculate the CR acceleration time to $E_{\max}$ in the [*amplified*]{} magnetic field from Bell’s instability, $\tau_{\rm acc, E_{\max}}$, and verify that it does not exceed the characteristic CR loss times $\tau_{\rm adiab}$, $\tau_{\rm pp}$, and $\tau_{\rm p\gamma}$, where $\tau_{\rm p\gamma}$ is the loss time due to pion production in inelastic $p\gamma$ collisions on the high-energy photons emitted by the CS. For $\tau_{\rm p\gamma}$, we calculate here a conservative (and pessimistic) lower value, $\tau_{\rm p\gamma, min}$, which assumes that the CS radiates all the energy it processes in photons with energies set to the threshold for pion production.
Results {#Sec_Results}
=======
{width="49.00000%"} {width="49.00000%"} {width="49.00000%"} {width="49.00000%"} {width="49.00000%"} {width="49.00000%"}
We present in Fig. \[Fig2\] results from our CR-radiation-hydrodynamics simulation of the explosion of the star, at three different times: Shortly before SB in the upper row, during SB in the middle row, and soon after SB in the lower row. In the left column, we show $U/c$, the velocity of the fluid normalized to $c$, as a function of the radius $R$, counted from the centre of the star. In the right column, we plot the electron temperature $T_{\rm e}$ (black lines), the fluid density $\rho$ (red), the radiation energy density $\eps_{\rm rad}$ (orange), and the CR energy density $\eps_{\rm CR}$ (green), as functions of $R$. See the keys in the Figure for the units and normalizations. In the initial profile, the optical depth is equal to 1 at $R \sim 5.5 \cdot 10^{11}$m. In the first row, the shock is still inside the optically thick layers of the star, and is radiation-mediated. It is located at $R \simeq 4.5 \cdot 10^{11}$m, see the vertical jump in the blue curve in the left panel. The shock thickness is several photon mean free paths, and the post-shock radiation is well confined, see the orange line in the right panel. In the middle row, the shock reaches the outer layers of the star. The radiation in the region immediately behind the shock starts to escape, and creates a flash of photons: The inflection around $R \simeq 1 \cdot 10^{12}$m in the curve for the radiation energy density (orange line in the right panel) corresponds to the front of SB photons which escape at the speed of light towards $R \rightarrow \infty$. The escaping radiation accelerates the circumstellar medium, which is visible in the left panel: The discontinuity in $U/c$ is now significantly wider, and spans from $R \simeq 6 \cdot 10^{11}$m to $R \simeq 10^{12}$m. In the lower row, the CS has formed: In the left panel, it appears as the abrupt discontinuity in $U/c$ at $R \simeq 1.05 \cdot 10^{12}$m, embedded within a significantly broader transition region smoothed by the radiation from SB. One can see in the right panel a spike in temperature ($\sim 100$keV), which corresponds to the shock-heated region in the downstream of the newly formed CS. CR acceleration has started at the CS, cf. the green curve.
{width="49.00000%"} {width="49.00000%"}
We show in Fig. \[Fig3\] our calculations of the typical timescales for particle acceleration and energy losses, as functions of time since core collapse. In the left panel, the solid red line corresponds to $\tau_{\rm acc, 1GeV}$, the CR acceleration time to 1GeV in an initial circumstellar magnetic field set to 1mG. It is plotted against the loss time for inelastic $pp$ collisions ($\tau_{\rm pp}$, solid blue line), as well as the adiabatic ($\tau_{\rm adiab}$, dotted black), and Coulomb loss times using the electron temperature in the CS upstream from the simulation ($\tau_{\rm Coul, T_{\rm e,u}}$, solid orange). For information, we also show the limiting Coulomb loss time one would have assuming that the electron temperature is zero ($\tau_{\rm Coul, 0}$, dashed orange). This plot shows that Coulomb losses are the limiting factor for the onset of CR acceleration here. In this simulation, the CS forms around $t \simeq 39000$s, and the red line drops below the solid orange one around $t \simeq 42000$s. This shows that Coulomb losses may delay the onset of CR acceleration after the formation of the CS by only $\sim 1$hour for this progenitor and a 1mG circumstellar magnetic field. In the right panel of Fig. \[Fig3\], we plot the CR acceleration time to $E_{\max}$ ($\tau_{\rm acc, E_{\max}}$, solid red line) and to 1TeV ($\tau_{\rm acc, 1TeV}$, dashed red) in the amplified magnetic field and assuming Bohm diffusion. We plot it against the $pp$ and adiabatic loss times, as well as the minimum $p\gamma$ loss time for CRs with energy $E_{\max}$ ($\tau_{\rm p\gamma, min}$, dashed blue line). Fig. \[Fig3\] (right panel) demonstrates that CR acceleration to $E_{\max}$ is not hindered by any of these losses.
{width="49.00000%"} {width="49.00000%"}
We plot in Fig. \[Fig4\] our calculations of the maximum CR energy at the forward shock, $E_{\max}$, versus time since core collapse. Results in the left panel are for a simulation in the average density profile of the SN progenitor and its surroundings. The solid red line is for the optimistic case of CR acceleration starting as soon as the CS is formed, i.e. for negligible Coulomb losses. This may be possible if the magnetic field strength around the progenitor star is $\gg 1$mG. The dashed line assumes that particle acceleration starts when $\tau_{\rm acc,1GeV} \leq \tau_{\rm Coul,T_{\rm e,u}}$. These results show that CRs reach $\geq 100$TeV energies typically $\sim 1$hour afer the onset of particle acceleration, and $E_{\max}$ starts to plateau to values of a few hundreds of TeV by the end of the first day following core collapse. The fact that both lines reach almost the same limiting value shows that Coulomb losses have little impact on $E_{\max}$, and only slightly delay the beginning of particle acceleration. The results in the right panel are for six density profiles along six given radial directions, denoted as D1, D2, ...,D6. The profiles D1 to D4 correspond to those plotted in Fig. \[Fig1\] (right panel) —the profiles D5 and D6 are not shown there due to limited space in the plot. The diversity of the curves in Fig. \[Fig4\] (right panel) provides an estimate of how the 3D geometry of the progenitor affects the results for $E_{\max}$ versus time. First, the times at which the CS forms and CR acceleration starts depend on the considered direction around the star. Indeed, the progenitor is not perfectly spherically symmetric, which implies that SB and thence the onset of CR acceleration occurs earlier in some directions than in others. However, we caution that a 3D simulation would be needed for a more watertight estimate of the spread in SB times: In 3D, the forward shock can get around dense clumps, which cannot be taken into account in our simulation. Second, the limiting value of $E_{\max}$ at which each curve plateaus depends on the direction. This means that the limiting $E_{\max}$ is not exactly the same in each region of the forward shock, at least during the first day of the SN. For example, the magenta curve reaches a value that is a few times larger than that of the black curve. Looking at Fig. \[Fig1\] (right panel), one can see that the circumstellar material at $R \geq 6 \cdot 10^{11}$m is about an order of magnitude denser in the direction D3 (magenta curve) than in the direction D1 (black one). These results are in line with the calculations of Ref. [@Bell:2013kq], where $E_{\max}$ is found to be larger in denser regions —at $U_{\rm s}$ fixed. All six curves nonetheless tend towards the same order-of-magnitude values of $E_{\max}$, which shows that taking the average density profile provides a reasonable first estimate.
Conclusions {#Sec_Conclusions}
===========
We study here the formation of a CS and the beginning of CR acceleration at a Type II SN, during the first day that follows core collapse. We present the CR-radiation-hydrodynamics simulation of the explosion of a red supergiant with a realistic density profile. We find that a CS soon forms after SB, and that CR acceleration starts soon after. For the progenitor considered here, Coulomb losses do not delay CR acceleration by more than an hour. The maximum CR energy at the forward shock quickly increases, and already reaches a few hundreds of TeV only hours after SB. It starts to plateau towards its maximum value by the end of the first day of the SN. The density of the circumstellar medium in the immediate surroundings of the progenitor depends on the considered radial direction around the star, and we find that CRs can be accelerated to higher energies in denser regions. Our results suggest that core-collapse SNe are plausible sources of very high-energy CRs, even in their earliest phases.
We thank Matthieu Renaud, Vincent Tatischeff, and Pierre Cristofari for useful discussions. This research collaboration is supported by a grant from the FACCTS program to the University of Chicago (PI: VVD). This work is supported by the ANR-14-CE33-0019 MACH project.
[99]{}
V. Tatischeff, *Radio emission and nonlinear diffusive shock acceleration of cosmic rays in the supernova SN 1993J*, *Astron. Astrophys.* [**499**]{} (2009) 191 \[[arXiv:0903.2944]{}\]. A. R. Bell, K. Schure, B. Reville, G. Giacinti, *Cosmic ray acceleration and escape from supernova remnants*, *MNRAS* [**431**]{} (2013) 415 \[[arXiv:1301.7264]{}\]. A. Marcowith, M. Renaud, V. Dwarkadas, V. Tatischeff, *Cosmic-ray acceleration and gamma-ray signals from radio supernov[æ]{}*, *Nucl. Phys. B Proc. Suppl.* [**256**]{} (2014) 94 \[[arXiv:1409.3670]{}\].
A. Marcowith, V. Dwarkadas, M. Renaud, V. Tatischeff, G. Giacinti, *Core collapse supernovae as Cosmic Ray sources*, *MNRAS* [**479**]{} (2018) 4470 \[[arXiv:1806.09700]{}\].
S. A. Colgate, *ApJ* [**187**]{} (1974) 321.
R. A. Chevalier, R. I. Klein, *Nonequilibrium processes in the evolution of type II supernovae*, *ApJ* [**234**]{} (1979) 597.
B. Freytag, M. Steffen, H.-G. Ludwig, S. Wedemeyer-B[ö]{}hm, W. Schaffenberger, O. Steiner, *Simulations of stellar convection with CO5BOLD*, *J. Comput. Phys.* [**231**]{} (2012) 919 \[[arXiv:1110.6844]{}\].
A. Chiavassa, B. Freytag, T. Masseron, B. Plez, *Radiative hydrodynamics simulations of red supergiant stars. IV. Gray versus non-gray opacities*, *Astron. Astrophys.* [**535**]{} (2011) A22 \[[arXiv:1109.3619]{}\].
B. Gustafsson, B. Edvardsson, K. Eriksson, U. G. J[ø]{}rgensen, Å Nordlund, B. Plez, *A grid of MARCS model atmospheres for late-type stars. I. Methods and general properties*, *Astron. Astrophys.* [**486**]{} (2008) 951 \[[arXiv:0805.0554]{}\].
K. Kravchenko, S. Van Eck, A. Chiavassa, A. Jorissen, B. Freytag, B. Plez, *Tomography of cool giant and supergiant star atmospheres. I. Validation of the method*, *Astron. Astrophys.* [**610**]{} (2018) A29 \[[arXiv:1711.08327]{}\].
G. Giacinti, A. R. Bell, *Collisionless shocks and TeV neutrinos before supernova shock breakout from an optically thick wind*, *MNRAS* [**449**]{} (2015) 3693 \[[arXiv:1503.04170]{}\]. G. Giacinti [*et al.*]{}, in preparation.
A. R. Bell, *Turbulent amplification of magnetic field and diffusive shock acceleration of cosmic rays*, *MNRAS* [**353**]{} (2004) 550.
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---
abstract: 'We study the behavior of zero-sets of the double zeta-function $\zeta_2(s_1,s_2)$ (and also of more general multiple zeta-function $\zeta_r(s_1,\ldots,s_r)$). In our former paper we studied the case $s_1=s_2$, but in the present paper we consider the more general two variable situation. We carry out numerical computations in order to trace the behavior of zero-sets. We observe that some zero-sets approach the points with $s_2=0$, while other zero-sets approach the points which are solutions of $\zeta(s_2)=1$. We give a theoretical proof of the latter fact, in the general $r$-fold setting.'
author:
- 'Kohji Matsumoto, Mayumi Sh[ō]{}ji'
title: 'Numerical computations on the zeros of the Euler double zeta-function II'
---
Introduction
============
The present paper is a continuation of the authors’ previous article [@MatSho14] on the study of the zeros of the Euler double zeta-function $$\begin{aligned}
\label{1-1}
\zeta_2(s_1,s_2)=\sum_{n_1=1}^{\infty}\sum_{n_2=1}^{\infty}
\frac{1}{n_1^{s_1}(n_1+n_2)^{s_2}},\end{aligned}$$ where $s_1,s_2$ are complex variables. This double series is convergent absolutely in the region defined by $\Re s_1+\Re s_2>2$ and $\Re s_2>1$, and can be continued meromorphically to the whole complex space $\mathbb{C}^2$ (see [@Mat02]).
This is the case $r=2$ of the more general Euler-Zagier $r$-fold sum $$\begin{aligned}
\label{1-2}
\zeta_r(s_1,\ldots,s_r)=\sum_{1\leq n_1<\cdots<n_r}\frac{1}{n_1^{s_1}\cdots
n_r^{s_r}},\end{aligned}$$ which has been investigated quite extensively from various aspects. However, the distribution of the zeros of has not been, except for the classical case of $r=1$ (that is the case of the Riemann zeta-function $\zeta(s)$), studied in detail. In order to understand the analytic properties of , it is very important to study the behavior of its zeros. The aim of the present series of papers is to study the behavior of the zeros of , the simplest case (except for the case $r=1$), from the viewpoint of numerical computations.
In [@MatSho14], we considered the situation when $s_1=s_2(=s)$. Then $\zeta_2(s,s)$ is a function of one variable, so we can study the distribution of the zeros of $\zeta_2(s,s)$ in a way analogous to the case of the Riemann zeta-function. Unlike the case of $\zeta(s)$, the function $\zeta_2(s,s)$ does not satisfy the analogue of the Riemann hypothesis. We found a lot of zeros in the strip $0\leq \Re s\leq 1$ off the line $\Re s=1/2$, or even outside that strip (see [@MatSho14 Observation 1] and [@MatSho14 Figure 1]). We pointed out that the distribution of those zeros is similar, not to that of $\zeta(s)$, but rather, to that of Hurwitz zeta-functions.
Inspired by [@MatSho14], Ikeda and Matsuoka proved several theoretical results on the distribution of the zeros of $\zeta_2(s,s)$ in [@IkeMat].
In the present paper, we consider the general situation, when $s_1$ and $s_2$ are moving independently. Then $\zeta_2(s_1,s_2)$ is a function of two variables, so its zeros are not isolated points, but they form analytic sets, which we call zero-sets. We carry out numerical computations in order to trace the behavior of such zero-sets. Our basic strategy is to begin with the zeros of $\zeta_2(s,s)$ discovered in [@MatSho14], and study the behavior of zero-sets of $\zeta_2(s_1,s_2)$ around those zeros. Then we consider the asymptotic situation of those zero-sets when, for example, $|s_1-s_2|$ becomes large, or $s_2\to 0$, or $|s_1| \to \infty$. We observe that some zero-sets approach the points with $s_2=0$, while other zero-sets approach the points which are solutions of $\zeta(s_2)=1$.
It seems that, in general, to give theoretical proofs for the observations described in the present paper is a very difficult task. However, at least, we can contruct a proof of the latest fact that some zero-sets approach the points satisfying $\zeta(s_2)=1$. In fact, we will prove a more general result for the $r$-fold sum in the last section.
The behavior of zero-sets
=========================
Let us begin with [@MatSho14 Figure 1], on which a lot of non-real zeros of $\zeta_2(s,s)$ are dotted. The values of some of which are given in the list written on [@MatSho14 pp.308-309], whose order is according to the magnitude of the imaginary parts of them. Denote those zeros by $a_1,a_2,a_3,\ldots$ and so on.
Since $\zeta_2(s_1,s_2)$ cannot have any isolated zero point, these $a_i$s are to be intersections of some zero-sets and the hyperplane $s_1=s_2$. Our first aim is to investigate the behavior of zero-sets near the points $a_i$.
Let $\delta$ be a positive number. We search for the zeros of $\zeta_2(s_1,s_2)$ around the point $a_i$ under the condition $|s_1-s_2|=\delta$. The method of computations is based on the Euler-Maclaurin formula, explained in [@MatSho14 Section 4]. (It is to be noted that the simple method using the harmonic product formula [@MatSho14 (2.1), (2.2)] cannot be applied to the present situation where $s_1\neq s_2$.) Figure \[Fig1-1\] describes the loci of the absolute values of zeros satisfying $|s_1-s_2|=\delta$ around $a_i$ ($5\leq i\leq 9$) for various values of $\delta$. We quote the values of those $a_i$s from [@MatSho14]: $$\begin{aligned}
&a_5=(0.12844956) + i(20.59707674),\\
&a_6=(0.08804454) + i(21.93232180),\\
&a_7=(1.10778631) + i(23.79708697),\\
&a_8=(0.27268471) + i(24.93425087),\\
&a_9=(-0.67413685) + i(26.88584448).\end{aligned}$$
The left one of Figure \[Fig1-1\] is the situation when $\delta=0.5$. In this figure we can observe that zero-sets around $a_{7}$ and $a_{8}$ become closer to each other, and it seems that these two zero-sets are connected in the central figure, when $\delta=1$. In the right figure, all the zero-sets around $a_5$ to $a_8$ seem connected. When $\delta$ becomes larger, more and more zero-sets seem to be connected with each other (see Figure \[Fig2-1\]).
![The loci of the absolute values of zeros of $\zeta_2(s_1,s_2)$ around $a_i$ ($5\leq i\leq 9$) with $|s_1-s_2|=\delta$, where $\delta=0.5$ (left), $1.0$ (center), $2.0$ (right). The horizontal axis represents $|s_1|$ and the vertical axis does $|s_2|$. The points indicated by the cross marks are $(|a_5|,|a_5|), (|a_6|,|a_6|), \ldots, (|a_9|,|a_9|)$ from the lower-left to the upper-right. []{data-label="Fig1-1"}](Figure1a.eps "fig:"){width="32.00000%"} ![The loci of the absolute values of zeros of $\zeta_2(s_1,s_2)$ around $a_i$ ($5\leq i\leq 9$) with $|s_1-s_2|=\delta$, where $\delta=0.5$ (left), $1.0$ (center), $2.0$ (right). The horizontal axis represents $|s_1|$ and the vertical axis does $|s_2|$. The points indicated by the cross marks are $(|a_5|,|a_5|), (|a_6|,|a_6|), \ldots, (|a_9|,|a_9|)$ from the lower-left to the upper-right. []{data-label="Fig1-1"}](Figure1b.eps "fig:"){width="32.00000%"} ![The loci of the absolute values of zeros of $\zeta_2(s_1,s_2)$ around $a_i$ ($5\leq i\leq 9$) with $|s_1-s_2|=\delta$, where $\delta=0.5$ (left), $1.0$ (center), $2.0$ (right). The horizontal axis represents $|s_1|$ and the vertical axis does $|s_2|$. The points indicated by the cross marks are $(|a_5|,|a_5|), (|a_6|,|a_6|), \ldots, (|a_9|,|a_9|)$ from the lower-left to the upper-right. []{data-label="Fig1-1"}](Figure1c.eps "fig:"){width="32.00000%"}
![The loci of the absolute values of zeros of $\zeta_2(s_1,s_2)$ around $a_i$ ($1\leq i\leq 13$) with $|s_1-s_2|=\delta$, where $\delta=5.0$ (left), and $10.0$ (right).[]{data-label="Fig2-1"}](Figure2a.eps "fig:"){width="45.00000%"} ![The loci of the absolute values of zeros of $\zeta_2(s_1,s_2)$ around $a_i$ ($1\leq i\leq 13$) with $|s_1-s_2|=\delta$, where $\delta=5.0$ (left), and $10.0$ (right).[]{data-label="Fig2-1"}](Figure2b.eps "fig:"){width="45.00000%"}
![The behavior of real parts and imaginary parts of zeros around $a_i$ ($5\leq i\leq 9$) when $\delta=1.0$. On the left figure, the horizontal axis represents $\Re s_1$, and the vertical axis does $\Re s_2$, while on the right figure they represent $\Im s_1$ and $\Im s_2$. Since the absolute value is almost determined by the imaginary part (because the real part is relatively small), the right figure is very similar to the central one of Figure 1. []{data-label="Fig2-2"}](Figure2-1.eps "fig:"){width="45.00000%"} ![The behavior of real parts and imaginary parts of zeros around $a_i$ ($5\leq i\leq 9$) when $\delta=1.0$. On the left figure, the horizontal axis represents $\Re s_1$, and the vertical axis does $\Re s_2$, while on the right figure they represent $\Im s_1$ and $\Im s_2$. Since the absolute value is almost determined by the imaginary part (because the real part is relatively small), the right figure is very similar to the central one of Figure 1. []{data-label="Fig2-2"}](Figure2-2.eps "fig:"){width="45.00000%"}
Since Figure \[Fig1-1\] only represents the behavior of absolute values, in order to make sure that the above zero-sets are indeed connected, it is necessary to investigate the values of real parts and imaginary parts of those. Figure \[Fig2-2\] gives such data. This figure shows that the loci of zeros around $a_7$ and $a_8$ are indeed connected.
From Figure \[Fig2-1\] it seems that all zero-sets appearing in this figure are connected. (The zero-sets including the points $a_1$ and $a_2$ are not connected to the other zero-sets on the figure, but further computations show that these zero-sets look connected also, when $\delta$ becomes larger.)
From this observation, perhaps we may expect that all $a_1,a_2,a_3,\ldots$ are lying on the same (unique?) zero-set. However, later in Section 4 we will see that the behavior of some zero-sets is rather different. Probably it is too early to raise any conjecture on the global behavior of zero-sets.
In [@MatSho14 p.308], it is noted that the left-most zero in [@MatSho14 Figure 1] is $$(-0.830372)+i(35.603804).$$ However, there is at least one zero of $\zeta_2(s,s)$ which is located more left. When the authors wrote [@MatSho14], they overlooked the following two zeros: $$\begin{aligned}
(-0.874058504) + i(44.93750365),\\
(-0.710036436) + i(53.91464901).\end{aligned}$$
Approaching the axis $s_2=0$
============================
From Figure \[Fig2-1\] we can observe that, when $\delta$ becomes larger, the curves consisting of zeros also becomes larger, and the bottoms of the curves look approaching to the horizontal axis (that is, the axis $s_2=0$). Figure \[Fig3\] describes the curves of the absolute values of zeros when $\delta=14.0$. In this case one curve indeed touches the horizontal axis.
![The loci of the absolute values of zeros of $\zeta_2(s_1,s_2)$ around $a_1$ and $a_2$ with $|s_1-s_2|=\delta$, where $\delta=14.0$. The horizontal axis represents $|s_1|$ and the vertical axis does $|s_2|$.[]{data-label="Fig3"}](Figure3.eps){width="50.00000%"}
![The absolute values of zeros satisfying $s_2=\eta s_1$, where $1\geq\eta\geq 0$. The horizontal axis represents $|s_1|$ (up to $60$) and the vertical axis does $|s_2|$. The cross marks represent the absolute values of zeros of $\zeta(s)$.[]{data-label="Fig4"}](Figure4.eps){width="90.00000%"}
How about the behavior of other curves? To investigate this point, now we use an alternative way of calculating zeros. Consider the equation $s_2=\eta s_1$. In [@MatSho14], we studied the zeros under the condition $\eta=1$. Starting with the data of those zeros, we search for the zeros for various values of $\eta$, $1\geq \eta\geq 0$. When $\eta\to 0$, we find that almost all curves consisting of zeros tend to the horizontal axis (see Figure \[Fig4\]).
= = $ (1.247595281) + i (14.14857043) $ $ (0.5) + i (14.13472514) $\
$ (1.279113136) + i (21.01244258) $ $ (0.5) + i (21.02203964) $\
$ (1.292752716) + i (25.03054326) $ $ (0.5) + i (25.01085758) $\
$ (1.304538379) + i (30.39099998) $ $ (0.5) + i (30.42487613) $\
$ (1.310484619) + i (32.97276761) $ $ (0.5) + i (32.93506159) $\
$ (1.319356152) + i (37.57182573) $ $ (0.5) + i (37.58617816) $\
$ (1.320370441) + i (40.90318345) $ $ (0.5) + i (40.91871901) $\
$ (1.328328378) + i (43.36573059) $ $ (0.5) + i (43.32707328) $\
$ (1.333209526) + i (47.94829355) $ $ (0.5) + i (48.00515088) $\
$ (1.330151768) + i (49.81347595) $ $ (0.5) + i (49.77383248) $\
$ (1.338852528) + i (52.98250152) $ $ (0.5) + i (52.97032148) $\
$ (1.341784096) + i (56.42981942) $ $ (0.5) + i (56.44624769) $\
$ (1.336083544) + i (59.30477438) $ $ (0.5) + i (59.34704400) $\
$ (1.347599692) + i (60.90235918) $ $ (0.5) + i (60.83177852) $\
$ (1.350310992) + i (65.06828349) $ $ (0.5) + i (65.11254405) $\
$ (1.343496503) + i (67.09040611) $ $ (0.5) + i (67.07981053) $\
$ (1.348806781) + i (69.55741541) $ $ (0.5) + i (69.54640171) $\
$ (1.354734356) + i (72.08670993) $ $ (0.5) + i (72.06715767) $\
$ (1.356705834) + i (75.63889778) $ $ (0.5) + i (75.70469069) $\
$ (1.344904017) + i (77.17673184) $ $ (0.5) + i (77.14484007) $\
$ (1.360065509) + i (79.37508073) $ $ (0.5) + i (79.33737502) $\
$ (1.360772059) + i (82.86977267) $ $ (0.5) + i (82.91038085) $\
$ (1.355304386) + i (84.75005063) $ $ (0.5) + i (84.73549298) $\
$ (1.353373038) + i (87.38999143) $ $ (0.5) + i (87.42527461) $\
$ (1.365347698) + i (88.87469754) $ $ (0.5) + i (88.80911121) $\
$ (1.367563235) + i (92.45429247) $ $ (0.5) + i (92.49189927) $\
$ (1.352878941) + i (94.60615786) $ $ (0.5) + i (94.65134404) $\
$ (1.364213735) + i (95.93955674) $ $ (0.5) + i (95.87063423) $\
$ (1.367791276) + i (98.82956582) $ $ (0.5) + i (98.83119422) $\
From Figure \[Fig4\] we observe that the points at which the loci touch the horizontal axis are almost the same as the absolute values of zeros of $\zeta(s)$. Let us list up those values. The left of Table \[table-1\] is the list of the values of $s_1$ of the points where the curves touch the horizontal axis. Comparing this table with the list of non-trivial zeros of $\zeta(s)$ (the right of Table \[table-1\]), we find:
The imaginary part of each $s_1$ in the left list of Table \[table-1\] is very close to an imaginary part of a value appearing in the right list of Table \[table-1\], that is, a non-trivial zero of $\zeta(s)$.
The following argument is not rigorous, but at least heuristically, explains this observation.
Recall the formula ([@MatSho14 (2.3)], originally in [@AET]): $$\begin{aligned}
\label{3-1}
&\zeta_2(s_1,s_2)=\frac{\zeta(s_1+s_2-1)}{s_2-1}-\frac{\zeta(s_1+s_2)}{2}\\
&\qquad+\sum_{q=1}^l (s_2)_q\frac{B_{q+1}}{(q+1)!}\zeta(s_1+s_2+q)
-\sum_{n_1=1}^{\infty}\frac{\phi_l(n_1,s_2)}{n_1^{s_1}},\notag\end{aligned}$$ where $(s_2)_q=s_2(s_2+1)\cdots(s_2+q-1)$, $B_{q+1}$ is the $(q+1)$-th Bernoulli number, and $$\begin{aligned}
\lefteqn{\phi_l(n_1,s_2)}\\
&=\sum_{k=1}^n\frac{1}{k^s}-\left\{\frac{n^{1-s}-1}{1-s}
+\frac{1}{2n^s}-\sum_{q=1}^l\frac{(s)_q B_{q+1}}{(q+1)! n^{s+q}}+\zeta(s)
-\frac{1}{s-1}\right\}\end{aligned}$$ (see [@MatSho14 (2.4)]). Put $s_2=0$ in . Since $\phi_l(n_1,0)=0$ (which can be seen by [@MatSho14 (2.5)]), we find that the two sums on the right-hand side of are both zero, so $$\begin{aligned}
\label{3-2}
\zeta_2(s_1,0)=-\zeta(s_1-1)-\frac{\zeta(s_1)}{2}.\end{aligned}$$ Now, let $s_1^*$ be one of the values in the left list of Table 1. Then $\zeta_2(s_1^*,0)=0$, so implies $$\begin{aligned}
\label{3-3}
\zeta(s_1^*)=-2\zeta(s_1^*-1).\end{aligned}$$
Let $$C(t)=\{\zeta(\sigma+it)\;|\;\sigma\in\mathbb{R}\}.$$ If $t$ is very close to the imaginary part of a non-trivial zero of $\zeta(s)$, the graph of the curve $C(t)$ passes very close to the origin when $\sigma=1/2$ (see Figure \[Fig5\]). As can be seen from Figure \[Fig5\], when $\sigma$ moves, the slope of the graph of $C(t)$ does not so rapidly change. This can be naturally expected, because the approximate functional equation of $\zeta(s)$ (see [@Tit51 Theorem 4.15]) implies that the behavior of $\zeta(s)$ is dominated by the terms of the form $n^{-s}=n^{-\sigma}e^{-it \log n}$, and if $t$ is fixed, then the “argument" part of these terms does not change.
![The first one is the locus of $\zeta(\sigma+it_0)$, where $t_0=14.13472514$ (= the first of the right list of Table \[table-1\]). Therefore this locus crosses the origin. The two black-dots represent the points $s(t_0)=(0.442629) + i(0.0469269)$ and $s(t_0-1)=(-0.218684) -i(0.0397677)$, with $\sigma(t_0)=1.2475$. This satisfies . The second one is the locus of $\zeta(\sigma+it_0^*)$, where $t_0^*=14.14857043$ (= the first of the left list of Table \[table-1\]).[]{data-label="Fig5"}](Figure5-1.eps){width="90.00000%"}
![The first one is the locus of $\zeta(\sigma+it_0)$, where $t_0=14.13472514$ (= the first of the right list of Table \[table-1\]). Therefore this locus crosses the origin. The two black-dots represent the points $s(t_0)=(0.442629) + i(0.0469269)$ and $s(t_0-1)=(-0.218684) -i(0.0397677)$, with $\sigma(t_0)=1.2475$. This satisfies . The second one is the locus of $\zeta(\sigma+it_0^*)$, where $t_0^*=14.14857043$ (= the first of the left list of Table \[table-1\]).[]{data-label="Fig5"}](Figure5-2.eps){width="90.00000%"}
Therefore we can find a number $\sigma(t)\in\mathbb{R}$ such that $$\begin{aligned}
\label{3-4}
|\zeta(s(t))|=2|\zeta(s(t)-1)|,\end{aligned}$$ where $s(t)=\sigma(t)+it$. We denote by $L(t)$ the segment joining $\zeta(s(t))$ and $\zeta(s(t)-1)$.
When $(1/2)+it_0$ is in the right list of Table \[table-1\], as in the upper one of Figure \[Fig5\] (where the case $t_0=14.13472514$ is described), $\arg\zeta(s(t_0)-1)$ is almost equal to $\arg\zeta(s(t_0))\pm\pi$, so $L(t_0)$ passes very close to the origin. (If $C(t_0)$ would be a straight line, then $L(t_0)$ could indeed cross the origin; but this is not the case.) Move the value of $t$ a little from $t_0$. Then the curve $C(t)$ also moves a little. Then, as in the lower one of Figure \[Fig5\], we may find a value of $t=t_0^*$, close to $t_0$, for which $L(t_0^*)$ indeed crosses the origin. This implies $\zeta(s(t_0^*))=-2\zeta(s(t_0^*)-1)$, that is, in view of , this $s(t_0^*)=\sigma(t_0^*)+it_0^*$ should be in the left list of Table \[table-1\].
It is also observed that the real parts of the points on the list of Table 1 are close to each other, around the value $1.3$. So far we have not found any theoretical reasoning of this phenomenon.
Approaching the zeros of $\zeta(s)=1$
=====================================
In Figure \[Fig4\], we can observe that almost all curves approach the horizontal axis. However, when we extend the range of computations, we find that there are curves which do not seem to approach the horizontal axis (Figure \[Fig4-2\]). Along these curves, it seems that $|s_1|$ becomes larger and larger.
![This figure represents the same computations as in Figure \[Fig4\], but the range of $|s_1|$ is up to $120$. The cross marks represent the absolute values of zeros of $\zeta(s)$. There appear two curves which do not approach the horizontal axis.[]{data-label="Fig4-2"}](Figure4-2.eps){width="90.00000%"}
Moreover, extending the range of computations, we can find more zero-sets, along them $|s_1|$ seems to tend to infinity (see Figure \[Fig7-1\]).
![The same computations as in Figure \[Fig4\], but the range of $|s_1|$ is up to $400$. Only the curves which do not approach the horizontal axis are drawn.[]{data-label="Fig7-1"}](Figure7-1.eps){width="60.00000%"}
The numerical data suggests that $\Re s_1$ tends to infinity along these curves (Figure \[Fig6\] and the left of Figure \[Fig7-2\]), while $\Re s_2$ remains finite (the right of Figure \[Fig7-2\]).
![The behavior of $s_1$ of the curves descrived in Figure \[Fig4-2\].[]{data-label="Fig6"}](Figure6.eps){height="0.9\textheight"}
![The behavior of $s_1$ (left) and $s_2$ (right) of curves described in Figure \[Fig7-1\]. The cross marks on the right figure are solutions of $\zeta(s_2)=1$. There may be more solutions, but we only mark the solutions we checked. []{data-label="Fig7-2"}](Figure7-2.eps "fig:"){width="45.00000%"} ![The behavior of $s_1$ (left) and $s_2$ (right) of curves described in Figure \[Fig7-1\]. The cross marks on the right figure are solutions of $\zeta(s_2)=1$. There may be more solutions, but we only mark the solutions we checked. []{data-label="Fig7-2"}](Figure7-3.eps "fig:"){width="45.00000%"}
What happens? We can prove the following facts.
\[prop1\] [(i)]{} [*Let $(s_1^{(m)},s_2^{(m)})$ $(m=1,2,\ldots)$ be a sequence of points on a zero-divisor of $\zeta_2(s_1,s_2)$. If $\sigma_1^{(m)}=\Re s_1^{(m)}$ tends to infinity and $|s_2^{(m)}|$ remains bounded as $m\to\infty$, then $s_2^{(m)}$ tends to a solution of $\zeta(s_2)=1$*]{}.
[(ii)]{} [*Conversely, for any solution $\rho_2$ of $\zeta(\rho_2)=1$ and any $\varepsilon>0$, we can find a zero of $\zeta_2(s_1,s_2)$ such that $|s_2-\rho_2|<\varepsilon$*]{}.
This result is due to Professor Seidai Yasuda (Osaka University). The authors express their sincere gratitude to him for the permission of including his result in the present paper.
In the next section we will prove general theorems on the asymptotic behavior of zero-sets of $r$-fold sum . The above proposition is just a special case of those theorems.
\[rem\_trivial\] The assertion (i) of the above proposition is, if $\Re s_2>1$, obvious. Because in this case we can use . Letting $\Re s_1\to\infty$ on , we see that the right-hand side tends to $\sum_{n_2=1}^{\infty}(1+n_2)^{-s_2}=\zeta(s_2)-1$.
The asymptotic behavior of zero-sets of the $r$-fold sum
========================================================
We study the behavior of $\zeta_r(s_1,\ldots,s_r)$ when $\sigma_k=\Re s_k$ for some $k$ tends to $+\infty$ while the other variables remain bounded. First, when $k\geq 2$, the conclusion is simple.
\[th-y-0\] Assume $r\geq 2$. When some $\sigma_k=\Re s_k$ $(2\leq k\leq r)$ tends to $+\infty$ while the other variables remain in a bounded region $E_{r-1}\subset\mathbb{C}^{r-1}$, which does not include singularities of relevant (multiple) zeta-functions, the value $\zeta_r(s_1,\ldots,s_r)$ tends to $0$, uniformly in $E_{r-1}$.
In the above statement, “relevant” means any (multiple) zeta-functions appearing not only in the statement, but also in the proof. We assume the same property for the region $D_{r-1}$ in the statement of the next theorem.
More interesting is the situation when $k=1$. Let $r\geq 1$, and define $$\begin{aligned}
\label{y-0}
F_{r-1}(s_2,\ldots,s_r)=\sum_{j=1}^{r-1}(-1)^{r-j+1}\zeta_j(s_{r-j+1},\ldots,s_r)-(-1)^r.\end{aligned}$$ [(]{}When $r=1$, we understand simply that $F_0=1$.[)]{} Denote by $\mathcal{H}_{r-1}$ the hypersurface in the space $\mathbb{C}^{r-1}$ defined by the equation $$\begin{aligned}
\label{y-1}
F_{r-1}(s_2,\ldots,s_r)=0.\end{aligned}$$
\[th-y-1\] Let $(s_1^{(m)},\ldots,s_r^{(m)})$ $(m=1,2,3,\ldots)$ be a sequence of points in $\mathbb{C}^r$. Assume that, as $m\to\infty$, $\sigma_1^{(m)}=\Re s_1^{(m)}$ tends to $+\infty$ while $(s_2^{(m)},\ldots,s_r^{(m)})$ remains in a bounded region $D_{r-1}$, which does not include singularities of relevant (multiple) zeta-functions. Then for any $\varepsilon>0$, we can choose a sufficiently large $M$, uniformly in $D_{r-1}$, for which $$\begin{aligned}
\label{y-1-1}
|\zeta_r(s_1^{(m)},\ldots,s_r^{(m)})-F_{r-1}(s_2^{(m)},\ldots,s_r^{(m)})|<\varepsilon\end{aligned}$$ holds for any $m\geq M$.
\[th-y-2\] Assume $r\geq 2$.
[(i)]{} Let $(s_1^{(m)},\ldots,s_r^{(m)})$ $(m=1,2,3,\ldots)$ be a sequence of points on a zero-set of $\zeta_r(s_1,\ldots,s_r)$. Assume that $\sigma_1^{(m)}=\Re s_1^{(m)}$ tends to $+\infty$ while $(s_2^{(m)},\ldots,s_r^{(m)})$ remains in $D_{r-1}$ (as in the statement of Theorem \[th-y-1\]). Then, uniformly on $D_{r-1}$, the points $(s_2^{(m)},\ldots,s_r^{(m)})$ is approaching the hypersurface $\mathcal{H}_{r-1}$. Therefore, we can find a subsequence $(s_2^{(k_m)},\ldots,s_r^{(k_m)})$ $(m=1,2,3,\ldots)$ which converges to a point $(\rho_2,\ldots,\rho_r)$ on $\mathcal{H}_{r-1}$.
[(ii)]{} Conversely, for any solution $(\rho_2,\ldots,\rho_r)$ of , we can find a sequence $(s_1^{(m)},\ldots,s_r^{(m)})$ $(m=1,2,3,\ldots)$ on a zero-set of $\zeta_r(s_1,\ldots,s_r)$ which converges to $(\rho_2,\ldots,\rho_r)$.
The case $r=2$ of Theorem \[th-y-2\] is exactly Proposition \[prop1\] given in the previous section.
We begin with the proof of Theorem \[th-y-0\].
When $r=2$, the double series $$\zeta_2(s_1,s_2)=\sum_{n_1=1}^{\infty}\sum_{n_2=1}^{\infty}n_1^{-s_1}(n_1+n_2)^{-s_2}$$ is absolutely convergent when $\sigma_2$ is sufficiently large. Since $n_1+n_2\geq 2$, all the terms on the right-hand side tend to 0 when $\sigma_2\to\infty$, uniformly in $s_1$ when $s_1\in E_1$. Hence the assertion for $r=2$ follows.
We prove the theorem by induction on $r$. Assume the theorem is true for $r-1$. The multiple series
$$\begin{aligned}
\label{y-2}
\zeta_r(s_1,\ldots,s_r)=\sum_{n_1=1}^{\infty}\cdots\sum_{n_r=1}^{\infty}n_1^{-s_1}
(n_1+n_2)^{-s_2}\cdots(n_1+\cdots+n_r)^{-s_r}\end{aligned}$$
is absolutely convergent when $\sigma_r$ is sufficiently large. Therefore the case when $\sigma_r\to\infty$ can be treated similarly to the above double zeta case.
However for the case $\sigma_k\to\infty$ $(2\leq k\leq r-1)$, this argument is not enough, because if $\sigma_r$ is not large, then the expression is not valid. Therefore we first carry out the meromorphic continuation.
Here, we apply the method of using the Mellin-Barnes integral formula $$\begin{aligned}
\label{y-3}
(1+\lambda)^{-s}=\frac{1}{2\pi i}\int_{(c)}\frac{\Gamma(s+z)\Gamma(-z)}{\Gamma(s)}
\lambda^z dz,\end{aligned}$$ where $s,\lambda\in\mathbb{C}$, $\Re s>0$, $\lambda\neq 0$, $|\arg\lambda|<\pi$, $-\Re s<c<0$, and the path of integration is the vertical line $\Re z=c$. The following argument was done (in a more general form) in [@Mat03]. Assume, at first, that $\Re s_k>1$ for all $1\leq k\leq r$. Then is absolutely convergent, and an application of yields $$\begin{aligned}
\label{y-4}
&\zeta_r(s_1,\ldots,s_r)=\frac{1}{2\pi i}\int_{(c)}\frac{\Gamma(s_r+z)\Gamma(-z)}{\Gamma(s_r)}\\
&\;\times\zeta_{r-1}(s_1,\ldots,s_{r-2},s_{r-1}+s_r+z)\zeta(-z)dz \notag\end{aligned}$$ (see [@Mat03 (12.3)]), where $-\Re s_r<c<-1$. Then we shift the path of integration to the line $\Re z=M-\varepsilon$, where $M$ is a large positive integer and $\varepsilon$ is a small positive number. Counting the relevant residues, we obtain $$\begin{aligned}
\label{y-5}
&\zeta_r(s_1,\ldots,s_r)=\frac{1}{s_r-1}\zeta_{r-1}(s_1,\ldots,s_{r-2},s_{r-1}+s_r-1)\\
&+\sum_{j=0}^{M-1}\binom{-s_n}{j}\zeta_{r-1}(s_1,\ldots,s_{r-2},s_{r-1}+s_r+j)\zeta(-j)\notag\\
&+\frac{1}{2\pi i}\int_{(M-\varepsilon)}\frac{\Gamma(s_r+z)\Gamma(-z)}{\Gamma(s_r)}\notag\\
&\quad\times\zeta_{r-1}(s_1,\ldots,s_{r-2},s_{r-1}+s_r+z)\zeta(-z)dz \notag\end{aligned}$$ (see [@Mat03 (12.7)]). Since the last integral is convergent in the wider region $$\begin{aligned}
\label{y-6}
\{(s_1,\ldots,s_r)\;|\;\Re(s_{r-j+1}+\cdots+s_r)>j-1-M+\varepsilon\; (1\leq j\leq r)\}\end{aligned}$$ (see [@Mat03 (12.9)]), and $M$ is arbitrary, implies the meromorphic continuation of $\zeta_r(s_1,\ldots,s_r)$ to the whole space $\mathbb{C}^r$.
Now we go back to the proof of the theorem. Consider the situation when some $\sigma_k$ $(2\leq k\leq r-1)$ tends to $+\infty$ while the other variables remain in the region $D_{r-1}$. Those $(s_1,\ldots,s_r)$’s are clearly included in the above region for sufficiently large $M$, so we can use the expression . By the induction assumption, we see that all the $\zeta_{r-1}(\cdot)$ factors on the right-hand side of tend to 0 when $\sigma_k\to\infty$. Moreover those convergences are uniform. This fact is clear for $\zeta_{r-1}(s_1,\ldots,s_{r-2},s_{r-1}+s_r+j)$ ($-1\leq j\leq M-1$) by the assumption. As for $\zeta_{r-1}(s_1,\ldots,s_{r-2},s_{r-1}+s_r+z)$, though $z$ is not restricted to a bounded region, we can also show the uniformity because this series is absolutely convergent by . Therefore the theorem is proved.
Next we proceed to the proof of Theorem \[th-y-1\]. The assertion (i) of Theorem \[th-y-2\] is clearly a special case of Theorem \[th-y-1\], that is, the case $\zeta_r(s_1^{(m)},\ldots,s_r^{(m)})=0$ in .
When $r=1$, the assertion of the theorem is $\zeta(s_1^{(m)})\to 1$ as $\sigma_1^{(m)}\to +\infty$, which is obvious. We assume that the theorem is true for $r-1$, and prove the theorem by induction. We use the following harmonic product formula. $$\begin{aligned}
\label{y-7}
&\zeta(s_1)\zeta_{r-1}(s_2,\ldots,s_r)\\
&\;=\sum_{n_1=1}^{\infty}n_1^{-s_1}\sum_{n_2<\cdots<n_r}n_2^{-s_2}\cdots n_r^{-s_r}\notag\\
&\;=\sum_{n_1<n_2<\cdots<n_r}+\sum_{n_1=n_2<n_3<\cdots<n_r}+\sum_{n_2<n_1<n_3<\cdots<n_r}\notag\\
&\quad+\sum_{n_2<n_1=n_3<\cdots<n_r}+\cdots+\sum_{n_2<\cdots<n_{r-1}<n_1<n_r}\notag\\
&\quad+\sum_{n_2<\cdots<n_{r-1}<n_1=n_r}+\sum_{n_2<\cdots<n_{r-1}<n_r<n_1}\notag\\
&\;=\zeta_r(s_1,s_2,\ldots,s_r)+\zeta_{r-1}(s_1+s_2,s_3,\ldots,s_r)
+\zeta_r(s_2,s_1,s_3,\ldots,s_r)\notag\\
&\quad+\zeta_{r-1}(s_2,s_1+s_3,\ldots,s_r)+\cdots+\zeta_r(s_2,\ldots,s_{r-1},s_1,s_r)\notag\\
&\quad+\zeta_{r-1}(s_2,\ldots,s_{r-1},s_1+s_r)+\zeta_r(s_2,\ldots,s_{r-1},s_r,s_1),\notag\end{aligned}$$ which is first valid in the region $\Re s_k>1$ ($1\leq k\leq r$), but then is valid in the whole space by meromorphic continuation.
Let $s_k=s_k^{(m)}$ ($1\leq k\leq r$) in , and consider the situation when $\sigma_1^{(m)}\to +\infty$ and $(s_2^{(m)},\ldots,s_r^{(m)})$ remains in $D_{r-1}$. Since $\zeta(s_1^{(m)})\to 1$ and $\zeta_{r-1}(s_2^{(m)},\ldots,s_r^{(m)})$ remains bounded, there exists a large $M_1=M_1(\varepsilon)$ for which $$\begin{aligned}
\label{y-7-1}
|\zeta(s_1)\zeta_{r-1}(s_2^{(m)},\ldots,s_r^{(m)})-\zeta_{r-1}(s_2^{(m)},\ldots,s_r^{(m)})|
<\varepsilon/3\end{aligned}$$ holds for any $m\geq M_1$.
On the right-hand side of , denote by $G(s_1,\ldots,s_r)$ the sum of all the terms, except for the first two terms. Then $G(s_1^{(m)},\ldots,s_r^{(m)})$ tends to 0, in view of Theorem \[th-y-0\]. That is, there exists $M_2=M_2(\varepsilon)$ for which $$\begin{aligned}
\label{y-7-2}
|G(s_1^{(m)},\ldots,s_r^{(m)})|<\varepsilon/3\end{aligned}$$ holds for any $m\geq M_2$.
We use the induction assumption to treat the second term on the right-hand side. We can find $M_3=M_3(\varepsilon)$ for which $$\begin{aligned}
\label{y-7-3}
|\zeta_{r-1}(s_1^{(m)}+s_2^{(m)},s_3^{(m)},\ldots,s_r^{(m)})-F_{r-2}(s_3^{(m)},\ldots,s_r^{(m)})|
<\varepsilon/3\end{aligned}$$ for any $m\geq M_3$.
Substituting , and into we find that $$\begin{aligned}
&|\zeta_r(s_1^{(m)},\ldots,s_r^{(m)})-
(\zeta_{r-1}(s_2^{(m)},\ldots,s_r^{(m)})
-F_{r-2}(s_3^{(m)},\ldots,s_r^{(m)}))|\\
&\qquad<\varepsilon/3+\varepsilon/3+\varepsilon/3,\end{aligned}$$ that is, $$\begin{aligned}
|\zeta_r(s_1^{(m)},\ldots,s_r^{(m)})-F_{r-1}(s_2^{(m)},\ldots,s_r^{(m)})|<\varepsilon.\end{aligned}$$ The uniformity of the convergence follows from the assertion on the uniformity in Theorem \[th-y-0\]. This completes the proof of Theorem \[th-y-1\], and hence the part (i) of Theorem \[th-y-2\].
(A generalization of Remark \[rem\_trivial\]) If $\Re s_r>1$, then is valid. In this case, when $\sigma_1\to +\infty$ and $(s_2^{(m)},\ldots,s_r^{(m)})$ tends to $(\rho_2,\ldots,\rho_r)$, from it is immediate that $\zeta_r(s_1^{(m)},\ldots,s_r^{(m)})$ tends to $$\begin{aligned}
\label{y-8}
\sum_{n_2=1}^{\infty}\cdots\sum_{n_r=1}^{\infty}(1+n_2)^{-\rho_2}\cdots(1+n_2+\cdots+n_r)^{-\rho_r}.\end{aligned}$$ Therefore in the region $\Re s_r>1$, the two expressions $F_{r-1}(\rho_2,\ldots,\rho_r)$ and should be equal to each other. In fact, putting $1+n_2=m_2$ in , we see that is equal to $$\begin{aligned}
&\sum_{m_2=2}^{\infty}\sum_{n_3=1}^{\infty}\cdots\sum_{n_r=1}^{\infty}
m_2^{-\rho_2}(m_2+n_3)^{-\rho_3}\cdots(m_2+n_3+\cdots+n_r)^{-\rho_r}\\
&=\zeta_{r-1}(\rho_2,\rho_3,\ldots,\rho_r)-\sum_{n_3=1}^{\infty}\cdots\sum_{n_r=1}^{\infty}
(1+n_3)^{-\rho_3}\cdots(1+n_3+\cdots+n_r)^{-\rho_r}.\end{aligned}$$ Next we put $1+n_3=m_3$ and argue similarly to obtain that the second sum on the right-hand side is $$=\zeta_{r-2}(\rho_3,\ldots,\rho_r)-\sum_{n_4=1}^{\infty}\cdots\sum_{n_r=1}^{\infty}
(1+n_4)^{-\rho_4}\cdots(1+n_4+\cdots+n_r)^{-\rho_r}.$$ Repeating this argument, we arrive at the expression $F_{r-1}(\rho_2,\ldots,\rho_r)$.
Lastly we prove the part (ii) of Theorem \[th-y-2\]. Let $(\rho_2,\ldots,\rho_r)$ be any solution of . We fix $\rho_2,\ldots,\rho_{r-1}$, and regard $F_{r-1}(\rho_2,\ldots,\rho_{r-1},s_r)$ as a function of one complex variable $s_r$ (and we denote it by $F(s_r)$ for brevity). Then $F(\rho_r)=0$, and since any zero of function of one complex variable is isolated, we can find a small positive $\varepsilon_0$, with the properties that $F(s_r)$ is holomorphic in $|s_r-\rho_r|<\varepsilon_0$ and $s_r=\rho_r$ is the only zero point in this region. Choose $0<\varepsilon<\varepsilon_0$, and put $$\begin{aligned}
\label{y-8-5}
m(\varepsilon)=\min_{|s_r-\rho_r|=\varepsilon}|F(s_r)|>0.\end{aligned}$$
Now, put $s_k=\rho_k$ ($2\leq k\leq r-1$) in . Also assume that $|s_r-\rho_r|\leq\varepsilon$. Then $|\zeta_{r-1}(\rho_2,\ldots,\rho_{r-1},s_r)|$ is bounded, so we have $$\begin{aligned}
\label{y-9}
|\zeta(s_1)\zeta_{r-1}(\rho_2,\ldots,\rho_{r-1},s_r)
-\zeta_{r-1}(\rho_2,\ldots,\rho_{r-1},s_r)|<m(\varepsilon)/3\end{aligned}$$ if $\sigma_1>M_1(\varepsilon)$ with a sufficiently large $M_1(\varepsilon)$.
By Theorem \[th-y-0\], there exists $M_2=M_2(\varepsilon)$ for which $$\begin{aligned}
\label{y-10}
|G(s_1,\rho_2,\ldots,\rho_{r-1},s_r)|<m(\varepsilon)/3\end{aligned}$$ for $\sigma_1>M_2(\varepsilon)$.
Lastly, by Theorem \[th-y-1\] we see that there exists $M_3=M_3(\varepsilon)$ for which $$\begin{aligned}
\label{y-11}
&\Bigl|\zeta_{r-1}(s_1+\rho_2,\rho_3,\ldots,\rho_{r-1},s_r)\Bigl.\\
&\bigl.-(\sum_{j=1}^{r-2}(-1)^{r-j}\zeta_j(\rho_{r-j+1},\ldots,\rho_{r-1},s_r)-(-1)^{r-1})\Bigr|
<m(\varepsilon)/3\notag\end{aligned}$$ for $\sigma_1>M_3(\varepsilon)$.
Combining , , and , and noting $$\begin{aligned}
&\zeta_{r-1}(\rho_2,\ldots,\rho_{r-1},s_r)-
(\sum_{j=1}^{r-2}(-1)^{r-j}\zeta_j(\rho_{r-j+1},\ldots,\rho_{r-1},s_r)-(-1)^{r-1})\\
&\quad=F_{r-1}(\rho_2,\ldots,\rho_{r-1},s_r)=F(s_r),\end{aligned}$$ we obtain $$\begin{aligned}
&|\zeta_r(s_1,\rho_2,\ldots,\rho_{r-1},s_r)
-F(s_r)|\\
&\qquad<m(\varepsilon)/3+m(\varepsilon)/3+m(\varepsilon)/3=m(\varepsilon)\end{aligned}$$ when $\sigma_1>\max\{M_1(\varepsilon),M_2(\varepsilon),M_3(\varepsilon)\}$. We fix such an $s_1$. From the above inequality and we obtain $$|\zeta_r(s_1,\rho_2,\ldots,\rho_{r-1},s_r)
-F(s_r)|
<|F(s_r)|.$$ Therefore by Rouch[é]{}’s theorem we find that the number of zeros of $\zeta_r(s_1,\rho_2,\ldots,\rho_{r-1},s_r)$ ( as a function in $s_r$) in the region $|s_r-\rho_r|<\varepsilon$ is equal to the number of zeros of $F(s_r)$ in the same region, but the latter is 1. This completes the proof of Theorem \[th-y-2\] (ii).
= = $ (1.247595281) + i (14.14857043) $ $ (0.5) + i (14.13472514) $\
$ (1.279113136) + i (21.01244258) $ $ (0.5) + i (21.02203964) $\
$ (1.292752716) + i (25.03054326) $ $ (0.5) + i (25.01085758) $\
$ (1.304538379) + i (30.39099998) $ $ (0.5) + i (30.42487613) $\
$ (1.310484619) + i (32.97276761) $ $ (0.5) + i (32.93506159) $\
$ (1.319356152) + i (37.57182573) $ $ (0.5) + i (37.58617816) $\
$ (1.320370441) + i (40.90318345) $ $ (0.5) + i (40.91871901) $\
$ (1.328328378) + i (43.36573059) $ $ (0.5) + i (43.32707328) $\
$ (1.333209526) + i (47.94829355) $ $ (0.5) + i (48.00515088) $\
$ (1.330151768) + i (49.81347595) $ $ (0.5) + i (49.77383248) $\
$ (1.338852528) + i (52.98250152) $ $ (0.5) + i (52.97032148) $\
$ (1.341784096) + i (56.42981942) $ $ (0.5) + i (56.44624769) $\
$ (1.336083544) + i (59.30477438) $ $ (0.5) + i (59.34704400) $\
$ (1.347599692) + i (60.90235918) $ $ (0.5) + i (60.83177852) $\
$ (1.350310992) + i (65.06828349) $ $ (0.5) + i (65.11254405) $\
$ (1.343496503) + i (67.09040611) $ $ (0.5) + i (67.07981053) $\
$ (1.348806781) + i (69.55741541) $ $ (0.5) + i (69.54640171) $\
$ (1.354734356) + i (72.08670993) $ $ (0.5) + i (72.06715767) $\
$ (1.356705834) + i (75.63889778) $ $ (0.5) + i (75.70469069) $\
$ (1.344904017) + i (77.17673184) $ $ (0.5) + i (77.14484007) $\
$ (1.360065509) + i (79.37508073) $ $ (0.5) + i (79.33737502) $\
$ (1.360772059) + i (82.86977267) $ $ (0.5) + i (82.91038085) $\
$ (1.355304386) + i (84.75005063) $ $ (0.5) + i (84.73549298) $\
$ (1.353373038) + i (87.38999143) $ $ (0.5) + i (87.42527461) $\
$ (1.365347698) + i (88.87469754) $ $ (0.5) + i (88.80911121) $\
$ (1.367563235) + i (92.45429247) $ $ (0.5) + i (92.49189927) $\
$ (1.352878941) + i (94.60615786) $ $ (0.5) + i (94.65134404) $\
$ (1.364213735) + i (95.93955674) $ $ (0.5) + i (95.87063423) $\
$ (1.367791276) + i (98.82956582) $ $ (0.5) + i (98.83119422) $\
[999]{}
, Analytic continuation of multiple zeta-functions and their values at non-positive integers, Acta Arith. [**98**]{} (2001), 107-116.
, On the zeros of the multiple zeta function, preprint.
, On the analytic continuation of various multiple zeta-functions, in “Number Theory for the Millennium II", M. A. Bennett et al. (eds.), A K Petres, 2002, pp.417-440.
, Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series, Nagoya Math. J. [**172**]{} (2003), 59-102.
, Numrical computations on the zeros of the Euler double zeta-function I, Moscow J. Combin. Number Theory [**4**]{} (2014), 295-313.
, The Theory of the Riemann Zeta-function, Oxford, 1951.
|
=0.5cm
[**$L^{2}$-HOMOLOGY FOR VON NEUMANN ALGEBRAS**]{}
Alain CONNES [^1] and Dimitri SHLYAKHTENKO [^2]
Introduction.
=============
The aim of this paper is to introduce a notion of $L^{2}$-homology in the context of von Neumann algebras. Finding a suitable (co)homology theory for von Neumann algebras has been a dream for several generations (see [@kad-ringI; @kad-ringII; @kad-ringIII; @sinclair-smith:cohomology] and references therein). One’s hope is to have a powerful invariant to distinguish von Neumann algebras. Unfortunately, little positive is known about the Kadison-Ringrose cohomology $H_{b}^{*}(M,M)$, except that it vanishes in many cases. Furthermore, there does not seem to be a good connection between the bounded cohomology theory of a group and of the bounded cohomology of its von Neumann algebra.
Our interest in developing an $L^{2}$-cohomology theory was revived by the introduction of $L^{2}$-cohomology invariants in the field of ergodic equivalence relations in the paper of Gaboriau [@gaboriau:ell2]. His results in particular imply that $L^{2}$-Betti numbers $\beta _{i}^{(2)}(\Gamma )$ of a discrete group are the same for measure-equivalent groups (i.e., for groups that can generate isomorphic ergodic measure-preserving equivalence relations). Parallels between the “worlds” of von Neumann algebras and measurable equivalence relations have been noted for a long time (starting with the parallel between the work of Murray and von Neumann [@vN] and that of H. Dye [@Dye]). Thus there is hope that an invariant of a group that “survives” measure equivalence will survive also “von Neumann algebra equivalence”, i.e., will be an invariant of the von Neumann algebra of the group.
The original motivation for our construction comes from the well understood analogy between the theory of II$_1$-factors and that of discrete groups, based on the theory of correspondences [@connes:correspondences; @connes:ncgeom]. This analogy has been remarkably efficient to transpose analytic properties such as “amenability" or “property $T$" from the group context to the factor context [@connesT] [@cj] and more recently in the breakthrough work of Popa [@popa] [@bbk]. We use the theory of correspondences together with the algebraic description of $L^{2}$-Betti numbers given by Luck. His definition involves the computation of the algebraic group homology with coefficients in the group von Neumann algebra. Following the guiding idea that the category of bimodules over a von Neumann algebra is the analogue of the category of modules over a group, we are led to the following algebraic definition of $L^{2}$-homology of a von Neumann algebra $M$:$$H_{k}^{(2)}(M)=H_{k}(M;M\bar{\otimes }M^{o}).$$ Here $H_{k}$ stands for the algebraic Hochschild homology of $M$. One is thus led to consider the $L^{2}$-Betti numbers,$$\beta _{k}^{(2)}(M)=\dim _{M\bar{\otimes }M^{o}}H_{k}^{(2)}(M)$$ (see Section \[sub:Fromgroupstovnalg\] for more motivation behind this definition).
The $L^2$ Betti numbers that we associate to a II$_1$ factor $M$ enjoy the following scaling property. If $p\in M$ is a projection of trace $\lambda$, then $$\beta_k^{(2)}(pMp) = \frac {1 }{\lambda ^2} \beta_k^{(2)} (M).$$ This behavior is a consequence of considering the “double” $M\otimes M^o$ of $M$. It is quite different from the behavior of $L^2$-Betti numbers associated to equivalence relations [@gaboriau:ell2], where the factor $\lambda^2$ is replaced by $\lambda$. Note that the behavior of our invariants is consistent with the formula for the “number of generators” of compressions of free group factors, where the same factor $\lambda^2$ appears. Indeed, for $p\in L(\mathbb{F}_t)
= L(\mathbb{F}_{(t-1)+1})$ a projection of trace $\lambda$, one has by the results of Voiculescu, Dykema and Radulescu (see e.g. [@dvv:book]) that $$pL(\mathbb{F}_t) p
\cong L(\mathbb{F}_{ ((t-1)\lambda ^{-2}) + 1}.$$ In this respect, our theory seems to be quite disjoint from a kind of relative theory of $L^2$-invariants for factors with $HT$-Cartan subalgebras discovered by Popa [@popa]. Indeed, the two theories have different scaling properties ($\lambda^{-2}$ versus $\lambda^{-1}$) under compression.
We have completely unexpectedly found a connection between our definition, involving algebraic homology, and the theory of free entropy dimension developed by Voiculescu and his followers ([@dvv:entropy1; @dvv:entropy2; @dvv:entropy3; @dvv:entropy5], see also [@dvv:entropysurvey] and references therein). Such a connection between $L^{2}$-Betti numbers and free entropy dimension parallels nicely the connections between free entropy dimension and cost for equivalence relations [@gaboriau:cost; @shlyakht:cost; @shlyakht:cost:micro].
Our definition seems to be robust with respect to modifications; for example, the connections with free probability theory do not seem to be accessible for certain variations of our definition.
This unexpected connection allows us to rely on several deep results in free probability theory, including the inequality between the microstates and microstates-free free entropy proved by recent fundamental work of Biane, Capitaine and Guionnet [@guionnet-biane-capitaine:largedeviations]. Drawing on these results, we obtain strong evidence that the first $L^{2}$-Betti number associated to the von Neumann algebra of a free group $\mathbb{F}_{n}$ on $n$ generators is bounded below by $n-1$ ($=\beta _{1}^{(2)}(\mathbb{F}_{n})$). Moreover, our inability to prove a similar inequality between free entropy dimension and $L^{2}$-Betti numbers obtained by varying our definition can be taken as evidence that the present definition is the correct one. Even more encouraging are the facts that the $L^{2}$-Betti numbers can be controlled from above at least in some cases (such as abelian von Neumann algebras, or von Neumann algebras with a diffuse center).
Our results also have several implications for the questions involving computation of free entropy dimension. In particular, we show that the modified free entropy dimension $\delta _{0}(\Gamma )$ of any discrete group $\Gamma $ is bounded from above by $\beta _{1}^{(2)}(\Gamma )-\beta
_{0}^{(2)}(\Gamma )+1$; a similar estimate holds for the various versions of the non-microstates free entropy dimension. If $\Gamma $ has property $T$ and if the von Neumann algebra $L(\Gamma )$ is diffuse, then $\delta _{0}(\Gamma )\leq 1$ (and $=1$ if $L(\Gamma )$ can be embedded into the ultrapower of the free group factor, as is the case when $\Gamma $ is residually finite). This generalizes a result of Voiculescu [@dvv:entropySLnZ] for the case that $\Gamma =SL(n,\mathbb{Z})$, $n\geq 3$.
Notation.
---------
Whenever possible, we shall use the following notation: the letter $A$ will stand for a tracial $\ast$-algebra with a fixed positive faithful trace $\tau $. The letter $M$ will stand for the von Neumann algebra generated by $A$ in the GNS representation associated to $\tau $. We write $L^{2}(M)$ for the associated representation space.
For a group $\Gamma $, we denote by $\mathbb{C}\Gamma $ its (algebraic) group algebra, and by $L(\Gamma )$ its group von Neumann algebra.
The tensor sign $\otimes $ will always refer to the algebraic tensor product. Tensor products that involve completions (such as the von Neumann algebra tensor product or the Hilbert space tensor product) will always be denoted by $\bar{\otimes }$.
We denote by $HS(L^{2}(M))$ the ideal of Hilbert-Schmidt operators on $L^{2}(M)$, and by $FR(L^{2}(M))$ the ideal of finite-rank operators on $L^{2}(M)$.
It will be useful to identify $HS(L^{2}(M))$ with $L^{2}(M)\bar{\otimes }L^{2}(M^{o})$ in the following way. Let $J:L^{2}(M)\to L^{2}(M)$ be the Tomita conjugation. Let $P_{1}\in L^{2}(M)$ be the projection onto the cyclic vector. Then consider the map $\Psi :M\otimes M^{o}\to FR(L^{2}(M))$ given by$$\Psi (x\otimes y^{o})(\xi )=x\ \tau (y\,\xi )\qquad x,y\in M,\ \xi \in
L^{2}(M).\label{eq:defofPsi}$$ The map $\Psi $ extends to an isometric isomorphism of $L^{2}(M\bar{\otimes }M^{o})$ with the Hilbert space $HS(L^{2}(M))$ of Hilbert-Schmidt operators in $L^{2}(M)$.
Note that $B(L^{2}(M))$ admits four commuting actions of $M$: $$\begin{aligned}
T & \mapsto & aT\\
T & \mapsto & Ta\\
T & \mapsto & Ja^{*}JT\\
T & \mapsto & TJa^{*}J.\end{aligned}$$ Here $a\in M$ and $T\in B(L^{2}(M))$. These four actions are intertwined by the map $\Psi $ with the four natural actions of $M$ on $L^{2}(M\bar{\otimes
}M^{o})$, listed in the corresponding order:$$\begin{aligned}
\label{bim}
x\otimes y^{o} & \mapsto & a\,x\otimes y^{o}\\
x\otimes y^{o} & \mapsto & x\otimes (y\,a)^{o} \nonumber\\
x\otimes y^{o} & \mapsto & x\,a\otimes y^{o}\nonumber\\
x\otimes y^{o} & \mapsto & x\otimes (a\,y)^{o},\nonumber\end{aligned}$$ where $a\in M$ and $x\otimes y^{o}\in M\otimes M^{o}$.
Definition of $L^{2}$-Betti numbers and $L^{2}$-homology.
=========================================================
Review of $L^{2}$-homology for groups.
--------------------------------------
### Co-compact actions.
The study of $L^{2}$-cohomology has been initiated by Atiyah in [@atiyah-L2], who considered $L^{2}$ deRham cohomology of a connected manifold $X$ endowed with a cocompact proper free action of a discrete group $\Gamma $. The $k$-th cohomology group is defined as the quotient of the space of closed $L^{2}$-integrable $k$-forms by the closure of the space of exact $L^{2}$ $k$-forms. In other words, one considers *reduced* $L^{2}$-cohomology (reduced means that one takes a quotient by the closure of the image of the boundary operator).
The cohomology groups in question can also be identified with the Hilbert spaces of $L^{2}$ harmonic forms, and are in a natural way modules over the group $\Gamma $. Furthermore, because of the properness and cocompactness assumptions, these spaces are actually modules over the group von Neumann algebra $L(\Gamma )$. Atiyah considered the Murray-von Neumann dimensions of these cohomology groups, and called them $L^{2}$-Betti numbers of the action. Remarkably, it turns out that these numbers are homotopy invariants; in particular, if the manifold $X$ is contractible, then the numbers depend only on the group $\Gamma $, and are called the $L^{2}$-Betti numbers of the group, $\beta _{i}^{(2)}(\Gamma )$. This definition extends to cover cocompact proper free actions of $\Gamma $ on a contractible simplicial complex, and gives an equivalent definition of the Betti numbers using $L^{2}$ singular homology. For example, if $\Gamma =\mathbb{F}_{n}$, the free group on $n$ generators, then $\beta _{i}^{(2)}(\Gamma )=0$ for $i\neq 1$, and $\beta _{1}^{(2)}(\Gamma )=n-1$.
### Cheeger-Gromov’s approach to non-cocompact actions.
Not every group can act on a contractible space in a proper free and co-compact way. For this reason, if one wants to obtain an $L^{2}$-homology theory for groups, one is forced to consider non-cocompact actions. This leads to difficulties of a technical nature. The problem is that because the spaces of $L^{2}$-(co)chains are now infinitely-dimensional modules over $L(\Gamma )$, one needs to be much more careful in taking the closure of the image of the boundary operator, and one may end up with having to consider dimensions of actions of $L(\Gamma )$ on quotients of Hilbert spaces by not necessarily closed subspaces.
This can be overcome in one of two ways. The first, which is the original approach of Cheeger and Gromov [@cheeger-gromov:l2], is to “approximate” the $L^{2}$-homology of a non-cocompact action on a contractible manifold $X$ by realizing the simplicial complex $(\Delta _{*},\partial _{*})$ underlying $X$ as an inductive limit of sub-complexes $(\Delta _{*}^{(k)},\partial
_{*}^{(k)})$, which correspond to co-compact (free) actions. Let $C_{*}^{(k)}(X)$ the the space of $L^{2}$-chains for $\Delta _{*}^{(k)}$ (i.e., the Hilbert space with orthonormal basis given by simplices in $\Delta _{*}^{(k)}$). Denote by $i_{k,l}$ the map from $C_{*}^{(k)}(X)$ to $C_{*}^{(l)}(X)$. Then the $n$-th $L^{2}$-Betti number is defined as$$\beta _{n}^{(2)}=\sup _{k\to \infty }\inf _{l>k}\dim _{L(\Gamma )}\frac{i_{k,l}(\ker
\partial _{n}^{(k)})}{\overline{i_{k,l}(\ker \partial _{n}^{(k)})\cap {\operatorname{im}}\partial _{n+1}^{(l)}}}.\label{eq:l2bettinoncocompactChGr}$$
### Luck’s approach to non-cocompact actions.
The second way, developed by Luck [@luck:foundations1], is to extend the notion of Murray-von Neumann dimension to algebraic modules over type II$_{1}$ von Neumann algebras. This way one can, for example, assign a dimension to the quotient of an $L(\Gamma )$ module by a non-closed submodule. Luck shows that such an extension indeed exists. In the case that $V$ is a finitely-generated module over $L(\Gamma )$, its dimension is just the supremum of the dimensions of *normal* $L(\Gamma )$ modules (i.e. finite projective modules) that can be embedded into $V$.
Returning briefly to the co-compact case, one can now consider *non-reduced* simplicial $L^{2}$ homology of a space $X$, defined as the quotient of the kernel of the boundary operator $\partial _{n}$ by the (non-closed) image of $\partial _{n+1}$. This results in $L(\Gamma )$ modules, which are not normal. However, due to the behavior of Luck’s extension of the Murray-von Neumann dimension, it turns out that the dimensions of these modules are the same as the Murray-von Neumann dimensions of the reduced homology groups (i.e., the $L^{2}$-Betti numbers).
In the case that $X$ is connected and contractible, its ordinary homology vanishes. This means that if we denote by $C_{k}^{(f)}$ the vector space with basis given by the $k$-chains on $X$, then the sequence $(C_{k}^{(f)},\partial _{k})$ is exact. Each $C_{k}^{(f)}$ is a flat module over $\Gamma $, and $C_{0}^{(f)}\cong \mathbb{C}$, since we assume that $X$ is connected. Thus $(C_{k}^{(f)},\partial _{k})$ forms a resolution of the trivial $\Gamma $-module $C_{0}^{(f)}$. Furthermore, the space of $L^{2}$ chains on $X$ is very roughly $L(\Gamma )\otimes _{\Gamma }C_{k}^{(f)}$.[^3] Thus the non-reduced $L^{2}$-homology of $X$ is nothing but the (algebraic) homology group$$H_{*}(\Gamma ;L(\Gamma ))$$ of the group $\Gamma $ with coefficients in $L(\Gamma )$ viewed as a right $\Gamma $-module. Since $L(\Gamma )$ is both a right $\Gamma $-module, and a left $L(\Gamma )$-module, the $H_{*}(\Gamma ;L(\Gamma ))$ are left $L(\Gamma )$-modules. Thus one can consider $\dim _{L(\Gamma )}H_{*}(\Gamma ;L(\Gamma ))$. However, as we pointed out, the dimensions of these non-reduced homology groups are precisely the $L^{2}$-Betti numbers of the group.
Thus even in the case that $\Gamma $ admits no proper free co-compact action on a contractible space, one can consider the numbers$$\beta _{*}^{(2)}(\Gamma )=\dim _{L(\Gamma )}H_{*}(\Gamma ;L(\Gamma
)),\label{eq:l2bettiLuck}$$ where $H_{*}$ stands for the algebraic group homology. Luck shows that a certain continuity property of his dimension leads to the formula$$\dim _{L(\Gamma )}H_{*}(\Gamma ;L(\Gamma ))=\sup _{k}\inf _{l>k}\dim _{L(\Gamma
)}i_{k,l}(H_{*}(C_{*}^{(k)}))$$ in the notation of the previous section and of equation (\[eq:l2bettinoncocompactChGr\]). Combined with (\[eq:l2bettinoncocompactChGr\]), this shows that this definition produces the same numbers as the one in equation (\[eq:l2bettinoncocompactChGr\]).
From groups to von Neumann algebras.\[sub:Fromgroupstovnalg\]
-------------------------------------------------------------
### Correspondences.
Let us recall the basis of the analogy between discrete groups and II$_1$-factors. A correspondence from II$_1$-factors is simply a Hilbert space endowed with a structure of (normal) bimodule. Correspondences on $M$ (i.e. $M$-$M$ Hilbert bimodules) play the role of unitary representations, and the dictionary begins as follows, $$\begin{matrix}
\hbox{Discrete Group $\Gamma$} &\hbox{II$_1$-Factor $M$} \cr
\cr
\hbox{Unitary Representation} &\hbox{$M$-$M$ Hilbert bimodule} \cr
\cr
\hbox{Trivial Representation} &\hbox{$L^2(M)$} \cr
\cr
\hbox{Regular Representation} &\hbox{Coarse Correspondence } \cr
\cr
\hbox{Amenability} &\hbox{$L^2(M)\subset_{\textrm{weakly}}L^{2}(M)\bar{\otimes }L^{2}(M^{o})$} \cr
\cr
\hbox{Property T} &\hbox{$L^2(M)$ isolated}
\cr
\end{matrix}$$
where the “coarse correspondence" is given by the bimodule $L^{2}(M)\bar{\otimes }L^{2}(M^{o})$ with bimodule structure given by the first two lines of (\[bim\]).
### Some homological algebra.
In order to find a suitable notion of $L^{2}$ Betti numbers of a tracial algebra $(M,\tau )$, we chose as our point of departure equation (\[eq:l2bettiLuck\]). It is perhaps best to rewrite the definition of group homology in the language of homological algebra see e.g. [@cartan-eilenberg:homalgebra]:$$H_{*}(\Gamma ;L(\Gamma ))={\operatorname{Tor}}_{*}^{Mod(\Gamma )}(\star ;L(\Gamma )).$$
This equation involves three objects: $Mod(\Gamma )$, $\star $ and $L(\Gamma )$. The role of $Mod(\Gamma )$ is to prescribe a suitable category; in this case, it is the category of left modules over $\Gamma $. The module $\star $ stands for the trivial left module. To compute the ${\operatorname{Tor}}$ functor, one must first choose a resolution of $\star $ by flat modules$$\star \leftarrow C_{0}\leftarrow C_{1}\leftarrow \cdots
.\label{eq:resolution}$$ This means that each $C_{i}$ is flat over $\Gamma $ (the definition of a flat module is not really needed here; free modules and, more generally, projective modules, are flat), and the sequence (\[eq:resolution\]) is exact.
Now $L(\Gamma )$ plays two roles. Its first role is that of a right module over $\Gamma $. To finish the computation of ${\operatorname{Tor}}$, one applies the functor $L(\Gamma )\otimes _{\Gamma }$ to the exact sequence (\[eq:resolution\]):$$L(\Gamma ) \otimes _{\Gamma }\star\leftarrow L(\Gamma
)\otimes _{\Gamma }C_{0}\leftarrow L(\Gamma
)\otimes _{\Gamma }C_{1}\leftarrow \cdots
.\label{eq:resolutiontensored}$$ This new sequence (\[eq:resolutiontensored\]) may well fail to be exact. Note, however, that it is a differential complex: the composition of any two consecutive arrows is zero (since the original sequences (\[eq:resolution\]) had this property). The value of ${\operatorname{Tor}}$ is precisely the homology of (\[eq:resolutiontensored\]), i.e., ${\operatorname{Tor}}_{k}$ is the kernel of the $k$-th map, divided by the image of the $k+1$-st. It is crucial that the value of the ${\operatorname{Tor}}$ functor is well-defined and is independent of the choice of the resolution (\[eq:resolution\]).
Finally, $L(\Gamma )$ plays its final role, which is that of not just a right $\Gamma $-module, but also of a left $L(\Gamma )$-module. It is crucial that the left action of $L(\Gamma )$ commutes with the right action of $\Gamma $. Because of this, (\[eq:resolutiontensored\]) is is a differential complex of left $L(\Gamma )$-modules. This makes it possible to take the dimension $$\beta _{*}^{(2)}(\Gamma )=\dim _{L(\Gamma )}{\operatorname{Tor}}_{*}^{Mod(\Gamma )}(\star ;L(\Gamma )),\label{eq:l2bettiLuck1}$$ as the definition of the Betti numbers (\[eq:l2bettiLuck\]).
### Analogs for tracial $\ast$-algebras.
Let now $(A,\tau )$ be a tracial $\ast$-algebra. We would like to make sense of $$\beta _{*}^{(2)}(A,\tau )=\dim _{X}{\operatorname{Tor}}_{*}^{Y}(Z,W),$$ where: $X$ is the analog of the group von Neumann algebra $L(\Gamma )$; $Y$ is the analog of the category of right $\Gamma $-modules; $Z$ is the analog of the trivial module; and $W$ is the analog of the $\Gamma ,L(\Gamma
)$-bimodule $L(\Gamma )$.
Let $M=W^{*}(A)$ in the GNS representation associated to $\tau $.
It is fortunate that the theory of correspondences furnishes perfect analogs for all of these objects. The analog of $Y$, i.e., of the category of left $\Gamma $-modules, is the category $Y$ of $A,A$-bimodules, or, better, of left modules over the algebraic tensor product $A\otimes A^{o}$ of $A$ by its opposite algebra $A^{o}$.
In particular the trivial $\Gamma $-module $Z$ is $A$ viewed as a bimodule over $A$ and since we want to view it as a [*left*]{} module over $A\otimes A^{o}$ we use $$\label{tri}
(m \otimes n^{o})\cdot \,a := m\, a \, n \,, \quad \forall m\,, n\,, a \in A$$
We next look for $X$, the analog of the group von Neumann algebra $L(\Gamma )$. Note that $L(\Gamma )$ is the von Neumann algebra generated by $\Gamma $ in the left regular representation. The analog of the left regular representation of $\Gamma $ on $\ell ^{2}(\Gamma )$ is the coarse correspondence, i.e. the representation of $A\otimes A^{o}$ on $L^{2}(M,\tau
)\bar{\otimes }L^{2}(M^{o},\tau )$ given by $$(m\otimes n^{o})\cdot(\sum a_{i}\otimes b_{i}^{o}) =\sum m\, a_{i}\, \otimes \,(b_{i}\,n)^{o},\qquad m\,,n\in
A\,.$$ i.e. the first two lines of (\[bim\]). Hence $X=W^{*}(M\otimes M^{o})$ in this representation, i.e., $X=M\bar{\otimes }M^{o}$ (von Neumann tensor product). Finally, $W=M\bar{\otimes }M^{o}$, but having two structures. One is that of a right $A\otimes A^{o}$-module, with the action $$\label{bim1}
(\sum a_{i}\otimes b_{i}^{o})\cdot (m\otimes n^{o})=\sum a_{i}\, m\otimes (n\,b_{i})^{o},\qquad m\,,n\in
A\,.$$ The other is that of an $M\bar{\otimes }M^{o}$-left module, given by left multiplication in $M\bar{\otimes }M^{o}$, i.e. $$(m\otimes n^{o})\cdot (\sum a_{i}\otimes b_{i}^{o})=\sum m\,a_{i}\otimes (b_{i}\,n)^{o}\,,\qquad
m,n\in M.$$ Note that the actions of $M\bar{\otimes }M^{o}$ and $A\otimes A^{o}$ on $W=M\bar{\otimes }M^{o}$ commute.
Thus we are led by our analogy to consider $$\begin{aligned}
H_{k}^{(2)}(A,\tau ) & = & {\operatorname{Tor}}_{k}^{A\otimes A^{o}}(A;M\bar{\otimes }M^{o}),\\
\beta _{k}^{(2)}(A,\tau ) & = & \dim _{M\bar{\otimes }M^{o}}H_{k}^{(2)}(A,\tau ).\end{aligned}$$ It turns out that in the category of bimodules over an algebra, ${\operatorname{Tor}}_{k}^{A\otimes
A^{o}}(A;M\bar{\otimes }M^{o})$ is exactly the $k$-th Hochschild homology of $A$ with coefficients in the bimodule $M\bar{\otimes }M^{o}$ of (\[bim1\]).
Let $(A,\tau )$ be a tracial $\ast$-algebra. Let $M=W^{*}(A)$ in the GNS representation associated to $\tau $. Then define the $k$-th $L^{2}$-homology group of $A$ to be the Hochschild homology group$$H_{k}^{(2)}(A,\tau )=H_{k}(A;M\bar{\otimes }M^{o}).$$ Also define the $k$-th $L^{2}$-Betti number of $A$ to be its extended Murray-von Neumann dimension (in the sense of Luck),$$\beta _{k}^{(2)}(A,\tau )=\dim _{M\bar{\otimes }M^{o}}H_{k}(A;M\bar{\otimes }M^{o}).$$
This definition of course depends on the trace that we choose on $A$.
The bar resolution.
-------------------
We will now give an “explicit” description of the $L^{2}$-homology group of $A$, which is at the same time the only description we have at present. The situation is somewhat analogous to the case of dealing with a general group $\Gamma $, for which one does not know whether or not one can choose a “nice” topological space on which $\Gamma $ can act properly and freely (which would give us a “nice” resolution with which to compute the homology). Thus one must instead resort to using the universal cover $E\Gamma $ of the universal classifying space $B\Gamma $ of $\Gamma $.
Let $C_{k}(A)=(A\otimes A^{o})\otimes A^{\otimes k}$, $k=0,1,\ldots $, viewed as a left $A\otimes A^{o}$-module via$$(m \otimes n^{o})\cdot((a\otimes b^{o})\otimes a_{1}\otimes \cdots \otimes a_{k})=(m\,a\,\otimes (b\,n)^{o})\otimes
a_{1}\otimes \cdots \otimes a_{k},$$ where $m,n,a,b,a_{1},\ldots ,a_{k}\in A$. Note that $C_{k}(A)$ is a free $A\otimes A^{o}$-module (typically on an infinite number of generators if $k>0$). Define$$\partial _{k}:C_{k}(A)\to C_{k-1}(A)$$ by the formula$$\begin{aligned}
\partial _{k}(T\otimes a_{1}\otimes \cdots \otimes a_{k}) & = & T\, (a_{1}\otimes 1)\, \otimes \cdots
\otimes a_{k}+\label{eq:differential}\\(-1)^{k}T\,(1 \otimes a_{k}^{o})\,\otimes a_{1}\otimes \cdots \otimes
a_{k-1}
&+ & \sum _{j=1}^{k-1}(-1)^{j}T\otimes a_{1}\otimes \cdots \otimes
a_{j}a_{j+1}\otimes \cdots \otimes a_{k};\nonumber \end{aligned}$$ where $T\in A\otimes A^{o}$, $a_{1},\ldots ,a_{k}\in A$ and we use the algebra structure of $A\otimes A^{o}$ to define $T\,(a_{1}\otimes 1)$ and $T\,(1 \otimes a_{k}^{o})$. Then $(C_{*}(A),\partial _{*})$ is exact [@cartan-eilenberg:homalgebra] and forms a resolution of the $A\otimes A^{o}$-left module $A$ of (\[tri\]) by $A\otimes A^{o}$-left modules, with the last map $C_{0}(A)=A\otimes A^{o}\mapsto A $ given by the multiplication $m$, $$\sum a_{i}\otimes b_{i}^{o} \rightarrow \sum a_{i}\, b_{i}$$
Let $C_{k}^{(2)}(A)=M\bar{\otimes }M^{o}\otimes _{A\otimes A^{o}}C_{k}=(M\bar{\otimes
}M^{o})\otimes A^{\otimes k}$. Let $\partial _{k}^{(2)}$ be given by the same formula as in (\[eq:differential\]), except that we now allow $T\in M\bar{\otimes }M^{o}$. Then $$H_{k}^{(2)}(A,\tau )=\frac{\ker \partial _{k}^{(2)}}{{\operatorname{im}}\partial _{k+1}^{(2)}}.$$ is by construction a left $M\bar{\otimes
}M^{o}$-module.
It is in general not clear how to compute these homology groups (or their dimensions over $M\bar{\otimes }M^{o}$). However, one can give a description of the $L^{2}$-Betti numbers of $A$ as a limit of finite numbers.
The bar resolution of $A$ can be written as an inductive limit of its sub-complexes $C_{*}^{(\ell)}$, $\ell\in I$, so that each $C_{*}^{(\ell)}$ is a finitely generated left module over $A\otimes A^{o}$. Moreover, if we denote by $i_{\ell,k}:C_{*}^{(\ell)}\to C_{*}^{(k)}$ the inclusion map, and by $H_{*}{(\ell)}$ the homology of the complex $(M\bar{\otimes }M)\otimes
_{A\otimes A^{o}}C_{*}^{(\ell)}$, then we have$$\begin{aligned}
H_{n}^{(2)}(A,\tau ) & = & {\lim_{\longrightarrow} }
(H_{n}{(\ell)},(i_{\ell,k})_{*}),\label{eq:homindlim}\\
\beta _{n}^{(2)}(A,\tau ) & = & \sup _{\ell}\inf _{k>\ell}\dim _{M\bar{\otimes
}M^{o}}(i_{\ell,k})_{*}H_{n}{(\ell)}.\label{eq:betaaslim}\end{aligned}$$
For each integer $n$, and finite subset $F\subset A$, let $V={\operatorname{span}}F$, and let $V_{0}=V^{\otimes n}$. Denote by $m$ the multiplication map $m:A\otimes A\to A$, and let$$\begin{aligned}
V_{1} & = & {\operatorname{span}}((m\otimes 1^{\otimes n-2})V_{0},(1\otimes m\otimes 1^{\otimes
n-3})V_{0},\ldots ,(1^{\otimes n-2}\otimes m)V_{0})\\
V_{2} & = & {\operatorname{span}}((m\otimes 1^{\otimes n-3})V_{1},(1\otimes m\otimes 1^{\otimes
n-4}V_{1},\ldots ,(1^{\otimes n-3}\otimes m)V_{1})\end{aligned}$$ and so on, where $1$ stands for the identity map $A \mapsto A$. Let$$C_{k}^{(F,n)}=\begin{cases} (A\otimes A^{o})\otimes V_{n-k}, & k\leq n\\
0, & k>n.\\ \end{cases}$$ Note that $C_{k}^{(F,n)}$ forms a sub-complex of the bar resolution of $A$.
Order the pairs $(F,n)$ by saying that $(F,n)<(F',n')$ if $F\subset F'$ and $n\leq n'$. If $(F,n)<(F',n')$, let$$i_{(F,n)}:C_{k}^{(F,n)}\to C_{k}^{(F',n')}$$ be the inclusion map. Then each $C_{k}^{(F,n)}$ is finitely-generated over $A\otimes A^{o}$ (it has at most $\dim V_{k}\,$ generators), and the bar resolution of $A$ is the inductive limit of the sub-complexes $C_{k}^{(F,n)}$.
It follows that the homology group $H_{n}(A;M\bar{\otimes }M^{o})$ is itself the inductive limit of the directed system $\{H_{n}{(\ell)},i_{\ell,k}\}$, so that (\[eq:homindlim\]) holds.
Because of the finite generation assumptions, we have that$$\dim _{M\bar{\otimes }M^{o}}(i_{\ell,k})_{*}H_{n}{(\ell)}<\infty ,\qquad \forall k>\ell.$$ Thus [@luck:foundations1 Theorem 2.9(2)] implies (\[eq:betaaslim\]).
A reader interested in an even more explicit description, valid for the first Betti number, is urged to look ahead to section \[sub:beta1inductive\].
Group algebras.
---------------
We have defined $L^{2}$-Betti numbers for any tracial $\ast$-algebra $(A,\tau )$. While the case of interest is when $A$ is a von Neumann algebra (so that $M=A$), we want to point out that the definition works well even in the purely algebraic setting.
Let $\Gamma $ be a discrete group, and denote by $\tau $ the von Neumann trace on the group algebra $\mathbb{C}\Gamma $. Let $\Gamma^{(2)}=\Gamma \times \Gamma^{o} $, with $\Gamma^{o} $ the opposite group, and view $\Gamma $ as a subgroup of $\Gamma^{(2)} $ via the diagonal inclusion map $\g \mapsto \Delta(\g)=(\g, (\g^{-1})^{o})$. Then$$\begin{aligned}
H_{k}^{(2)}(\mathbb{C}\Gamma ,\tau ) & = & L(\Gamma^{(2)} )\otimes _{L(\Gamma
)}H_{k}^{(2)}(\Gamma ),\label{eq:algvsgroup1}\\
\beta _{k}^{(2)}(\mathbb{C}\Gamma ,\tau ) & = & \beta _{k}^{(2)}(\Gamma
).\label{eq:algvsgroup2}\end{aligned}$$
Note that $L(\Gamma )\bar{\otimes }L(\Gamma^{o} )=L(\Gamma \times \Gamma^{o} )=L(\Gamma^{(2)} )$. By [@cartan-eilenberg:homalgebra], we have that$$H_{k}(\mathbb{C}\Gamma ;L(\Gamma )\bar{\otimes }L(\Gamma^{o} ))=H_{k}(\Gamma ,L(\Gamma^{(2)}
)).$$ Since $L(\Gamma^{(2)} )=L(\Gamma^{(2)} )\otimes _{L(\Gamma )}L(\Gamma )$ as $\Gamma $-modules, and the functor $L(\Gamma^{(2)} )\otimes _{L(\Gamma )}$ is flat [@luck:foundations1 Theorem 3.3(1)], it follows that$$H_{k}(\mathbb{C}\Gamma ;L(\Gamma )\bar{\otimes }L(\Gamma^{o} ))=L(\Gamma^{(2)} )\otimes
_{L(\Gamma )}H_{k}(\Gamma ;L(\Gamma ))=L(\Gamma^{(2)} )\otimes _{L(\Gamma
)}H_{k}^{(2)}(\Gamma ).$$ Equation (\[eq:algvsgroup2\]) now follows from [@luck:foundations1 Theorem 3.3(2)].
Compressions of von Neumann algebras.
-------------------------------------
As was shown by Gaboriau in the context of measurable measure-preserving equivalence relations [@gaboriau:ell2], $L^{2}$-Betti numbers behave well under restrictions of equivalence relations. More precisely, if an ergodic equivalence relation $R$ is restricted to subset $X$ of measure $\lambda $, then one has$$\beta _{n}^{(2)}(R_{\lambda })=\frac{1}{\lambda }\beta _{n}^{(2)}(R).$$ The analogue of this fact is given by the following theorem. It should be noted that the factor $1/\lambda $ in Gaboriau’s result is replaced in our context by the factor of $1/\lambda ^{2}$. This is explained by the fact that in constructing our $L^{2}$-homology, we have passed to the category of bimodules, so the natural object that we are working with is $M\otimes M^{o}$ (and not $M$). Compressing $M$ to a projection of trace $\lambda $ amounts to compressing $M\otimes M^{o}$ by a projection of trace $\lambda ^{2}$.
Let $M$ be a factor and let $p\in M$ be a projection of trace $\lambda $. Then$$\beta _{n}^{(2)}(pMp,\frac{1}{\tau (p)}\tau (p\cdot p))=\frac{1}{\lambda ^{2}}\beta
_{n}^{(2)}(M,\tau ).\label{eq:bettascale}$$
Let $(C_{*}(M), \partial_{*})$ be the bar resolution of $M$,$$\begin{aligned}
C_{k}(M) & = & (M\otimes M^{o})\otimes M^{\otimes k},\qquad k\geq 0\end{aligned}$$ with $\partial_{*}$ as in \[eq:differential\]. Let $N = M\otimes M^{o}, q=p\otimes p^{o}$ which is an idempotent in $N$. The reduced algebra $N_q= q\,N\,q$ is $pMp\otimes (pMp)^{o}$.
Let $F$ be the functor $V \mapsto F(V)=q\, V$ from the category of left $N$-modules to that of left $N_q$-modules. It is an exact functor since for $T : V\mapsto W\, , \;x\in {\operatorname{im}}T \cap q\, W$ one has $$x = T y = q T y = T q y \in {\operatorname{im}}(T/qV)\,.$$ Note that in our case $qN$ is a projective left module over $N_q$ since $pM$ is a projective left module over $pMp$ and similarly for the opposite algebras. Thus $F$ maps projective modules to projective modules. Moreover when we apply $F$ to the “trivial" $N$-module $M$ of \[tri\] we get $F(M)= pMp$, the “trivial" $pMp\otimes (pMp)^{o}$-module.
This shows that $F(C_{*}(M), \partial_{*})$ is a projective resolution of $pMp$ and hence that, $$\begin{aligned}
H_{*}^{(2)}(pMp,\frac{1}{\tau (p)}\tau (p\cdot p)) & = & (p\otimes p^{o})H_{*}^{(2)}(M,\tau ).\end{aligned}$$
Equation (\[eq:bettascale\]) now follows from the fact that for an $M\bar{\otimes }M^{o}$ left module $V$ and the projection $q=p\otimes p^{o}\in
M\bar{\otimes }M^{o}$, we have$$\dim _{pMp\bar{\otimes (}pMp)^{o}}qV=\dim _{q(M\bar{\otimes
}M^{o})q}qV=\frac{1}{(\tau \otimes \tau )(q)}\dim _{M\bar{\otimes }M^{o}}V.$$
Direct sums.
------------
$L^{2}$-homology behaves well with respect to direct sums:
Let $(A,\tau )=\bigoplus _{i}(A_{i},\tau _{i})$ (finite direct sum), so that the trace on $A$ decomposes as $\bigoplus _{i}\alpha _{i}\tau _{i}$ in terms of normalized traces $\tau _{i}$ on $A_{i}$.
Then$$H_{k}^{(2)}(A,\tau )\cong \bigoplus _{i}H_{k}^{(2)}(A_{i},\tau _{i})$$ and$$\beta _{k}^{(2)}(A,\tau )=\sum _{i}\alpha _{i}^{2}\beta _{k}^{(2)}(A_{i},\tau _{i}).$$
Let $C_{\ast}^{(i)}$ be the bar resolution of $A_{i}$ with its differential $d_{\ast}^{(i)}$, and let$$C_{k}=\bigoplus _{i}C_{k}^{(i)},\qquad d_{k}=\bigoplus _{i}d_{k}^{(i)}.$$ Then each $C_{k}$, $k\geq 1$ is a projective module over $A\otimes A^{o}$. This is because $\bigoplus _{i}A_{i}\otimes A_{i}^{o}$ is a direct summand of $A\otimes A^{o}=\bigoplus _{i,j}A_{i}\otimes A_{j}^{o}$. Thus $(C_{k},d_{k})$ is a projective resolution of $A$. Using this resolution to compute the $L^{2}$-homology of $A$ we obtain $$H_{k}^{(2)}(A,\tau )=\bigoplus _{i}H_{k}^{(2)}(A_{i},\tau _{i}).$$
Let $M_{i}=W^{*}(A_{i})$, $M=W^{*}(A)$ (each time in the GNS representation associated to $\tau $). The formula for Betti numbers is now immediate, once we remark that if $V_{i}$ is a module over $M_{i}\bar{\otimes }M_{i}^{o}$, then$$\dim _{M\bar{\otimes }M^{o}}(\bigoplus _{i}V_{i})=\sum \alpha _{i}^{2}\dim
_{M_{i}\bar{\otimes }M_{i}^{o}}V_{i},$$ the factor $\alpha _{i}^{2}$ coming from the fact that$$M\bar{\otimes }M^{o}=\bigoplus _{i,j}M_{i}\otimes M_{j}^{o},\qquad \tau \otimes \tau
=\bigoplus _{i,j}\alpha _{i}\alpha _{j}\tau _{i}\otimes \tau _{j}.$$
Zeroth Betti number and zeroth $L^{2}$-homology for von Neumann algebras.
-------------------------------------------------------------------------
Let $M$ be a von Neumann algebra. By definition $$H_{0}^{(2)}(M,\tau )
=\frac{\ker \partial _{0}^{(2)}}{{\operatorname{im}}\partial _{1}^{(2)}}$$ is the quotient of $M\bar{\otimes }M^{o}$ by the left ideal $L$ generated by $$V=\{1\otimes n^{o}-n\otimes 1:n\in M\}.$$ or in other words $$H_{0}^{(2)}(M,\tau )=(M\bar{\otimes }M^{o})\otimes _{M\otimes M^{o}}M.$$
Let $M$ be a II$_1$-factor. Then $H_{0}^{(2)}(M,\tau )\neq 0$ if and only if $M$ is hyperfinite.
Since as a left $M\bar{\otimes }M^{o}$-module, $H_{0}^{(2)}(M)$ is clearly generated by the class in the quotient of the element $1\otimes 1$, $H_{0}^{(2)}(M)=0$ if and only if $[1\otimes 1]=0$.
By ([@connes:injective]) $M$ is hyperfinite if and only if the trivial correspondence is weakly contained in the coarse correspondence. That is to say, there exists a nonzero non-normal state $\theta :M\bar{\otimes }M^{o}\rightarrow \mathbb{C}$ with the property that $$\label{hyper}
\theta (m\otimes n^{o})=\tau (m\, n),\qquad \forall m,n\in M.$$
Assume first that $M$ is hyperfinite. Then $\theta(\,x^\ast \,x)=0$ for $x\in V$. Thus $\theta |_{J}=0$, so that $\theta $ factors through to a non-zero linear functional on $H_{0}^{(2)}(M)$ and $H_{0}^{(2)}(M)\neq 0$.
Conversely, assume that $H_{0}^{(2)}(M)\neq 0$. Then for any $n$ unitaries $u_{i}\in M$, the operator $$T=\frac{1}{n}\sum u_{i}\otimes u_{i}^{*o}-1,$$
belongs to the left ideal $L$ so that $T$ and hence $T^{*}T$ are not invertible, for any $u_{i}$. Thus one can find a non-normal state $\phi $ on $M\bar{\otimes }M^{o}$, for which $\phi (T^{*}T)=0$ for any such $T$. Denoting by $\xi _{\phi }$ the associated cyclic vector, we get that $T\xi _{\phi }=0$ and so$$(u_{i}\otimes u_{i}^{*o})\,\xi _{\phi }=\xi _{\phi }.$$ But then we have$$(u_{i}\otimes 1)\,\xi _{\phi }=(1\otimes u_{i}^{o})\,\xi _{\phi }$$ and hence$$(n\otimes 1)\,\xi _{\phi }=(1\otimes n^{o})\,\xi _{\phi }$$ for all $n\in M$. Hence$$(m\,n\otimes 1)\,\xi _{\phi }=(m \otimes 1)(n\otimes 1)\,\xi _{\phi }=(m\otimes n^{o})\,\xi _{\phi },$$ so that $\phi (m\otimes n^{o})=\phi (m\,n\otimes 1)$. Lastly,$$\begin{aligned}
\phi (n\,m\otimes 1) &=\langle (n\otimes m^{o})\,\xi _{\phi },\xi _{\phi }\rangle \\
& =\langle (1\otimes m^{o})\cdot (n\otimes 1)\,\xi _{\phi },\xi _{\phi }\rangle \\
& =\langle (n\otimes 1)\,\xi _{\phi },(1\otimes m^{*o})\xi _{\phi }\rangle \\
& =\langle (n\otimes 1)\,\xi _{\phi },(m^{*}\otimes 1)\,\xi _{\phi }\rangle \\
& =\langle (m\, n\otimes 1)\,\xi _{\phi },\xi _{\phi }\rangle = \phi (m\,n\otimes 1),\end{aligned}$$ so that $\phi |_{M\otimes 1}$ is a trace, and hence the unique trace $\tau $ on $M\cong M\otimes 1$, thus $\phi$ fulfills \[hyper\].
We get as consequence of Luck’s Theorem 1.8 [@luck:foundations1], his definition of the projective part of a module and of Theorem 0.6 in [@luck:foundations1], and of the fact that $H_{0}^{(2)}(A,\tau
)=M\bar{\otimes }M^{o}\otimes _{A\otimes A^{o}}A$, $M=W^{*}(A)$ is finitely (in fact, singly) generated as an $M\bar{\otimes }M^{o}$ module, that$$\beta _{0}(A,\tau )=\dim _{M\bar{\otimes }M^{o}}\textrm{Hom}(M\bar{\otimes
}M^{o}\otimes _{A\otimes A^{o}}A,M\bar{\otimes }M^{o})$$
\[pro:centerbetti0\]If $(A,\tau )$ contains an element $x$, whose distribution with respect to $\tau $ is non-atomic, then $\beta
_{0}^{(2)}(A,\tau )=0$.
Assume $\beta _{0}^{(2)}(A,\tau )\neq 0$, let $\phi \neq 0$, $\phi \in {\operatorname{Hom}}(M\bar{\otimes }M^{o}\otimes _{A\otimes A^{o}}A,M\bar{\otimes }M^{o})$. Denote by $[1\otimes 1]$ the class of $1\otimes 1$ in $M\bar{\otimes }M^{o}\otimes
_{A\otimes A^{o}}A$. Let $\xi =\phi ([1\otimes 1])\in M\bar{\otimes }M^{o}$. Since $M\bar{\otimes }M^{o}\otimes _{A\otimes A^{o}}A$ is generated by $[1\otimes 1]$, $\phi \neq 0$ implies that $\xi \neq 0$.
We thus have a vector $\xi \neq 0$ in $L^{2}(M\bar{\otimes }M^{o})$ with the property that $(m\otimes 1-1\otimes m^{o})\,\xi =0$ for all $m\in A$ and hence for all $m\in A''=M$. Identify $L^{2}(M\bar{\otimes }M^{o})$ with the space of Hilbert-Schmidt operators on $L^{2}(M)$, by the map $\Psi$ of (\[bim\]). Then $\Psi(\xi) $ is a non-zero Hilbert-Schmidt operator, commuting with $M$ by (\[bim\]). But this is impossible, since $M$ contains an element with a diffuse spectrum.
If $M$ is a type II$_{1}$ factor, then $\beta _{0}^{(2)}(M)=0$.
Let $M$ be a finite-dimensional von Neumann algebra with a faithful trace $\tau $. Decompose $M=\oplus M_{i}$ into factors with $M_{i}\cong
M_{k_{i}\times k_{i}}$(the algebra of $k_{i}\times k_{i}$ matrices), and let $\lambda _{i}$ be the trace of the minimal central projection in $M$ corresponding to the $i$-th summand.
Then $$\begin{aligned}
\beta _{0}^{(2)}(M) & = & \sum _{i}\frac{\lambda _{i}^{2}}{k_{i}^{2}},\\
\beta _{k}^{(2)}(M) & = & 0,\qquad k\geq 1.\end{aligned}$$
Since there is no difference between $M\otimes M$ and $M\bar{\otimes }M$ in the finite-dimensional case, $\beta _{k}^{(2)}(M)=0$ if $k>0$. For $\beta _{0}^{(2)}$ we find easily that $\beta _{0}^{(2)}(\mathbb{C})=1$; the compression formula then gives $\beta _{0}^{(2)}(M_{k\times k})=\frac{1}{k^{2}}$, and the direct sum formula gives us finally the desired expression.
$L^{2}$-Betti numbers for bimodule maps.
----------------------------------------
### Betti numbers for group module maps.
$L^{2}$-Betti numbers can be more generally defined for maps between group modules. Let us for definiteness consider a module map $f$ between two free left $\Gamma$-modules:$$f:(\mathbb{C}\Gamma )^{n}\to (\mathbb{C}\Gamma )^{m}.$$ Thus $f$ is given by an $n\times m$ matrix of right-multiplication operators in $\mathbb{C}\Gamma $. Consider now$$f^{(p)}:(\ell ^{p}(\Gamma ))^n\to (\ell ^{p}(\Gamma ))^m,$$ given by the same matrix. The kernel of $f^{(p)}$ may be larger than the $\ell ^{p}$-closure of the kernel of $f$. To measure the difference, consider for $p\leq 2$$$\beta ^{(2,p)}(f)=\dim _{L(\Gamma )}\frac{\overline{\ker f^{(p)}}^{\ell
^{2}}}{\overline{\ker f}^{\ell ^{2}}}.$$ Set$$\beta ^{(2)}(f)=\beta ^{(2,2)}(f)$$ for convenience.
Note that if $\Gamma $ acts freely and cocompactly on some contractible simplicial complex $X$, then $\beta _{j}^{(2)}(\Gamma )$ is then just $\beta ^{(2)}(\partial _{j})$, where $\partial _{j}$ is the boundary operator of $X$. Indeed, contractibility implies that $\ker \partial _{k}={\operatorname{im}}\partial
_{k+1}$, so that the closures of these two subspaces of $(\ell ^{2})^N$ are the same.
### Betti numbers for bimodule maps.
Let $(A,\tau )$ be a tracial $\ast$-algebra. Let $F:(A\otimes A^{o})^{n}\rightarrow
(A\otimes A^{o})^{m}$ be a left $A\otimes A^{o}$-module map (or, equivalently, an $A,A$-bimodule map). Then $F$ is given by a matrix$$F=\left(\begin{array}{ccc}
F_{11} & \cdots & F_{1n}\\
\vdots & \ddots & \vdots \\
F_{m1} & \cdots & F_{mn}\\ \end{array}
\right),$$ where $F_{ij}\in A\otimes A^{o}$ and the action is given by right multiplication.
Let $M=W^{*}(A)$ in the GNS representation associated to $\tau $.
The right multiplication by $F_{ij}$ admits a unique continuous extension to $L^{2}(M\bar{\otimes }M^{o})$. Thus $F$ admits a unique continuous extension to a left $M\bar{\otimes }M^{o}$-module map from $(L^{2}(M\bar{\otimes }M^{o}))^{n}$ to $(L^{2}(M\bar{\otimes }M^{o}))^{m}$, which we denote by $F^{(2)}$.
We note that $\ker F\subset \overline{\ker F}\subset \ker F^{(2)}$ (where $\overline{\cdot }$ refers to closure in $L^{2}$-norm). By analogy with the group case, we make the following definition (compare [@luck:hilbertmodules], Definition 5.1).
The $L^{2}$ Betti number of $F$ is the Murray-von Neumann dimension$$\beta ^{(2)}(F)=\dim _{M\bar{\otimes }M^{o}}\frac{\ker F^{(2)}}{\overline{\ker
F}}=\dim _{M\bar{\otimes }M^{o}}\ker F^{(2)}-\dim _{M\bar{\otimes
}M^{o}}\overline{\ker F}.$$
\[lem:l2closure\] One has $$\dim _{M\bar{\otimes }M^{o}}\frac{\ker F^{(2)}}{\overline{\ker F}^{L^{2}}}=\dim
_{M\bar{\otimes }M^{o}}\frac{\ker F^{vN}}{(M\bar{\otimes }M^{o})\ker F},$$ where $F^{vN}$ is the restriction of $F^{(2)}$ to $(M\bar{\otimes
}M^{o})^{n}\subset (L^{2}(M\bar{\otimes }M^{o}))^{n}$, and $(M\bar{\otimes }M^{o})\cdot \ker F$ denotes the saturation of $\ker F$ under the action of $M\bar{\otimes }M^{o}$.
The proof is almost verbatim the argument on the bottom of page 158 and top of page 159 of [@luck:foundations1], see also Theorem 5.4 of [@luck:hilbertmodules].
Note that $F^{vN}$ is exactly the map$$1\otimes F:(M\bar{\otimes }M^{o})\otimes _{A\otimes A^{o}}(A\otimes
A^{o})^{n}\rightarrow (M\bar{\otimes }M^{o})\otimes _{A\otimes A^{o}}(A\otimes
A^{o})^{m},$$ if we identify $(M\bar{\otimes }M^{o})\otimes _{A\otimes A^{o}}(A\otimes A^{o})$ with $M\bar{\otimes }M^{o}$. Thus in particular,$$\beta ^{(2)}(F)=\dim _{M\bar{\otimes }M^{o}}\frac{\ker (1\otimes F)}{(M\bar{\otimes
}M^{o})\cdot \ker F}.$$
### Homological algebra interpretation.
Let $F:(A\otimes A^{o})^{n}\to (A\otimes A^{o})^{m}$ be a bimodule map, as above. Put $V=(A\otimes A^{o})^{n}$, $W=(A\otimes A^{o})^{m}$.
Consider the exact sequence$$V\stackrel{F}{\longrightarrow }W\to {\operatorname{im}}F\to 0.$$ Since the $A\otimes A^{o}$- left modules $V$ and $W$ are projective (in fact, free), this sequence is the beginning of a projective resolution of ${\operatorname{im}}F$. More precisely, one can choose projective modules $V_{1},V_{2},\ldots $ and morphisms $F_{1},F_{2},\ldots$ so that the following sequence is exact:$$\cdots \to V_{2}\stackrel{F_{2}}{\longrightarrow
}V_{1}\stackrel{F_{1}}{\longrightarrow }V\stackrel{F}{\longrightarrow }W\to {\operatorname{im}}F\to 0.$$ Note that ${\operatorname{im}}F_{1}=\ker F$. Consider the differential complex$$\cdots \to (M\bar{\otimes }M^{o})\otimes _{A\otimes A^{o}}V_{1}\stackrel{1\otimes
F_{1}}{\longrightarrow }(M\bar{\otimes }M^{o})\otimes _{A\otimes
A^{o}}V\stackrel{1\otimes F}{\longrightarrow }(M\bar{\otimes }M^{o})\otimes
_{A\otimes A^{o}}W\to \cdots .$$ By definition,$${\operatorname{Tor}}_{1}^{A\otimes A^{o}}({\operatorname{im}}F,M\bar{\otimes }M^{o})=\frac{\ker (1\otimes
F)}{{\operatorname{im}}(1\otimes F_{1})}.$$ Since ${\operatorname{im}}(1\otimes F_{1})=(M\bar{\otimes }M^{o})\cdot {\operatorname{im}}F_{1}=(M\bar{\otimes }M^{o})\ker F$, we conclude that $$\label{hom1}
\beta ^{(2)}(F)=\dim _{M\bar{\otimes }M^{o}}({\operatorname{Tor}}_{1}^{A\otimes A^{o}}({\operatorname{im}}F,M\bar{\otimes }M^{o})).$$
### Examples of Betti numbers.
The following statement gives many examples of bimodule maps over von Neumann algebras, for which the $L^{2}$ Betti numbers are non-zero. For a group module map $f:\mathbb{C}\Gamma ^{n}\rightarrow \mathbb{C}\Gamma ^{m}$, denote by $f^{(1)}$ its extension to $\ell ^{1}(\Gamma )^{n}$. Denote by $\beta ^{(2,1)}(f)$ the dimension $$\beta ^{(2,1)}(f)=\dim _{L(\Gamma )}(\ker f^{(2)})-\dim _{L(\Gamma )}\overline{\ker
f^{(1)}}.$$
\[Prop:sameAsInGroupCase\]Let $\Gamma $ be a discrete group, $n,m<\infty $ and let $f:\mathbb{C}\Gamma ^{n}\rightarrow \mathbb{C}\Gamma ^{m}$ be a $\Gamma$-left module map given by a matrix with entries $f_{ij}\in \mathbb{C}\Gamma $. Let $A=M=L(\Gamma )$.
Let $F_{ij}=\Delta(f_{ij})\in M\bar{\otimes }M^{o}$ be the images of $f_{ij}$ under the canonical diagonal inclusion $\Delta(\g)=(\g, (\g^{-1})^{o})$ of $\mathbb{C}\Gamma $ into $M\otimes M^{o}=L(\Gamma )\otimes L(\Gamma^{o} )$. Let $F:(M\otimes
M^{o})^{n}\rightarrow (M\otimes M^{o})^{m}$ be given by the matrix whose entries are right multiplications by $F_{ij}$. Then,
$$\qquad \beta ^{(2)}(f)\geq \beta ^{(2)}(F)\geq \beta ^{(2,1)}(f)$$
Note that the statement is not automatic, since we are comparing$${\operatorname{Tor}}_{1}^{\Gamma }({\operatorname{im}}f,L(\Gamma ))\qquad \textrm{with}\qquad {\operatorname{Tor}}_{1}^{L(\Gamma )\otimes L(\Gamma^{o} )}({\operatorname{im}}F,L(\Gamma )\bar{\otimes }L(\Gamma^{o} ))$$ and not$${\operatorname{Tor}}_{1}^{\Gamma }({\operatorname{im}}f,L(\Gamma ))\qquad \textrm{with}\qquad {\operatorname{Tor}}_{1}^{\mathbb{C}\Gamma \otimes \mathbb{C}\Gamma^{o} }({\operatorname{im}}F,L(\Gamma )\bar{\otimes
}L(\Gamma^{o} ))$$
Let $\Delta_*$ be the induction functor from left $\Gamma$-modules to left $M\otimes M^{o}$-modules associated to the morphism $\Delta:\mathbb{C}\Gamma
\mapsto M\otimes M^{o}$, $$\Delta_*(X):= (M\otimes M^{o})\otimes _{\mathbb{C}\Gamma }X$$ where $M\otimes M^{o}$ is viewed as a right $\mathbb{C}\Gamma $-module using $\Delta$, then the $M\otimes M^{o}$-module $(M\otimes M^{o})^{n}$ is induced from $\mathbb{C}\Gamma ^{n}$ while $F$ is induced from $f$, or in short $F=1\otimes f.$\
It is sufficient to prove that $\dim _{M\bar{\otimes }M^{o}}\ker F^{(2)}=\dim
_{L(\Gamma )}\ker f^{(2)}$, $\dim _{M\bar{\otimes }M^{o}}\overline{\ker F}\geq \dim _{L(\Gamma )}\overline{\ker f}$ and $\dim _{M\bar{\otimes }M^{o}}\overline{\ker F}\leq \dim _{L(\Gamma )}\overline{\ker
f^{(1)}}$. Here $f^{(1)}$ stands for the extension of $f$ to $\ell ^{1}(\Gamma )$.
The morphism $\Delta$ preserves the trace and extends to an inclusion of von Neumann algebras $\Delta:
L(\Gamma )\mapsto M\bar{\otimes }M^{o}$. Denote by $F^{*}$ the “adjoint” of $F$ (with $i,j$-th entry $F_{ji}^{*}$). Then $\ker F^{(2)}=\ker ((F^{(2)})^{*}F^{(2)})$. Regard $T=(F^{(2)})^{*}F^{(2)}$ as an element of the algebra of $n\times n$ matrices over $M\bar{\otimes }M^{o}$. Then the dimension of the kernel of $F^{(2)}$ is precisely the non-normalized trace, computed in $M_{n\times
n}(M\bar{\otimes }M^{o})$, of the spectral projection $P$ of $T$ corresponding to the eigenvalue $0$. But $F= \Delta(f)$, and hence $P= \Delta(p)$ where $p$ is the spectral projection $p$ corresponding to the eigenvalue $0$ of the element $t=(f^{(2)})^{*}f^{(2)}\in M_{n\times n}(L(\Gamma ))$. The trace of $p$ is exactly $\dim _{L(\Gamma )}\ker f^{(2)}$. Thus $$\dim _{M\bar{\otimes
}M^{o}}\ker F^{(2)}=\dim _{L(\Gamma )}\ker f^{(2)}$$
Consider now the orthogonal projection $E:L^{2}(M\bar{\otimes }M^{o})\rightarrow
\Delta(L^{2}(M))$. It defines a conditional expectation $E: M\bar{\otimes }M^{o}
\rightarrow M$ where we identify $\Delta M$ with $M$. Note that if $\eta\in M\otimes M^{o}\subset L^{2}(M\bar{\otimes }M^{o})$, then $E(\eta)\in \ell ^{1}(\Gamma )$. To prove this, note first that it is sufficient to prove that $E(\eta)\in \ell ^{1}(\Gamma )$ whenever $\eta$ is a simple tensor of the form $\xi \otimes \zeta $, $\xi \in L^{2}(M)$, $\zeta \in L^{2}(M^{o})$. For $\gamma \in \Gamma $, let $u_{\gamma }$ be the corresponding unitary in $M$. Let $\xi =\sum \alpha _{\gamma }u_{\gamma }$ and $\zeta =\sum \beta _{\gamma }u_{\gamma ^{-1}}^{o}$ (where the sums are in $L^{2}$ sense). Then $E(\xi \otimes \zeta )=\sum \alpha _{\gamma }\,\beta
_{\gamma }\,\Delta(u_{\gamma })\sim \sum \alpha _{\gamma }\,\beta
_{\gamma }\,u_{\gamma }$. Since the sequences $\{\alpha _{\gamma }\}$ and $\{\beta _{\gamma }\}$ are in $\ell ^{2}(\Gamma )$, their product lies in $\ell ^{1}(\Gamma )$, and $E(\eta)\in \ell ^{1}(\Gamma )$.
Denote by $E_{\gamma }$ the map $\eta\mapsto E((u_{\gamma ^{-1} }\otimes 1)\cdot \eta$). Then any $\eta\in L^{2}(M\bar{\otimes }M^{o})$ is the $L^{2}$-sum $\eta=\sum
(u_{\gamma }\otimes 1)\cdot E_{\gamma
}(\eta)$. We extend $E$ and $E_{\gamma }$ (componentwise) to maps of direct sums of $L^{2}(M\bar{\otimes }M^{o})$ and denote them with the same letter. Since $E$ is a conditional expectation and $F$ acts by right multiplication by the $\Delta(f_{ij})$ one has $E\circ F^{(2)}=f^{(2)}\circ E$ and also $$E_{\gamma }\circ
F^{(2)}=f^{(2)}\circ E_{\gamma }\, \quad \forall \gamma \in \Gamma$$
Let now $\eta\in \ker F$. Then $\eta\in (M\otimes M^{o})^n$, and $E_{\gamma }(\eta)\in (\ell
^{1}(\Gamma ))^{n}$, for all $\gamma $. Denote by $f^{(1)}$ the restriction of $f^{(2)}$ to $(\ell ^{1}(\Gamma ))^{n}$. Then $f^{(1)}\circ E_{\gamma }(\eta)
=E_{\gamma }\circ F^{(2)}(\eta)=0$, so that $E_{\gamma }(\eta)\in \ker f^{(1)}$. Denote by $q$ the projection in $M_{n\times n}(M)$ corresponding to the invariant subspace $\overline{\ker
f^{(1)}}$. Let $Q =\Delta(q)$ be the image of $q$ under the inclusion map of $M_{n\times n}(M)\subset
M_{n\times n}(M\bar{\otimes }M^{o})$ induced by $\Delta$. Since $E_{\gamma }(\eta)\in \ker f^{(1)}$ for all $\gamma $, it follows easily that $\eta=\sum
(u_{\gamma }\otimes 1)\cdot E_{\gamma
}(\eta)$ is in the range of $Q$. Thus we have proved that $\ker F$ is contained in the range of $Q$, so that $\dim _{M\bar{\otimes }M^{o}}\overline{\ker F}\leq \dim
_{M\bar{\otimes }M^{o}}{\operatorname{im}}Q=\dim _{M}{\operatorname{im}}q=\dim _{M}\overline{\ker f^{(1)}}$.\
Finally, if we are given a finite sequence $t_{\gamma }\in \ker f$, then $\sum
(u_{\gamma }\otimes 1)\cdot t_{\gamma }\in \ker F$. Thus the induced module from $\overline{\ker f}$ is contained in $\overline{\ker F}$. This shows that $\dim _{M}\overline{\ker f}\leq \dim
_{M\bar{\otimes }M^{o}}\overline{\ker F}$. Thus we have$$\dim _{M}\overline{\ker f}\leq \dim _{M\bar{\otimes }M^{o}}\overline{\ker F}\leq
\dim _{M}\overline{\ker f^{(1)}}.$$
It would be interesting to know exactly when $\beta ^{(2)}(F)=\beta ^{(2)}(f)$. Note that by the results of [@mineev:vanishing], if $\Gamma $ is a “combable group” (in particular, a finitely generated hyperbolic group), and $f:C_{n}(X)\rightarrow C_{n-1}(X)$ is the boundary homomorphism of a contractible chain complex with a cocompact free action of $\Gamma $, then $\beta ^{(2)}(f)=\beta ^{(2,1)}(f)$. Indeed, it it proved in [@mineev:vanishing] that any element in $\ker f^{(1)}$ can be approximated in $\ell ^{1}$ (and hence $\ell ^{2}$) by elements from $\ker f$. This implies the following fact:
\[thm:combableBetti\]Let $\Gamma $ be a discrete combable group acting freely and co-compactly on a contractible chain complex $C_{*}(X)$ of a topological space $X$. Let $D_{*}=(L(\Gamma )\otimes L(\Gamma^{o} ))\otimes
_{\Gamma }C_{*}(X)$, where $\Gamma $ acts on $L(\Gamma )\otimes L(\Gamma^{o} )$ by $\Delta$. Then for each $k$,$$\begin{aligned}
\dim _{L(\Gamma )\bar{\otimes }L(\Gamma^{o} )}H_{k}((L(\Gamma )\bar{\otimes }L(\Gamma^{o}
))\otimes _{L(\Gamma )\otimes L(\Gamma^{o} )}D_{*}(X)) & = & \dim _{L(\Gamma
)}H_{k}(L(\Gamma )\otimes _{\Gamma }C_{*}(X))\\
& = & \beta _{k}^{(2)}(\Gamma ).\end{aligned}$$
This theorem is of interest in conjunction with equation (\[eq:betaaslim\]), since $D_{*}(X)$ (is homotopic to a a sub-complex that) occurs among the approximating sub-complexes of the bar resolution of $L(\Gamma )$.
Dual definition of Betti numbers for bimodule maps.
---------------------------------------------------
For the remainder of this section, we shall concentrate on bimodule maps over a von Neumann algebra; i.e., we assume that $A=M$ is a von Neumann algebra with a fixed trace $\tau $.
Let $f:(M\otimes M^{o})^{n}\rightarrow (M\otimes M^{o})^{m}$ be a left $M\otimes M^{o}$-module map. We are interested in the size of the kernel $\ker f$ in $(M\otimes M^{o})^{n}$. As before, denote by $f^{vN}$ and $f^{(2)}$, respectively, the extensions of $f$ to $(M\bar{\otimes }M^{o})^{n}$ and $L^{2}(M\bar{\otimes }M^{o})^{n}$.
Let us consider the algebraic tensor product $FR=L^{2}(M)\otimes L^{2}(M^{o})$ as a subset of $L^{2}(M)\bar{\otimes }L^{2}(M^{o})=L^{2}(M\bar{\otimes }M^{o})$ in the natural way. Note that $M\otimes M^{o}\subset FR$.
Note also that in the identification $\Psi $ of $L^{2}(M\bar{\otimes }M^{o})$ with the space $HS=HS(L^{2}(M))$ of Hilbert-Schmidt operators on $L^{2}(M)$ (see (\[eq:defofPsi\])), the set $FR$ corresponds precisely to the subset of $HS$ consisting of finite-rank operators.
We begin with a lemma, which shows that it does not matter for the purposes of $L^{2}$-closure whether we compute the kernel of $f^{(2)}$ in $(M\otimes M^{o})^{n}$ or $FR^{n}$.
\[lem:algsameasFR\]$\ker f^{(2)}\cap FR^{n}$ has the same $L^{2}$-closure in $L^{2}(M\bar{\otimes }M^{o})^{n}$ as $\ker f=\ker f^{(2)}\cap (M\otimes M^{o})^{n}$. Thus$$\beta ^{(2)}(f)=\dim _{M\bar{\otimes }M^{o}}\ker f^{(2)}-\dim _{M\bar{\otimes }M^{o}}(\overline{\ker f^{(2)}\cap FR^{n}})\label{eq:betaFR}$$
We view elements of $L^{2}(M)$ as unbounded operators (of left multiplication) on $L^{2}(M)$, affiliated with $M$.
Given any finite subset $K \subset L^{2}(M)$, and $\epsilon >0$ there exists a projection $e \in M$ such that $$e \,\xi \in M \,,\quad \Vert e \xi - \xi \Vert < \epsilon \,,\quad \forall \xi \in K$$ where $e \xi \in M$ means that the a priori unbounded operator of left multiplication by $e \xi $ is bounded. This is proved for a single $\xi$ using the polar decomposition $\xi=bu$ and a suitable spectral projection of the unbounded self-adjoint operator $b$. One controls moreover the trace $\tau(1-e)<\epsilon$. Taking the intersection $f$ of the projections $e_\xi, \xi \in K$ one gets $\tau(1-f)<n \epsilon$, $n= {\rm card}(K)$. This gives a sequence of projections such that $f_k \,\xi$ is bounded $\forall \xi \in K $ and $f_k \rightarrow 1$ in $L^2$ and hence strongly, which gives the answer. This shows that for any element $\xi \in FR^{n}$ there exists a projection $e \in M$ such that $(e \otimes e^{o})\,\xi \in (M\otimes M^{o})^{n}$ and $ \Vert (e \otimes e^{o})\,\xi - \xi \Vert < \epsilon$. Since $\ker f^{(2)}\cap FR^{n}$ is a left module over $M\otimes M^{o}$ the conclusion of the lemma follows.
As consequence, we have the following description of the dimension of the kernel of a left $M\otimes M^{o}$-module map $f:(M\otimes M^{o})^{n}\rightarrow (M\otimes M^{o})^{m}$. Extend $f$ to a map (still denoted by $f$) from $FR^{n}\rightarrow FR^{m}$ as in the previous Lemma. Identify $FR$ with the set of finite-rank operators on $L^{2}(M)$ using the identification $\Psi $. Let $B$ be the von Neumann algebra of all bounded operators on $L^{2}(M)$. Denote by $\langle \cdot ,\cdot \rangle $ the canonical pairing between $FR^{n}$ and $B^{n}$ given by $$\langle (T_{1},\ldots ,T_{n}),(S_{1},\ldots ,S_{n})\rangle =\sum _{j=1}^{n}{\operatorname{Tr}}(T_{j}S_{j}).$$ Denote by $f^{\,t}$ the map $$f^{\,t}:B^{m}\rightarrow B^{n}$$ uniquely determined by $$\langle f^{\,t}(T),S\rangle =\langle T,f(S)\rangle .$$ The map $f^{\,t}$ is well-defined and is in fact given by right multiplication by the matrix $f_{ji}^{\,t}$ where for $h =\sum m_i \otimes n_i^{o} \in M\otimes M^{o}$ one lets $$h^{\,t}:= \sum n_i \otimes m_i^{o} \,.$$ The right multiplication by $a\otimes b^{o}\in M\otimes M^{o}$ acts in the obvious way on $(M\otimes M^{o})^{n}$ and becomes, after the identification $\Psi $, using (\[bim\]) $$\label{rm}
T \cdot (a\otimes b^{o})=Ja^{*}J\,T\,Jb^{*}J \,,\quad \forall T \in B\,.$$
\[lem:dualbetti\] We have $$\dim _{M\bar{\otimes }M^{o}}((M\bar{\otimes }M^{o})\cdot \ker f)=n-\dim _{M\bar{\otimes }M^{o}}(\overline{f^{\,t}(B^{m})}^{w}\cap HS^{n}),$$ where the closure is taken with respect to the weak operator topology. In particular, $$\beta ^{(2)}(f)=\dim _{M\bar{\otimes }M^{o}}\left(\frac{\overline{f^{\,t}(HS^{m})}^{w}\cap HS^{n}}{\overline{f^{\,t}(HS^{m})}^{HS}}\right).$$
We note that $FR$ is the dual of $B(H)$ with respect to the weak topology. By duality, $\overline{f^{\,t}(B^{m})}^{w}$ is the annihilator of $\ker f\subset FR^{n}$. Let us show that $\overline{f^{\,t}(B^{m})}^{w}\cap HS$ is the annihilator of $\overline{\ker f}^{HS}$ in $HS^{n}$. The answer then follows by duality in $HS^{n}$ whose dimension over $M\bar{\otimes }M^{o}$ is equal to $n$.
Note that the two pairings are compatible. By continuity of the pairing in $HS^{n}$, $\overline{f^{\,t}(B^{m})}^{w}\cap HS^{n}$ is perpendicular to $\overline{\ker f}^{HS}$.
Since the Hilbert-Schmidt topology is stronger than the weak topology on $HS$, the subspace $\overline{f^{\,t}(B^{m})}^{w}\cap HS^{n}$ is already closed in the Hilbert-Schmidt topology.
Assume that $\xi \in HS^{n}$ belongs to $(\overline{\ker f}^{HS})^{\perp }$. Then $\xi \perp \ker f$ and viewing $\xi \in HS^{n}\subset B^{n}$, as an element of $B^{n}$ we find using the compatibility of the pairings that $\xi $ is in the co-kernel of $f$, so that $\xi \in \overline{f^{\,t}(B^{m})}^{w}$ and $\xi \in \overline{f^{\,t}(B^{m})}^{w}\cap HS^{n}$ as claimed.
Finally note that $\overline{f^{\,t}(HS^{m})}^{w}=
\overline{f^{\,t}(B^{m})}^{w}$, because $HS$ is weakly-dense in $B$ and $f^{\,t}$ is weakly continuous.
First Betti number and $\Delta $.
=================================
In this section, we concentrate on the first $L^{2}$-Betti number.
$\beta _{1}^{(2)}$ as a limit.\[sub:beta1inductive\]
----------------------------------------------------
### Sub-complexes associated to a set of generators.
We recall that all $L^{2}$-Betti numbers can be represented as limits, as described by equation (\[eq:betaaslim\]). We particularize to the first Betti number.
Let $M$ be a von Neumann algebra with a faithful trace-state $\tau $.
Let $F=\{X_{1},\ldots ,X_{n}\}\,,X_{j}\in M$ be a self-adjoint set of elements in $M$; that is, we assume that $X^* \in F$ whenever $X\in F$. Assume further that $F$ generates $M$ as a von Neumann algebra. Let$$C_{1}(F)=(M\otimes M^{o})\otimes {\operatorname{span}}F\cong (M\otimes M^{o})^{\dim {\operatorname{span}}F}$$ and consider$$\partial _{F}:C_{1}(F)\to M\otimes M^{o},$$ given by $$\partial_{F} (a\otimes b^{o}\otimes X)=a\,X\otimes b^{o}-a\otimes (X\,b)^{o},\qquad a\otimes b^{o}\in M\otimes M^{o},\ X\in F.
\label{df}$$ Then$$\ker \partial _{F}\to C_{1}(F)\stackrel{\partial _{F}}{\longrightarrow }M\otimes M^{o}\to M\to 0\label{eq:weakresgenerators}$$ is a sub-complex of the bar resolution of $M$. The sequence above is not exact, and the quotient of $M\otimes M^{o}$ by the image of $\partial _{F}$ is the left $M\otimes M^{o}$-module $(M\otimes M^{o})\otimes_{A\otimes A^{o}} A$, where $A$ is the algebra generated by $F$. When viewed as a bimodule over $M$ this left $M\otimes M^{o}$-module can be identified by the map $x\otimes y^{o}\mapsto x\otimes y$ with $M\otimes_{A} M$, where $(m\otimes n^{o})\cdot( x\otimes_{A}y) = m\,x\otimes_{A}y\,n$. We shall thus use the notation, $$M\otimes_{A} M :=(M\otimes M^{o})\otimes_{A\otimes A^{o}} A$$ The sequence $$\ker \partial _{F}\to C_{1}(F)\stackrel{\partial _{F}}{\longrightarrow }M\otimes M^{o}\to M\otimes _{A}M\to 0
\label{res1}$$ is exact. Applying the induction functor $(M\bar{\otimes }M^{o})\otimes_{M\otimes M^{o}}$ one gets the map$$1\otimes \partial _{F}:(M\bar{\otimes }M^{o})\otimes {\operatorname{span}}F\to M\bar{\otimes }M^{o},$$ given as in (\[df\]) by right multiplication by $X\otimes 1-1\otimes X^{o}$. Then the first homology of the induced complex from (\[res1\]) (or (\[eq:weakresgenerators\])) is given by$$H(F)=\frac{\ker (1\otimes \partial _{F})}{(M\bar{\otimes }M^{o})\cdot \ker \partial _{F}}.$$ In other words,
\[lem:HofFasTor\]Let $A$ be the algebra generated by $F$. Then $$H(F)={\operatorname{Tor}}_{1}^{M\otimes M^{o}}(M\otimes _{A}M,M\bar{\otimes }M^{o}).$$ In particular, $H(F)$ depends only on the inclusion $A\subset M$ and not on $F$.
We let $
\beta (F)=\dim _{M\bar{\otimes }M^{o}}H(F)$ so that by Lemma \[lem:l2closure\] one has $$\beta (F) = \beta^{(2)} (\partial _{F})$$ Note that if $F\subset F'$, then there is a natural inclusion map $i_{F',F}:C_{1}(F)\to C_{1}(F')$. This map induces a homomorphism$$(i_{F',F})_{*}:H(F)\to H(F').$$
For $F\subset F'$ two finite subsets, let$$H(F:F')=\frac{i_{F',F}(\ker (1\otimes \partial _{F}))}{i_{F',F}(\ker (1\otimes \partial _{F}))\cap (M\bar{\otimes }M^{o})\cdot \ker \partial _{F'}}=(i_{F',F})_{*}H(F).$$ Note that $i_{F',F}(\ker (1\otimes \partial _{F}))\cap (M\bar{\otimes }M^{o})\cdot \ker \partial _{F'}$ is exactly the intersection of $(M\bar{\otimes }M^{o})\otimes {\operatorname{span}}F$ with $(M\bar{\otimes }M^{o})\cdot \ker \partial _{F'}$.
Then (\[eq:betaaslim\]) implies that$$\beta _{1}^{(2)}(M)=\sup _{F}\inf _{F'\supset F}\dim _{M\bar{\otimes }M^{o}}H(F:F').$$ We are thus led to the natural question of the computation of$$\beta (F:F')=\dim _{M\bar{\otimes }M^{o}}H(F:F')$$ and, in particular, of $
\beta (F)=\dim _{M\bar{\otimes }M^{o}}H(F)$ (which corresponds to the case that $F=F'$).
Note in particular that$$\beta (F:F')\leq \beta (F)$$ (since the dimension of the image via $(i_{F',F})_{*}$ is not larger that the dimension of the domain), and also that $$\label{betafprime}
\beta (F:F')\leq \beta (F')$$ (since the image via $(i_{F',F})_{*}$ is a sub-module of $H(F')$).
Let $F=(X_{1},\ldots ,X_{n})$, $F'=F\cup (Y_{1},\ldots ,Y_{m})$, and assume for simplicity that $n=\dim {\operatorname{span}}F$. Denote by $\partial _{F}^{(2)}$ the extension of $\partial _{F}$ to $(L^{2}(M)\bar{\otimes }L^{2}(M^{o}))^{n}\cong L^{2}(M\bar{\otimes }M^{o})\otimes {\operatorname{span}}F$ obtained by continuity. By Lemma \[lem:algsameasFR\], equation (\[eq:betaFR\]), we have that$$\beta (F)=\beta ^{(2)}(\partial _{F})=\dim _{M\bar{\otimes }M^{o}}\ker \partial _{F}^{(2)}-\dim _{M\bar{\otimes }M^{o}}\overline{\ker \partial _{F}^{(2)}\cap FR^{n}}.$$
\[lem:kerPartial2\]Let $F\subset F'$ be self-adjoint sets of elements in $M$, as before. If $F$ generates $M$ as a von Neumann algebra, then$$\dim _{M\bar{\otimes }M^{o}}i_{F',F}\ker \partial _{F}^{(2)}=n-(1-\beta _{0}^{(2)}(M,\tau )),$$ and$$\begin{aligned}
\beta (F) & = & n-(1-\beta _{0}^{(2)}(M,\tau ))-\dim _{M\bar{\otimes }M^{o}}\overline{(\ker \partial _{F}^{(2)}\cap FR^{n})},\\
\beta (F:F') & = & n-(1-\beta _{0}^{(2)}(M,\tau ))-\dim _{M\bar{\otimes }M^{o}}\overline{\ker \partial _{F'}}\cap i_{F',F}\ker \partial _{F}^{(2)}.\end{aligned}$$
We just need to prove the first statement. The inclusion map $i_{F',F}:C_{1}(F)\to C_{1}(F')$ is injective, so that we need only to consider the case that $F=F'$. As explained in the proof of Proposition \[pro:centerbetti0\], $\beta _{0}^{(2)}(M,\tau )$ is $1-\dim _{M\bar{\otimes }M^{o}}(\overline{{\operatorname{im}}(\partial_1^{(2)})})$ where $\partial_1^{(2)}$ comes from the bar resolution.
Considering the kernel and cokernel of the morphism$$M\bar{\otimes }M^{o}\otimes F\stackrel{\partial _{F}^{(2)}}{\longrightarrow }M\bar{\otimes }M^{o},\label{eq:d1}$$ it is enough to show that $$\overline{{\operatorname{im}}(\partial_{F}^{(2)})}=\overline{{\operatorname{im}}(\partial_1^{(2)})}\,.$$ By construction $\overline{{\operatorname{im}}(\partial_1^{(2)})}$ is the strong closure in $M\bar{\otimes }M^{o}$ of the left ideal $L$ generated by $$V=\{n\otimes 1-1\otimes n^{o}: n\in M\}.$$
We use the map $\Psi :M\bar{\otimes }M^{o}\to HS=HS(L^{2}(M))$ (of (\[eq:defofPsi\])); one has by (\[bim\]) or (\[rm\]) $$\label{psit}
\Psi(x (n\otimes 1-1\otimes n^{o}))=[Jn^*J,\, \Psi(x)]\,,\quad \forall x\in M\bar{\otimes }M^{o}\,,
n\in M.$$ We adopt the following notation for any bounded operator $T$ in $L^{2}(M)$, $$\label{ttt}
\sigma(T):= T^\sigma=JT^*J \,,\quad \forall T\in B\,.$$ This gives an antiautomorphism of $B$ which restricts to the canonical antiisomorphism $\sigma:M\rightarrow M'$. We thus see that the closure of $L$ in $L^{2}(M\bar{\otimes }M^{o})$ can be identified with the subspace$$\overline{[HS,M']},$$ which is the closure of the linear span of commutators of $M'$ with $HS$.
Similarly the closure of ${\operatorname{im}}(\partial_{F}^{(2)})$ is the subspace $$\overline{[HS,F^\sigma]}\,,\quad F^\sigma:=\sigma(F).$$
We just need to show that $[HS,F^\sigma]$ is dense in $[HS,M']$ in the $HS$-topology. Then the algebra $A$ generated by $F$ is $\ast$-strongly dense in $M$ by hypothesis. For fixed $T \in HS$ the map $x \mapsto [T,x]$ is continuous on bounded sets, from $B$ endowed with the strong topology to $HS$. Thus $[HS,\sigma(A)]$ is dense in $[HS,M']$.
It remains to sow that $[HS,\sigma(A)]=[HS,F^\sigma]$, which follows from the indentities, $$\begin{aligned}
[T,X_1 X_2\ldots X_n] & = & [X_2\ldots X_n \,T,\,X_1] +[X_3\ldots X_n \,T\,X_1,\,X_2]\\
& + & \ldots+[X_{j+1}\ldots X_n \,T\,X_1\ldots X_{j-1},\,X_j]\\
& + & \ldots+ [T\,X_1\ldots X_{n-1},\,X_n] \end{aligned}$$ and the fact that $HS$ is a two sided ideal.
### $\Delta (F)$ and $\Delta (F:F')$.
It is thus of interest to consider the quantities:$$\begin{aligned}
\Delta (F) & = & n-\dim _{M\bar{\otimes }M^{o}}\overline{(\ker \partial _{F}^{(2)}\cap FR^{n})}=n-\dim _{M\bar{\otimes }M^{o}}\overline{\ker \partial _{F}},\label{eq:deltaF}\\
\Delta (F:F') & = & n-\dim _{M\bar{\otimes }M^{o}}\overline{\ker \partial _{F'}}\cap i_{F',F}\ker \partial _{F}^{(2)},\nonumber \\
\Delta (M,\tau ) & = & \sup _{F\textrm{ s.t. }M=W^{*}(F)}\inf _{F'\supset F}\Delta (F;F'),\nonumber \end{aligned}$$ where in the last equation we require that $F$ generates $M$.
Explicitly, if $F=(X_{1},\ldots ,X_{n})$, $F'=F\cup (Y_{1},\ldots ,Y_{m})$, we have:$$\begin{aligned}
\label{explicit}
\Delta (F) & = & n-\dim _{M\bar{\otimes }M^{o}}\overline{\{(T_{1},\ldots ,T_{n})\in FR^{n}:\sum [T_{j},X_{j}^\sigma]=0\}}^{HS},\\
\Delta (F:F') & = & n-\dim _{M\bar{\otimes }M^{o}}\Big (HS^{n}\oplus 0\ \nonumber \\
& & \cap \ \overline{\{(T_{1},\ldots ,T_{n},S_{1},\ldots ,S_{m})\in FR^{n+m}}\nonumber\\
& & \qquad \overline{:\sum [T_{j},X_{j}^\sigma]+\sum [S_{j},Y_{j}^\sigma]=0\}}^{HS}\Big ),\nonumber\end{aligned}$$ where we used the map $\Psi :M\bar{\otimes }M^{o}\to HS=HS(L^{2}(M))$. Note that the $X_j, Y_k$ are moved to the commutant $M'$ of $M$ by the map $\sigma$, as follows from (\[psit\]), it is thus clear that the subspaces of $HS^n$ involved in the above equations are $M$-bimodules. In either equation above, $FR^{n}$ can be replaced by $\Psi (M\otimes M^{o})^{n}\subset FR^{n}$.
Furthermore, we have by Lemma \[lem:kerPartial2\]:$$\beta _{1}^{(2)}(M,\tau )=\Delta (M,\tau )-(1-\beta _{0}^{(2)}(M,\tau )).\label{eq:beta1MdeltaM}$$
Note that if $F=(X_{1},\ldots ,X_{n})$, then $\partial _{F}:FR^{n}=FR\otimes {\operatorname{span}}F\to FR$ is given by$$\partial _{F}(T_{1},\ldots ,T_{n})=-\sum [T_{i},X_{i}^\sigma].$$ The transpose of $\partial _{F}$ is the map$$\partial _{F}^{\,t}:B(L^{2}(M))\to B(L^{2}(M))^{n}$$ given by$$\partial _{F}^{\,t}(D)=([D,X_{1}^\sigma],\ldots ,[D,X_{n}^\sigma]).\label{eq:partialstarF}$$ In view of Lemma \[lem:dualbetti\], we have the following description of $\Delta (F)$:$$\Delta (F)=\dim _{M\bar{\otimes }M^{o}}\overline{\partial _{F}^{\,t}(B(L^{2}(M))}^{w}\cap HS^{n}.\label{eq:Deltadualdescription}$$ Similarly, if $F'=F\cup \{Y_{1},\ldots ,Y_{m}\}$, then$$\Delta (F:F')=\dim _{M\bar{\otimes }M^{o}}\pi _{n}(\overline{\partial _{F'}^{\,t}(B(L^{2}(M))}^{w}\cap HS^{n+m}),\label{eq:dualRelDelta}$$ where $\pi _{n}:HS^{n+m}\to HS^{n}$ denotes the orthogonal projection onto the first $n$ coordinates.
Properties of $\Delta $.
------------------------
We record the following properties of $\Delta $:
Let $X_{1},\ldots ,X_{n}\in (M,\tau )$ be a fixed self-adjoint set of elements. Then we have:
\(a) $\Delta (X_{1},\ldots ,X_{n})\leq n$.
\(b) $\Delta (X_{1},\ldots ,X_{n})$ depends only on the pair $(A,\tau |_{A})$, where $A$ is the algebra generated by $X_{1},\ldots ,X_{n}$
\(c) Let $\Gamma $ be a finitely generated group, and let $X_{1},\ldots ,X_{n}\in L(\Gamma )$ be a family of unitaries associated to a symmetric set of generators of $\Gamma $. Then$$\Delta (X_{1},\ldots ,X_{n})\leq \beta _{1}^{(2)}(\Gamma )-\beta _{0}^{(1)}(\Gamma )+1.$$ If in addition $\Gamma $ is combable, we have that$$\Delta (X_{1},\ldots ,X_{n})=\beta _{1}^{(2)}(\Gamma )-\beta _{0}^{(2)}(\Gamma )+1.$$
\(d) For all $m<n$,$$\Delta (X_{1},\ldots ,X_{n})\leq \Delta (X_{1},\ldots ,X_{m})+\Delta (X_{m+1},\ldots ,X_{n}).$$
\(e) Let $1<m<n$, and assume that the families $X_{1},\ldots ,X_{m}$, $X_{m+1},\ldots ,X_{n}$ are free. Then$$\Delta (X_{1},\ldots ,X_{n})=\Delta (X_{1},\ldots ,X_{m})+\Delta (X_{m+1},\ldots ,X_{n}).$$
\(a) follows immediately from the definition of $\Delta $.
\(b) Let $A$ be the algebra generated by $X_{1},\ldots ,X_{n}$, and let $N$ be the von Neumann algebra generated by $A$ inside of $M$. Let $F=(X_{1},\ldots ,X_{n})$. By the obvious variant of Lemma \[lem:kerPartial2\] for non generating sets, we find that$$\Delta (F)=\beta _{1}^{(2)}(F)+(1-\beta _{0}^{(2)}(N)).$$ where $\beta _{1}^{(2)}(F)$ is computed inside $(M,\tau )$. Moreover, by Lemma \[lem:HofFasTor\], we have that$$\beta _{1}^{(2)}(F)=\dim _{M\bar{\otimes }M^{o}}{\operatorname{Tor}}_{1}^{M\otimes M^{o}}(M\otimes _{A}M,M\bar{\otimes }M^{o}).$$ Since the functors$$M\otimes _{N}\cdot ,\qquad \cdot \otimes _{N^{o}}M^{o},\qquad M\bar{\otimes }M^{o}\otimes _{N\bar{\otimes }N^{o}}$$ are flat [@luck:hilbertmodules], it follows that$${\operatorname{Tor}}_{1}^{M\otimes M^{o}}(M\otimes _{A}M,M\bar{\otimes }M^{o})=M\bar{\otimes }M^{o}\otimes _{N\bar{\otimes }N^{o}}{\operatorname{Tor}}_{1}^{N\otimes N^{o}}(N\otimes _{A}N,N\bar{\otimes }N^{o}).$$ Finally, since$$\dim _{M\bar{\otimes }M^{o}}(M\bar{\otimes }M^{o}\otimes _{N\bar{\otimes }N^{o}}W)=\dim _{N\bar{\otimes }N^{o}}W,$$ we find that$$\Delta (F)=\dim _{N\bar{\otimes }N^{o}}{\operatorname{Tor}}_{1}^{N\otimes N^{o}}(N\otimes _{A}N
,N\bar{\otimes }N^{o})+1-\beta _{0}^{(2)}(N),$$ which depends only on $A$ and $\tau |_{A}$.
\(c) The inequality follows from Theorem \[Prop:sameAsInGroupCase\]. The equality in the combable case follows from Theorem \[thm:combableBetti\].
\(d) Let $F_{1}=(X_{1},\ldots ,X_{m})$ and $F_{2}=(X_{m+1},\ldots ,X_{n})$, $F=F_{1}\cup F_{2}$. Let $V_{i}={\operatorname{span}}F_{i}$, $C_{i}=(M\bar{\otimes }M^{o})\otimes V_{i}$, $i=1,2$. Put $C=(M\bar{\otimes }M^{o})\otimes {\operatorname{span}}(V_{1},V_{2})$. Consider$$\partial _{F_{i}}:C_{i}\to M\otimes M^{o}$$ given by$$\partial _{F_{i}}(a\otimes b^{o}\otimes x)=a\,x\otimes b^{o}-a\otimes (x\,b)^{o}.$$ Then$$(\ker \partial _{F_{1}})\oplus (\ker \partial _{F_{2}})\subset \ker \partial _{F}\cap C.$$ Thus$$\dim _{M\bar{\otimes }M^{o}}\overline{\ker \partial _{F}}\geq \dim _{M\bar{\otimes }M^{o}}\overline{\ker \partial _{F_{1}}}+\dim _{M\bar{\otimes }M^{o}}\overline{\ker \partial _{F_{2}}}.$$ In view of (b), this implies the desired inequality for $\Delta $.
\(e) Let $M_{1}=W^{*}(X_{1},\ldots ,X_{m})$ and $M_{2}=W^{*}(X_{m+1},\ldots ,X_{n})$. By Remark 13.2(e) of [@dvv:entropy6], there exist operators $D_{1}$, $D_{2}$ in $B(L^{2}(M))$ so that $$\label{dual}
[D_{i},M_{k}]=\{0\},\quad [D_{i},m]=[m,\Psi (1\otimes 1)],$$ for all $i\neq k$ and $m\in M_{i}$. These operators are denoted by $T_{j}$ in [@dvv:entropy6] and are called a dual system to $M_{1},M_{2},\mathbb{C}1$. It is worth mentioning in conjunction with Def. 13.1 of [@dvv:entropy6] that a single algebra $A$ always has a dual system relative to $\mathbb{C}1$, namely the operator of orthogonal projection onto $\mathbb{C}1$ in $L^{2}(A)$.
One can in fact explicitly describe these operators. Denote by $H_{i}^{0}$ the space $L^{2}(M_{i})\ominus \mathbb{C}1$. Then since $M=M_{1}*M_{2}$, $$L^{2}(M)=\mathbb{C}1\oplus \bigoplus _{k}\bigoplus _{i_{1}\neq i_{2},i_{2}\neq i_{3},\ldots i_{k-1}\neq i_{k}}H_{i_{1}}^{0}\otimes \cdots \otimes H_{i_{k}}^{0}.$$ We refer the reader to [@dvv:book] for more details and the definition of the action of $M_{j}$ on this space. The operator $D_{k}$ is then given by$$D_{k}1=1$$ and$$D_{k}\xi _{1}\otimes \cdots \otimes \xi _{r}=\begin{cases} 0, & \xi _{r}\in H_{k}^{o}\\
\xi _{1}\otimes \cdots \otimes \xi _{r}, & \textrm{otherwise.}\\ \end{cases}$$
Let $F=(X_{1},\ldots ,X_{n})$. Assume now that$$\sum _{i=1}^{n}T_{i}\otimes X_{i}\in \ker \partial _{F},\qquad T_{i}\in M\otimes M^{o}.$$ Then $$\sum _{i=1}^{n}[\Psi (T_{i}),X_{i}^\sigma]=0.$$ Let $k$ be equal to $1$ or $2$, and write $I_{1}=\{1,\ldots ,m\}$, $I_{2}=\{m+1,\ldots ,n\}$. Then for all $a,b\in M$, $$\begin{aligned}
0 & = & \sum _{i}{\operatorname{Tr}}([\Psi (T_{i}),X_{i}^\sigma]\,a\, D_{k}^\sigma\, b)\\
& = & \sum _{i}{\operatorname{Tr}}(\Psi (T_{i})[X_{i}^\sigma,\,a\, D_{k}^\sigma\, b])\\
& = & \sum _{i}{\operatorname{Tr}}(\Psi (T_{i})\,a\,[X_{i}^\sigma,\, D_{k}^\sigma]\, b)\\
& = & \sum _{i\in I_{k}}{\operatorname{Tr}}(\Psi (T_{i})[\Psi (a\otimes b^{o}),X_{i}^\sigma])\\
& = & -\sum _{i\in I_{k}}{\operatorname{Tr}}([\Psi (T_{i}),X_{i}^\sigma]\Psi (a\otimes b^{o})).\end{aligned}$$ where we used (\[dual\]) and the equality $\Psi (1\otimes 1)^\sigma=\Psi (1\otimes 1)$.
It follows (since $a,b\in M$ were arbitrary) that $$\sum _{i=1}^{m}[\Psi (T_{i}),X_{i}^\sigma]=\sum _{i=m+1}^{n}[\Psi (T_{i}),X_{i}^\sigma]=0.$$
It follows that $\sum _{i=1}^{m}T_{i}\otimes X_{i}\in \ker \partial _{F}$, and $\sum _{i=m+1}^{n}T_{i}\otimes X_{i}\in \ker \partial _{F}$.
Let $F_{1}=(X_{1},\ldots ,X_{m})$, $F_{2}=(X_{m+1},\ldots ,X_{n})$. If we denote by $V_{j}$ the span of $F_{j}$ and let $V={\operatorname{span}}(V_{1},V_{2})$, then we have shown that $$\ker \partial _{F}\subset \ker \partial _{F}\cap M\otimes M^{o}\otimes V_{1}+\ker \partial _{F}\cap M\otimes M^{o}\otimes V_{2},$$ so that$$\ker \partial _{F}\subset \ker \partial _{F_{1}}+\ker \partial _{F_{2}}.$$
Thus$$\dim _{M\bar{\otimes }M^{o}}(\overline{\ker \partial _{F}})\leq \sum _{k=1}^{2}\dim _{M\bar{\otimes }M^{o}}(\overline{\ker \partial _{F_{k}}}).$$
From this and part (b) we conclude that $$\Delta (X_{1},\ldots ,X_{n})\geq \Delta (X_{1},\ldots ,X_{m})+\Delta (X_{m+1},\ldots ,X_{n}).$$
Since we have proved the opposite inequality in part (c), the desired equality now follows.
$\Delta $ and diffuse center.
-----------------------------
\[lem:diffuseCenter\]Let $F=(X,X_{1},\ldots ,X_{n})$, and assume that $[X,X_{j}]=0$ for all $j$. Assume furthermore that the spectrum of $X$ is diffuse. Then $\Delta (F)=1$.
Let $M=W^{*}(F)$. Since $\Delta (F)=\beta _{1}^{(2)}+(1-\beta _{0}^{(2)}(M))$, we find that$$\Delta (F)\geq 1-\beta _{0}^{(2)}(M).$$ Since $M$ contains a diffuse von Neumann algebra (namely, $W^{*}(X)$), it follows from Proposition \[pro:centerbetti0\] that $\beta _{0}^{(2)}(M)=0$. Thus $\Delta (F)\geq 1.$
For the opposite inequality, let $FR$ be the ideal of finite-rank operators on $L^{2}(M)$, and assume that $Q_{1},\ldots ,Q_{n}\in FR^{n}$ are arbitrary. Let$$\begin{aligned}
T_{j} & = & [Q_{j},X^\sigma],\\
T & = & -\sum _{j=1}^{n}[Q_{j},X_{j}^\sigma].\end{aligned}$$ Then since $[X_{j}^\sigma,X^\sigma]=0$ for all $j$, we have that$$\begin{aligned}
[T,X^\sigma]+\sum _{j=1}^{n}[T_{j},X_{j}^\sigma] & = &
\sum _{j=1}^{n}-[[Q_{j},X_{j}^\sigma],X^\sigma]+[[Q_{j},X^\sigma],X_{j}^\sigma]=0\,\\\end{aligned}$$ by the Jacobi identity. Thus the image of the map$$FR^{n}\ni (Q_{1},\ldots ,Q_{n})\mapsto (T,T_{1},\ldots ,T_{n})$$ lies inside $\ker \partial _{F}$ (we identify as usual $M\otimes M^{o}$ with a subset of $FR$ via the map $\Psi $; see Lemma \[lem:algsameasFR\]). It follows that the closure of $\ker \partial _{F}$ in the Hilbert-Schmidt norm contains the image of the map$$\phi :HS^{n}\ni (Q_{1},\ldots ,Q_{n})\mapsto (-\sum _{j=1}^{n}[Q_{j},X_{j}^\sigma],[Q_{1},X^\sigma],\ldots ,[Q_{n},X^\sigma]).$$ Since $X$ has diffuse spectrum, the commutant of $W^{*}(X^\sigma)$ in $B(L^{2}(M))$ does not intersect compact (and hence Hilbert-Schmidt) operators. Thus the map $\phi $ is injective. Hence by Luck’s results on additivity of dimension for weakly exact sequences [@luck:foundations1] we conclude that$$\dim _{M\bar{\otimes }M^{o}}\overline{\ker \partial _{F}}^{HS}\geq \dim _{M\bar{\otimes }M^{o}}{\operatorname{im}}\phi =\dim _{M\bar{\otimes }M^{o}}HS^{n}=n.$$ Thus$$\Delta (F)=n+1-\dim _{M\bar{\otimes }M^{o}}\overline{\ker \partial _{F}}^{HS}\leq 1.$$ We conclude that $\Delta (F)=1$.
If $(M,\tau )$ is a von Neumann algebra, and $M$ has a diffuse center, then $\beta _{1}^{(2)}(M,\tau )=0$ and $\Delta (M,\tau )=1$.
Let $X$ be a generator of the center of $M$. Then for any finite subset $F$ of self-adjoint elements of $M$, we have that$$\Delta (F\cup \{X\})=1,$$ by Lemma \[lem:diffuseCenter\]. Hence if $F$ generates $M$, $F'\supset F$ and $X\in F'$, then by (\[betafprime\]) $$\begin{aligned}
\Delta (F:F') & = & \beta _{1}^{(2)}(F:F')+(1-\beta _{0}^{(2)}(M))\\
& \leq & \beta _{1}^{(2)}(F')+(1-\beta _{0}^{(2)}(M))\\
& = & \Delta (F')=1.\end{aligned}$$
Thus for any $F$ generating $M$, we have that$$\inf _{F'\supset F}\Delta (F:F')\leq \Delta (F:F\cup \{X\})=1,$$ so that $\Delta (M,\tau )\leq 1$. Since $M$ contains a diffuse von Neumann algebra (namely, $W^{*}(X)$), it follows that $\Delta (M,\tau )=1$, and that $\beta _{1}^{(2)}(M)=0$.
$\Delta $ and free entropy dimension.
=====================================
Free entropy dimension.
-----------------------
The properties of $\Delta (X_{1},\ldots ,X_{n})$ seem very similar to those enjoyed by the various versions of Voiculescu’s free entropy dimension. We therefore are interested in connections between the two quantities.
### Non-microstates entropy dimension.
We consider the free entropy dimensions defined in terms of the non-microstates free entropy and the non-microstates free Fisher information. Let $S_{1},\ldots ,S_{n}$be a free semicircular family, free from $(X_{1},\ldots ,X_{n})$. Then set$$\delta ^{*}(X_{1},\ldots ,X_{n})=n-\liminf _{\varepsilon \to 0}\frac{\chi ^{*}(X_{1}+\sqrt{\varepsilon }S_{1},\ldots ,X_{n}+\sqrt{\varepsilon }S_{n})}{\log \varepsilon ^{1/2}}$$ and$$\delta ^{\star }(X_{1},\ldots ,X_{n})=n-\liminf _{\varepsilon \to 0}\varepsilon\, \Phi ^{*}(X_{1}+\sqrt{\varepsilon }S_{1},\ldots ,X_{n}+\sqrt{\varepsilon }S_{n}).$$ Note that $\delta ^{\star }$ is obtained from $\delta ^{*}$ by formally applying L’Hopital’s rule to the limit. We will also use the microstates free entropy dimension $\delta $ and $\delta _{0}$, which were introduced in [@dvv:entropy2; @dvv:entropy3]. Here$$\Phi ^{*}(X_{1}+\sqrt{\varepsilon }S_{1},\ldots ,X_{n}+\sqrt{\varepsilon }S_{n})=\sum _{i=1}^{n}\Vert \xi _{i}^{\varepsilon }\Vert _{2}^{2},$$ where $\xi _{i}^{\varepsilon }\in L^{2}(W^{*}(X_{1},\ldots ,X_{n}))$ are the conjugate variables$$\xi _{i}^{\varepsilon }=J(X_{i}+\sqrt{\varepsilon }S_{i}:X_{1}+\sqrt{\varepsilon }S_{1},\ldots ,\widehat{X_{i}+\sqrt{\varepsilon }S_{i}},\ldots ,X_{n}+\sqrt{\varepsilon }S_{n})$$ (here $\hat{\cdot }$ denotes omission).
Let$$E_{\varepsilon }=E_{W^{*}(X_{1}+\sqrt{\varepsilon }S_{1},\ldots ,X_{n}+\sqrt{\varepsilon }S_{n})}$$ be the unique conditional expectation. By [@dvv:entropy5], one has:$$\xi _{i}^{\varepsilon }=\frac{1}{\sqrt{\varepsilon }}E_{\varepsilon }(S_{i}).$$ Thus$$\delta ^{\star }=n-\liminf _{\varepsilon \to 0}\sum _{i=1}^{n}\Vert E_{\varepsilon }(S_{i})\Vert _{2}^{2}.$$
\[lemma:StarvsStar\]$\delta ^{\star }(X_{1},\ldots ,X_{n})\geq \delta ^{*}(X_{1},\ldots ,X_{n})$.
Assume that $\delta ^{\star }(X_{1},\ldots ,X_{n})< n-C$. Thus$$\liminf _{\varepsilon \to 0}\varepsilon \,\Phi ^{*}(X_{1}+\sqrt{\varepsilon }S_{1},\ldots ,X_{n}+\sqrt{\varepsilon }S_{n})> C.$$ Then for some $\varepsilon _{0}>0$ and all $0<\varepsilon <\varepsilon _{0}$, we have that$$\varepsilon \,\Phi ^{*}(X_{1}+\sqrt{\varepsilon }S_{1},\ldots ,X_{n}+\sqrt{\varepsilon }S_{n})\geq C,$$ so that$$\Phi ^{*}(X_{1}+\sqrt{\varepsilon }S_{1},\ldots ,X_{n}+\sqrt{\varepsilon }S_{n})\geq \frac{C}{\varepsilon }.$$ Thus for all $0<\varepsilon <\varepsilon _{0}$, $$\begin{aligned}
\frac{1}{2}\int _{\varepsilon }^{\varepsilon _{0}}\Phi ^{*}(X_{1}+\sqrt{t}S_{1},\ldots ,X_{n}+\sqrt{t}S_{n})\,dt & \geq & \frac{1}{2}\int _{\varepsilon }^{\varepsilon _{0}}\frac{C}{\ t}dt\\
& = & C(\log \varepsilon _{0}^{1/2}-\log \varepsilon ^{1/2}).\end{aligned}$$ Now $\chi _{\varepsilon }=\chi ^{*}(X_{1}+\sqrt{\varepsilon }S_{1},\ldots ,X_{n}+\sqrt{\varepsilon }S_{n})$ is given by (cf [@dvv:entropy5])$$\begin{aligned}
\chi _{\varepsilon } & =\frac{1}{2} & \int _{0}^{\infty }\left(\frac{n}{1+t}-\Phi ^{*}(X_{1}+\sqrt{t+\varepsilon }\,S_{1},\ldots ,X_{n}+\sqrt{t+\varepsilon }\,S_{n}\right)dt\\
& = & \frac{1}{2}\int _{\varepsilon }^{\infty }\left(\frac{n}{1+t-\varepsilon }-\Phi ^{*}(X_{1}+\sqrt{t}S_{1},\ldots ,X_{n}+\sqrt{t}S_{n}\right)dt\\
& \leq & K+\frac{1}{2}\int _{\varepsilon }^{\varepsilon _{0} }\left(\frac{n}{1+t-\varepsilon }-\Phi ^{*}(X_{1}+\sqrt{t}S_{1},\ldots ,X_{n}+\sqrt{t}S_{n}\right)dt,\end{aligned}$$ for some constant $K$ depending only on $\varepsilon _{0}$ and $X_{1},\ldots ,X_{n}\,.$
Thus $$\chi _{\varepsilon }\leq K+\frac{n}{2}\log \left(1+\varepsilon _{0}-\varepsilon \right)+C(\log \varepsilon ^{1/2}-\log \varepsilon _{0}^{1/2})$$ Since for small $\varepsilon $, $\log \varepsilon $ is negative, it follows that$$\liminf _{\varepsilon \to 0}\frac{\chi ^{*}(X_{1}+\sqrt{\varepsilon }S_{1},\ldots ,X_{n}+\sqrt{\varepsilon }S_{n})}{\log \varepsilon ^{1/2}}\geq \liminf _{\varepsilon \to 0}\frac{C\log \varepsilon ^{1/2}}{\log \varepsilon ^{1/2}}=C.$$ Thus $\delta ^{*}(X_{1},\ldots ,X_{n})\leq n-C$. Since $C$ is arbitrary, we get that $\delta ^{*}(X_{1},\ldots ,X_{n})\leq \delta ^{\star }(X_{1},\ldots ,X_{n})$.
The inequality $\Delta \geq \delta ^{\star }$.
----------------------------------------------
In preparation for the next theorem, we need to set up some notation.
Let $S_{1},\ldots ,S_{n}$ be a free semicircular system, free from $(X_{1},\ldots ,X_{n})$. Let $X_{j}({\varepsilon })=X_{j}+\sqrt{\varepsilon }\,S_{j}$, $j=1,\ldots ,n$. Let also $M_{\varepsilon }=W^{*}(X_{1}({\varepsilon }),\ldots ,X_{n}({\varepsilon }))$, $N=W^{*}(X_{1},\ldots ,X_{n},S_{1},\ldots ,S_{n})$, $H=L^{2}(N)$. (We recall in Appendix II the details of this standard construction (cf. [@dvv:entropy5])) Thus $M_{\varepsilon }\subset N\subset B(H)$. Let $E_{\varepsilon }$ be the orthogonal projection from $H$ onto $L^{2}(M_{\varepsilon })$. We denote by the same symbol the conditional expectation from $N$ onto $M_{\varepsilon }$. Note that $P_{1}=E_{\varepsilon }P_{1}=P_{1}E_{\varepsilon }$.
Let $D_{j}({\varepsilon })$, $j=1,\ldots ,n$, be a dual system to $(X_{1}({\varepsilon }),\ldots ,X_{n}({\varepsilon }))$ on $B(H)$. That is, we require that $$\label{dual2}
D_{j}({\varepsilon })=E_{\varepsilon }D_{j}({\varepsilon })E_{\varepsilon }\,,\quad
[D_{j}({\varepsilon }),X_{i}({\varepsilon })]=\delta _{ij}P_{1}$$ Such a system is always possible to find: one can set $$\label{dual3}
D_{j}({\varepsilon })=E_{\varepsilon }\frac{1}{\sqrt{\varepsilon }}Q_{j}E_{\varepsilon }$$ where $Q_{j}$ is a right creation operator (cf. Appendix II). Note that one has the property that $\Vert D_{j}({\varepsilon })\Vert _{\infty }\leq 1/\sqrt{\varepsilon }$.
Identify now $M\otimes M^{o}$ via the map $\Psi $ with a subspace of the space of finite-rank operators on $H$. Let $T_{1},\ldots ,T_{n}\in \Psi (M\otimes M^{o})$ be so that $\sum [T_{i},X_{i}^{\sigma}(\varepsilon)]=0$. To be explicit, let $T_{i}=\sum _{k}a_{k}^{i}\,P_{1}\,b_{k}^{i}$, $a_{k}^{i},b_{k}^{i}\in M$.
We first need a few lemmas.
\[Lemma:TandTprime\]Let $\delta >0$. Then there exist $x_{k}^{i}(\varepsilon),
y_{k}^{i}(\varepsilon)\in M_{\varepsilon }$ and $\varepsilon _{0}>0$ such that, $$\Vert T_{i}'(\varepsilon)-T_{i}\Vert _{HS}\leq \Vert T_{i}'(\varepsilon)-T_{i}\Vert _{1}<\delta \,,\quad \forall \varepsilon <\varepsilon _{0}$$ where $$T_{i}'(\varepsilon)=\sum _{k}\,x_{k}^{i}(\varepsilon)\,P_{1}\,y_{k}^{i}(\varepsilon)\,.$$
(of Lemma). It is sufficient to prove the statement for single rank-one operator $T_{a,b}=\,a\,P_{1}\,b\,$. Note that $\Vert T_{a,b}\Vert _{1}=\sup _{\Vert S\Vert _{\infty }=1}|\langle a,Sb\rangle |=\Vert a\Vert _{2}\Vert b\Vert _{2}$. We can assume that $\Vert a\Vert _{2}=\Vert b\Vert _{2}=1$. Choose non-commutative polynomials $p$ and $q$ so that$$\Vert p(X_{1},\ldots ,X_{n})-a\Vert _{2}\leq \delta /4,\ \Vert q(X_{1},\ldots ,X_{n})-b\Vert _{2}<\delta /4.$$ Let $\varepsilon _{0}>0$ so that whenever $X_{j}'\in N$ and $\Vert X_{j}-X_{j}'\Vert _{\infty }<2\sqrt{\varepsilon _{0}}$, we have$$\Vert p(X_{1},\ldots ,X_{n})-p(X_{1}',\ldots ,X_{n}')\Vert _{\infty }<\delta /4,\ \Vert q(X_{1},\ldots ,X_{n})-q(X_{1}',\ldots ,X_{n}')\Vert _{\infty }<\delta /4.$$ Set $x(\varepsilon)=p(X_{1}(\varepsilon),\ldots ,X_{n}(\varepsilon))$, $y(\varepsilon)=q(X_{1}(\varepsilon),\ldots ,X_{n}(\varepsilon))$. Let $0<\varepsilon <\varepsilon _{0}$. Then $\,\Vert \,x(\varepsilon)-a\Vert _{2}\leq \delta /2$ and $\Vert y(\varepsilon)-b\Vert _{2}\leq \delta /2$ which gives the answer.
The following lemma is implicit in [@dvv:entropy5], but we restate it for convenience.
\[lemma:DvsJ\]Let $N$ be von Neumann algebra, and let $\tau $ be a faithful normal trace on $N$. Let $H=L^{2}(N,\tau )$, and let $J$ be the Tomita conjugation associated to $N$. Denote by $P_{1}$ the orthogonal projection onto $1\in H$.
Let $Q\in B(H)$ and $Z\in N$. Then$${\operatorname{Tr}}(P_{1}[Q,Z^{*}])=\langle (JQ^{*}J-Q)1,Z1\rangle .$$
Using $\langle J \,\xi,\,\eta \rangle=\langle J \,\eta,\,\xi \rangle$ we get: $$\begin{aligned}
{\operatorname{Tr}}(P_{1}[Q,Z^{*}]) & = & \langle QZ^{*}1,1\rangle -\langle Z^{*}Q1,1\rangle \\
& = & \langle Z^{*},Q^{*}1\rangle -\langle Q1,Z\rangle \\
& = & \langle JQ^{*}J1,Z\rangle -\langle Q1,Z\rangle \\
& = & \langle (JQ^{*}J-Q)1,Z\rangle ,\end{aligned}$$ which is the desired identity.
\[thm:DeltaGEQdeltastar\]$\Delta (X_{1},\ldots ,X_{n})\geq \delta ^{\star }(X_{1},\ldots ,X_{n}).$
Let $T_{1}^{j},\ldots ,T_{n}^{j}$, $j=1,\ldots ,n$ be in $\Psi (M\otimes M^{o})$ and such that $\sum _{i}[T_{i}^{j},X_{i}^\sigma]=0$ for all $j$. Let $$T_{i}^{j}=\sum _{k}\,a_{k}^{ij}\,P_{1}\,b_{k}^{ij}\,.$$ Then using (\[dual2\]), $\sum _{i}[T_{i}^{j},X_{i}^\sigma]=0$, and $\sigma(P_{1})=P_{1}$, we get, $$\begin{aligned}
\sum _{j}\langle T_{j}^{j},P_{1}\rangle & = & \sum _{ij}{\operatorname{Tr}}(T_{i}^{j}\,[X_{i}^\sigma(\varepsilon),D^\sigma_{j}(\varepsilon)])\\
& = & \sum _{ij}{\operatorname{Tr}}([T_{i}^{j},\,X_{i}^\sigma(\varepsilon)]D^\sigma_{j}(\varepsilon))\\
& = & \sum _{ij}{\operatorname{Tr}}([T_{i}^{j},\,\sqrt{\varepsilon }\,S_{i}^\sigma]\,D^\sigma_{j}(\varepsilon))\\
& = & \sum _{ij}{\operatorname{Tr}}(T_{i}^{j}[S_{i}^\sigma,\,\sqrt{\varepsilon }\,D^\sigma_{j}(\varepsilon)]).\end{aligned}$$ Now, fix $\kappa >0$ and let $\delta =\kappa /8n^{2}$. Choose $\varepsilon _{0}$ and $T_{ij}'(\varepsilon)$ as in Lemma \[Lemma:TandTprime\], so that $$\label{approx}
\Vert T_{i}^{j}-T_{ij}'(\varepsilon)\Vert _{1}<\delta .$$ Then for all $\varepsilon <\varepsilon _{0}$, since $\Vert [\sqrt{\varepsilon }D_{j}(\varepsilon),S_{i}]\Vert _{\infty }\leq 4$, we find that $$\label{inf1}
\vert\sum _{j}\langle T_{j}^{j},P_{1}\rangle \vert \leq \kappa /2+\vert\sum _{ij}{\operatorname{Tr}}(T_{ij}'(\varepsilon)[\sqrt{\varepsilon }\,D_{j}^\sigma(\varepsilon),S_{i}^\sigma])\vert .$$ Since $$T_{ij}'(\varepsilon)=\sum \,x_{k}^{ij}(\varepsilon)\,P_{1}\,y_{k}^{ij}(\varepsilon)
\,,\qquad x_{k}^{ij}(\varepsilon),y_{k}^{ij}(\varepsilon)\in M_{\varepsilon }$$ and $P_{1}=E_{\varepsilon }P_{1}E_{\varepsilon }$, we have that $$T_{ij}'(\varepsilon)
=E_{\varepsilon }T_{ij}'(\varepsilon)E_{\varepsilon }.$$ Let $\xi _{i}(\varepsilon)=E_{\varepsilon }(\frac{1}{\sqrt{\varepsilon }}S_{i})$ (cf.[@dvv:entropy5]) then $\xi _{i}(\varepsilon)\in M_{\varepsilon }$ and $\Vert \sqrt{\varepsilon }\,\xi _{i}(\varepsilon)\Vert _{\infty }\leq \Vert S_{i}\Vert _{\infty }\leq 2$, in fact $$E_{\varepsilon }S_{i} E_{\varepsilon }= \sqrt{\varepsilon }\,\xi _{i}(\varepsilon)$$ Note that $E_{\varepsilon }^\sigma=E_{\varepsilon }$ and $\sqrt{\varepsilon }\,D_{j}^\sigma(\varepsilon)=E_{\varepsilon } Q_{j}^\sigma E_{\varepsilon }$ by (\[dual3\]). One has$$\begin{aligned}
\label{inf2}
\sum _{ij}{\operatorname{Tr}}(T_{ij}'(\varepsilon)[\sqrt{\varepsilon }\,D_{j}^\sigma(\varepsilon),S_{i}^\sigma])& = & \sum _{ij}{\operatorname{Tr}}(T_{ij}'(\varepsilon)[Q_{j}^\sigma,E_{\varepsilon }S_{i}^\sigma E_{\varepsilon }])\\
& = & \sum _{ij}{\operatorname{Tr}}(T_{ij}'(\varepsilon)[Q_{j}^\sigma,\sqrt{\varepsilon }\,\xi _{i}(\varepsilon)^\sigma
])\nonumber \end{aligned}$$ We thus get, using (\[approx\]), the fact that $\,a_{k}^{ij}\,$, $\,b_{k}^{ij}\,$ commute with $\xi _{i}(\varepsilon)^\sigma$ and the inequality $\Vert \sqrt{\varepsilon }\, \xi _{i}(\varepsilon)\Vert _{\infty }\leq \Vert S_{i}\Vert _{\infty }\leq 2$:$$\begin{aligned}
\vert\sum _{ij}{\operatorname{Tr}}(T_{ij}'(\varepsilon)[Q_{j}^\sigma,\sqrt{\varepsilon }\,\xi _{i}(\varepsilon)^\sigma
])\vert
& \leq & \kappa/2 +|\sum _{ij}{\operatorname{Tr}}(T_{i}^{j}
[Q_{j}^\sigma,\sqrt{\varepsilon }\,\xi _{i}(\varepsilon)^\sigma
])\\
& = & \kappa/2 +|\sum _{ij}{\operatorname{Tr}}(P_{1}\ [\sum _{k}\,b_{k}^{ij}\,Q_{j}^\sigma\,a_{k}^{ij}\,,\ \sqrt{\varepsilon }\,\xi _{i}(\varepsilon)^\sigma
])|\\
& = & \kappa/2 +|\sum _{ij}\langle (Y_{ij}-JY_{ij}^{*}J)1,
\sqrt{\varepsilon }\,\xi _{i}(\varepsilon)^\sigma
\rangle _{H}|,\end{aligned}$$ where $Y_{ij}=\sum _{k}\,b_{k}^{ij}\,Q_{j}^\sigma\,a_{k}^{ij}\,$, and the last equality is by Lemma \[lemma:DvsJ\].
Combining this with (\[inf1\]) and (\[inf2\]) we get $$\label{inf3}
\vert\sum _{j}\langle T_{j}^{j},P_{1}\rangle \vert \leq \kappa +|\sum _{ij}\langle (Y_{ij}-JY_{ij}^{*}J)1,
\sqrt{\varepsilon }\,\xi _{i}(\varepsilon)^\sigma
\rangle _{H}|.$$
Let $\eta _{ij}=(Y_{ij}-JY_{ij}^{*}J)1 \in H$. Computing explicitly, we get that (cf. Appendix II) $$\label{comp}
\eta _{ij}=-\sum _{k}b_{k}^{ij}S_{j}a_{k}^{ij}$$ and in particular $\Vert \sum _{j}\eta _{ij}\Vert _{2}^{2}=\sum _{j}\Vert T_{i}^{j}\Vert _{2}^{2}$, since the subspaces $MS_{j}M$ are orthogonal for $j=1,\ldots ,n$, and the map $\sum x_{k}\otimes y_{k}\to \sum x_{k}S_{j}y_{k}$ is an isometry from $L^{2}(M)\bar{\otimes }L^{2}(M)$ into $L^{2}(N)$, for each $j$ (cf. Appendix II). We thus conclude that$$\begin{aligned}
|\sum _{j}\langle T_{j}^{j},P_{1}\rangle | & \leq & \kappa +|\sum _{i}\langle \sum _{j}\eta _{ij},\sqrt{\varepsilon }\,\xi _{i}(\varepsilon)^\sigma
\rangle |\\
& \leq & \kappa +(\sum _{i}\Vert \sum _{j}\eta _{ij}\Vert ^{2})^{1/2}(\varepsilon \sum _{i}\Vert \xi _{i}(\varepsilon)\Vert ^{2})^{1/2}\\
& = & \kappa +(\sum _{ij}\Vert T_{i}^{j}\Vert _{2}^{2})^{1/2}(\varepsilon \Phi ^{*}(X_{1}(\varepsilon),\ldots ,X_{n}(\varepsilon)))^{1/2}.\end{aligned}$$ since the free Fisher information is defined as $\Phi ^{*}(X_{1}(\varepsilon),\ldots ,X_{n}(\varepsilon))=\sum _{i=1}^{n}\Vert \xi _{i}(\varepsilon)\Vert _{2}^{2}$. Passing to $\liminf _{\varepsilon \to 0}$ and noticing that $\kappa $ is arbitrary finally gives us: $$\label{est}
|\sum _{j}\langle T_{j}^{j},P_{1}\rangle |\leq (n-\delta ^{\star }(X_{1},\ldots ,X_{n}))^{1/2}\left(\sum _{ij}\Vert T_{i}^{j}\Vert _{2}^{2}\right)^{1/2}.$$
The conclusion of the proof of the theorem now follows from the next lemma (cf. [@shlyakht:qdim Lemma 2.9]) applied to the von Neumann algebra $M\bar{\otimes }M^{o}$ and the subspace $K$ of $L^{2}(M\bar{\otimes }M^{o})^n=HS^n$ closure of the space $\{(T_{1},\ldots ,T_{n})\in \Psi (M\otimes M^{o})^{n}:\sum [T_{i},X_{i}^\sigma]=0\}$.
\[Lemma:distAndDim\]Let $N$ be a finite von Neumann algebra with a faithful normal trace $\tau $. Let $n$ be a finite integer, and let $H=L^{2}(N,\tau )^{n}$ viewed as a left module over $N$. Denote by $\Omega \in L^{2}(N,\tau )$ the GNS vector associated to $\tau $.
Let $K\subset H$ be a closed $N$-invariant subspace of $H$. Endow $M_{n\times n}(L^{2}(N))$ with the norm$$\Vert h\Vert _{M_{n}}^{2}=\sum _{ij=1}^{n}\Vert h_{ij}\Vert ^{2}.$$ Let $A(K)=\{T\in M_{n\times n}(N):TH\subset K\}\cong K^{n}$. Then we have:$$\dim _{N}K={\rm Sup}\vert\langle T, I\rangle \vert^2 /\Vert T \Vert^2\,,\quad T \in A(K),$$ where $I\in M_{n}(H)$ denotes the matrix $I_{ij}=\delta _{ij}\Omega $
(of Lemma). We identify the commutant $N'$ of $N$ acting on $H$ with the algebra of $n\times n$ matrices $M_{n}(N)$. Endow this algebra with the non-normalized trace ${\operatorname{Tr}}$, defined by the property that ${\operatorname{Tr}}(1)=n$, where $1\in M_{n}(N)$ denote the identity matrix. Let $e_{K}\in N'$ be the orthogonal projection from $H$ onto $K$. Then$$\dim _{N}K={\operatorname{Tr}}(e_{K}).$$ Now, $L^{2}(M_{n}(N),{\operatorname{Tr}})=M_{n}(H)$ isometrically. Moreover, the orthogonal projection of $I$ onto $A(K)$ is $e_{K}\in A(K)$, since $1-e_{K}$ is orthogonal to $A(K)=e_{K}\,M_{n}(N)$. The above supremum is thus reached for $T=e_{K}$ and its value is ${\operatorname{Tr}}(e_{K})$ which gives the result.
### Some consequences for $\Delta $.
We have$$\Delta (X_{1},\ldots ,X_{n})\geq \delta ^{\star }(X_{1},\ldots ,X_{n})\geq \delta ^{*}(X_{1},\ldots ,X_{n})\geq \delta (X_{1},\ldots ,X_{n})\geq \delta _{0}(X_{1},\ldots ,X_{n}).$$
This is immediate from the preceding discussion and the work of Biane, Guionnet and Capitaine [@guionnet-biane-capitaine:largedeviations].
The following corollary gives a strong indication that the first $L^{2}$-Betti number of a free group factor does not vanish (compare with equations (\[eq:beta1MdeltaM\]) and (\[eq:deltaF\])).
Let $F=(X_{1},\ldots ,X_{n})$ be a self-adjoint finite subset of $M$, and assume that $F$ generates $M$.
Assume that the microstates free entropy $\chi (X_{1},\ldots ,X_{n})$ is finite.
Then for any seld-adjoint subset $F'$ of $M$, we have$$\Delta (F\cup F')\geq n.$$
Let $F'=(Y_{1},\ldots ,Y_{n})$. Then$$\Delta (F\cup F')\geq \delta (X_{1},\ldots ,X_{n},Y_{1},\ldots ,Y_{m})\geq n$$ where the second inequality follows from [@dvv:entropy2].
It is of course of interest if one has $\Delta =\delta ^{\star }$. In conjunction with this, we note the following. Let $F=(X_{1},\ldots ,X_{n})$ be a finite self-adjoint subset of $M$. Consider as in (\[eq:partialstarF\])$$\partial _{F}^{\,t}:B(L^{2}(M))\to B(L^{2}(M))^{n}$$ given by$$\partial _{F}^{\,t}(D)=([D,X_{1}^\sigma],\ldots ,[D,X_{n}^\sigma]).$$
One has$$\dim _{M\bar{\otimes }M^{o}}\overline{\partial _{F}^{\,t}(B(L^{2}(M))\cap HS^{n}}^{HS}\leq \delta ^{*}(X_{1},\ldots ,X_{n})$$ $$\begin{aligned}
\delta ^{\star }(X_{1},\ldots ,X_{n}) & \leq & \dim _{M\bar{\otimes }M^{o}}\overline{\partial _{F}^{\,t}(B(L^{2}(M))}^{w}\cap HS^{n}=\Delta (F).\end{aligned}$$
The first inequality is the statement of Corollary 2.12 in [@shlyakht:qdim]. The second inequality is the statement of Theorem \[thm:DeltaGEQdeltastar\], together with the “dual” description of $\Delta (F)$ given in equation (\[eq:Deltadualdescription\]).
### Some consequences for free entropy dimension.
Let $C(\Gamma)$ be the cost of a discrete group $\Gamma$ in the sense of [@gaboriau:cost].
Let $\Gamma $ be a finitely generated group with a symmetric set of generators $\gamma _{1},\ldots ,\gamma _{n}$. Denote by $u_{i}=\lambda (\gamma _{i})\in L(\Gamma )$ the corresponding unitaries in the left regular representation. Let $X_{i}=u_{i}+u_{i}^{*}$, $Y_{i}=i(u_{i}-u_{i}^{*})$. Then$$\delta ^{*}(X_{1},\ldots ,X_{n},Y_{1},\ldots ,Y_{n})\leq \beta _{1}^{(2)}(\Gamma )-\beta _{0}^{(2)}(\Gamma )+1 \leq C(\Gamma).$$
The inequality between the cost and $\beta_1 - beta_0 + 1$ is due to Gaboriau [@gaboriau:ell2]. The rest of the inequalities follow immediately from the corresponding estimate for $\Delta $.
Let $\Gamma $ be a discrete group with Kazhdan’s property (T). Let $\gamma _{1},\ldots ,\gamma _{n}$ be a symmetric set of generators of $\Gamma $, and let $u_{j}=\lambda (\gamma _{j})\in L(\Gamma )$ be the associated unitaries in the left regular representation. Let $X_{i}=u_{i}+u_{i}^{*}$, $Y_{i}=i(u_{i}-u_{i}^{*})$. Then$$\delta _{0}(\Gamma )\leq \delta (X_{1},\ldots ,X_{n},Y_{1},\ldots ,Y_{n})\leq \delta ^{\star }(X_{1},\ldots ,X_{n},Y_{1},\ldots ,Y_{n})\leq 1.$$ If moreover $L(\Gamma )$ is diffuse, one has$$\delta ^{*}(X_{1},\ldots ,X_{n},Y_{1},\ldots ,Y_{n})=\delta ^{\star }(X_{1},\ldots ,X_{n},Y_{1},\ldots ,Y_{n})=1.$$ If $L(\Gamma )$ is diffuse and moreover $L(\Gamma )$ can be embedded into the ultrapower of the hyperfinite II$_{1}$-factor, one has$$\delta _{0}(\Gamma )=1.$$
The upper estimates are a consequence of the fact that if $\Gamma $ has property (T), then $\beta _{1}^{(2)}(\Gamma )=0$ (see e.g. [@cheeger-gromov:l2; @bekka-valette:l2cohomology]). Thus$$\Delta (\Gamma )\leq \beta _{1}^{(2)}(\Gamma )-\beta _{0}^{(2)}(\Gamma )+1=1-\beta _{0}^{(2)}(\Gamma )\leq 1.$$ The lower estimate for $\delta ^{*}$ is a consequence of [@shlyakht:qdim Theorem 2.13]. The corresponding estimate for $\delta _{0}$ is a consequence of hyperfinite monotonicity of [@jung-freexentropy].
$\Delta (F)$ and $\Delta (F:F')$.
---------------------------------
The results of the previous section are insufficient to give a lower bound for $\Delta (M,\tau )$ and thus for $\beta _{1}^{(2)}(M,\tau )$. We show, however, that under certain smoothness conditions on the families $F$ and $F'$, $\Delta (F,F')\geq \Delta (F)$.
Let $F=(X_{1},\ldots ,X_{n})$ be a self-adjoint family of generators of $M$ and let $F'=F\cup (Y_{1},\ldots ,Y_{m})$. Let $D_{1},\ldots ,D_{n}$ be a dual system to $X_{1},\ldots ,X_{n}$ in the sense of [@dvv:entropy5]; thus $D_{j}\in B(L^{2}(M))$ satisfy$$[D_{j},X_{i}]=\delta _{ji}P_{1},$$ where $P_{1}$ denotes the projection onto the trace vector in $L^{2}(M)$. Assume that $[D_{j},Y_{i}]$ is a Hilbert-Schmidt operator for all $i$ and $j$. Then$$\Delta (X_{1},\ldots ,X_{n}:X_{1},\ldots ,X_{n},Y_{1},\ldots ,Y_{m})=\Delta (X_{1},\ldots ,X_{n})=n.$$
Note that for each $j$, the $n+m$-tuple$$(0,\ldots ,P_{1},\ldots ,0,[Y_{1},D_{j}]^\sigma,\ldots ,[Y_{m},D_{j}]^\sigma)$$ ($P_{1}$ in the $j$-th place) lies in $\partial _{F\cup F'}^{\,t}(B(L^{2}(M)))$, in the notation of (\[eq:partialstarF\]). Thus$$\xi _{j}=(0,\ldots ,P_{1},\ldots ,0)$$ ($P_{1}$ in the $j$-th place) lies in$$K=\pi _{n}(\overline{\partial _{F\cup F'}^{\,t}(B(L^{2}(M))}^{w}\cap HS^{n+m}),$$ in the notation of (\[eq:dualRelDelta\]). Since $(\xi _{1},\ldots ,\xi _{n})$ clearly densely generate $HS^{n}$ as an $M,M$-bimodule, it follows that the dimension of $K$ over $M\bar{\otimes }M^{o}$ is exactly $n$. Thus $\Delta (F:F')=n$. Applying this to the case that $m=0$ gives also the estimate for $\Delta (F:F)=\Delta (F)$.
An important case of existence of a dual system is when $X_{1},\ldots ,X_{n}$ are free semicircular variables; see [@dvv:entropy5].
Appendix I: Abelian von Neumann algebras.
=========================================
The following theorem is the analog of [@luck:foundations1 Theorem 5.1], which makes one suspect that its statement should hold more generally if $A$ is hyperfinite. We were unable to prove this, however. If the statement holds for $A$ hyperfinite, it would be interesting if it can be used as a characterization of hyperfinite algebras (see Remark 5.13 in [@luck:foundations1]).
\[pro:TorAvanishes\]Let $A$ be a commutative von Neumann algebra and $\tau$ a normal faithful trace on $A$.
\(i) Let $f:(A\otimes A^{o})^{n}\rightarrow (A\otimes A^{o})^{m}$ be a left $A\otimes A^{o}$-module map, then $\beta ^{(2)}(f)=0$.
\(ii) Let $W$ be an arbitrary $A\otimes A^{o}$-module. Then for all $p\geq 1$,$$\dim _{A\bar{\otimes }A^{o}}{\operatorname{Tor}}_{p}^{A\otimes A^{o}}(W,\,A\bar{\otimes }A^{o})=0,$$
Let us first prove a simple lemma,
\[sup\] Let $f\in A\otimes A^{o}$, then the spectral projection $p$ of $f^*\,f$ corresponding to $\ker f$ is the supremum of the projections $e \leq p : e \in A\otimes A^{o}$.
We can assume $A$ is diffuse and identify $A$ with $L^{\infty }([0,1])$, $\tau$ with the Lebesgue measure $\lambda$ and $L^{2}(A)$ with $L^{2}([0,1])$. We drop the distinction between $A$ and $A^{o}$.
Let $f=\sum g_{i}\otimes h_{i}$ and consider $f$ as the function$$f(x,y)=\sum _{i=1}^{k}g_{i}(x)\,h_{i}(y)\,,\quad x\,,y\in [0,1].$$ Then the projection $p\in A\bar{\otimes }A^{o}=L^{\infty }([0,1]\times [0,1])$ is given by the zero set$$Z=\{(x,y):f(x,y)=0\}.$$ Recall that a point $z\in Z$ is called a point of density of $Z$ if the proportion of $Z$ in squares $S= I\times J$, ($I$ and $J$ intervals of equal length) with center $z$ tends to $1$ when their size tends to $0$. By Lebesgue’s a.e. differentiability theorem, the set $Z$ differs by a set of measure zero from its set of points of density. We thus only need to prove the following, with $k$ as above,
Let $I,J\subset [0,1]$ be intervals, and $\delta<k^{-1}$ with $$\frac{\lambda ^{\times 2}((I\times J)\cap Z)}{\lambda ^{\times 2}(I\times J)}>1-\delta^{2}.$$ Then there are measurable subsets $E\subset I$, $F\subset J$, such that $E\times F\subset Z$ and$$\frac{\lambda ^{\times 2}(E\times F)}{\lambda (I\times J)}\geq (1-k\,\delta )^{2}.$$
To prove the claim, let $g:I\rightarrow \mathbb{C}^{k}$, $h:J\rightarrow \mathbb{C}^{k}$ be given by $
(g(x))_{i}=g_{i}(x)$, $ h(x))_{j}=h_{j}(x)$ so that,$$f(x,y)=g(x)\cdot h(y)\,,\quad \forall x\,,y\in [0,1],$$ where $\cdot $ denotes the standard scalar product on $\mathbb{C}^{k}$.
Let $$E:= \{x\in I :\lambda \{y:(x,y)\in Z\}>(1-\delta) \lambda (J)\}.$$ Then by Fubini’s Theorem $\lambda (E)>(1-\delta) \lambda (I)$. Denote by $V(x)$ the subspace of $\mathbb{C}^{k}$ spanned by $g(x)$. Let $V={\operatorname{span}}(V(x):x\in E)$. Since the dimension of $V$ is at most $k$, we can choose $x_{1},\ldots ,x_{l}\in E$, $l\leq k$, so that $V={\operatorname{span}}(g(x_{1}),\ldots ,g(x_{l}))$. For each $1\leq j\leq l$, the set $F_{j}$ of $y\in J$ for which $h(y)$ is perpendicular to $g(x_{j})$ (i.e. $(x_{j},y)\in Z$) has measure at least $(1-\delta )\lambda (J)$. Thus the measure of $F=\bigcap F_{j}$ is at least $(1-l\delta )\lambda (J)\geq (1-k\,\delta)\lambda (J)$. But then for all $y\in F$ and $x\in E$, $g(x)\in V$ and $h(y)\perp V$, so that $f(x,y)=0$. It follows that $E\times F\subset Z$.
(of Theorem \[pro:TorAvanishes\]). First the above lemma implies (i) for $n=m=1$. Indeed let $f\in A\otimes A^{o}$, and $p$ the spectral projection of $f^*\,f$ corresponding to $\ker f$. The subspace $(A\otimes A^{o}) \cdot p $ is dense in $\ker f^{(2)}$. Thus since $p$ is a strong limit of projections $e_j \in A\otimes A^{o}$, $p\,e_j=e_j\, p=e_j$, one gets $(A\otimes A^{o})\cdot e_j\subset \ker(f)$ and the required density of $\ker f$ in $\ker f^{(2)}$.
Let now $n$ and $m$ be arbitrary, and reduce to $n=m$ by e.g. replacing $f$ with $f^{*}f$. Let $F(x,y)$ be the matrix with entries $f_{ij}(x,y)$ and,
$$F{i_{1}\, \, i_{2}\, \, \cdots \, \, i_{k} \choose j_{1}\, \, j_{2}\, \, \cdots \, \, j_{k}}$$ the $k\times k$ minor of $F$ obtained by keeping the $i_{1},\ldots ,i_{k}$-th rows and $j_{1},\ldots ,j_{k}$-th columns of $F$. Let $Z{i_{1}\cdots i_{k} \choose j_{1}\cdots j_{k}}$ be the zero set of $F{i_{1}\cdots i_{k} \choose j_{1}\cdots j_{k}}$. Since $F{i_{1}\cdots i_{k} \choose j_{1}\cdots j_{k}}$ is a polynomial expression in the entries of $F$, it belongs to $A\otimes A^{o}$.
Let $t\in [0,1]^{2}$ be such that the minors $F{1 \choose 1}$, $\ldots $, $F{1\cdots r \choose 1\cdots r}$ are all non-zero, while the minors $F{1\cdots r+1 \choose 1\cdots r+1}$, $\ldots $, $F{1\cdots n \choose 1\cdots n}$ are zero.
In this case, the equation $F\xi =0$, $\xi =(\xi _{1},\ldots ,\xi _{n})$ after performing Gaussian elimination has the form$$\begin{aligned}
a_{11}^{(0)}\xi _{1}+a_{12}^{(0)}\xi _{2}+\cdots +\cdots +a_{1n}^{(0)}\xi _{n} & = & 0\\
a_{22}^{(1)}\xi _{2}+\cdots +\cdots +a_{2n}^{(1)}\xi _{n} & = & 0\\
\cdots & & \\
a_{rr}^{(r-1)}\xi _{r}+\cdots +a_{rn}^{(r-1)}\xi _{n} & = & 0\end{aligned}$$ where$$a_{ik}^{(p)}=\frac{F{1\cdots p\, \, i \choose 1\cdots p\, \, k}}{F{1\cdots p \choose 1\cdots p}}$$ (see pp. 24–25 in [@gantmaher]). Thus a basis for the null space of $F$ consists of the vectors $\eta ^{(k,r)}$, $k=1,\ldots ,n-r$, with coordinates$$\begin{aligned}
\eta _{t}^{(k,r)} & = & 0,\quad t>r,\, \, t\ne k+r\\
\eta _{r}^{(k,r)} & = & -\frac{a_{rk}^{(r-1)}}{a_{rr}^{(r-1)}},\\
\eta _{r-1}^{(k,r)} & = & -\frac{a_{r-1\, \, k}^{(r-2)}+a_{r-1\, \, r}^{(r-2)}\eta _{r}^{(k)}}{a_{r-1\, \, r-1}^{(r-2)}}\\
& \cdots & \\
\eta _{1}^{(k,r)} & = & -\frac{a_{1k}^{(0)}+a_{12}^{(0)}\eta _{2}^{(k)}+\cdots +a_{1r}^{(0)}\eta _{r}^{(k)}}{a_{11}^{(0)}}.\end{aligned}$$ If we set $\xi ^{(k,r)}$ to be the product of $\eta ^{(k,r)}$ by a sufficiently high power of the (nonzero) expression$$F{1 \choose 1}\cdots F{1\cdots r \choose 1\cdots r},$$ we get that for each $k$, the $\xi _{j}^{(k,r)}$ are polynomials in the entries of $F$, and the vectors $\xi^{(k,r)}$ span the kernel of $F$.
The polynomial expressions $\xi _{j}^{(k,r)}$ in the entries of $F$ make sense without any assumptions on $F$. If $F{1\cdots r+1 \choose 1\cdots r+1}$, $\ldots $, $F{1\cdots n \choose 1\cdots n}$ are zero, the vectors $\xi ^{(k,r)}$ lie in the kernel of $F$ (although they will no longer span the kernel unless $F{1 \choose 1}$, $\ldots $, $F{1\cdots r \choose 1\cdots r}$ are all nonzero).
By construction $\xi _{j}^{(k,r)}\in A\otimes A^{o}$ (as polynomial functions in $F$). Since $\xi ^{(k,r)}\in \ker F(z)$ for all $z$ such that $F{1\cdots r+1 \choose 1\cdots r+1}$, $\ldots $, $F{1\cdots n \choose 1\cdots n}$ are zero, using Lemma \[sup\] we therefore get that, $$\zeta ^{(k,r)}(F)=\xi ^{(k,r)}\chi _{\left\{ z:F{1\cdots r+1 \choose 1\cdots r+1}(z)=\cdots =F{1\cdots n \choose 1\cdots n}(z)=0\right\} }\in \overline{\ker f}$$ (here $\chi $ denotes the characteristic function of the given set).
Applying this result to the matrix $F^{\sigma ,\sigma '}$ obtained from $F$ by permuting rows via a permutation $\sigma $ and columns via a permutation $\sigma '$, we obtain vectors $$\zeta ^{(k,r,\sigma ,\sigma ')}=\sigma ^{-1}(\xi ^{(k,r)}(F^{\sigma ,\sigma '}))
\in \overline{\ker f}\,.$$
For each $z$, let $r$ be the rank of $F(z)$, we can find $\sigma ,\sigma '$ so that the ${1 \choose 1}$, $\ldots $, ${1\cdots r \choose 1\cdots r}$-minors of $F^{\sigma ,\sigma '}$ are non-zero and the ${1\cdots r+1 \choose 1\cdots r+1}$, $\ldots $, ${1\cdots n \choose 1\cdots n}$ minors are zero. Thus $\{\zeta ^{(k,r,\sigma ,\sigma ')}(z):1\leq k\leq n-r\}$ (and hence $\{\zeta ^{(k,r,\sigma ,\sigma '}(z)\}_{k,r,\sigma ,\sigma '}$ span the kernel of $F$ at $z$. It thus follows that $\ker f$ is dense in $\ker f^{(2)}$ which proves (i).\
Finally the proof of (ii) follows verbatim the argument of Luck (Th. 5.1 [@luck:foundations1]).
Let $A$ be an abelian von Neumann algebra. Then for all $k\geq 1$,$$\beta _{k}^{(2)}(A)=0.$$
By definition,$$\beta _{k}^{(2)}=\dim _{A\bar{\otimes }A^{o}}{\operatorname{Tor}}_{k}^{A\otimes A^{o}}(A,A\bar{\otimes }A^{o}),$$ which is zero by the main result of this section.
Appendix II: Dual Systems.
==========================
We recall in this appendix the construction of the dual system in the framework of section 4, and give the details of the proof of (\[comp\]).
Let $M$ be the von Neumann algebra generated by $X_{1},\ldots ,X_{n}$, and let $\Omega \in L^{2}(M)$ be the trace vector. We start by explicitly constructing the standard form of the von Neumann algebra $N$ obtained by adjoining the free semicircular variables $S_{1},\ldots ,S_{n}$.\
Consider the vector space $V=L^{2}(M)\otimes L^{2}(M)\oplus \cdots \oplus L^{2}(M)\otimes L^{2}(M)$ $=(L^{2}(M)\otimes L^{2}(M))^n$, and let$$H=L^{2}(M)\oplus V\oplus (V\otimes _{M}V)\oplus (V\otimes _{M}V\otimes _{M}V)\oplus \cdots$$
Note that $V\otimes _{M}V\cong (L^{2}(M)\otimes L^{2}(M)\otimes L^{2}(M))^{n^{2}}$.
Then $M$ acts on $H$ both on the right and on the left in the obvious way, acting on the leftmost or rightmost tensor copy of $V$ each time. Denote by $\phi _{i}$ the inclusion map from $L^{2}(M)\otimes L^{2}(M)$ into $V$, which places $L^{2}(M)\otimes L^{2}(M)$ as the $i$-th direct summand.
Let $\omega _{i}$ be the $i$-th copy of $1\otimes 1$ in $V$. Denote by $L_{i}$ and $R_{i}$ the following operators on $H$:$$\begin{aligned}
L_{i}m & = & \phi _{i}(1\otimes m)\,,\quad \forall m \in L^{2}(M)\\
L_{i}v_{1}\otimes \cdots \otimes v_{n} & = & \omega _{i}\otimes v_{1}\otimes \cdots \otimes v_{n}
\,,\quad \forall v_j \in V\\
R_{i}m & = & \phi _{i}(m\otimes 1)\,,\quad \forall m \in L^{2}(M)\\
R_{i}v_{1}\otimes \cdots \otimes v_{n} & = & v_{1}\otimes \cdots \otimes v_{n}\otimes \omega _{i}
\,,\quad \forall v_j \in V.\end{aligned}$$ Formally, these are the left and right tensor multiplications by $\omega _{i}$.
It is not hard to check that if we denote by $\lambda $ the left action of $M$ on $H$, then we have$$L_{i}^{*}\lambda (m)L_{j}=\delta _{ij}\tau (m),\qquad \forall m\in M.\label{eq:Ls}$$ Similarly, if we denote by $\rho $ the right action of $M$ on $H$, then we have$$R_{i}^{*}\rho (m)R_{j}=\delta _{ij}\tau (m),\qquad \forall m\in M.$$ In particular, $L_{i}^{*}L_{i}=R_{i}^{*}R_{i}=1$, and these operators have norm one.
Furthermore,$$[R_{i},\lambda (m)]=[R_{i}^{*},\lambda (m)]=0,\qquad \forall m\in M.$$ Consider on $B(H)$ the vector state $\psi =\langle \Omega ,\cdot \,\Omega \rangle $, where $\Omega \in L^{2}(M)$ is regarded as a vector in $H$. By a result from [@shlyakht:freeness] it follows that if we let$$S_{i}=L_{i}+L_{i}^{*},$$ then $S_{1},\ldots ,S_{n}$ are a family of free semicircular variables, free in $(B(H),\psi )$ from $\lambda (M)$. Furthermore, the von Neumann algebra generated by $M$ (which we identify with $\lambda (M)$) and $S_{1},\ldots ,S_{n}$ is in standard form, and the operator $J$ is given by$$\begin{aligned}
J(v_{1}\otimes \cdots \otimes v_{n}) & = & v_{n}^{s }\otimes \cdots \otimes v_{1}^{s },\\
J|_{L^{2}(M)} & = & J_{M},\end{aligned}$$ where $s (x\otimes y)=Jy\otimes Jx$, and $J_{M}$is the Tomita conjugation associated to $M$.
Moreover, one has$$[R_{i},S_{j}]=\delta _{ij}P_{\Omega },$$ which means that $Q_{j}=R_{j}$ is the desired conjugate system to $S_{1},\ldots ,S_{n}$ relative to $X_{1},\ldots ,X_{n}$, i.e., it satisfies:$$[R_{i},X_{j}]=0,\qquad [R_{i},S_{j}]=\delta _{ij}P_{\Omega }.$$ This way, if we set $D_{j}(\varepsilon)=E_{W^{*}(X_{1}(\varepsilon),\ldots ,X_{n}(\varepsilon))}\frac{1}{\sqrt{\varepsilon}}Q_{j}E_{W^{*}(X_{1}(\varepsilon),\ldots ,X_{n}(\varepsilon))},$ we get that $$[D_{j}(\varepsilon),X_{i}(\varepsilon)]=\delta _{ij}\,P_{\Omega }$$ which thus gives the desired dual system.
Moreover, one sees that $J(\sigma (a)Q_{j}\sigma (b))^{*}J1=0$, since $\sigma (a)^{*}J1$ lies in $L^{2}(M)\subset H$ and $Q_{j}^{*}L^{2}(M)=0$. On the other hand,$$\begin{aligned}
\sigma (a)Q_{j}\sigma (b)\cdot 1 & = & \sigma (a)Q_{j}\cdot b=\sigma (a)\cdot (\phi _{j}(b\otimes 1))\\
& = & J\lambda (a^{*})J\cdot (\phi _{j}(b\otimes 1))\\
& = & J\lambda (a^{*})\cdot \phi _{j}(1\otimes b^{*})\\
& = & J\phi _{j}(a^{*}\otimes b^{*})\\
& = & \phi _{j}(b\otimes a)\\
& = & (bS_{j}a).\end{aligned}$$ which gives (\[comp\]).
[CFW81]{}
M. F. Atiyah. Elliptic operators, discrete groups and von [N]{}eumann algebras. In [*Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974)*]{}, pages 43–72. Astérisque, No. 32–33. Soc. Math. France, Paris, 1976.
P. Biane, M. Capitaine, and A. Guionnet. Large deviation bounds for matrix [B]{}rownian motion. , 152(2):433–459, 2003.
M. Bekka and A. Valette. Group cohomology, harmonic functions and the first [$L\sp 2$]{}-[B]{}etti number. , 6(4):313–326, 1997.
H. Cartan and S. Eilenberg. . Princeton University Press, Princeton, N. J., 1956.
J. Cheeger and M. Gromov. -cohomology and group cohomology. , 25(2):189–215, 1986.
A. Connes. Theory of Correspondences. Unpublished notes.
A. Connes. Classification of injective factors. [C]{}ases [$II\sb{1},$]{} [$II\sb{\infty },$]{} [$III\sb{\lambda },$]{} [$\lambda \not=1$]{}. , 104(1):73–115, 1976.
A. Connes, A type II$_1$ factor with countable fundamental group. 4 (1980), 151-153.
A. Connes, V.F.R. Jones, Property T for von Neumann algebras. 17 (1985), 57-62.
A. Connes. . Academic Press, 1994.
A. Connes. Nombres de Betti $L^2$ et facteurs de type II${}_1$, d’après D. Gaboriau et S. Popa. . 55ème année, 2002-2003.
H. Dye. On groups of measure preserving transformations I, II. 81 (1959), 119–159, and 85 (1963), 551–576.
D. Gaboriau. Coût des relations d’équivalence et des groupes. 139 (2000), 41–98.
D. Gaboriau. Invariants $\ell^2$ de relations d’équivalence et de groupes. 95 (2002), 93–150.
F.R. Gantmaher. , volume 1. Chelsea Publ. Co., New York, NY, 1960.
B. E. Johnson, R. V. Kadison, and J. R. Ringrose. Cohomology of operator algebras. [III]{}. [R]{}eduction to normal cohomology. , 100:73–96, 1972.
K. Jung. The free entropy dimension of hyperfinite von neumann algebras. Preprint, math.OA/0112039, 2001.
R. V. Kadison and J. R. Ringrose. Cohomology of operator algebras. [I]{}. [T]{}ype [$I$]{} von [N]{}eumann algebras. , 126:227–243, 1971.
R. V. Kadison and J. R. Ringrose. Cohomology of operator algebras. [II]{}. [E]{}xtended cobounding and the hyperfinite case. , 9:55–63, 1971.
W. L[ü]{}ck. Hilbert modules and modules over finite von [N]{}eumann algebras and applications to [$L\sp 2$]{}-invariants. , 309(2):247–285, 1997.
W. L[ü]{}ck. Dimension theory of arbitrary modules over finite [von Neumann]{} algebras and ${L^2}$-[Betti]{} numbers [I]{}: foundations. , 495:135–162, 1998.
I. Mineev. $\ell_1$-homology of combable groups a nd $3$-manifold groups. preprint.
F. Murray, J. von Neumann. Rings of operators IV, 44 (1943), 716-808.
S. Popa, On a class of type II$_1$ factors with Betti numbers invariants. preprint.
D. Shlyakhtenko. Free [F]{}isher information with respect to a completely positive map and cost of equivalence relations. , 218(1):133–152, 2001.
D. Shlyakhtenko. Some applications of freeness with amalgamation. , 500, 191–212.
D. Shlyakhtenko. Microstates free entropy and cost of equivalence relations. , 118:375–425, 2003.
D. Shlyakhtenko. Some estimates for non-microstates free entropy dimension, with applications to $q$-semicircular families. preprint, 2003.
Allan M. Sinclair and Roger R. Smith. , volume 203 of [*London Mathematical Society Lecture Note Series*]{}. Cambridge University Press, Cambridge, 1995.
D.-V. Voiculescu, K. Dykema, and A. Nica. , volume 1 of [*CRM monograph series*]{}. American Mathematical Society, 1992.
D.-V. Voiculescu. The analogues of entropy and of [Fisher’s]{} information measure in free probability theory [I]{}. , 155:71–92, 1993.
D.-V. Voiculescu. The analogues of entropy and of [Fisher’s]{} information measure in free probability theory [II]{}. , 118:411–440, 1994.
D.-V. Voiculescu. The analogues of entropy and of [Fisher]{}’s information measure in free probability theory, [III]{}. , 6:172–199, 1996.
D.-V. Voiculescu. The analogues of entropy and of [Fisher]{}’s information measure in free probabilility, [V]{}. , 132:189–227, 1998.
D.-V. Voiculescu. The analogues of entropy and of [Fisher’s]{} information measure in free probability, [VI]{}. , 146(2):101–166, 1999.
D.-V. Voiculescu. Free entropy dimension $\leq 1$ for some generators of property $T$ factors of type ${\rm II}\sb 1$. , 514:113–118, 1999.
D.-V. Voiculescu. Free entropy. , 34(3):257–278, 2002.
[^1]: Collège de France, 3 rue d’Ulm, 75 005 Paris,\
I.H.E.S. and Vanderbilt University, connes$@$ihes.fr
[^2]: Department of Mathematics University of California, Los Angeles, CA 90095, USA\
shlyakht@math.ucla.edu. Reserach supported by the NSF and the Sloan Foundation
[^3]: More precisely, the space of $L^{2}$-chains is $\ell ^{2}(\Gamma )\otimes
_{\Gamma }C_{k}^{(f)}=\ell ^{2}(\Gamma )\otimes _{L(\Gamma )}L(\Gamma )\otimes
_{\Gamma }C_{k}^{(f)}$. However, since by [@luck:foundations1], the functor $\ell ^{2}\otimes _{L(\Gamma
)}\cdot $ in the category of finitely-generated $L(\Gamma )$-modules is flat, the nuance between $\ell ^{2}(\Gamma )\otimes _{\Gamma }C_{k}^{(f)}$ and $L(\Gamma )\otimes _{\Gamma }C_{k}^{(f)}$ is irrelevant in the foregoing.
|
---
abstract: 'The acceleration of thermal solar wind protons at spherical interplanetary shocks driven by coronal mass ejections is investigated. The solar wind velocity distribution is represented using $\kappa$-functions, which are transformed in response to simulated shock transitions in the fixed-frame flow speed, plasma number density, and temperature. These heated solar wind distributions are specified as source spectra at the shock from which particles with sufficient energy can be injected into the diffusive shock acceleration process. It is shown that for shock-accelerated spectra to display the classically expected power-law indices associated with the compression ratio, diffusion length scales must exceed the width of the compression region. The maximum attainable energies of shock-accelerated spectra are found to be limited by the transit times of interplanetary shocks, while spectra may be accelerated to higher energies in the presence of higher levels of magnetic turbulence or at faster-moving shocks. Indeed, simulations suggest fast-moving shocks are more likely to produce very high-energy particles, while strong shocks, associated with harder shock-accelerated spectra, are linked to higher intensities of energetic particles. The prior heating of the solar wind distribution is found to complement shock acceleration in reproducing the intensities of typical energetic storm particle events, especially where injection energies are high. Moreover, simulations of $\sim$0.2 to 1 MeV proton intensities are presented that naturally reproduce the observed flat energy spectra prior to shock passages. Energetic particles accelerated from the solar wind, aided by its prior heating, are shown to contribute substantially to intensities during energetic storm particle events.'
author:
- 'P. L. Prinsloo and R. D. Strauss'
- 'J. A. le Roux'
title: Acceleration of solar wind particles by traveling interplanetary shocks
---
Introduction {#sec:intro}
============
Energetic particle enhancements observed at Earth are often associated with the passage of interplanetary (IP) shocks, driven by e.g. coronal mass ejections (CMEs), during energetic storm particle (ESP) events [@Bryantetal1962; @LarioDecker2002; @Hoetal2009; @HuttunenHeikinmaaValtonen2009; @Makelaetal2011]. These enhancements are thought to occur as a result of particle acceleration at the shock, for which diffusive shock acceleration [DSA, @Axfordetal1977; @Krymsky1977; @Bell1978a; @BlandfordOstriker1978] continues to be regarded a viable mechanism, often coupled with self-generated turbulence in the form of magnetohydromagnetic waves [@Lee1983]. See also the relevant discussions of [@DesaiGiacalone2016] on ESPs and DSA. More recently, attention has been drawn to the notable flattening of proton energy spectra in observations directly preceding the passage of IP shocks at Earth [@Larioetal2018]. This is attributed to follow as a result of the propagation of accelerated particles from the shock to the observer, but is not fully explained. Moreover, while CME shock properties such as speed and compression ratio have been linked to the efficiency of particle acceleration [@Larioetal2005b; @Giacalone2012], shock obliquity [@Ellisonetal1995] and the nature of the seed particles [@Desaietal2006] have also been identified as important factors.
It has been suggested that IP shocks can accelerate particles from thermal energies [e.g. @Giacalone2005]. Indeed, for quasi-parallel shocks, where particles can repeatedly cross the shock front along magnetic field lines, injection energies are assumed to be small, and Maxwellian-like distributions are proposed to provide adequate seed particles for DSA [@Giacaloneetal1992; @NeergaardParkerZank2012]. By contrast, injection energies are generally assumed to be high for perpendicular shocks, because the perpendicular diffusion coefficient is small for particles interacting resonantly with micro-scale turbulence. However, the injection energy is considerably lowered when particles experience perpendicular diffusion along intermediate-scale meandering magnetic field lines. The injection speed at perpendicular shocks can become comparable to those at parallel shocks, and DSA thereby more effective, if the amplitude of field line meandering is large enough [@Giacalone2005]. On the other hand, the shock front itself may be rippled: while an idealized shock propagating through an approximately radial magnetic field may vary from being quasi-perpendicular at the CME flanks to quasi-parallel near the nose, realistically, local geometries may resemble any obliquity [see @KleinDalla2017].
Aside from conducive shock geometry, injection into the DSA process is facilitated by the formation of non-thermal seed particles [@NeergaardParkeretal2014; @Zank2017], which has also recently been investigated using kinetic hybrid simulations [@Capriolietal2015; @Sundbergetal2016]. Suprathermal tails are often observed in solar wind (SW) velocity distributions [@Collieretal1996; @Maksimovicetal1997; @Chotooetal2000; @Qureshietal2003] and are considered conducive features for the injection of particles into DSA [@Desaietal2006; @Kangetal2014]. To parameterize these distributions, [@Vasyliunas1968] introduced the Kappa($\kappa$)- distribution function, which characterizes both the Maxwellian core and the suprathermal tail at higher energies. Refer to [@PierrardLazar2010] and [@LivadiotisMcComas2013] for complete reviews on the theory and applications of $\kappa$-functions. These functions characterise SW distributions using only three parameters, namely, the $\kappa$-parameter, which relates to the spectral index of the high-energy tail, and the equivalent temperature and number density of the plasma or particle species considered [@Formisanoetal1973; @ChateauMeyerVernet1991]. Of course, quantities such as the plasma density and temperature are observed to change during the passage of an IP shock [e.g. @LarioDecker2002], which also transforms the $\kappa$-function and thereby the properties of the potential DSA seed population it represents.
In this study, two broad scientific questions are addressed, namely, how the prior heating of the SW distribution affects acceleration at IP shocks, and what role these shock-accelerated particles play in producing the spectral features observed during ESP events. The acceleration of SW particles is modelled by solving a set of stochastic differential equations (SDEs; introduced in Section \[sec:ModelSDEs\]), equivalent to the [@Parker1965] transport equation, in spherical symmetry. The applications are limited to halo CME shocks expanding radially at constant speeds. In this spherically symmetrical scenario, the shock normal is radially aligned so that the shock obliquity can be approximated using the [@Parker1958] spiral angle.
The manner in which the shock passage affects the energy distribution of SW particles is investigated both before and after injection into the DSA process: It is firstly considered how the initial SW distribution transforms in response to changes in the plasma properties (Section \[sec:SWheating\]), and later how the resulting distribution affects the subsequent DSA process as a seed population (Section \[sec:SWtoESPs\]). The classical spectral characteristics of DSA for travelling shocks, and the comparative efficiency of DSA at fast and strong shocks, are revisited in Sections \[sec:spectral\_features\] and \[sec:strongvsfast\], respectively. In addition to reproducing these more typical DSA features, the model is also applied to investigate the spectral flattening reported by [@Larioetal2018] ahead of IP shock passages (Section \[subsec:spec\_evolution\]), as well as to identify the original spatial distributions and energies of seed particles (Section \[subsec:accsites\]) that would eventually contribute to intensities during ESP events.
Modelling CME shock-induced changes in the solar wind {#sec:SWheating}
=====================================================
To describe both the thermal and suprathermal velocity distributions observed in the SW, the $\kappa$-function is implemented in terms of particle speed $v$ as $$\label{eqn:kappafunc}
f_{\kappa} = A_{\kappa}\left( 1+\frac{v^2}{\kappa v_{\kappa}^2} \right)^{-\kappa -1} \text{,}$$ where $v_{\kappa}^2$ is the generalised thermal speed given by $$\label{eqn:vthermal}
v_{\kappa}^2 = \left(2\kappa - 3 \right)\frac{k_b T}{\kappa m_p} \text{,}$$ with $k_{B}$ the Boltzmann constant and $m_{p}$ the proton mass, and where $$\label{eqn:kappanormconst}
A_{\kappa}=\frac{n}{\left( \pi \kappa v_{\kappa}^2\right)^{3/2}} \frac{\Gamma\left( \kappa+1 \right)}{\Gamma\left(\kappa-\frac{1}{2} \right)}$$ is the normalization constant obtained when setting $\int f_{\kappa} d^3 v = n$, with $n$ taken as the SW number density and where $\Gamma$ is the Gamma function. This normalization is discussed in some detail in Appendix \[subsec:on\_kappa\_funcs\].
The $\kappa$-parameter is related to the power-law index of the suprathermal tail. Note that Eq. \[eqn:kappafunc\] is defined for $\kappa > 3/2$, and that if $\kappa \rightarrow \infty$, it reduces to a Maxwellian. For the correct interpretation of Eq. \[eqn:vthermal\] and the associated temperature $T$, it is instructive to consider the discussion by [@Hellbergetal2009]: $v_{\kappa}$ is introduced by [@Vasyliunas1968] as the most probable particle speed, associated with the non-relativistic kinetic energy of $E_{\kappa} = m_{p}v_{\kappa}^2/2$. Evaluating the second moment of $f_{\kappa}$ yields a total energy of $NE_{N} = \frac{3}{2}NE_{\kappa} \kappa/(\kappa-3/2)$, where $E_{N}$ and $N$ are the mean energy per particle and the total number of particles, respectively. Eq. \[eqn:vthermal\] then follows upon the introduction of the plasma temperature $T$ [originally by @Formisanoetal1973] through the invocation of the equipartition theorem, $E_{N}=\frac{3}{2}k_{B}T$, for a monatomic gas. Although this temperature definition and the foregoing assumption of the equipartition of energy are not strictly valid for non-Maxwellian distributions, their use in this manner has become standard practice and is generally considered appropriate [see @Hellbergetal2009 and the references therein].
Changes in plasma properties across the shock {#subsec:plasmaprop}
---------------------------------------------
An objective of this study is to consider how the $\kappa$-function changes during the passage of an IP shock, and to implement this transformed distribution as a seed-particle spectrum for DSA. To this end, while the $\kappa$-index is assumed constant across the shock, it is necessary to model the change in $n$ and $T$, upon which $f_{\kappa}$ depends [see also @Livadiotis2015]. For consistency, transitions across the shock are modelled to correspond to that of the flow speed as would be observed by e.g. a spacecraft in Earth’s orbit. The SW is consequently modelled to transition across the shock between up- and downstream flow speeds in the spacecraft (or fixed) frame, that is, $V_1^{\prime}$ and $V_2^{\prime}=\left(V_{sh}\left( s-1 \right)+V_1^{\prime}\right)/s$, respectively, according to $$\begin{aligned}
\label{eqn:Vtransition}
V_{sw}^{\prime} &= \frac{1}{2s}\left(V_1^{\prime}\left(s+1\right)+V_{sh}\left(s-1\right)\right) \nonumber\\
&-\frac{1}{2s}\left(\left(s-1\right)\left(V_{sh}-V_1^{\prime}\right) \tanh\left(\frac{r-r_{sh}}{L}\right)\right) \text{,} \end{aligned}$$ where $s$ is the shock compression ratio (or ratio of up- and downstream flow velocities in the shock frame), $V_{sh}$ is the shock speed, $r-r_{sh}$ is the radial position relative to the position of the shock $r_{sh}$, and $L$ is a characteristic length used to specify the broadness of this transition. Figure \[fig:flowdensity\] illustrates this transition for the reference parameters listed in Table \[tab:kappaparams\]. Should $V_{sh}=0$, Eq. \[eqn:Vtransition\] reduces to an expression [used by e.g. @leRouxetal1996] for a stationary shock.
![Top: The transition of the fixed-frame SW flow speed $V_{sw}^{\prime}$, as viewed by an observer at the Earth, as a function of the shock position relative to that of the Earth ($r_{sh}-r_0$). $L\ (=0.005$ AU) and $\Delta x_{sh}$ are as defined for Eq. \[eqn:Vtransition\] and Eq. \[eqn:L\_relation\], respectively. Bottom: The SW number density profile corresponding to the flow-speed profile shown above. \[fig:flowdensity\]](flowdensity.pdf)
Note furthermore that $L$ is not a direct measure of the shock width. Its extent can however be approximated from Eq. \[eqn:Vtransition\] as $\triangle x_{sh} \approx 2r_{c}$, with $$\label{eqn:L_relation}
r_{c} = \tanh^{-1}\left( \frac{2sV_{c}-V_1^{\prime}(s+1)-V_{sh}(s-1)}{(1-s)(V_{sh}-V_1^{\prime})}\right) L \text{,}$$ where a fraction $\mathcal{C} \in [0,0.5]$ is chosen such that $V_{c} = V_1^{\prime} + \mathcal{C}(V_2^{\prime}-V_1^{\prime})/2$ is the flow speed at some distance $r_c$ from the shock. Typically, to approximate the shock width, $\mathcal{C} =0.01$ so that $V_{c}\approx V_1^{\prime}$. It is illustrated in Figure \[fig:flowdensity\] how the lengths $\triangle x_{sh}$, $L$, and the actual shock width compare.
With the flow speed transition given, the change in number density $n$ across the shock follows from the continuity equation. Of course, the total factor by which the number density jumps across the shock must be equal to the compression ratio. In terms of $V_{sw}^{\prime}$ it follows, in spherical coordinates, that $$\label{eqn:ntransition}
n = n_0\frac{V_{sh}-V_1^{\prime}}{V_{sh}-V_{sw}^{\prime}} \left(\frac{r_0}{r}\right)^2 \text{,}$$ where $n_0$ is the number density at some reference position such as the Earth ($r_0 = $ 1 AU), and where the flow speeds are transformed back to the shock frame, where their ratio is equal to $s$. This transition is also shown in Figure \[fig:flowdensity\].
To estimate the jump in temperature across the shock, instead of implementing the relevant hydromagnetic Rankine-Hugoniot jump condition, a simpler approach capturing the essential physics is used. The magnetic energy is expected to be much smaller than the thermal energy in the flow-dominated region considered in this study. It is therefore assumed that the change in kinetic energy density $d\mathcal{E}_{sw}$ of the bulk solar wind flow, as a function of position through the shock, is converted to thermal energy. That is, $$\label{eqn:dE_transition}
d\mathcal{E}_{sw} = \frac{1}{2} m_p \left( n_0 \left( V_1^{\prime} - V_{sh}\right)^2 - n \left( V_{sw}^{\prime} - V_{sh}\right)^2 \right) \text{,}$$ which is related through the equipartition theorem to the change in temperature $dT_{sw}$ by $$\label{eqn:dT_transition}
dT_{sw} = \frac{2}{\mathcal{N}_{df}} \frac{d\mathcal{E}_{sw}}{n k_b} \text{,}$$ where $\mathcal{N}_{df}=2/(\gamma_c-1)$ is the number of degrees of freedom and $\gamma_c$ is the ratio of specific heats. $dT_{sw}$ is subsequently added to the unshocked temperature value, that is, $T = T_0 +dT$, where $$\label{eqn:T_unshocked}
T_0 = T_e \left( r/r_0 \right)^{2\left(1-\gamma_c\right)}$$ and $T_e$ is the plasma temperature at the Earth. For a monatomic gas for which $\gamma_c=5/3$ it follows that $T_0\propto r^{-4/3}$. It is assumed that the temperature gain during the shock passage is not significant enough to disassociate molecules, and hence $\gamma_c$ remains constant as for a calorically ideal gas. Furthermore, given the multi-species composition of the SW plasma, $\gamma_{c}$ may deviate from the monatomic value of $5/3$. It is noted that while the hydromagnetic treatment of the shock jump conditions may break down in the transition region itself, it approximates the up- and downstream quantities adequately. Since DSA requires particles to interact with the shock on length scales exceeding its width [e.g. @JonesEllison1991; @KruellsAchterberg1994], this description is sufficient for the current study. See also Section \[subsec:fractional\_compression\].
-------------------- ---------------- ------------------- -------------
$V_{sw}^{\prime}$: $V_1^{\prime}$ 600 km s$^{-1}$
$V_{sh}$ 2400 km s$^{-1}$
$s$ 3.0
$L$ 0.005 AU
$n$: $n_0$ 6.0 cm$^{-3}$
$T$: $T_e$ $5 \times 10^{5}$ K
$\gamma_c$ 3.5/3
$f_{\kappa}$: $\kappa$ 2.25
-------------------- ---------------- ------------------- -------------
: Reference configuration of the parameters needed to define the fixed-frame flow speed $V_{sw}^{\prime}$, number density $n$, temperature $T$ and $\kappa$-function as discussed in Section \[subsec:plasmaprop\].[]{data-label="tab:kappaparams"}
The shock transitions of $V_{sw}^{\prime}$, $n$ and $T$ are displayed in Figure \[fig:plasmakappas\] as viewed by an observer at the position of the Earth. The parameters used to define these quantities are listed in Table \[tab:kappaparams\] and are mostly informed by observations of CMEs during the 2003 Halloween epoch [@Skougetal2004; @Richardsonetal2005; @Gopalswamyetal2005; @Larioetal2005a; @Wuetal2005]. In particular, SW flow speeds in excess of 1800 km s$^{-1}$ [@Skougetal2004] and shock speeds of $\sim$2400 km s$^{-1}$ [@Gopalswamyetal2005; @Wuetal2005] are noted. Since density measurements during some larger events were uncertain [e.g. @Skougetal2004; @Wuetal2005], the compression ratio for such events can be calculated using the ratios of shock-frame flow velocities. For example, $s = (V_{sh}-V_1^{\prime})/(V_{sh}-V_{2}^{\prime})=3$ for $V_{sh} =$ 2400 km s$^{-1}$, $V_1^{\prime}=$ 600 km s$^{-1}$ and $V_{2}^{\prime}=$ 1800 km s$^{-1}$.
Evolution of the $\kappa$-function during the shock passage {#subsec:kappa_evolution}
-----------------------------------------------------------
Aside from $\kappa$, which is kept fixed, $f_{\kappa}$ is shown above to depend on parameters $n$ and $T$, both of which change during the shock passage. Additionally, to illustrate how the SW distribution shifts when the flow speed is shocked to higher values, it is necessary to express the particle speed in $f_{\kappa}$ relative to flow speed in the spacecraft frame, that is, $v:\rightarrow v-V_{sw}^{\prime}$ [see also @Leubner2004; @Kongetal2017]. The SW distribution is therefore expected to be transformed by the shock in at least three ways, depending on how $V_{sw}^{\prime}$, $n$ and $T$ change. To illustrate this transformation, the SW distribution, as represented by $f_{\kappa}$ and viewed by an observer at Earth, is shown at different stages during the shock passage on the right-hand side of Figure \[fig:plasmakappas\] for parameters as specified in Table \[tab:kappaparams\]. Bear in mind these distributions have not yet been injected into the DSA process, and that these transformations occur solely because the shock had heated the SW plasma. Also note that for consistency with the units in which observations are typically presented, $f_{\kappa}$ is converted throughout this study into units of differential intensity (as discussed in Appendix \[subsec:toward\_j\]) and is subsequently denoted as $j_{\kappa}$.
With the shock still 0.02 AU away from the Earth, Figure \[fig:plasmakappas\] shows that the SW distribution at Earth is as of yet unchanged, since none of the plasma quantities have been shocked at this point. When the shock passes Earth, that is, when $r_{sh}=r_0$, the temperature is shown to have increased substantially, accompanied by more modest increases in the flow speed and density. The most obvious changes the SW distribution incurred at this point is that it notably broadened as a result of the temperature increase, and that the thermal peak shifted to higher energies as a result of the increased flow speed. Note that because $f_{\kappa}$ is normalised to the number density, the conservation of particles demands that the peak intensity of the distribution decreases when it broadens. Considering, finally, how the distribution changes when the shock has moved 0.02 AU beyond the Earth, it shifts further, to higher energies, as the flow speed attains its full downstream value, while increasing overall intensities due to the increase in number density. At this point the temperature had already plateaued and hence the distribution shows no appreciable broadening from when the shock passed Earth’s position.
These SW distributions are specified as source spectra for DSA in Section \[subsec:sourcefunction\].
Modelling energetic particle acceleration at CME shocks {#sec:ModelSDEs}
=======================================================
The events of interest in this study are large ESP events with small associated anisotropies. These events are typically associated with fast-moving shocks driven by halo CMEs [@Makelaetal2011], that is, those propagating radially outward and approximately centred on the solar disk. Therefore, to describe the transport and DSA of energetic particles associated with such events, it sufficient to solve the [@Parker1965] transport equation (TPE) for a single spatial dimension. See also the motivation offered by [@Giacalone2015] in this regard. Hence, in radial coordinates, the TPE is written as $$\begin{aligned}
\label{eqn:TPE}
\frac{\partial f}{\partial t^{\prime}} &= \left(\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \kappa_{rr}\right)-V_{sw}^{\prime}\right) \frac{\partial f}{\partial r} + \kappa_{rr}\frac{\partial^2 f}{\partial r^2} \nonumber \\ &+ \frac{1}{3r^2}
\frac{d \left(r^2 V_{sw}^{\prime}\right)}{dr}E\ \xi(E)\frac{\partial f}{\partial E} + Q \text{,} \end{aligned}$$ with $\xi(E)=(E+2E_p)/(E+E_p)$, where $E_p$ is the proton rest-mass energy. The TPE contains the relevant particle transport processes such as SW convection, spatial diffusion, and energy changes due to transport in regions of compressing or expanding SW flows. The last-mentioned process implicitly simulates DSA for regions such as shocks with negative SW velocity divergences. $Q$ represents a particle source function. Note that the TPE is solved for the pitch-angle averaged, omni-directional distribution function $f=f_0 \left(r,p,t \right)$, which is only a function of position, scalar momentum and time; see Appendix \[sec:countingparticles\]. As before, when presented, the distribution function is converted to units of differential intensity, that is, $j = p^2 f$.
The diffusion coefficient considered is the effective radial diffusion coefficient $\kappa_{rr}$, which relates to the equivalent mean free path (MFP) $\lambda_{rr}$ according to $$\label{eqn:difcoef}
\kappa_{rr} = \frac{v}{3}\lambda_{rr} \text{,}$$ where $v$ is the particle speed and where $\lambda_{rr}$ is given by $$\label{eqn:MFPdef}
\lambda_{rr} = \lambda_0 \left(\frac{R}{R_0}\right)^{1/3}$$ where $R = \sqrt{E^2+2EE_p}$ is the particle rigidity and where $\lambda_0$ is a reference MFP defined at $R_0 \equiv$ 1 GeV. Given that the parallel diffusion coefficient is much larger than the perpendicular coefficient, and magnetic field lines are largely radial for the fast flow speeds considered, diffusion along field lines is assumed to dominate. The rigidity dependence implemented here is hence chosen to emulate that predicted for parallel diffusion by quasi-linear theory [@Jokipii1966] for a Kolmogorov-distributed turbulence power spectrum. [@Kallenrodeetal1992] reports no pronounced variation in MFPs with radial distance between the Sun and 1 AU for diffusive events. As such, a radial dependence is omitted in Eq. \[eqn:MFPdef\]. Furthermore, radial MFPs of 0.02 to 0.15 AU are reported for 0.3 to 0.8 MeV electrons [@Kallenrodeetal1992], and the ratio of MFPs for 18 MeV protons to 1 MeV electrons as $1.6\pm 0.9$ [@Kallenrode1993]. Taking the lower limits of the aforementioned quantities into account, the corresponding lower limit of $\lambda_0$ (defined at 1 GeV) for protons is estimated from Eq. \[eqn:MFPdef\] as $\sim$0.05 AU. $\lambda_0$ is varied in Section \[sec:spectral\_features\] and chosen as 0.06 AU elsewhere, using the above lower limit and the results of the aforementioned section as guidelines.
The numerical model: SDEs {#subsec: SDEs}
-------------------------
The TPE specified in multiple computational dimensions typically requires numerical methods to solve. However, instead of solving Eq. \[eqn:TPE\] using finite-difference methods [e.g. @Giacalone2015], the transport and DSA of energetic particles are simulated here by solving an equivalent set of SDEs [see also @KruellsAchterberg1994; @MarcowithKirk1999; @Zhang2000]. The aspects of the SDE approach that are important for this study are detailed below. Refer to [@StraussEffenberger2017] for a comprehensive review.
Eq. \[eqn:TPE\] is conveniently written in the form of the time-backward Kolmogorov equation $$\begin{aligned}
\label{eqn:timeback_kolmgorov}
-\frac{\partial f}{\partial t} = \mu_r \frac{\partial f}{\partial r} + \mu_E \frac{\partial f}{\partial E} + \frac{1}{2} \sigma_r^2\ \frac{\partial^2 f}{\partial r^2} \text{,}\end{aligned}$$ from which the SDEs can be cast into the form of the It[ô]{} equation [see e.g. @Zhang1999] $$\begin{aligned}
\label{eqn:genSDE_r}
dr &= \mu_r dt + \sigma_r dW \\
dE &= \mu_E dt \text{,} \label{eqn:genSDE_E}\end{aligned}$$ where $dW = \sqrt{dt}\ \Lambda(t)$ represents the Wiener process and $\Lambda(t)$ is a simulated Gaussian-distributed pseudo-random number. Note that the forward time $t^{\prime}$ as used in Eq. \[eqn:TPE\] is related to the backward time $t$ according to $t=t_T-t^{\prime}$, where $t_T$ is the total simulation time. It thus follows that $\partial/\partial t^{\prime}=-\partial/\partial t$ in Eq. \[eqn:timeback\_kolmgorov\]. Substituting the coefficients corresponding to $\mu_r$, $\mu_E$, and $\sigma_r$ from Eq. \[eqn:TPE\] into Eqs. \[eqn:genSDE\_r\] and \[eqn:genSDE\_E\] yields $$\begin{aligned}
\label{eqn:rad_SDE}
dr &= \left( \frac{1}{r^2}\frac{\partial}{\partial r}\left( r^2 \kappa_{rr} \right) - V_{sw}^{\prime} \right)dt + \sqrt{2 \kappa_{rr} dt}\ \Lambda(t) \\
\label{eqn:energy_SDE}
dE &= \frac{1}{3r^2}
\frac{d \left(r^2 V_{sw}^{\prime}\right)}{dr} E\ \xi(E)\ dt \text{,}\end{aligned}$$ which are equivalent to the TPE in Eq. \[eqn:TPE\].
These SDEs are integrated in a time-backwards fashion: Starting from an observational point $(r_{\text{obs}},E_{\text{obs}})$ at which the value of the distribution function is sought, Eqs. \[eqn:rad\_SDE\] and \[eqn:energy\_SDE\] are solved iteratively using the Euler-Maruyama numerical scheme [@Maruyama1955] for a finite time step $\Delta t$. The coordinates $r$ and $E$ are updated upon each iteration until $t^{\prime}=0$, or equivalently, until $t=t_T$. When simulating ESP events, $t_T$ is chosen as the transit time of the shock travelling (in a time-backwards fashion) from $r_{\text{obs}}$ to the inner modulation boundary near the Sun ($r_{\text{min}}=5R_{\odot}$), that is, $t_T=(r_{\text{obs}}-r_{\text{min}})/V_{sh}$. The shock position $r_{sh}$ is therefore updated in step with $r$ and $E$, according to $$\label{eqn:rshock}
r_{sh} = r_{\text{obs}}-V_{sh}t$$ where $0<t\le t_T$. A constant shock speed is assumed, since the mean acceleration of CME-driven shocks associated with ESP events is reportedly close to zero [@Makelaetal2011]. Note that the time is not incremented by the same amount during each iteration. It is instead specified to vary, depending on the dominant transport process at the current position, to maintain a fixed step length, that is, $$\label{eqn:dtvar}
\Delta t = \text{min}\left\lbrace
\dfrac{0.1L}{\text{max}\left(\left\lvert
\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 \kappa_{rr} \right)
\right\rvert, V_{sw}^{\prime}\right)},\frac{0.1L^2}{\kappa_{rr}}
\right\rbrace \text{.}$$ This has at least two advantages: Firstly, limiting the step length to some fraction of the length scale $L$, which is associated with the shock, ensures that the shock structure is properly resolved. Secondly, scaling $\Delta t$ in this manner saves computation time, since instead of applying a very small fixed time step, $\Delta t$ will only be small when it is required. As a result of this variable time step, computation times are typically longer for larger values of $\kappa_{rr}$, such as at higher energies, and for narrower shocks (smaller $L$-values).
The above time-integration is repeated $N_p$ (typically, $\sim 10^{6}$) times for each observational point $(r_{\text{obs}},E_{\text{obs}})$, thereby tracing out $N_p$ trajectories in $r$ and $E$ of phase-space density elements, conventionally referred to as pseudo particles. The source function $Q$ in Eq. \[eqn:TPE\] is handled in the SDE approach as a correction term [@StraussEffenberger2017]. Expressing it as a rate of contribution to the distribution function allows the contribution per pseudo particle (or its *amplitude*) to be calculated iteratively along the integration trajectory. That is, $Q = d\alpha/dt \implies \Delta\alpha = Q\Delta t$, from which it follows that $$\label{eqn:particleweight}
\alpha_i(t-\Delta t) = \alpha_i(t) + Q\Delta t \text{.}$$ The source contribution is then tallied for $N_p$ pseudo particles such that $$\label{eqn:sourcecontribution}
f_\text{obs} \left( r_{\text{obs}},E_{\text{obs}} \right) = \frac{1}{N_p} \sum^{N_p}_{i=1} \alpha_i$$ gives the value of the distribution function at the observational point at time $t^{\prime}=t_T\ (t=0)$. Section \[subsec:sourcefunction\] discusses how the source function itself is specified. Furthermore, physical particle distributions can also be obtained by tallying the amplitude-weighted flux contributions of pseudo particles within appropriately sectored spatial and energy intervals (or *bins*) at any time during the simulation, divided by the number of pseudo particles counted within each sector. This method is used in Section \[subsec:accsites\] to trace likely acceleration sites and seed-particle energies in a time-backwards fashion.
Finally, a reflective inner boundary is implemented such that if $r<r_{\text{min}} \implies r = 2r_{\text{min}}-r$, while at the outer boundary, if $r>r_{\text{max}}=1.4$ AU, the time-integration routine is interrupted and that pseudo-particle’s contribution is discarded, defining an absorbing boundary condition.
Modelling the shock source function {#subsec:sourcefunction}
-----------------------------------
![Top: The transition of the fixed-frame SW flow speed $V_{sw}^{\prime}$ as function of heliocentric distance $r$ with the shock centred at Earth, that is, $r_{sh}=$ 1 AU. Bottom: Absolute value of the gradient of $V_{sw}^{\prime}$ corresponding to the profile shown above. $L\ (=0.005$ AU) and $\Delta x_{sh}$ are as defined for Eq. \[eqn:Vtransition\] and Eq. \[eqn:L\_relation\], respectively. \[fig:source\_width\]](source_width.pdf)
As per Eq. \[eqn:particleweight\], when a pseudo-particle is traced back to a point where it interacts with the shock, its amplitude is attributed a value that depends on the product of the source function $Q$ and the time $\Delta t$ it spends interacting with the shock. Indeed, considering Eq. \[eqn:TPE\], $Q$ should have units corresponding to that of the distribution function per unit time. Otherwise, the specification of $Q$ is largely arbitrary: e.g. [@Giacalone2015] employs delta functions, others [e.g. @leRouxetal1996] specify distribution functions to represent seed populations, while [@MalkovVoelk1995] also consider shock-heated seed populations. Drawing on aspects of the aforementioned examples, the following source function is proposed: $$\label{eqn:sourcefunction}
Q = \frac{f_{\kappa}}{m_p^3}\ \left| \frac{dV_{sw}^{\prime}}{dr} \right| \ H\left(\frac{E-E_{\text{inj}}}{E_{\text{inj}}}\right) \text{.}$$ Here, $f_{k}$ is the $\kappa$-function representing the SW distribution. As discussed in Appendix \[subsec:on\_kappa\_funcs\], the $m_p^3$ factor is included for dimensional consistency between $f_{\kappa}$ and the distribution function in the TPE. Since $f_{\kappa}$ is already normalised to the number density, scaling the SDE solutions is not necessary. The dimensionless Heaviside function $H\left((E-E_{\text{inj}})/E_{\text{inj}}\right)$ ensures that only the contributions of particles with energies larger than the injection energy $E_{\text{inj}}$ are included. Note that $E_{\text{inj}}=$ 60 keV unless stated otherwise; see also Section \[subsec:Einjvar\]. The absolute value of the SW velocity gradient $\left| dV_{sw}^{\prime}/dr \right|$ is included to provide a spatial region with a width and position that is self-consistently associated with the shock and that has a reasonable probability of being sampled by pseudo particles. Figure \[fig:source\_width\] shows how $\left| dV_{sw}^{\prime}/dr \right|$ is representative of the width of the shock. While it is sufficient for a physical particle to pass between the upstream and downstream media in order to gain energy, using this time-backward SDE method, a finite-width region has to be sampled by pseudo particles to register flux emanating from the shock.
Advantages and limitations of the SDE approach {#subsec:prosandcons}
----------------------------------------------
The SDE approach has been successfully applied in many instances to simulate space particle transport [@Zhang1999; @Peietal2010; @Straussetal2011; @Straussetal2013; @Molotoetal2018] and DSA in particular [@KruellsAchterberg1994; @MarcowithKirk1999; @Zhang2000; @Zuoetal2011; @Huetal2017]. The time-backward approach is favoured here due to its efficiency, since every simulated pseudo-particle contributes to intensities at the desired observational point. Additionally, in a similar fashion to how the boundary interactions of pseudo particles are used to trace the most probable points of entry into the heliosphere for cosmic rays [e.g. @Straussetal2011], this backward tracing of phase-space trajectories is similarly utilised in this study to map probable acceleration sites and seed-particle energies; see Section \[subsec:accsites\]. Since these pseudo-particle trajectories are solved entirely independent of each other, this approach is also conducive to the utilisation of parallel computing platforms.
However, this mutual independence of the SDE solutions also limits applications to the test-particle case. Non-linear effects of shock acceleration such as self-generated turbulence [@Lee1983; @leRouxArthur2017] or particle mediation of shock structures [@Mostafavietal2017] can therefore not be considered. Moreover, any process that requires the calculation of particle intensity gradients becomes computationally expensive [e.g. @Molotoetal2018]. Nevertheless, the SDE approach remains suitable to investigate the intricacies of classical DSA, without the limitations imposed by a numerical grid, such as instabilities involving the large gradients that are typically encountered at shocks. It is shown in Appendix \[sec:benchmark\] that the SDEs produce appreciably similar results to a finite-difference approach in reproducing ESP events for the same parameter set. Furthermore, as opposed to finite-differences schemes, the SDE model introduced in this study can seamlessly be expanded to a larger number of computational dimensions [e.g. @Peietal2010] as required to study particle acceleration at more complicated shock geometries.
Spectral features and diffusion dependence of shock-accelerated particles. {#sec:spectral_features}
==========================================================================
The most distinctive characteristics of shock-accelerated particles are observed in their energy spectra, which are considered here at the time the shock passes an observer, e.g. a spacecraft, near the Earth. Implementing the model configuration discussed in Sections \[sec:SWheating\] and \[sec:ModelSDEs\], and scaling the diffusion coefficient by varying the value of $\lambda_0$ in Eq. \[eqn:MFPdef\], the dependence of shock-accelerated spectral features on this transport process is investigated. The resultant spectra are shown in Figure \[fig:spectra\_lambdavar\].
![Modelled energy spectra at the Earth ($r=$ 1 AU) at the time of the shock passage for different values of $\lambda_0$. Step-like lines represent SDE solutions, while the smooth solid lines are corresponding fits of Eq. \[eqn:genDSAform\] using the parameters listed in Table \[tab:bestfit\]. Also included are the heated $\kappa$-distribution (dashed grey line), specified as a source function on the shock, and the $E^{-1.25}$ power law associated with a shock compression ratio of $s=3$ (dotted line). \[fig:spectra\_lambdavar\]](spectra_lambdavar.pdf)
Qualitatively, the typical DSA-associated features are evident: A power-law distribution at lower energies transitioning to an exponential-like decrease at higher energies [@EllisonRamaty1985]. This distribution can be described using a simple function in the form of $$\label{eqn:DSAform}
j_{\text{DSA}} = j_0 \left(\frac{E}{E_0}\right)^{\gamma_s} \mathrm{e}^{ -\left(E/E_{\text{cut}} \right)^2} \text{,}$$ where $j_0$ is a differential intensity defined at some reference energy $E_0$, $E_{\text{cut}}$ is the cut-off (or roll-over) energy above which the distribution begins to decrease exponentially, and where $$\label{eqn:specInd_DSA}
\gamma_s = \frac{s+2}{2-2s}$$ is the spectral index (for $E \ll E_p$) associated with the compression ratio $s$ of the shock [see also @Ellisonetal1990]. For $s=3$, as specified for these solutions, it is therefore expected that $\gamma_s=-1.25$. However, not all of the spectra in Figure \[fig:spectra\_lambdavar\] appear to follow a $E^{-1.25}$ power law. This is accentuated in Figure \[fig:specind\_lambdavar\], which shows spectral indices for solutions with smaller $\lambda_0$-values actually varying with energy even before the exponential decreases ensue. To describe this behaviour, it is useful to generalise Eq. \[eqn:DSAform\] as follows: $$\label{eqn:genDSAform}
j_{\text{DSA}} = j_0 \left(\frac{E}{E_0}\right)^{\gamma_a} \left(\frac{E^{\xi}+E_{\text{tr}}^{\xi}}{E_0^{\xi}+E_{\text{tr}}^{\xi}} \right)^{\left(\gamma_b-\gamma_a\right)/\xi} \mathrm{e}^{ -\left(E/E_{\text{cut}} \right)^2} \text{.}$$ This describes a function transitioning between power-law indices $\gamma_a$ and $\gamma_b$ about $E_{\text{tr}}$, with $\xi$ specifying the smoothness of this transition, and which rolls over into an exponential decrease above $E_{\text{cut}}$. Note that if $\gamma_a=\gamma_b$, this expression simplifies to the form of Eq. \[eqn:DSAform\]. Setting $\gamma_b=\gamma_s$, where $\gamma_s=-1.25$ for $s=3$, and choosing $\xi=0.8$, the function given in Eq. \[eqn:genDSAform\] is fitted to the solutions in Figure \[fig:spectra\_lambdavar\] with parameters as presented in Table \[tab:bestfit\].
------- --------- ------ ------
0.005 $-2.5$ 1.5 30.0
0.01 $-2.4$ 0.5 11.0
0.02 $-2.0$ 0.2 5.0
0.035 $-1.25$ 0.08 2.5
0.06 $-1.25$ - 1.5
0.1 $-1.25$ - 0.9
------- --------- ------ ------
: Parameters used to fit Eq. \[eqn:genDSAform\] to the corresponding model solutions shown in Figure \[fig:spectra\_lambdavar\]. The $\lambda_0$-values are constants used to scale the MFPs in Eq. \[eqn:MFPdef\]. $E_{\text{tr}}$ and $E_{\text{cut}}$ are energies at which spectra transition to power-law indices associated with the full shock compression and roll over into exponential decreases, respectively. $\gamma_a$ is the spectral index for $E < E_{\text{tr}}$. Note that $E_{\text{tr}}$-values are omitted for the two largest $\lambda_0$-values, because $\gamma_a=\gamma_b=-1.25$, which reduces Eq. \[eqn:genDSAform\] to the form of Eq. \[eqn:DSAform\]. []{data-label="tab:bestfit"}
Both $\gamma_a$ and $E_{\text{cut}}$ respond to varying the value of $\lambda_0$, affecting both the hardness of the spectra and the energies up to which they are accelerated. The behaviour of these two sets of features and the underlying physics are discussed separately in the subsections below.
Fractional compression sampling: On shock widths and diffusion length scales {#subsec:fractional_compression}
----------------------------------------------------------------------------
![Spectral indices for the corresponding fits of Eq. \[eqn:genDSAform\] shown in Fig. \[fig:spectra\_lambdavar\]. The spectral index $\gamma_s=-1.25$ for a shock with $s=3$ is also shown. \[fig:specind\_lambdavar\]](specind_lambdavar.pdf)
Classically, the spectral index associated with a DSA-produced spectrum is a function of the shock compression ratio alone. The behaviour observed in Figure \[fig:spectra\_lambdavar\], where shock-accelerated spectra become softer, displaying smaller spectral indices, for smaller diffusion coefficients, is therefore not theoretically expected. Note, however, the solutions for $\lambda_0 \geq 0.035$ AU do follow the theoretically predicted power law of $E^{-1.25}$ for $s=3$. Figure \[fig:specind\_lambdavar\] shows the spectral indices of the solutions presented in Figure \[fig:spectra\_lambdavar\]. Here, spectral indices are also shown to be equal to $-1.25$ for large $\lambda_0$-values at low energies. Indices become progressively smaller for smaller $\lambda_0$-values, but increase toward higher energies. Consider that particles with $\lambda_0 \lesssim 0.035$ AU may be sampling only a fraction of the total compression of the shock. The spectra harden toward higher energies, because the MFPs themselves increase with energy and progressively greater fractions of the total compression are sampled.
![Diffusion length scales at the shock, calculated using Eq. \[eqn:diflenscl\] and corresponding to the solutions shown in Figure \[fig:spectra\_lambdavar\]. Varying $\lambda_0$-values scales the energy profiles of $\kappa_{rr}/V_{sw}$ uniformly. Markers indicate the values of $\kappa_{rr}/V_{sw}$ at the corresponding cut-off energies ($E_{\text{cut}}$) listed in Table \[tab:bestfit\]. Arrows indicate the energies $E_{\text{tr}}$ (also listed in Table \[tab:bestfit\]) at which the respective $\kappa_{rr}/V_{sw}$ profiles are equal to the shock width (dashed horizontal line), which is calculated using Eq. \[eqn:L\_relation\]. \[fig:difscl\_lambdavar\]](diflengthscl_lambdavar.pdf)
This effect is illustrated in Figure \[fig:difscl\_lambdavar\]. Recall from Eq. \[eqn:difcoef\] that $\kappa_{rr} \propto \lambda_{rr}$, which is scaled using $\lambda_0$. Since $\kappa_{rr}$ does not have dimensions of length, when comparing to other length scales it is useful to define the diffusion length scale $\kappa_{rr}/V_{sw}$, which is often expressed as the sum of the down- and upstream values at the shock [e.g. @SteenbergMoraal1999]. That is, $$\begin{aligned}
\label{eqn:diflenscl}
\left(\frac{\kappa_{rr}}{V_{sw}}\right)_{\left( r=r_{sh} \right)} &= \left(\frac{\kappa_{rr}}{V_{sw}}\right)_{1} + \left(\frac{\kappa_{rr}}{V_{sw}}\right)_{2}\nonumber\\ &= \kappa_{rr}\left( \frac{1}{V_{1}}+\frac{1}{V_{2}} \right) \text{,}\end{aligned}$$ where $(\kappa_{rr})_1=(\kappa_{rr})_2=\kappa_{rr}$, since $\kappa_{rr}$ is assumed not to change across the shock, with subscripts 1 and 2 denoting up- and downstream values, respectively. Here, $V_{1}=V_{sh}-V_1^{\prime}$ and $V_{2}=(V_{sh}-V_1^{\prime})/s$ are the up- and downstream flow speeds in the shock frame. This quantity is plotted as a function of energy in Figure \[fig:difscl\_lambdavar\], along with the shock width, calculated using Eq. \[eqn:L\_relation\] for $L = 0.005$ AU. Note that for $\lambda_0=0.06$ and $0.1$ AU, for which spectra are aligned with $E^{-1.25}$, the diffusion length scales are greater than the shock width for all energies in the considered domain. Those particles therefore sample the full compression ratio, and hence their energy distributions display the power law associated with $s=3$. A similar effect is reported for the heliospheric termination shock [e.g. @ArthurLeRoux2013]. For $\lambda_0<0.06$ AU, particles sample fractional compression ratios up to the energies where their respective diffusion length scales begin to exceed the shock width. Indeed, the energies at which $\kappa_{rr}/V_{sw}$ and the shock width intersect provide good estimates for $E_{\text{tr}}$ in Table \[tab:bestfit\].
Since the prediction of DSA given in Eq. \[eqn:specInd\_DSA\] is only observed where $\kappa_{rr}/V_{sw}$ is larger than the width of the shock, these results are in agreement with the length scale hierarchy of $\kappa_{rr}/V_{sw} \gg \lambda_{rr} \gg \Delta x_{sh}$ required for classical DSA to be valid [@BlandfordOstriker1978; @JonesEllison1991]. Given the $L$-dependence of the variable time step (Eq. \[eqn:dtvar\]), to limit computation times, the shock width in this study is chosen much broader ($L=0.005$ AU) than typical IP shock widths [$L\sim 10^{-6}$ AU; e.g. @Sapunovaetal2017]. Nevertheless, the results of the model should remain valid as long as the aforementioned length scale hierarchy is observed.
Finite-time acceleration and the termination of shock-accelerated spectra {#subsec:acctime_cutoffs}
-------------------------------------------------------------------------
The second set of features considered entails the highest energies attained by shock-accelerated spectra before intensities begin to decrease exponentially. Figure \[fig:spectra\_lambdavar\] illustrates that spectra for smaller $\lambda_0$-values are accelerated to higher energies before terminating. These energies, represented by $E_{\text{cut}}$, are listed in Table \[tab:bestfit\]. Since they notably respond to varying $\lambda_0$, it seems reasonable to expect that this spectral transition also occurs due to $\kappa_{rr}/V_{sw}$ attaining some characteristic length. Shock-accelerated spectra have previously been reported to roll over due to diffusion length scales becoming comparable to system sizes or some related length of the shock geometry [@EllisonRamaty1985; @SteenbergMoraal1999]. Indeed, Figure \[fig:difscl\_lambdavar\] shows the diffusion length scales corresponding to each of the cut-off energies of the solutions in Figure \[fig:spectra\_lambdavar\] are reasonably similar, distributed between 0.1 and 0.4 AU. However, these do not resemble any obvious length scale in either the physical system or within the numerical model. Simulations where the position of modulation boundaries were varied yielded negligible effects on energy spectra, suggesting that system size is not the limiting factor in this instance.
![Acceleration time as function of kinetic energy, calculated using Eq. \[eqn:acctime\], for the corresponding diffusion length scale profiles shown in Figure \[fig:difscl\_lambdavar\]. Markers indicate the time required to accelerate spectra up to the corresponding cut-off energies ($E_{\text{cut}}$) listed in Table \[tab:bestfit\]. The simulation time (horizontal line) is the time taken by the shock to travel from near the Sun to Earth at 2400 km s$^{-1}$. \[fig:acctime\_lambdavar\]](acctime_lambdavar.pdf)
Another instructive quantity that is related to the diffusion length scale is the acceleration time [@Drury1983; @Ellisonetal1990], which is the time required to accelerate particles from a particular energy (or equivalent momentum) to another at a planar shock. This is expressed in terms of the diffusion length scale as $$\label{eqn:acctime}
\tau_a = \frac{3}{V_{1}-V_{2}} \int\limits_{p_{\text{inj}}}^{p}\kappa_{rr}\left(\frac{1}{V_{1}}+\frac{1}{V_{2}}\right) \frac{dp^{\prime}}{p^{\prime}} \text{,}$$ where $V_{1}$ and $V_{2}$ are as defined for Eq. \[eqn:diflenscl\] and $p_{\text{inj}}$ is the momentum equivalent to $E_{\text{inj}}$. Note that momentum is generally related to kinetic energy according to $p=(1/c)\sqrt{E^2+2EE_{p}}$ (for protons), with $E_{p}$ the proton rest-mass energy. Eq. \[eqn:acctime\] is used in Figure \[fig:acctime\_lambdavar\] to approximate the time required to accelerate spectra up to different energies for each of the length scale profiles shown in Figure \[fig:difscl\_lambdavar\]. Also shown in Figure \[fig:acctime\_lambdavar\] as symbols are the times required to accelerate spectra up to the observed cut-off energies listed in Table \[tab:bestfit\] for each respective $\kappa_{rr}/V_{sw}$-profile. Similar to the diffusion length scales corresponding to the cut-off energies, the associated acceleration times also have similar values. These $\tau_a$-values also compare well to the total duration of the simulation, which is equal to the time taken by the shock to travel from near to the Sun to Earth at 2400 km s$^{-1}$, that is, $\sim$17.3 hours. This suggests that for each of the spectra presented in Figure \[fig:spectra\_lambdavar\], $E_{\text{cut}}$ is the highest energy that could be attained in the available time [see also @Channoketal2005]. At $E > E_{\text{cut}}$, the rate of escape of particles from the shock begins to exceed the acceleration rate, which leads to the observed exponential intensity decreases. In this context, smaller $\lambda_0$-values, and consequently smaller diffusion coefficients, serve to better confine particles near the shock. This, in turn, reduces the time required to accelerate particles up to a particular energy, or stated differently, allows particles to be accelerated to higher energies within the available time frame.
Spectral features and acceleration efficiency {#subsec:acc_efficiency}
---------------------------------------------
The preceding discussions reveal that the characteristics of shock-accelerated spectra are sensitive to the value of diffusion length scales in two opposing ways: Diffusion length scales should exceed shock widths for spectra to display DSA-predicted power-law indices, but accelerated spectra terminate at lower energies for larger diffusion length scales. The top panel of Figure \[fig:fluenceE\_lambdavar\] illustrates this dichotomy. The energies above which $\kappa_{rr}/V_{sw}$ exceeds the shock width, that is, $E_{\text{tr}}$, become smaller with increasing $\lambda_0$-values, implying greater overall spectral hardening, while similarly decreasing $E_{\text{cut}}$-values imply that spectra terminate at lower energies. Harder spectra yield larger intensities of energetic particles, but so do higher cut-off energies. Yet, with regards to diffusion properties, they are attained in opposite ways. Neither of these therefore necessarily provide meaningful measures of acceleration efficiency on their own.
![Top: Energies $E_{\text{tr}}$ and $E_{\text{cut}}$ at which spectra for different values of $\lambda_0$ transition to power-law indices associated with the full compression and roll over into exponential decreases, respectively. Bottom: The energy-integrated distributions for different values of $\lambda_0$ as a measure of how efficiently DSA produces particles above the injection energy and 1 MeV, respectively. \[fig:fluenceE\_lambdavar\]](fluenceE_lambdavar.pdf)
To evaluate which combination of spectral features yield greater intensities of energetic particles, it is useful to consider particle fluences. These are calculated by integrating differential intensities over energy as follows: $$\label{eqn:fluence}
I_{E_{l}} = \int\limits^{\infty}_{E_{l}} j(E)\ dE \text{.}$$ This integral converges, since $j$ fortuitously always decreases as $E \rightarrow \infty$. Eq. \[eqn:fluence\] is evaluated for each of the spectra shown in Figure \[fig:spectra\_lambdavar\] with two different lower limits, namely $E_{l} = E_{\text{inj}}$ and 1 MeV. The resulting energy-integrated intensities are shown in the bottom panel of Figure \[fig:fluenceE\_lambdavar\]. The total intensity for all particles with $E>E_{\text{inj}}$ peaks at $\lambda_0\sim$ 0.035 AU, where the power-law segments of spectra have reached, or nearly reached, their maximum hardness. $E_{\text{cut}}$ becomes more important when considering higher-energy particles. For intensities integrated upward from 1 MeV, fluences peak between $\lambda_0= $ 0.02 and 0.035 AU and notably decrease for larger $\lambda_0$-values due to spectra rolling over at lower energies.
It is revealed that both spectral hardness and the maximum energies spectra attain contribute appreciably to the number of energetic particles the DSA process is able to produce. Of course, the energies up to which spectra extend become increasingly important for the intensities of higher-energy particles.
Strong shocks versus fast shocks: Which is the more efficient particle accelerator? {#sec:strongvsfast}
===================================================================================
Fast-moving CME shocks with large compression ratios are reported to be more efficient at accelerating energetic particles [@Larioetal2005b; @Makelaetal2011; @Giacalone2012]. How conducive each of these shock properties are to producing large numbers of high-energy particles, especially as opposed to each other, raises an interesting subject for investigation. Both the shock speed $V_{sh}$ and the compression ratio $s$ affect how particle distributions evolve, and often in contradicting (or even self-contradicting) ways. Prior to injection, both properties affect the heating of the SW, while the hardness of shock-accelerated spectra is already shown to depend on the compression ratio. To explore and compare the effects of these two parameters, they are varied in three different ways: $V_{sh}$ and $s$ are each varied separately with the other remaining fixed, and they are varied together such that the factor by which the fixed-frame flow speed jumps across the shock remains constant. For ease of reference, these three sets of parameter configurations are shown in Table \[tab:shspeedV\], along with the corresponding downstream flow speeds in both the fixed frame and the shock frame, denoted $V_{2}^{\prime}$ and $V_{2}$, respectively.
Varying $V_{sh}$ and $s$ affects the shock transitions of the SW flow speed, number density, and temperature, and thereby also influences how the energy distribution of SW particles changes during the passage of the shock. Each of the aforementioned parameters’ shock transitions are shown in Figure \[fig:plasma\_shspeed\] for the configurations listed in Table \[tab:shspeedV\]. The fixed-frame flow speed, that is, the flow speed as viewed by an observer stationed at Earth, increases with both $V_{sh}$ and $s$ when each is varied separately. The results, as shown in the top row of frames in Figure \[fig:plasma\_shspeed\], yield larger jumps in flow speed should either $V_{sh}$ or $s$ be increased. As intended, when $V_{sh}$ and $s$ are varied together, the downstream flow speed remains constant in the fixed frame. Hence the factor by which the flow speed increases is the same for these instances. The corresponding shock transitions of the number density and temperature in Figure \[fig:plasma\_shspeed\] follow from Eqs. \[eqn:ntransition\] and \[eqn:dT\_transition\]. Figure \[fig:kappas\_shspeed\] shows how the SW energy distributions change in response to these shock transitions. It appears the SW energy distribution is heated by comparable amounts for varying shock speeds and strengths. However, $j_{\kappa}$ appears marginally more sensitive to changes in temperature, which in turn is affected most appreciably by changes in $V_{sh}$. Therefore, it can be argued that $V_{sh}$ contributes more towards heating the SW distribution, as opposed to $s$, considering the large shifts in energy and broadening of $j_{\kappa}$ that follows from varying it.
------ ------ ----- ------ ------ ------
1: 3000 3 800 2000 3.67
2800 3 733 2067 3.44
2600 3 667 1933 3.22
2280 3 560 1720 2.87
2200 3 533 1667 2.78
2: 2400 2 900 1500 2.5
2400 2.2 818 1581 2.64
2400 2.5 720 1680 2.8
2400 3.5 514 1886 3.14
2400 4 450 1950 3.25
3: 3000 2 1200 1800 3
2800 2.2 1000 1800 3
2600 2.5 800 1800 3
2280 3.5 480 1800 3
2200 4 400 1800 3
ref: 2400 3 600 1800 3.0
------ ------ ----- ------ ------ ------
: Configurations used throughout Section \[sec:strongvsfast\], in which 1: the shock speed $V_{sh}$ is varied, 2: the compression ratio $s$ is varied, and 3: $V_{sh}$ and $s$ are varied together such that $V_{2}^{\prime}/V_1^{\prime}$ remains constant. Here, $V_1^{\prime}=$ 600 km s$^{-1}$ and $V_{2}^{\prime}=(V_{sh}(s-1)+V_1^{\prime})/s$ are the fixed-frame flow speeds, respectively up- and downstream of the shock. $V_{2}=(V_{sh}-V_1^{\prime})/s$ is the downstream flow speed in the shock frame. The last line contains the reference configuration.[]{data-label="tab:shspeedV"}
The heated SW distributions discussed above essentially serve as input spectra, from which particles are injected into the DSA process at the shock for $E > E_{\text{inj}}$. The shock-accelerated spectra obtained by solving the SDEs of Section \[subsec: SDEs\] for each of the configurations in Table \[tab:shspeedV\] are presented in Figure \[fig:spectra\_Vshvar\]. The analyses of these spectra are analogous to those of spectra presented in Figure \[fig:spectra\_lambdavar\], while the discussions that follow draw on concepts introduced in Section \[sec:spectral\_features\]. Accordingly, functions are fitted to the SDE solutions and the parameters tabulated for each configuration in Table \[tab:bestfit\_shspeedV\]. The value of $\lambda_0$ is chosen as 0.06 AU throughout this section. With reference to Figure \[fig:difscl\_lambdavar\], this implies the diffusion length scale exceeds the shock width for all considered energies and that the full compression ratio is sampled. It is therefore sufficient to fit the single power-law function of Eq. \[eqn:DSAform\], with spectral indices as predicted by DSA.
The shock-accelerated spectra presented in Figure \[fig:spectra\_Vshvar\] share the same general features as those encountered before: a power-law segment extending from near the injection energy up to where it rolls over into an exponential decrease. The following discussion considers how these features change as a result of varying the shock speed and compression ratio. Firstly, varying $V_{sh}$, while keeping the compression ratio fixed at $s=3$, produces spectra as shown in the left column of Figure \[fig:spectra\_Vshvar\]. They are power-law distributed with a spectral index of $-1.25$, as illustrated in the accompanying frame below. This is the index expected from DSA for $s=3$ (see Eq. \[eqn:specInd\_DSA\]). The sizeable differences in the intensities of these three spectra are due in large part to the heated SW spectra, which themselves are notably affected by varying $V_{sh}$. When varying $s$ instead and keeping the shock speed fixed at $V_{sh}=$ 2400 km s$^{-1}$ as shown in the right-most column of Figure \[fig:spectra\_Vshvar\], intensities differ mostly due to changes in the power-law indices of shock-accelerated spectra. These indices, shown in the accompanying frame below as $-2$, $-1.25$ and $-1$, correspond to those expected from Eq. \[eqn:specInd\_DSA\] for $s=2$, 3, and 4, respectively.
The middle column of Figure \[fig:spectra\_Vshvar\] illustrates the case where both $V_{sh}$ and $s$ are varied so that $V_{2}^{\prime}/V_1^{\prime}=3$. It is worth drawing attention to the fact that the spectral indices these spectra display are not the index associated with $V_{2}^{\prime}/V_1^{\prime}=3$, even though this would be the jump factor in the flow speed observed by spacecraft during the passage of the shock. The power-law indices of all shock-accelerated spectra are consistently those associated with $s$. The observed factor by which flow speeds change across a shock should not be confused with the actual compression ratio, which is the factor by which either the density or the flow speeds in the shock frame change across the shock. This is important when calculating the power-law index of the DSA-predicted spectrum for comparison against an observed energy spectrum.
![Cut-off energies, where spectra roll over into exponential decreases, as functions of shock speed $V_{sh}$ and compression ratio $s$, for configurations where $V_{sh}$ and $s$ are varied both separately and together as specified in Table \[tab:bestfit\_shspeedV\]. \[fig:Ecut\_shspeed\]](Ecut_shspeed.pdf)
------ ------ ----- ------- ------
1: 3000 3 -1.25 2.0
2800 3 -1.25 1.9
2600 3 -1.25 1.8
2280 3 -1.25 1.4
2200 3 -1.25 1.35
2: 2400 2 -2.0 1.4
2400 2.2 -1.75 1.5
2400 2.5 -1.5 1.5
2400 3.5 -1.1 1.7
2400 4 -1.0 1.7
3: 3000 2 -2.0 1.65
2800 2.2 -1.75 1.6
2600 2.5 -1.5 1.6
2280 3.5 -1.1 1.65
2200 4 -1.0 1.55
ref: 2400 3 -1.25 1.5
------ ------ ----- ------- ------
: Parameters used to fit Eq. \[eqn:DSAform\] to the solutions of Figure \[fig:spectra\_Vshvar\] and corresponding to the configurations of shock speed $V_{sh}$ and compression ratio $s$ introduced in Table \[tab:shspeedV\]. Here, $\gamma(s)$ is the spectral index associated with $s$ as given by Eq. \[eqn:specInd\_DSA\] and $E_{\text{cut}}$ is the energy at which spectra roll over into exponential decreases. The last line contains the reference configuration. []{data-label="tab:bestfit_shspeedV"}
The spectra shown in the left and right-hand sides of Figure \[fig:spectra\_Vshvar\] extend to higher energies for both faster and stronger shocks, where $V_{sh}$ and $s$ are each varied separately with the other held constant. Spectra extend up to similar energies where $V_{sh}$ and $s$ are varied together. This is also illustrated in Figure \[fig:Ecut\_shspeed\], where the cut-off energies of the aforementioned spectra are plotted against both $V_{sh}$ and $s$, showing almost no reponse to changing shock conditions. Similar to Section \[subsec:acctime\_cutoffs\], the acceleration times corresponding to each of the spectra in Figure \[fig:spectra\_Vshvar\] are calculated and compared to the simulation times, as shown in Figure \[fig:acctime\_shspeed\]. As before, the maximum attainable energies for shock-accelerated spectra are determined predominantly by the shock transit time.
Acceleration efficiency: Strong versus fast shocks {#subsec:acc_efficiency_svsv}
--------------------------------------------------
The preceding sections analyse the collated effects of varying $V_{sh}$ and $s$ during the complete acceleration process, including the heating of the SW energy distribution and the subsequent shock acceleration. It can be argued that the shock speed has the largest effect, since varying $V_{sh}$ yields pronounced changes in both the heated SW distributions and in reducing acceleration times, thereby extending spectra to higher energies. On the other hand, the most notable effect of varying $s$ is the changes in the power-law indices displayed by shock-accelerated spectra, which also affects energetic particle intensities considerably.
Implementing the same technique used in Section \[subsec:acc\_efficiency\], particle fluences are calculated using Eq. \[eqn:fluence\]. The resulting energy-integrated intensities are shown in the top frames of Figure \[fig:fluences\_shspeed\] as functions of both $V_{sh}$ and $s$. To constrain the intensities resulting solely as a result of DSA and not including those of the input spectra, fluences are also calculated for the heated SW distributions ($I_{\text{kappa}}$) and subtracted from those calculated for the shock-accelerated spectra ($I_{\text{total}}$). These two additional sets of energy-integrated intensities, namely $I_{\text{kappa}}$ and $I_{\text{DSA}}=I_{\text{total}}-I_{\text{kappa}}$, are also shown in Figure \[fig:fluences\_shspeed\]. Note that $I_{\text{kappa}}$ is roughly an order of magnitude smaller than $I_{\text{total}}$ on average, with comparably weak dependences on $V_{sh}$ and $s$. $I_{\text{DSA}}$ therefore closely resembles $I_{\text{total}}$. Considering particle fluences as functions of $V_{sh}$ and $s$, where each is varied alone, it is apparent that both faster and stronger shocks yield greater numbers of energetic particles, whether considered for $E>E_{inj}$ or for $E>$ 1 MeV. Since compression ratios for shocks in the SW do not typically exceed 4, and there is no hard limit on the speed shocks can attain, faster shocks arguably have greater potential as particle accelerators.
It is insightful to consider fluences for the configurations where $V_{sh}$ and $s$ are varied together, since this is essentially comparing the acceleration efficiencies of strong slower-moving shocks against weaker fast-moving shocks. For these cases, fluences remain fairly constant for all combinations of $V_{sh}$ and $s$ for $E>E_{\text{inj}}$. However, if only the number of higher-energy particles is considered, that is, for $E >$ 1 MeV, the results are more interesting: As a function of compression ratio, the fluences increase as shocks become stronger, despite simultaneously becoming slower. As a function of the shock speed, these fluences become smaller for faster shocks for which the accompanying compression ratios are smaller. This implies that strong slower-moving shocks are able to produce a greater number of energetic particles than weak fast-moving shocks, at least within the parameter ranges considered and for shocks with $V_2^{\prime}/V_1^{\prime}=3$.
From the solar wind to energetic storm particles {#sec:SWtoESPs}
================================================
In this section, the development of particles injected from the suprathermal SW into energetic particles associated with ESP events is considered in greater detail. The underlying question of how SW particles can be accelerated to the much higher energies at which ESPs are observed is visually represented in Figure \[fig:heatingratio\]. The intensities observed during the *Halloween* ESP event of 2003 October 29 are shown as an example of typical energy spectra observed during such events. Note this particular spectrum is reproduced well by the power law associated with $s=4$ [e.g. @Giacalone2015].
The acceleration process is initiated with the heating of the SW energy distribution, where the $\kappa$-function describing it is allowed to change in response to changes in flow speed, density and temperature across the shock as shown in Section \[sec:SWheating\]. The importance of this initial heating cannot be understated: consider, firstly, a spectrum shock-accelerated from the original, undisturbed SW distribution. It would likely not extend up to the energies of observed ESPs because of the limited time available for acceleration. Secondly, assuming the injection threshold for DSA is much larger than thermal energies, the larger intensities associated with the heated SW could allow shock-accelerated intensities to reproduce those of typical ESPs. Indeed, the bottom panel of Figure \[fig:heatingratio\] illustrates that the heated SW distribution increases intensities of potential DSA seed particles by up to a factor of $10^{4}$, while increasing suprathermal tail intensities a hundredfold at least. Interestingly, observations suggest that peak fluxes during ESP events display some dependence on CME sheath temperature [@Dayehetal2018], while the broadening of SW distributions (which influence eventual intensities of shock-accelerated particles) is predominantly dictated by temperature changes across the shock.
![Top: The undisturbed and heated SW energy distributions at Earth, with the latter corresponding to the time at which the shock passes the Earth’s position. For reference purposes, the intensities observed by *ACE*/EPAM LEMS30/120 during an energetic particle event on 2003 October 29 is also shown. These observed intensities can be fitted with the $E^{-1}$ power law associated with a strong shock of $s=4$. Bottom: Intensity ratios of the heated and undisturbed SW distributions shown above. \[fig:heatingratio\]](heatingratio.pdf)
---- --------- ---- -----
10 $-2.0$ 30 0.7
30 $-1.8$ 30 1.0
60 $-1.25$ - 1.5
---- --------- ---- -----
: Parameters used to fit Eq. \[eqn:DSAform\] to the solutions of Figure \[fig:spectra\_einjvar\] for different injection energies $E_{\text{inj}}$. All quantities are as described for Table \[tab:bestfit\]. []{data-label="tab:bestfit_Einj"}
Injection from the heated SW distribution {#subsec:Einjvar}
-----------------------------------------
In the simplest terms, the injection threshold for DSA can be considered the energy at which particles should at least propagate upwind in order to make repeated shock crossings. This injection energy can be inferred by matching theoretically expected shock-accelerated spectra with observations [e.g. @NeergaardParkerZank2012]. It can also be estimated using the argument that the particle anisotropy must be small in order for DSA to be valid [@GiacaloneJokipii1999; @Zanketal2006]. For the typical values implemented in this study ($V_1^{\prime}=$ 600 km s$^{-1}$, $s=3$), injection energies are in the order of a few keV for parallel shocks, for which injection from an unheated Maxwellian seed particle distribution would indeed be possible [@NeergaardParkerZank2012]. However, for quasi-perpendicular shocks, and in the absence of magnetic field-line wandering, it is estimated that $E_{\text{inj}}\gtrsim$ 0.1 MeV [see also @Li2017]. The prior heating of the SW distribution becomes especially useful in this situation. See also the simplified method used by [@Huetal2017] to estimate injection energies.
![Modelled shock-accelerated spectra at the Earth at the time of the shock passage for different injection energies. Step-like lines represent SDE solutions while the solid lines are fits of Eq. \[eqn:genDSAform\] with parameters as listed in Table \[tab:bestfit\_Einj\]. The dashed grey line represents the heated SW distribution. The ESP observations are the same as shown in Figure \[fig:heatingratio\]. \[fig:spectra\_einjvar\]](spectra_einjvar.pdf)
The halo-CME shocks considered in this study are mostly quasi-parallel if a [@Parker1958] spiral is assumed for the magnetic field. Here, $E_{\text{inj}}$ is varied between 10 and 60 keV, spanning an energy domain extending from the thermal peak to the suprathermal tail of the heated SW distribution, and bounded by the aforementioned estimates for parallel and quasi-perpendicular shocks. The effects of these varied injection thresholds on shock-accelerated spectra are shown in Figure \[fig:spectra\_einjvar\]. As before, Eq. \[eqn:genDSAform\] is fitted for each case and the parameters listed in Table \[tab:bestfit\_Einj\]. The reference configuration as specified in Table \[tab:shspeedV\] is used, with $\lambda_0=$ 0.06 AU, $V_{sh}=$ 2400 km s$^{-1}$, and $s=3$. The standard features are visible: the spectra are distributed according to $E^{-1.25}$ as expected for $s=3$ and rolls over exponentially at higher energies. Spectra appear to terminate at similar energies for the injection thresholds considered, as this is presumably governed by the acceleration time.
It can be seen from Figure \[fig:spectra\_einjvar\] that injecting particles from the heated SW distribution yields high intensities in the energy domain typically associated with ESPs. In the case of $E_{\text{inj}}=$ 60 keV, the shock-accelerated intensities are quite similar to those observed for the ESP event included in Figure \[fig:spectra\_einjvar\] as an example. Should the injection speed be defined as the minimum possible speed a particle needs to stay ahead of the shock, whilst following a Parker field line with a 45$^{\circ}$ angle ahead of the shock nose, the injection speed can be written from focused transport theory as $v_{\text{inj}}=\lvert V_1^{\prime} - V_{sh} \rvert/ \cos\left(45^{\circ}\right)$. For a fast halo-CME shock with $V_{sh}= 2400$ km s$^{-1}$ and a more typical upstream flow speed of 400 km s$^{-1}$, the injection speed at 1 AU in the upstream flow frame is $v_{\text{inj}}=$ 2828 km s$^{-1}$, which corresponds to an injection energy of $E_{\text{inj}} \sim 50$ keV. This is reasonably similar to the 60 keV injection energy required to fit the presented observations using the heated $\kappa$-distribution as a source.
Note the presented simulations are not intended to reproduce observations exactly, nor is it posited that the spectrum observed during the presented ESP event is the unambiguous result of particles accelerated from the SW. The aforementioned results do however demonstrate that the prior heating of the SW plasma and energy distribution complements the shock acceleration process well and might even be necessary when considering shocks with large injection energies.
The evolution of shock-accelerated distributions during the shock passage {#subsec:spec_evolution}
-------------------------------------------------------------------------
Particles of sufficient energy can be injected into the DSA process at any time during a CME shock’s passage between the Sun and the Earth. However, ESP events entail the local enhancement of particle intensities as viewed by spacecraft at Earth. The largest enhancement is naturally expected when the shock reaches Earth. Solving the time-backward SDEs for the reference configuration and different starting positions of the shock, the energy spectra and intensity profiles of shock-accelerated particles are simulated for the approach and aftermath of the shock’s passage at Earth. These are shown in Figures \[fig:spectra\_shockapproach\] and \[fig:profiles\_shockapproach\], respectively.
![Modelled shock-accelerated spectra at the Earth at different times during the approach of the shock. Step-like lines represent SDE solutions, dashed lines indicate flat segments, and the thick solid lines indicate power law segments that are distributed as $E^{-1.25}$ as expected from a shock with $s=3$. \[fig:spectra\_shockapproach\]](spectra_shockapproach.pdf)
The time-evolution of energy spectra during the approach of the shock is considered first. From Figure \[fig:spectra\_shockapproach\], the most recognizable aspect is of course the power-law form of the spectrum at the time of the shock’s arrival at Earth, that is, where $r_{sh}=$ 1 AU, followed by a cut-off at higher energies. However, this spectrum appears notably different when the shock is just 0.01 AU away, which for $V_{sh}=2400$ km s$^{-1}$ is a few minutes before its arrival at Earth. The differences, which become more stark for larger distances between the shock and the Earth, include lower overall intensities and a downturn in the spectrum at low energies. The progressively lower intensities follow merely as a result of the source of energetic particles (i.e. the shock) being further away. The low-energy downturns are more severe when the shock is far from the Earth, but manifest as flattened spectra for smaller distances from the Earth. This likely follows because shock-accelerated particles are adiabatically cooled in the expanding SW while they propagate ahead of the shock toward Earth. These steep spectra with positive power-law indices at lower energies are known spectral characteristics of adiabatic energy losses [e.g. @MoraalPotgieter1982; @Straussetal2011].
![Profiles of energetic particle intensities at different energies as viewed by an observer at Earth during the passage of the shock. The vertical dotted line represents the moment of the shock’s arrival at Earth. The shaded region accentuates where energy spectra are approximately flat. \[fig:profiles\_shockapproach\]](profiles_shockapproach.pdf)
A particularly interesting feature visible in Figure \[fig:spectra\_shockapproach\] is the flattening of spectra during the approach of the shock. Similar flattening has recently been reported in observations of proton spectra between 50 keV and 1 MeV prior to the passage of an IP shock at Earth [@Larioetal2018]. It is conceivable that this is the result of the competing effects of shock acceleration and adiabatic cooling of particles. Whereas shock acceleration tends to distribute particles according to $E^{\gamma_s}$, where $\gamma_s=-1.25$ for $s=3$ according to Eq. \[eqn:specInd\_DSA\], cooling tends to force protons into a characteristic $E^{+1}$ spectrum at Earth. When the shock is nearer to Earth, the shock-accelerated component dominates, whereas particles transported from the shock while it is still further upwind have been cooled to a greater extent. Where these effects balance, the spectrum flattens. As a result of progressively stronger cooling, Figure \[fig:spectra\_shockapproach\] shows the flattened segments narrow and move to higher energies for larger distances between the shock and the Earth. Figure \[fig:profiles\_shockapproach\] shows intensity profiles similar to how the observations of [@Larioetal2018] are presented. The flat energy spectra can be discerned from the coinciding intensity profiles of 0.25 to 1 MeV particles. This energy range can be broadened in the simulations by specifying a lower injection energy or assuming a stronger shock, e.g. $s=4$, for which the shock-accelerated spectrum will be less steep. These flat segments are visible for shock positions up to at least 0.15 AU away from the Earth, that is, for nearly three hours prior to its arrival. Note the shock speed of 2400 km s$^{-1}$ implemented in these simulations is very fast. If a more typical shock speed of e.g. 1000 km s$^{-1}$ [e.g. @Makelaetal2011] is assumed, this effect would be visible for a longer time prior to the shock’s arrival at Earth.
The intensity profiles shown in Figure \[fig:profiles\_shockapproach\] largely resemble that of typical ESP events: a large onset of particle intensities over a relatively short time is visible before the arrival of the shock, followed by a more gradual decline of intensities after it has passed. Note that the peaks of the simulated profiles, corresponding to the arrival of the IP shock at Earth, are not as sharp as observed intensities tend to show. These peaks are often associated with large anisotropies [e.g. @Larioetal2005a], which can be more appropriately modelled using focused transport models [@Zuoetal2011; @leRouxWebb2012]. The TPE in this study is solved for an omni-directional distribution function (as discussed in Appendix \[sec:countingparticles\]) and describes only the near-isotropic particle component.
Seed-particle energies and initial positions {#subsec:accsites}
--------------------------------------------
It only remains to be investigated where and from which energies SW particles contributing to any given observational point are accelerated. To do this, the binning technique discussed in Section \[subsec: SDEs\] is implemented. Firstly, pseudo particles are traced time-backwards from the observational point $(r_{\text{obs}},E_{\text{obs}}) = (\text{1 AU, 1 MeV})$, whereafter the average particle amplitudes are calculated for each $(r,E)$-bin at $t^{\prime}=0$. As before, the simulation is run for the reference parameters listed in Table \[tab:shspeedV\]. The result is shown as a colour-scaled plot in Figure \[fig:contours1MeV\].
![Initial radial and kinetic energy distribution of SW particles contributing to 1 MeV intensities at Earth during an ESP event. The color scale indicates the average particle amplitudes at each $(r,E)$-point, where the maximum amplitude has been normalised to unity. The observational point, from where pseudo-particle trajectories are traced in a time-backwards fashion, is shown as a marker at the intersection of vertical and horizontal dashed lines indicating Earth’s position and 1 MeV, respectively. *The online version shows the evolution of this plot as the shock progresses toward Earth, where the position of the shock front is indicated using a solid vertical line.* \[fig:contours1MeV\]](sf_1MeV_contours.pdf)
This plot essentially maps the relative contributions of particles from different initial positions and energies to the intensities of 1 MeV particles at Earth’s position at the time of the shock’s passage. As such, they reveal a great deal of insight into the probable original energies and locations of seed particles. Figure \[fig:contours1MeV\] shows the undisturbed distribution (that is, before the departure of the shock from near to the Sun) of SW particles that would eventually contribute to 1 MeV intensities at Earth. It is illustrated that the largest contributions are from particles initially located upwind from Earth and accelerated from energies ranging from tens to a few hundred keV. The lower limit of this range corresponds to the 60 keV injection energy implemented in this study. The contributions of particles either cooled to 1 MeV from higher energies or convected to Earth without incurring energy changes are minuscule by comparison. During the shock transit, the distribution of contributing particles will move closer to the observational point, until, at the time of the shock passage at Earth, only that point will be populated on the plot.
The much larger relative contribution of particles accelerated to $E_{\text{obs}}$ from lower energies illustrates that DSA at the travelling IP shock is the chief contributor to intensities during the simulated ESP event.
Discussion and conclusions {#sec:conclusions}
==========================
In this study, the acceleration of SW particles at halo CME-driven IP shocks is investigated. The thermal and suprathermal components of the SW velocity (or equivalent energy) distributions are collectively described using $\kappa$-functions. Furthermore, these SW distributions are transformed in response to simulated shock transitions in plasma properties upon which they depend, such as the flow speed, number density, and temperature. These transformed (or heated) distributions are consequently specified as source spectra, from which particles with sufficient energy can be injected into the DSA process at the shock. To model this acceleration process, SDEs equivalent to the Parker TPE are solved in a time-backwards fashion. Using the combined approach of the pre-injection heating of the SW distribution and DSA, simulations reveal a number of noteworthy results with regards to the particle acceleration at IP shocks and particularly of SW particles.
The simulations are shown to produce the classical spectral features of DSA. However, in cases where diffusion length scales are small relative to the shock width, shock-accelerated spectra do not display the spectral indices associated with the full compression ratio. Such situations may arise in the case of high levels of magnetic turbulence, which can decrease diffusion length scales, or in the case of broader compressions in the SW [e.g. @Giacaloneetal2002]. At any rate, the softer power-law distributions that are attained as a result notably reduces the overall intensities of shock-accelerated particles. Furthermore, it is shown that the highest attainable energies of shock-accelerated spectra are limited by the transit time of the shock. Reduced diffusion length scales may serve to better confine particles near the shock, thereby allowing spectra to be accelerated to higher energies within the available time. Intensities of higher-energy particles ($E>$ 1 MeV) are shown to be particularly sensitive to this time limit.
The dependence of the acceleration process on shock properties such as its speed and compression ratio is also investigated. Fast shocks contribute appreciably to the heating of the SW distribution. While fast-moving shocks have shorter associated transit times, allowing less time for particle acceleration, larger shock speeds have a net positive effect on particle acceleration. Through their contribution to larger flow speeds they reduce diffusion length scales, improving particle confinement at the shock. They also reduce the time required for acceleration: since larger shock speeds imply larger differences of flow speeds across the shock, a particle scattered across it experiences a larger mean energy gain per crossing; the scattering centres can also be thought of converging on the shock at a higher rate for fast shocks.
The compression ratio, as measure of shock strength, also affects the magnitude of flow speed transitions across the shock, providing the actual factor by which both the shock-frame flow speed and number density change across the shock. In particular, larger compression ratios increase intensities of the SW distribution across the shock, since the $\kappa$-function is normalised to the number density. While it has a more modest effect on acceleration times than the shock speed, its effect on overall particle intensities is significant, since the spectral indices of the shock-accelerated spectra depend directly on the compression ratio. When strong and fast shocks are compared as particle accelerators, it is found that strong slower-moving shocks produce larger numbers of energetic particles than weak fast-moving shocks. The compression ratio, being directly associated with the steepness of accelerated energy distributions, is therefore identified to be the greater limiting factor between these two shock properties.
With regards to simulating ESP events, the prior heating of the SW distribution during the shock passage is found to complement the DSA process well: it provides greater intensities of potential seed particles for DSA, especially where large injection energies are considered, and allows shock-accelerated spectra to achieve large enough intensities at sufficiently high energies to reproduce typical ESP events. This result is consistent with observations reporting larger peak particle fluxes during ESP events for warmer CME sheath temperatures [@Dayehetal2018]. Furthermore, simulations of shock-accelerated intensities at Earth reveal significant flattening of spectra forming ahead of the shock during its approach. This is found to result from shock-accelerated particles experiencing adiabatic cooling in the expanding SW while propagating toward Earth ahead of the shock. This provides a potential explanation for similar features recently reported in observations [@Larioetal2018].
Finally, taking advantage of the time-backward tracing of phase-space density elements, with their flux contributions weighted according to particle amplitude, it is revealed that most SW particles contributing to intensities during a simulated ESP event are transported to Earth from upwind and are accelerated from energies ranging from tens to a few hundred keV. Due to the outsized fraction of particles accelerated to 1 MeV from lower energies, it can be concluded that DSA is indeed a chief contributor to intensities at Earth during ESP events.
It is ultimately concluded that with the combination of prior heating and shock acceleration, energetic particles can be accelerated from the suprathermal SW at IP shocks, and that these particles may make appreciable contributions to intensities during observed ESP events. Also, the reproduction of the observed flat energy spectra ahead of shock passages [@Larioetal2018], resulting from the competing processes of DSA and adiabatic cooling, illustrates the advantage of studying particle acceleration in association with more general transport processes. It is furthermore demonstrated that the SDE approach, while shown in Appendix \[sec:benchmark\] to yield comparable results to finite-difference methods, can be used to reveal unique physical insights with regards to particle transport and acceleration at IP shocks. Along with its computational advantages, the SDE model developed in this study can be expanded to include more spatial dimensions, providing a means to explore more complicated shock geometries. Further research using this SDE approach may also include more refined characterisations of transport coefficients, the solution of focused transport equations to study large-anisotropy events, and the reproduction of various shock-related particle events, such as the acceleration of particles associated with co-rotating interactions regions.
PLP and RDS acknowledges the financial support of the South African National Research Foundation (NRF). Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the NRF.
Counting particles {#sec:countingparticles}
==================
The distribution function and number density {#subsec: def_f}
--------------------------------------------
![Left: Representations of volumes in real and phase space, $d^3x$ and $d^3p$, for which the distribution function is defined according to Eq. \[eqn:distfunc\_def\]. $d^3x$ contains all the particles at position $\vec{r}$ with a momentum $\vec{p}$ in $d^3p$. Right: A spherical momentum coordinate system, where $p_z$ is along the magnetic field line and the particle momentum $\vec{p}$ is defined in terms of the scalar momentum $p$, gyro-phase angle $\phi$ and pitch angle $\theta$. \[fig:def\_dist\_func\]](dist_func_def.pdf "fig:") ![Left: Representations of volumes in real and phase space, $d^3x$ and $d^3p$, for which the distribution function is defined according to Eq. \[eqn:distfunc\_def\]. $d^3x$ contains all the particles at position $\vec{r}$ with a momentum $\vec{p}$ in $d^3p$. Right: A spherical momentum coordinate system, where $p_z$ is along the magnetic field line and the particle momentum $\vec{p}$ is defined in terms of the scalar momentum $p$, gyro-phase angle $\phi$ and pitch angle $\theta$. \[fig:def\_dist\_func\]](momentum_coord.pdf "fig:")
Consider a number of particles $d \mathcal{N}$ in a small volume $d^3x$ at a position $\vec{x} = (x,y,z)$ and with momentum in $d^3p$ around $\vec{p} = (p_x, p_y, p_z)$ as illustrated in Figure \[fig:def\_dist\_func\] (left). The distribution function $f\left(\vec{x},\vec{p},t\right)$ is introduced such that $$\label{eqn:distfunc_def}
d\mathcal{N} = f\left(\vec{x},\vec{p},t\right) d^3x\ d^3p \text{.}$$ This can be related to the ordinary number density $n$ by integrating over all momentum space, $$\label{eqn:volint_f_numdense}
n = \int f\left(\vec{x},\vec{p},t\right) d^3p \text{.}$$The above volume integral can be re-written by considering a momentum coordinate system as shown in Figure \[fig:def\_dist\_func\] (right). Assuming the gyro-centre at the origin and the magnetic field line about which the particle gyrates along $p_z$, that is $\vec{B}=B\hat{z}$, the azimuthal angle $\phi$ represents the particle’s gyrophase and the polar angle $\theta$ its pitch angle.
This allows the momentum-space volume element to be expressed in spherical coordinates as $d^3p = p^2\ d\Omega\ dp$, where $d\Omega = \sin\theta\ d\theta\ d\phi$ is a solid-angle element about $\vec{p}$ and $p = |\ \vec{p}\ |$ is the particle’s scalar momentum. Noting that $\vec{p}$ can hence be expressed in terms of the coordinates $(p,\ \phi,\ \theta)$, Eq. \[eqn:volint\_f\_numdense\] can be rewritten as $$\begin{aligned}
\label{eqn:volint_f_numdense_spherical}
n &= \int\limits_0^{\infty} \int\limits_{0}^{2\pi} \int\limits_{0}^{\pi} p^2\ f\left(\vec{x},p,\theta,\phi,t\right) \sin\theta\ d\theta\ d\phi\ dp \nonumber \\
&= \int\limits_0^{\infty}p^2\ \int\limits_{\Omega} f\left(\vec{x},p,\theta,\phi,t\right) d\Omega\ dp \text{,}\end{aligned}$$ where $\int_\Omega$ represents the integral over all solid angles. However, since particle detectors typically cannot resolve statistically significant numbers of particles incident from only one particular direction, the average number of particles per unit solid angle in momentum space is considered. Accordingly, the omni-directional distribution function (that is, averaged over all directions in $\vec{p}$-space for a fixed $p$) is defined as $$\label{eqn:omni_f}
f_0\left(\vec{x},p,t\right) = \dfrac{\int\limits_{\Omega} f\left(\vec{x},\vec{p},t\right) d\Omega}{\int_{\Omega} d\Omega} = \frac{1}{4\pi} \int\limits_{\Omega} f\left(\vec{x},\vec{p},t\right) d\Omega\ \text{,}$$ since $\int_{\Omega} d\Omega = \int\limits_0^{2\pi} d\phi \int\limits_0^\pi \sin\theta d\theta = 4\pi$. Note that $f_0\left(\vec{x},p,t\right)$ is the distribution function solved for in the [@Parker1965] transport equation. Note furthermore from Eq. \[eqn:omni\_f\] that $f\left(\vec{x},\vec{p},t\right)$ can be taken out of the integral over solid angle if it is assumed to be gyro- and isotropic, that is, independent of both $\phi$ and $\theta$. In this case it follows that $f\left(\vec{x},\vec{p},t\right)=f_0\left(\vec{x},p,t\right)$. From Eqs. \[eqn:volint\_f\_numdense\_spherical\] and \[eqn:omni\_f\] the number density can be expressed as $$\begin{aligned}
\label{eqn:omni_numdense}
n &= \int\limits_0^{\infty} 4\pi p^2\ f_0\left(\vec{x},p,t\right) dp \\ &= \int\limits_0^{\infty} U_{p}\left(\vec{x},p,t\right) dp \text{,} \label{eqn:omni_difdense}\end{aligned}$$ where $U_{p}\left(\vec{x},p,t\right)=4\pi p^2\ f_0\left(\vec{x},p,t\right)$ is known as the differential number density. Here, $U_p$ represents the number density between two spherical shells in $\vec{p}$-space with radii of $p$ and $p+dp$, respectively. While spacecraft cannot resolve volume densities, $U_p$ does prove useful to relate quantities they measure to those introduced above.
Toward the differential intensity {#subsec:toward_j}
---------------------------------
Spacecraft essentially measure the flux of particles observed for a particular viewing direction within a particular momentum (or energy) band. This is known as the differential intensity $j_p$, and is the number of particles detected per momentum interval per unit time per unit area of observed space per unit solid angle. Dimensionally, this can be attained by the product of the differential number density and the speed at which particles move toward the detector through a surface perpendicular to their motion, averaged over all solid angles. That is, $$\begin{aligned}
\label{eqn:j_difnumdense_fo}
j_p \left(\vec{x},p,t \right) &= \frac{v\ U_p\left(\vec{x},p,t \right)}{4\pi} \nonumber \\
&= v\ p^2\ f_0\left(\vec{x},p,t\right) \text{,} \end{aligned}$$ which, in turn, relates the differential intensity to the distribution function. It is furthermore useful to express the differential intensity in terms of kinetic energy to compare with spacecraft observations. The conversion of $j_p$, expressed per momentum interval $dp$, to $j_E$, expressed per interval of kinetic energy $dE$, is carried out by noting that $j_p\ dp = j_E\ dE$ due to the conservation of particles. It hence follows that $$\label{eqn:j_E_j_p}
j_E = j_p \frac{dp}{dE} \text{.}$$Note that since $p^2 = \left( E_T^2 - E_0^2 \right)/c^2$, where $E_T = E + E_0$ with $E_T$, $E$ and $E_0$ denoting the total, kinetic and rest-mass energies, respectively, it follows that $$\label{eqn:dp_dE}
\frac{dp}{dE} = \frac{E+E_0}{pc^2} = \frac{E_T}{pc^2} = \frac{\gamma m_0 c^2}{\gamma m_0 v c^2}=\frac{1}{v} \text{,}$$ where $\gamma$ is the Lorentz factor and $m_0$ is the rest mass. Then, using Eq. \[eqn:j\_E\_j\_p\], $j_E$ can be related to the distribution function as follows $$\label{eqn:j_E_f0}
j_E = \frac{j_p \left(\vec{x},p,t \right)}{v} = p^2\ f_0\left(\vec{x},p,t\right) \text{.}$$
On kappa velocity distribution functions {#subsec:on_kappa_funcs}
----------------------------------------
The normalization constant $A_{\kappa}$ of the standard $\kappa$-function introduced in Eq. \[eqn:kappafunc\] is obtained by setting its integral over all phase space equal to the number density of the solar wind. Assuming an omni-directional (or three-dimensional) $\kappa$-function, averaged as per Eq. \[eqn:omni\_f\], the expression given in Eq. \[eqn:omni\_numdense\] can be used to calculate the normalization constant. Note, however, because $f_{\kappa}$ is often expressed as a function of velocity instead of momentum, the analogous expression in velocity coordinates, namely $$\label{eqn:omni_numdense_v}
n = \int\limits_0^{\infty} 4\pi v^2\ f_{\kappa}\left(v\right) dv \text{,}$$ is used to calculate the number density. Bear in mind that for use in the transport equation, e.g. when specifying $f_s$ in the source function, the conversion $f_s = f_{\kappa}/m_p^3$ applies, because $\int 4\pi p^2\ f_s\ dp = \int 4\pi (m_p v)^2\ f_s\ d(m_p v) \equiv \int 4\pi v^2\ f_{\kappa}\ dv $. Here, only the proton rest mass $m_p$ is used, since this study is concerned with solar wind protons with kinetic energies typically much smaller than their rest-mass energy. A non-relativistic description is therefore applicable. Note furthermore that Eq. \[eqn:omni\_numdense\_v\] is also known as the zeroth-order velocity moment of $f_{\kappa}$, and that it can be cast into a more general form for the $n^{\text{th}}$-order moment, namely $$\label{eqn:nth_order_moment}
{\langle v^n \rangle} = \int\limits_0^{\infty} 4\pi v^{n+2}\ f_{\kappa}\left(v\right) dv \text{,}$$ which, upon substituting Eq. \[eqn:kappafunc\], becomes $$\label{eqn:nth_order_kappa}
{\langle v^n \rangle} = 4\pi A_{\kappa} \int\limits_0^{\infty} v^{n+2}\ \left( 1+\frac{v^2}{\kappa v_{\kappa}^2} \right)^{-(\kappa+1)} dv \text{,}$$ from which the normalization constant $A_{\kappa}$ can be calculated. The integral in Eq. \[eqn:nth\_order\_kappa\] is evaluated by applying the transformation $v^2 / (\kappa v_{\kappa}^2) :\rightarrow \eta$, which allows the expression to be rewritten as $$\label{eqn:nth_order_kappa_transformed}
{\langle v^n \rangle} = 2\pi A_{\kappa}\ (\kappa v_{\kappa}^2)^{(n+3)/2} \int\limits_0^{\infty} \eta^{(n+1)/2}\ \left( 1+\eta \right)^{-(\kappa+1)} d\eta \text{.}$$ Note that the integral is now in the form of the $\beta$-function, given by $$\label{eqn:beta_function}
\beta(x,y) = \int\limits_0^{\infty} \eta^{x-1}\ (1+\eta)^{-(x+y)} d\eta = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \text{.}$$ Recognizing that $x = (n+3)/2$, $y = \kappa - (n+1)/2$, and $x+y = \kappa +1$, the integral in Eq. \[eqn:nth\_order\_kappa\_transformed\] can be solved by invoking the identity given in Eq. \[eqn:beta\_function\], so that $$\label{eqn:nth_order_kappa_solved}
{\langle v^n \rangle} = 2\pi A_{\kappa}\ (\kappa v_{\kappa}^2)^{(n+3)/2}\ \frac{\Gamma((n+3)/2)\Gamma(\kappa - (n+1)/2)}{\Gamma(\kappa +1)} \text{,}$$ from which the normalization constant can be determined as $$\label{eqn:A_kappa}
A_{\kappa} = \frac{n_{sw}}{2\pi(\kappa v_{\kappa}^2)^{3/2}}\ \frac{\Gamma(\kappa+1)}{\Gamma(3/2)\Gamma(\kappa-1/2)}$$ by setting ${\langle v^{n=0} \rangle}=n_{sw}$, yielding the same expression given in Eq. \[eqn:kappanormconst\]. The standard kappa function is hence given by $$\label{eqn:A_kappa_final}
f_{\kappa} = \frac{n_{sw}}{2\pi(\kappa v_{\kappa}^2)^{3/2}}\ \frac{\Gamma(\kappa+1)}{\Gamma(3/2)\Gamma(\kappa-1/2)}\ \left( 1+\frac{v^2}{\kappa v_{\kappa}^2} \right)^{-\kappa -1} \text{.}$$
Equivalence of SDE and finite-difference numerical methods: Application to an event during the 2003 Halloween epoch {#sec:benchmark}
===================================================================================================================
In a similar study, where DSA is also assumed as an acceleration mechanism in simulating ESP events, [@Giacalone2015] utilizes a finite-difference numerical scheme to solve the [@Parker1965] TPE in a single spatial dimension for a strong fast-moving shock ($V_{sh}=$ 1900 km s$^{-1}$, $s=4$). In that study, the energetic particle event, or *Halloween* event, of 2003 October 29 is considered as an application for which observed energy spectra and temporal profiles are reproduced. To demonstrate their equivalence as numerical methods, the SDE approach of this study is implemented to reproduce particle intensities for the same event using a similar parameter configuration to that of [@Giacalone2015]. The source function in that study is bound to a numerical grid, which does not exist in the SDE approach. For the purposes of this application, the source function is instead specified at the shock as a very soft power law in momentum ($p^{-7}$), where simulated intensities are retroactively normalised to observed intensities. Furthermore, the diffusion coefficient is implemented as specified in that study, that is, of order $10^{19}$ cm$^2$ s$^{-1}$ near the Sun and with a momentum dependence of $\sim p^{1.5}$ [refer to @Giacalone2015 for further details].
 
The resultant energy spectra and time profiles of energetic particle intensities are shown in Figure \[fig:halloween\]. Overall, the finite-difference and SDE solutions are similar. Recall that the SDEs must be solved for a large number of pseudo particles for each energy, position or point in time at which intensities are sought. The resolution of SDE solutions therefore merely depends on the number of times the simulation is repeated and how the observational points are distributed, as opposed to the fixed grid resolution of finite-difference schemes. For example, in Figure \[fig:halloween\], a greater number of more tightly spaced observational points are chosen in energy than in time, as shown in the left and right-hand side panels, respectively. Note that the SDE solutions tend to undershoot the finite-difference solutions where steeper gradients exist. This follows because there is no passage of information between pseudo particles and their associated observational points as would exist between adjacent points in finite-difference schemes. The SDEs typically yield less gradual intensity gradients as a result. This is useful to bear in mind when fitting functions to shock-accelerated spectra simulated using these two different approaches, since the high-energy cut-off is likely to be more abrupt for the SDEs than for finite-difference solutions.
Note, as an aside, that simulated intensities of either method reproduce observations better near the shock than away from it. This follows since the only source of particles in the simulations is situated on the shock itself, while the observed intensities include contributions from other sources. In any event, the comparison of the simulations generated using the two different numerical methods demonstrate the equivalence of their results.
natexlab\#1[\#1]{}
, A. D., & [le Roux]{}, J. A. 2013, , 772, L26
, W. I., [Leer]{}, E., & [Skadron]{}, G. 1977, ICRC, 11, 132
, A. R. 1978, , 182, 147
, A. R. 1978, , 182, 443
, R. D. and [Ostriker]{}, J. P. 1978, , 221, L29
, D. A., [Cline]{}, T. L., [Desai]{}, U. D., & [McDonald]{}, F. B. 1962, , 67, 4983
, D., [Pop]{}, A. R., & [Spitkovsky]{}, A. 2015, , 798, L28
, C., [Ruffolo]{}, D., [Desai]{}, M. I., & [Mason]{}, G. M. 2005, , 633, L53
, Y. F., & [Meyer-Vernet]{}, N. 1991, , 96, 5825
, K., [Schwadron]{}, N. A., [Mason]{}, G. M., [et al.]{} 2000, , 105, 23107
, M. R., [Hamilton]{}, D. C., [Gloeckler]{}, G., [Bochsler]{}, P., & [Sheldon]{}, R. B. 1996, , 23, 1191
, M. I., & [Giacalone]{}, J. 2016, LRSP, 13, 3
, M. I., [Mason]{}, G. M., [Mazur]{}, J. E., & [Dwyer]{}, J. R. 2006, , 124, 261
, M. A, [Desai]{}, M. I., [Ebert]{}, R. W., [et al.]{} 2018, J. Phys. Conf. Ser., 1100, 012008
, L. O. 1983, Rept. Progr. Phys., 46, 973
, D. C. and [Ramaty]{}, R. 1985, , 298, 400
, D. C., [Jones]{}, F. C., & [Reynolds]{}, S. P. 1990, , 360, 702
, D. C., [Baring]{}, M. G., & [Jones]{}, F. C. 1995, , 453, 873
, V., [Moreno]{}, G., [Palmiotto]{}, F., & [Hedgecock]{}, P. C. 1973, , 78, 3714
, J. 2005, , 628, L37
, J. 2012, , 761, 28
, J. 2015, , 799, 80
, J., [Burgess]{}, D., [Schwartz]{}, S. J., & [Ellison]{}, D. C. 1992, , 19, 433
, J., & [Jokipii]{}, J. R. 1999, , 520, 204
, J., [Jokipii]{}, J. R., & [K[ó]{}ta]{}, J. 2002, , 573, 845
, N., [Yashiro]{}, S., [Liu]{}, Y., [et al.]{} 2005, , 110, A09S15
, M. A., [Mace]{}, R. L., [Baluku]{}, T. K., [Kourakis]{}, I., & [Saini]{}, N. S. 2009, Phys. Plasmas, 16, 094701
, G. C. and [Lario]{}, D. and [Decker]{}, R. B. 2009, AIP Conf. Proc, 1183, 19
, J., [Li]{}, G., [Ao]{}, X., [Zank]{}, G. P., & [Verkhoglyadova]{}, O. 2017, , 122, 10
, K. & [Valtonen]{}, E. 2009, AnGeo, 27, 767
, J. R. 1966, , 146, 480
, F. C., & [Ellison]{}, D. C. 1991, , 58, 259
, M.-B. 1993, , 98, 19
, M.-B., [Wibberenz]{}, G., & [Hucke]{}, S. 1992, , 394, 351
, H., [Petrosian]{}, V., [Ryu]{}, D., & [Jones]{}, T. W. 2014, , 788, 142
, K. L., & [Dalla]{}, S. 2017, , 212, 1107
, F.-J., [Qin]{}, G., [Zhang]{}, L.-H. 2017, , 845, 43
, W. M & [Achterberg]{}, A. 1994, , 286, 314
, G. F. 1977, DoSSR, 234, 1306
, D. & [Decker]{}, R. B. 2002, , 29, 1393
, D., [Decker]{}, R. B., [Livi]{}, S., [et al.]{} 2005a, , 110, A09S11
, D., [Hu]{}, Q., [Ho]{}, G. C., [Decker]{}, R. B., [Roelof]{}, E. C., & [Smith]{}, C. W. 2005b, ESASP, 592, 81L
, D., [Berger]{}, L., [Wilson]{}, L. B. III, [et al.]{} 2018, J. Phys. Conf. Ser., 1100, 012014
, M. A. 1983, , 88, 6109
, M. P. 2004, Phys. Plasmas, 11, 1308
, J. A. & [Arthur]{}, A. D. 2017, J. Phys.: Conf. Ser., 900, 012013
, J. A., [Potgieter]{}, M. S., & [Ptuskin]{}, V. S. 1996, , 101, 4791
, J. A. & [Webb]{}, G. M. 2012, , 746, 104
, G. 2017, Science China Earth Sciences, doi: 10.1007/s11430-017-9083-y
, G. 2015, , 809, 111
, G., & [McComas]{}, D. J. 2013, , 175, 183
, P., [Gopalswamy]{}, N., [Akiyama]{}, S., [Xie]{}, H., & [Yashiro]{}, S. 2011, , 116, A08101
, M., [Pierrard]{}, V., & [Riley]{}, P. 1997, , 24, 1151
, M. A., & [V[ö]{}lk]{}, H. J. 1995, , 300, 605
, A. & [Kirk]{}, J. G. 1999, , 347, 391
, G. 1955, Rend. Circolo Mat. Palermo, 4, 48
, K. D., [Engelbrecht]{}, N. E., [Strauss]{}, R. D., [Moeketsi]{}, D. M., & [van den Berg]{}, J. P. 2019, Adv. Space Res., 63, 626
, H. & [Potgieter]{}, M. S. 1982, , 84, 519
, P., [Zank]{}, G. P., & [Webb]{}, G. M. 2017, , 841, 4
, L., & [Zank]{}, G. P. 2012, , 757, 97
, L., [Zank]{}, G. P., & [Hu]{}, Q. 2014, , 782, 52
, E. N 1958, , 128, 664
, E. N 1965, , 13, 9
, V., & [Lazar]{}, M. 2010, , 267, 153
, C., [Bieber]{}, J. W., [Burger]{}, R. A., & [Clem]{}, J. 2010, , 115, A12107
, M. N. S., [Pallocchia]{}, G., [Bruno]{}, R., [et al.]{} 2003, AIP Conf. Proc, 679, 489
, J. D., [Wang]{}, C., [Kasper]{}, J. C, & [Liu]{}, Y. 2005, , 32, L03S03
, O. V., [Borodkova]{}, N. L., [Eselevich]{}, V. G., [Zastenker]{}, G. N., & [Yermolaev]{}, Y. I. 2017, Cosmic Research, 55, 396
, R. M., [Gosling]{}, J. T., [Steinberg]{}, J. T., [et al.]{} 2004, , 109, A09102
, C. D, & [Moraal]{}, H. 1999, , 104, 24879
, R. D., & [Effenberger]{}, F. 2017, , 212, 151
, R. D., [Potgieter]{}, M. S., [Kopp]{}, A., & [B[ü]{}sching]{}, I., 2011, , 116, A12105
, R. D., [Potgieter]{}, M. S., [Ferreira]{}, S. E. S., [Fichtner]{}, H., & [Scherer]{}, K. 2013, , 765, L18
, T., [Haynes]{}, C. T., [Burgess]{}, D., & [Mazelle]{}, C. X. 2013, , 820, 21
, V. M. 1968, , 73, 2839
, C.-C., [Wu]{}, S. T., [Dryer]{}, M., [et al.]{} 2005, , 110, A09S17
, G. P. 2017, in Kappa Distributions: Theory and Applications in Plasmas (Elsevier), 609
, G. P., [Li]{}, G., [Florinski]{}, V., [et al.]{} 2006, , 111, A06108
, M. 1999, , 513, 409
, M. 2000, , 541, 428
, Z., [Zhang]{}, M., [Gamayunov]{}, K., [Rassoul]{}, H., & [Luo]{}, X. 2011, , 738, 168
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---
abstract: 'We present a multiple instance learning class activation map (MIL-CAM) approach for pixel-level minirhizotron image segmentation given weak image-level labels. Minirhizotrons are used to image plant roots *in situ*. Minirhizotron imagery is often composed of soil containing a few long and thin root objects of small diameter. The roots prove to be challenging for existing semantic image segmentation methods to discriminate. In addition to learning from weak labels, our proposed MIL-CAM approach re-weights the root versus soil pixels during analysis for improved performance due to the heavy imbalance between soil and root pixels. The proposed approach outperforms other attention map and multiple instance learning methods for localization of root objects in minirhizotron imagery.'
author:
- 'Guohao Yu[[](https://orcid.org/0000-0002-6850-7241)]{}'
- 'Alina Zare[[](https://orcid.org/0000-0002-4847-7604)]{}'
- 'Weihuang Xu[[](https://orcid.org/0000--0001-8463-9319)]{}'
- 'Roser Matamala [[](https://orcid.org/0000-0001-5552-9807)]{}'
- 'Joel Reyes-Cabrera [[](https://orcid.org/0000-0002-3535-8665)]{}'
- 'Felix B. Fritschi [[](https://orcid.org/0000-0003-0825-6855)]{}'
- 'Thomas E. Juenger [[](https://orcid.org/0000-0001-9550-9288)]{}'
bibliography:
- 'egbib.bib'
title: 'Weakly Supervised Minirhizotron Image Segmentation with MIL-CAM'
---
Introduction
============
Minirhizotron (MR) imaging plays an important role in plant root studies. It is a widely-used non-destructive root sampling method used to monitor root systems over extended periods of time without repeatedly altering critical soil conditions or root processes [@johnson2001advancing; @bates1937device; @waddington1971observation; @rewald2013minirhizotron]. Yet, a significant bottleneck which impacts the value of MR systems is the analysis time needed to process collected imagery. Standard analysis approaches require manual root tracing and labeling of root characteristics. Manually tracing roots collected with MR systems is very tedious, slow, and error prone. Thus, MR image analysis would greatly benefit from automated methods to segment and trace roots. There have been advancements made in this area [@wang2019segroot; @xu2019overcoming; @yu2019root]. However, the effective automated methods still require a manually-labeled training set. Although these approaches provide a reduction in effort needed over hand-tracing an entire collection of data, the generation of these training sets is still time consuming and labor intensive. In this paper, we propose a weakly supervised MR image segmentation method that relies only on image-level labels.
By relying only on weak image level labels (e.g., this image does/does not contain roots), the time and labor needed to generate a training set is drastically reduced [@papandreou2015weakly; @pinheiro2015image; @zhou2016learning; @roy2017combining; @wei2018revisiting; @tang2018normalized]. It is also much easier and less error prone to identify when an image does or does not contain roots as opposed to correctly labeling every pixel in an image. However, current weakly-supervised methods used to infer pixel-levels labels do not perform as well as semantic image segmentation methods leveraging full annotation [@long2015fully; @ronneberger2015u; @lin2016efficient; @chen2017deeplab; @badrinarayanan2017segnet].
[0.15]{} ![Example attention maps from different methods of semantic segmentation of an MR image. () Original MR Image. () CAM Result () Grad-CAM Result () Grad-CAM++ Result () SMOOTHGRAD Result () Result of proposed method,**** MIL-CAM.[]{data-label="fig:1_1"}](Fig1image_T112_L5_FRMI_3_2.jpg "fig:"){width="1\linewidth"}
[0.15]{} ![Example attention maps from different methods of semantic segmentation of an MR image. () Original MR Image. () CAM Result () Grad-CAM Result () Grad-CAM++ Result () SMOOTHGRAD Result () Result of proposed method,**** MIL-CAM.[]{data-label="fig:1_1"}](Fig1cam_T112_L5_FRMI_3_2.jpg "fig:"){width="1\linewidth"}
[0.15]{} ![Example attention maps from different methods of semantic segmentation of an MR image. () Original MR Image. () CAM Result () Grad-CAM Result () Grad-CAM++ Result () SMOOTHGRAD Result () Result of proposed method,**** MIL-CAM.[]{data-label="fig:1_1"}](Fig1gradcam_T112_L5_FRMI_3_2.jpg "fig:"){width="1\linewidth"}
[0.15]{} ![Example attention maps from different methods of semantic segmentation of an MR image. () Original MR Image. () CAM Result () Grad-CAM Result () Grad-CAM++ Result () SMOOTHGRAD Result () Result of proposed method,**** MIL-CAM.[]{data-label="fig:1_1"}](Fig1gradcamplus_T112_L5_FRMI_3_2.jpg "fig:"){width="1\linewidth"}
[0.15]{} ![Example attention maps from different methods of semantic segmentation of an MR image. () Original MR Image. () CAM Result () Grad-CAM Result () Grad-CAM++ Result () SMOOTHGRAD Result () Result of proposed method,**** MIL-CAM.[]{data-label="fig:1_1"}](Fig1smoothcam_T112_L5_FRMI_3_2.jpg "fig:"){width="1\linewidth"}
[0.15]{} ![Example attention maps from different methods of semantic segmentation of an MR image. () Original MR Image. () CAM Result () Grad-CAM Result () Grad-CAM++ Result () SMOOTHGRAD Result () Result of proposed method,**** MIL-CAM.[]{data-label="fig:1_1"}](Fig1milcam_T112_L5_FRMI_3_2.jpg "fig:"){width="1\linewidth"}
Attention or class activation maps have been widely used to infer pixel-level labels from training data with weak image-level labels [@zhou2016learning; @selvaraju2017grad; @chattopadhay2018grad; @smilkov2017smoothgrad]. However, existing attention map approaches have been found to be inaccurate in identifying and delineating roots in MR imagery. For example, CAM (the class activation maps) approach [@zhou2016learning] overestimates the size of the roots as shown in Fig.\[fig:1\_1b\]. The Gradient-weighted Class Activation Mapping (Grad-CAM) [@selvaraju2017grad] approach incorrectly identifies the background soil as the root target as shown in Fig.\[fig:1\_1c\]. Grad-CAM++ [@chattopadhay2018grad] and SMOOTHGRAD [@smilkov2017smoothgrad] shown in Fig.\[fig:1\_1d\] and Fig.\[fig:1\_1e\], respectively, result in maps with poor contrast between roots and soil and, thus, many false alarms.
In this paper, we propose the multiple instance learning CAM (MIL-CAM) approach to address root segmentation in MR imagery given weak image-level labels. MIL-CAM is outlined in Section \[sec3\]. In Section \[sec4\], we compare MIL-CAM approach results to existing approaches with both weak- and full-annotation on an MR dataset collected from switchgrass.
Related Work
============
[0.18]{} ![Examples of the gradients with respect to the image classification score of the target root class using various individual feature maps. () Original MR image. () Feature map of Fig.\[fig:8\_1a\] which highlights the root area. () Gradients of feature map in Fig.\[fig:8\_1b\] with respect to the root class score. () Gradients in Fig.\[fig:8\_1c\] around the masked root regions shown in Fig.\[fig:8\_1e\]. The red lines indicate the boundary of root object. () Cropped area from full image as mask in Fig.\[fig:8\_1d\] and Fig.\[fig:8\_1h\]. () Feature map of Fig.\[fig:8\_1a\] which highlights soil. () Gradients of feature map in Fig.\[fig:8\_1g\]. () Gradients in Fig.\[fig:8\_1g\] around root object in Fig.\[fig:8\_1e\]. The red lines indicate the boundary of the root object. Fig.\[fig:8\_1d\] and Fig.\[fig:8\_1h\] are rescaled to match the size of other subfigure. () Colorbar used in for images in Fig. 2.[]{data-label="fig:8_1"}](Fig8orgImage.jpg "fig:"){width="1\linewidth"}
[0.18]{} ![Examples of the gradients with respect to the image classification score of the target root class using various individual feature maps. () Original MR image. () Feature map of Fig.\[fig:8\_1a\] which highlights the root area. () Gradients of feature map in Fig.\[fig:8\_1b\] with respect to the root class score. () Gradients in Fig.\[fig:8\_1c\] around the masked root regions shown in Fig.\[fig:8\_1e\]. The red lines indicate the boundary of root object. () Cropped area from full image as mask in Fig.\[fig:8\_1d\] and Fig.\[fig:8\_1h\]. () Feature map of Fig.\[fig:8\_1a\] which highlights soil. () Gradients of feature map in Fig.\[fig:8\_1g\]. () Gradients in Fig.\[fig:8\_1g\] around root object in Fig.\[fig:8\_1e\]. The red lines indicate the boundary of the root object. Fig.\[fig:8\_1d\] and Fig.\[fig:8\_1h\] are rescaled to match the size of other subfigure. () Colorbar used in for images in Fig. 2.[]{data-label="fig:8_1"}](Fig8feature_pos.png "fig:"){width="1\linewidth"}
[0.18]{} ![Examples of the gradients with respect to the image classification score of the target root class using various individual feature maps. () Original MR image. () Feature map of Fig.\[fig:8\_1a\] which highlights the root area. () Gradients of feature map in Fig.\[fig:8\_1b\] with respect to the root class score. () Gradients in Fig.\[fig:8\_1c\] around the masked root regions shown in Fig.\[fig:8\_1e\]. The red lines indicate the boundary of root object. () Cropped area from full image as mask in Fig.\[fig:8\_1d\] and Fig.\[fig:8\_1h\]. () Feature map of Fig.\[fig:8\_1a\] which highlights soil. () Gradients of feature map in Fig.\[fig:8\_1g\]. () Gradients in Fig.\[fig:8\_1g\] around root object in Fig.\[fig:8\_1e\]. The red lines indicate the boundary of the root object. Fig.\[fig:8\_1d\] and Fig.\[fig:8\_1h\] are rescaled to match the size of other subfigure. () Colorbar used in for images in Fig. 2.[]{data-label="fig:8_1"}](Fig8gradientFull_pos.png "fig:"){width="1\linewidth"}
[0.18]{} ![Examples of the gradients with respect to the image classification score of the target root class using various individual feature maps. () Original MR image. () Feature map of Fig.\[fig:8\_1a\] which highlights the root area. () Gradients of feature map in Fig.\[fig:8\_1b\] with respect to the root class score. () Gradients in Fig.\[fig:8\_1c\] around the masked root regions shown in Fig.\[fig:8\_1e\]. The red lines indicate the boundary of root object. () Cropped area from full image as mask in Fig.\[fig:8\_1d\] and Fig.\[fig:8\_1h\]. () Feature map of Fig.\[fig:8\_1a\] which highlights soil. () Gradients of feature map in Fig.\[fig:8\_1g\]. () Gradients in Fig.\[fig:8\_1g\] around root object in Fig.\[fig:8\_1e\]. The red lines indicate the boundary of the root object. Fig.\[fig:8\_1d\] and Fig.\[fig:8\_1h\] are rescaled to match the size of other subfigure. () Colorbar used in for images in Fig. 2.[]{data-label="fig:8_1"}](Fig8gradient_pos.png "fig:"){width="1\linewidth"}
\
[0.18]{} ![Examples of the gradients with respect to the image classification score of the target root class using various individual feature maps. () Original MR image. () Feature map of Fig.\[fig:8\_1a\] which highlights the root area. () Gradients of feature map in Fig.\[fig:8\_1b\] with respect to the root class score. () Gradients in Fig.\[fig:8\_1c\] around the masked root regions shown in Fig.\[fig:8\_1e\]. The red lines indicate the boundary of root object. () Cropped area from full image as mask in Fig.\[fig:8\_1d\] and Fig.\[fig:8\_1h\]. () Feature map of Fig.\[fig:8\_1a\] which highlights soil. () Gradients of feature map in Fig.\[fig:8\_1g\]. () Gradients in Fig.\[fig:8\_1g\] around root object in Fig.\[fig:8\_1e\]. The red lines indicate the boundary of the root object. Fig.\[fig:8\_1d\] and Fig.\[fig:8\_1h\] are rescaled to match the size of other subfigure. () Colorbar used in for images in Fig. 2.[]{data-label="fig:8_1"}](Fig8cropImage.jpg "fig:"){width="1\linewidth"}
[0.18]{} ![Examples of the gradients with respect to the image classification score of the target root class using various individual feature maps. () Original MR image. () Feature map of Fig.\[fig:8\_1a\] which highlights the root area. () Gradients of feature map in Fig.\[fig:8\_1b\] with respect to the root class score. () Gradients in Fig.\[fig:8\_1c\] around the masked root regions shown in Fig.\[fig:8\_1e\]. The red lines indicate the boundary of root object. () Cropped area from full image as mask in Fig.\[fig:8\_1d\] and Fig.\[fig:8\_1h\]. () Feature map of Fig.\[fig:8\_1a\] which highlights soil. () Gradients of feature map in Fig.\[fig:8\_1g\]. () Gradients in Fig.\[fig:8\_1g\] around root object in Fig.\[fig:8\_1e\]. The red lines indicate the boundary of the root object. Fig.\[fig:8\_1d\] and Fig.\[fig:8\_1h\] are rescaled to match the size of other subfigure. () Colorbar used in for images in Fig. 2.[]{data-label="fig:8_1"}](Fig8feature_neg.png "fig:"){width="1\linewidth"}
[0.18]{} ![Examples of the gradients with respect to the image classification score of the target root class using various individual feature maps. () Original MR image. () Feature map of Fig.\[fig:8\_1a\] which highlights the root area. () Gradients of feature map in Fig.\[fig:8\_1b\] with respect to the root class score. () Gradients in Fig.\[fig:8\_1c\] around the masked root regions shown in Fig.\[fig:8\_1e\]. The red lines indicate the boundary of root object. () Cropped area from full image as mask in Fig.\[fig:8\_1d\] and Fig.\[fig:8\_1h\]. () Feature map of Fig.\[fig:8\_1a\] which highlights soil. () Gradients of feature map in Fig.\[fig:8\_1g\]. () Gradients in Fig.\[fig:8\_1g\] around root object in Fig.\[fig:8\_1e\]. The red lines indicate the boundary of the root object. Fig.\[fig:8\_1d\] and Fig.\[fig:8\_1h\] are rescaled to match the size of other subfigure. () Colorbar used in for images in Fig. 2.[]{data-label="fig:8_1"}](Fig8gradientFull_neg.png "fig:"){width="1\linewidth"}
[0.18]{} ![Examples of the gradients with respect to the image classification score of the target root class using various individual feature maps. () Original MR image. () Feature map of Fig.\[fig:8\_1a\] which highlights the root area. () Gradients of feature map in Fig.\[fig:8\_1b\] with respect to the root class score. () Gradients in Fig.\[fig:8\_1c\] around the masked root regions shown in Fig.\[fig:8\_1e\]. The red lines indicate the boundary of root object. () Cropped area from full image as mask in Fig.\[fig:8\_1d\] and Fig.\[fig:8\_1h\]. () Feature map of Fig.\[fig:8\_1a\] which highlights soil. () Gradients of feature map in Fig.\[fig:8\_1g\]. () Gradients in Fig.\[fig:8\_1g\] around root object in Fig.\[fig:8\_1e\]. The red lines indicate the boundary of the root object. Fig.\[fig:8\_1d\] and Fig.\[fig:8\_1h\] are rescaled to match the size of other subfigure. () Colorbar used in for images in Fig. 2.[]{data-label="fig:8_1"}](Fig8gradient_neg.png "fig:"){width="1\linewidth"}
[0.058]{} ![Examples of the gradients with respect to the image classification score of the target root class using various individual feature maps. () Original MR image. () Feature map of Fig.\[fig:8\_1a\] which highlights the root area. () Gradients of feature map in Fig.\[fig:8\_1b\] with respect to the root class score. () Gradients in Fig.\[fig:8\_1c\] around the masked root regions shown in Fig.\[fig:8\_1e\]. The red lines indicate the boundary of root object. () Cropped area from full image as mask in Fig.\[fig:8\_1d\] and Fig.\[fig:8\_1h\]. () Feature map of Fig.\[fig:8\_1a\] which highlights soil. () Gradients of feature map in Fig.\[fig:8\_1g\]. () Gradients in Fig.\[fig:8\_1g\] around root object in Fig.\[fig:8\_1e\]. The red lines indicate the boundary of the root object. Fig.\[fig:8\_1d\] and Fig.\[fig:8\_1h\] are rescaled to match the size of other subfigure. () Colorbar used in for images in Fig. 2.[]{data-label="fig:8_1"}](Fig8colorbar.png "fig:"){width="1\linewidth"}
Attention Maps for Semantic Segmentation
----------------------------------------
CAM [@zhou2016learning] is one of the earliest methods showing that attention maps can localize the discriminative image components for classification. CAM uses a network structure consisting of a block of fully convolutional layers followed by a global average pooling layer and a single fully connected layer. The block of fully convolutional layers extract image features. These extracted features are combined linearly using the weights of the final fully connected layer to define the CAM. However, the CAM from this approach often has a low resolution, making it challenging to infer pixel-level labels precisely for semantic segmentation. Grad-CAM [@selvaraju2017grad] is an extension of CAM which can estimate higher resolution attention maps by using features from any convolutional layer in the network. Specifically, Grad-CAM estimates the weights used for combining features as the average of gradients of the image classification score with respect to each value in the corresponding feature maps. Following the introduction of Grad-CAM, many methods were proposed to attempt to improve the quality of attention maps generated. Grad-CAM++ [@chattopadhay2018grad] takes a weighted average of only the positive gradients of the image classification score with respect to each feature map. SMOOTHGRAD [@smilkov2017smoothgrad] averages over several Grad-CAM estimated attention maps with added zero-mean Gaussian noise with the aim of reducing sensitivity to feature map noise. Smooth Grad-CAM++ [@omeiza2019smooth] mimics SMOOTHGRAD but applies the approach to Grad-CAM++ estimated attention maps. Score-CAM [@wang2019score] attempts to improve Grad-CAM by weighting feature maps based on a metric which measures the increase of confidence for a class associated with the inclusion of each feature map. In application, all of these methods have been found to be either imprecise or sensitive to imbalanced data sets. Specifically in our application, soil pixels having complex gradients (i.e., both positive and negative gradients) which has a huge impact on the weights. Consider the example shown in Fig.\[fig:8\_1\]. Gradients across the feature maps have differing signs as shown in Fig.\[fig:8\_1d\] and Fig.\[fig:8\_1h\] and, thus, when averaged over the map may cancel each other out. Given this, the standard Grad-CAM approach is ineffective since the average of the gradients over the feature map is used to compute the attention map.
Weakly Supervised Learning
--------------------------
Weakly supervised and multiple instance learning (MIL) algorithms for image segmentation do not require precise pixel-level labels. Under MIL, a set of samples (e.g., an image) is labeled as either “positive” or “negative.” Positively labeled images are assumed to have at least one pixel corresponding to the target class (i.e., in our case, roots). The number of target pixels in positively labeled images are unknown. Negatively labeled images are composed of only non-target class (i.e., soil) pixels. Often, MIL approaches iteratively estimate the likelihood each pixel is a target and, using these values, update classifier parameters (and, then, subsequently update likelihood values again) [@oquab2015object; @durand2017wildcat]. Similarly, the pixels with the lowest target likelihood in each positively labeled image is also commonly assumed to be from the non-target class [@durand2017wildcat; @durand2016weldon]. In contrast to methods that select likely target and non-target pixels, some methods have been proposed which consider all pixels in an image as equally contributing to the image-level label [@zhou2016learning]. The Log-Sum-Exp (LSE) algorithm uses a hyper-parameter which trades off between selecting a single pixel as the target representative and considering all pixels in an image as target with equal contribution [@pinheiro2015image]. Global weighted rank pooling (GWRP) is another way to generalize number of pixels identified as targets [@kolesnikov2016seed]. In all of these approaches, it is difficult to select a fixed number of pixels to identify as targets representatives.
One reason that identifying the number of target pixels is challenging is that the size of the target class objects vary across images (e.g., some images contain only very few thin, fine roots whereas others are filled with roots of varying diameter). To alleviate this challenge, some approaches identify target pixels by adapting a threshold [@wei2018revisiting; @huang2018weakly; @ahn2019weakly; @lee2019ficklenet]. In [@wei2018revisiting], pixels with target class scores larger than a predefined threshold are labeled as targets and pixels with low saliency values are considered background. However, in this approach pixels are often unassigned to either target or background classes, pixels may be assigned to multiple target labels, or pixels may be assigned to background despite being surrounded in their neighborhood by target pixels. In [@wei2017object], pixels in all of these cases are ignored. The approach outlined in [@huang2018weakly] attempts to deal with these ignored pixels using deep seeded region growing (DSRG). DSRG proposes to propagate labels from labeled pixels to unlabeled pixels. The method presented in [@lee2019ficklenet] extends the DSRG approach [@huang2018weakly] by thresholding aggregated localization maps to improve delineation of target regions and adapts their algorithm to accommodate semi-supervised segmentation. Following an initial segmentation, some approaches apply post-processing steps to smooth and improve segmentation labels. These include conditional random fields (CRF) and the GWRP approach [@krahenbuhl2011efficient; @ahn2019weakly; @kolesnikov2016seed].
MR Image Segmentation
---------------------
Several methods have been developed for automated minirhizotron image segmentation [@zeng2006detecting; @heidari2014new; @zeng2010rapid; @rahmanzadeh2016novel]. Currently, supervised deep learning approaches are the methods that are achieving the state-of-art results in MR image segmentation [@xu2019overcoming; @wang2019segroot; @yasrab2019rootnav; @smith2020segmentation]. Yet, deep learning methods require a large collection of precisely traced root images for training the networks. A small number of approaches have been investigated for weakly supervised MR image segmentation [@yu2019root]. Yu, et al. [@yu2019root] studied the application of three MIL algorithms: multiple instance adaptive cosine coherence estimator (MI-ACE) [@zare2017discriminative], multiple instance support vector machine (miSVM) [@andrews2003support], and multiple instance learning with randomized trees (MIForests) [@leistner2010miforests] for application to MR imagery. These methods, however, did not do feature learning and, so, the authors manually identified color features to be used during segmentation.
MIL-CAM Methodology {#sec3}
===================
Semantic segmentation from weak labels using MIL-CAM is achieved in two training stages. The first stage, outlined in Alg. \[alg:MILCAM\], estimates the set of parameters needed to compute an attention map $\textbf{S}^c \in \mathbb{R}^{M \times N} $ for a class $c$ where $M$ and $N$ are the numbers of rows and columns of the input image, respectively. The attention map is estimated using the softmax output of a weighted linear combination feature maps extracted from the various layers of a trained CNN as described in Eq. \[eq:1\],
$$\mathbf{S}^c = \displaystyle \frac{exp(\sum_{j} w^c_j y(\mathbf{F}_j) + b^c)}{\sum_{q}exp(\sum_{j} w^q_j y(\mathbf{F}_j) + b^q)}.
\label{eq:1}$$
where $q$ is an index over all output classes, $w^q_j \in \mathbb{R}$ is the weight estimated for class $q$ and feature map $\mathbf{F}_j \in \mathbb{R}^{A_j \times B_j}$, $A_j$ and $B_j$ are the number of rows and columns in the $j^{th}$ feature map, $b^q \in \mathbb{R}$ is an estimated bias term, and $y(\cdot)$ is an interpolation function to scale an input to the size of $M \times N$.
Once attention maps are obtained using Alg. \[alg:MILCAM\], a segmentation network is then trained as outlined in Alg. \[alg:MILSeg\]. After training, the segmentation network maps input test imagery to get pixel level segmentation outputs.
Attention Map Estimation
------------------------
MIL-CAM estimates the set of parameters needed to obtain attention maps and compute Eq. \[eq:1\] using the combination of three key components: (a) a pixel-level feature extraction component; (b) a pixel sampling component used to form a bag for each image for MIL analysis; and (c) a linear model that performs the MIL-based segmentation. The sampled pixels with features extracted from the image classification network are used to train the linear model. The approach is illustrated in Fig. \[fig:1\_6\] and outlined in the following sub-sections.
![Architecture of MIL-CAM. GAP represents a global average pooling layer and fc represents a fully connected layer. cls loss represents the loss for image classification into positive (i.e., containing roots) or negative (i.e., does not contain roots). pix loss represents the loss for pixel level classification into root vs. soil.[]{data-label="fig:1_6"}](Fig6MILCAM.jpg){width="1\linewidth"}
### Feature Extraction and Interpolation
An image-level CNN classification network is first trained to extract coarse feature maps for each image. The training data set, $\{(\mathbf{I}_1, l_1),...(\mathbf{I}_k, l_k),...(\mathbf{I}_K, l_K) \}$, consists of $K$ images where each image $\mathbf{I}_k \in \mathbb{R}^{3 \times M \times N}$ is paired with image label $l_k \in \left\{0,1\right\}$ where 0 represents a negative image (i.e., does not contain roots) and 1 represents a positive image (i.e., contains roots). Using this training data an image-level classification network is trained by optimizing the cross-entropy loss as shown in Eq. 2, $$\displaystyle \min_{\boldsymbol{\theta}_0} \displaystyle\sum_{k=1}^K L_{cls-loss}(\mathbf{I}_{k}; \boldsymbol{\theta}_0, \mathbf{l}_{k}) = \frac{-1}{K} \sum_{k=1}^K\sum_{q=1}^2l_{kq}\log {f_{q}(\mathbf{I}_{k};\boldsymbol{\theta}_0)}
\label{eq:2}$$ where $\mathbf{l}_{k} = [l_{k1}, l_{k2}]$ is the one-hot encoded label of the image $\mathbf{I}_{k}$ and $f_{q}(\mathbf{I}_{k};\boldsymbol{\theta}_0)$ is the $q^{th}$ element of the softmax output layer of the the image classification network defined by parameters $\boldsymbol{\theta}_0$. The assumption is, provided an effective image-level classification network can be trained, that the network is extracting features that are useful for the semantic segmentation problem and these useful features are encoded in the CNN feature maps. Once the classification network is trained, then the coarse CNN feature maps are upsampled using bilinear interpolation to match the size of the input image. Each pixel is represented by the corresponding feature vector obtained from the collection of upsampled feature maps.
Train the image classification network with $(\mathbf{I}, \mathbf{l})$ Extract feature maps $\mathbf{F}$ from image classification network for $(\mathbf{I}, \mathbf{l})$ Interpolate the CNN feature maps $y(\mathbf{F})$ for $(\mathbf{I}, \mathbf{l})$ Sample instances and construct bags, $\{(\mathbf{B}_1, l_1),...(\mathbf{B}_k, l_k),...(\mathbf{B}_K, l_K)\}$
Initialize each instance label with the label of its corresponding bag
### Instance Sampling
In order to address some of the imbalance in the data set (i.e., there are many more soil pixels than root pixels), a sampling approach is used to identify representative pixels from each image. The green band of the RGB minirhizotron image is used for instance sampling. The approach draws a single pixel to represent the set of pixels from each possible 8-bit value from the green band in the image. In other words, a 256 bin histogram is built using the values of the green band of the MR imagery. For each non-empty bin, a uniform random draw is used to identify a representative pixel for that green-level. In our application, we found this to be an effective approach to re-balance root-vs-nonroot pixels in positively labeled imagery (given that pixel level labels are unavailable). The sampled pixels are organized into a set of bags, $\{(\mathbf{B}_1, l_1),...(\mathbf{B}_k, l_k),...(\mathbf{B}_K, l_K) \}$. Each bag, $\mathbf{B}_{k} = \left\{\mathbf{x}_{k}^{1},\mathbf{x}_{k}^{2},\dotsc,\mathbf{x}_{k}^{N_{k}}\right\}$, corresponds to one image $\mathbf{I}_k$ with image label $l_K$ and is composed of $N_k$ instances. The instance $\mathbf{x}_{k}^{n} \in \mathbb{R}^{J}$ is the feature vector for the $n^{th}$ instance in the $k^{th}$ bag where $J$ is the number of feature maps used to construct the feature vectors.
Update parameters $\boldsymbol{\theta}_1$ for data set with pixel labels $\mathbf{l}_{k}^0(m,n)$ for a fixed number of epochs
### Estimated Weights and Biases
After instance sampling, the weights and biases used to compute the attention maps as defined in Eq. \[eq:1\] are estimated by optimizing the cross-entropy loss shown in Eq. \[eq:4\] given the MIL constraints that for each positive bag, at least one instance must be labeled as root and all instances in every negative bag are labeled as non-root, $$\displaystyle \min_{\mathbf{l}_{k}^{n}} \min_{\mathbf{w},\mathbf{b}}
\displaystyle\sum_{\mathbf{x}_{k}^{n}} L_{pix-loss}( \mathbf{x}_{k}^{n};\mathbf{w},\mathbf{b}, \mathbf{l}_{k}^{n}) = \frac{-1}{\sum_{k}{N_k}}\sum_{\mathbf{x}_{k}^{n}}\sum_{q}l_{kq}^n\log {g_{q}(\mathbf{x}_{k}^n;\mathbf{w},\mathbf{b})}
\label{eq:4}$$ where $\mathbf{l}_{k}^n$ is the one-hot encoded label of the instance $\mathbf{x}_{k}^{n}$ and $g_{q}(\mathbf{x}_{k}^n;\mathbf{w},\mathbf{b})$ is the $q^{th}$ element of the softmax output of the MIL-CAM with parameters $(\mathbf{w},\mathbf{b})$. The loss is updated iteratively as outlined in Alg. \[alg:MILCAM\]. During the initial epoch, each instance is labeled the same label as its bag. In all subsequent epochs, the probability that an instance belongs to the target class, $p_{k}^{n} = g(\mathbf{x}_{k}^n;\mathbf{w},\mathbf{b})$, is predicted by the linear model trained from the previous epoch. Then for each positive bag, a threshold $p_{t}$ is computed using Otsu’s threshold [@otsu1979threshold] and all instances greater than the threshold are labeled as target whereas all others are labeled as non-target. For negative bags, all instances are labeled as non-target.
Training the Image Segmentation Network
---------------------------------------
Once MIL-CAM attention maps can be estimated, an image segmentation network is trained as outlined in Alg.\[alg:MILSeg\]. First, target class attention maps for positively labeled images are estimated and thresholded using Otsu’s threshold to obtain a label for each pixel. All pixels in negatively labeled images are given a non-target label. These labels are used to estimate the parameters for the U-Net [@ronneberger2015u] architecture illustrated in lower branch of Fig. \[fig:1\_5\]. After initially training the U-Net with labels obtained from the attetnion maps, the U-Net is iteratively fine-tuned. A score-map, $\mathbf{P}_k \in \mathbb{R}^{M \times N}$, is computed using the soft-max output of the U-Net. The score-map of positively-labeled is thresholded using a fixed (large) threshold parameter, $s_t$, to obtain updated pixel level labels which highlight more likely positive samples. The updated labels are iteratively used to fine-tune the parameters of the U-Net.
![Architecture of segmentation U-Net with MIL training branch. The bottom branch is the U-Net. The top branch is used to infer label of training data.[]{data-label="fig:1_5"}](Fig5MILUNET.jpg){width="1\linewidth"}
Experiments {#sec4}
===========
Data Description
----------------
For our experiments, we used a switchgrass (*Panicum virgatum* L.) MR imagery dataset consisting of 561 training images with image-level labels and 30 test and validation images with pixel-level labels. Each image was $2160 \times 2550$ in size and was divided into sub-images of size $720 \times 510$. 500 sub-images containing roots and 500 sub-images containing only soil were randomly selected as training data for estimating attention map parameters. 1500 root sub-images and 1500 soil sub-images were randomly selected as training data for the U-Net segmentation network. The 30 images with pixel-level labels were randomly divided into 10 validation images and 20 test images.
Architecture
------------
Our experiments use U-Net [@ronneberger2015u] with layer depth of 5 as backbone for MR image segmentation. The feature extraction network used to estimate attention map parameters was a 2-class convolutional neural network with the encoder of the U-Net, followed by a global average pooling layer and a fully connected layer. We extract $1024\times 46 \times 33$ feature maps and vectorize the feature maps to classify each image into 2 classes with a fully connected layer. The feature extraction net is trained using SGD at a learning rate of 0.0001 and momentum of 0.8 in the online mode to minimize the cross entropy loss. The MIL-CAM attention map module extracts a 64-dimensional feature for each sampled instance from the fourth layer of the encoder of the feature extraction network. Then, classifies each sampled instance into one of two classes using a fully connected layer. The MIL-CAM attention map module is trained using SGD at a learning rate of 0.001 and momentum of 0.5 in the online mode to minimize the cross entropy loss.
The image segmentation network was a U-Net of depth 5 and a MIL training branch. The MIL training branch extracts $64 \times 720 \times510$ features from the first layer of the encoder of the feature extraction network and compute a $720 \times 510$ score-map of target class for each training image. The threshold parameter $s_t$ was set to 0.9 to estimate pixel label from the score-map. The U-Net was first initialized for 10 epochs using Adam at learning rate of 0.0001 in the online mode to minimize the cross entropy loss where the root class was weighted by 50 using the labels produced by the attention maps. Then, during iterative fine-tuning, the network parameters were also updated using Adam with learning rate of 0.0001 in the online mode to minimize the cross entropy loss with the root class having an additional weight of 50. The weight on root class addressed the imbalance issue between root class and soil class.
Experiments: MIL-CAM Attention Maps
-----------------------------------
[0.15]{} ![Attention maps of different methods. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:3_1"}](Fig3_1_6_org.png "fig:"){width="1\linewidth"}
[0.15]{} ![Attention maps of different methods. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:3_1"}](Fig3_1_2cam.png "fig:"){width="1\linewidth"}
[0.15]{} ![Attention maps of different methods. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:3_1"}](Fig3_1_3_gradcam.png "fig:"){width="1\linewidth"}
[0.15]{} ![Attention maps of different methods. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:3_1"}](Fig3_1_4_gradpp.png "fig:"){width="1\linewidth"}
[0.15]{} ![Attention maps of different methods. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:3_1"}](Fig3_1_5_smooth.png "fig:"){width="1\linewidth"}
[0.15]{} ![Attention maps of different methods. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:3_1"}](Fig3_1_1_milcam.png "fig:"){width="1\linewidth"}
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[0.15]{} ![Attention maps of different methods. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:3_1"}](Fig3_2_6_org.png "fig:"){width="1\linewidth"}
[0.15]{} ![Attention maps of different methods. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:3_1"}](Fig3_2_2_cam.png "fig:"){width="1\linewidth"}
[0.15]{} ![Attention maps of different methods. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:3_1"}](Fig3_2_3_gradcam.png "fig:"){width="1\linewidth"}
[0.15]{} ![Attention maps of different methods. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:3_1"}](Fig3_2_4_gradpp.png "fig:"){width="1\linewidth"}
[0.15]{} ![Attention maps of different methods. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:3_1"}](Fig3_2_5_smooth.png "fig:"){width="1\linewidth"}
[0.15]{} ![Attention maps of different methods. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:3_1"}](Fig3_2_1_milcam.png "fig:"){width="1\linewidth"}
\
[0.15]{} ![Attention maps of different methods. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:3_1"}](Fig3_3_6_org.png "fig:"){width="1\linewidth"}
[0.15]{} ![Attention maps of different methods. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:3_1"}](Fig3_3_2_cam.png "fig:"){width="1\linewidth"}
[0.15]{} ![Attention maps of different methods. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:3_1"}](Fig3_3_3_gradcam.png "fig:"){width="1\linewidth"}
[0.15]{} ![Attention maps of different methods. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:3_1"}](Fig3_3_4_gradpp.png "fig:"){width="1\linewidth"}
[0.15]{} ![Attention maps of different methods. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:3_1"}](Fig3_3_5_smooth.png "fig:"){width="1\linewidth"}
[0.15]{} ![Attention maps of different methods. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:3_1"}](Fig3_3_1_milcam.png "fig:"){width="1\linewidth"}
The attention maps of MIL-CAM were first qualitatively compared with attention maps of other methods as shown in Fig.\[fig:3\_1\]. As can be seen, MIL-CAM results shown in Fig.\[fig:3\_6f\] more accurately indicate root locations as compared to the attention maps produced by CAM in Fig.\[fig:3\_2b\]. This difference in performance is largely due to the fact that CAM requires interpolating a low resolution attention map to the size of the input image resulting in blurred, oversized detection regions. Grad-CAM in Fig.\[fig:3\_3c\]. fails to correctly identify roots and, instead, highlights soil. Furthermore, MIL-CAM produced attention maps with higher contrast between root pixels and background than those Grad-CAM ++ in Fig.\[fig:3\_4d\] and SMOOTHGRAD in Fig.\[fig:3\_5e\].
[0.15]{} ![Thresholded attention maps. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:7_1"}](Fig3_1_6_org.png "fig:"){width="1\linewidth"}
[0.15]{} ![Thresholded attention maps. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:7_1"}](Fig7_1_2cam.jpg "fig:"){width="1\linewidth"}
[0.15]{} ![Thresholded attention maps. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:7_1"}](Fig7_1_3_gradcam.jpg "fig:"){width="1\linewidth"}
[0.15]{} ![Thresholded attention maps. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:7_1"}](Fig7_1_4_gradpp.jpg "fig:"){width="1\linewidth"}
[0.15]{} ![Thresholded attention maps. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:7_1"}](Fig7_1_5_smooth.jpg "fig:"){width="1\linewidth"}
[0.15]{} ![Thresholded attention maps. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:7_1"}](Fig7_1_1_milcam.jpg "fig:"){width="1\linewidth"}
\
[0.15]{} ![Thresholded attention maps. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:7_1"}](Fig3_2_6_org.png "fig:"){width="1\linewidth"}
[0.15]{} ![Thresholded attention maps. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:7_1"}](Fig7_2_2_cam.jpg "fig:"){width="1\linewidth"}
[0.15]{} ![Thresholded attention maps. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:7_1"}](Fig7_2_3_gradcam.jpg "fig:"){width="1\linewidth"}
[0.15]{} ![Thresholded attention maps. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:7_1"}](Fig7_2_4_gradpp.jpg "fig:"){width="1\linewidth"}
[0.15]{} ![Thresholded attention maps. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:7_1"}](Fig7_2_5_smooth.jpg "fig:"){width="1\linewidth"}
[0.15]{} ![Thresholded attention maps. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:7_1"}](Fig7_2_1_milcam.jpg "fig:"){width="1\linewidth"}
\
[0.15]{} ![Thresholded attention maps. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:7_1"}](Fig3_3_6_org.png "fig:"){width="1\linewidth"}
[0.15]{} ![Thresholded attention maps. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:7_1"}](Fig7_3_2_cam.jpg "fig:"){width="1\linewidth"}
[0.15]{} ![Thresholded attention maps. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:7_1"}](Fig7_3_3_gradcam.jpg "fig:"){width="1\linewidth"}
[0.15]{} ![Thresholded attention maps. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:7_1"}](Fig7_3_4_gradpp.jpg "fig:"){width="1\linewidth"}
[0.15]{} ![Thresholded attention maps. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:7_1"}](Fig7_3_5_smooth.jpg "fig:"){width="1\linewidth"}
[0.15]{} ![Thresholded attention maps. () Original Image. () Result of CAM. () Result of Grad-CAM. () Result of Grad-CAM++. () Result of SMOOTHGRAD. () Result of MIL-CAM.[]{data-label="fig:7_1"}](Fig7_3_1_milcam.jpg "fig:"){width="1\linewidth"}
[|l|\*[11]{}[c|]{}]{} Method & Precision & Recall & F1 score & mIoU\
CAM & $0.045 \pm0.0053$ & $ \textbf{0.931} \pm \textbf{0.0459}$ & $0.085 \pm 0.0098$ & $0.045 \pm0.0053$\
Grad-CAM & $0.003\pm0.0012$ & $0.229\pm0.0939$ & $0.006\pm0.0024 $ & $0.003\pm0.0012 $\
Grad-CAM++ & $0.015 \pm 0.0084$ & $0.550\pm0.1951$ & $0.030\pm0.0159$ & $0.015\pm0.0083$\
SMOOTHGRAD&$0.033 \pm 0.0028$ & $0.782 \pm 0.0191$ & $0.064 \pm 0.0052$ & $0.033 \pm 0.0028$\
MIL-CAM & $\textbf{0.248} \pm \textbf{0.1870}$ & $0.536\pm0.1450$ & $\textbf{0.289}\pm \textbf{0.1814} $ & $\textbf{0.177}\pm \textbf{0.1190}$\
Fig.\[fig:7\_1\] compares attention maps from a selection of approaches after thresholding with Otsu’s threshold. Table \[table:inference\] lists the average and standard devation for precision, recall and F1 score of three training runs of the various approaches to compare the quality these thresholded results. The proposed MIL-CAM method has a significantly higher F1 score among all those compared. The precision of MIL-CAM is an order of magnitude better than the comparison methods without a significant loss in recall as compared with the gains of precision. Although other methods except Grad-CAM have a better recall, the low precision scores of these methods indicate a large amount of background pixels are mislabeled as root pixels. This can be visualized in Fig.\[fig:7\_1\].
Experiments: Semantic Segmentation
----------------------------------
[|l|\*[11]{}[c|]{}]{} Method & Label & Precision & Recall & F1 score & mIoU\
U-Net [@xu2019overcoming] & pixel &$0.307$ &$\textbf{0.913}$&$0.459$ & $0.298$\
MI-ACE[@yu2019root] & image & $0.130\pm 0.0010$ & $0.775\pm0.0067$ & $0.223\pm0.0017$ & $0.125\pm0.0011$\
miSVM[@yu2019root] & image & $0.134\pm0.0015$ & $0.798\pm0.0104$ & $0.229\pm0.0026$ & $0.129\pm0.0017$\
MIForests[@yu2019root] & image & $0.101\pm0.0104$ & $0.582\pm0.0664$ & $0.172\pm0.0180$ & $0.094\pm0.0108$\
MIL-CAM Th & image & $0.145\pm0.0050 $ & $ 0.878\pm0.0341$ & $0.249\pm0.0088$ & $0.142\pm0.0057$\
argmax MIL-CAM & image & $0.186\pm 0.0278$ &$0.859\pm0.0423$ &$0.304\pm0.0364$ & $0.180\pm0.0251$\
MIL-CAM + CRF & image & $\textbf{0.667} \pm \textbf{0.0257}$ &$0.692 \pm 0.0267$&$\textbf{0.678}\pm \textbf{0.0058}$& $\textbf{0.513} \pm \textbf{0.0066}$\
We also compared the performance of our final MIL segmentation network (i.e., MIL-CAM Th in the table) against other MIL methods (MI-ACE[@yu2019root], miSVM[@yu2019root], and MIForest[@yu2019root]). The average and standard deviation of three runs of the precision, recall, F1 score and mIoU were compared at false positive rate (FPR) is 0.03 in Table \[table:post\]. Our proposed approach outperformed all other MIL methods. The proposed MIL-CAM Th method (i.e., the thresholded MIL-CAM result) achieved recall$= 0.878$. The recall of MIL-CAM Th was 10% better than miSVM which was the second best. MIL-CAM Th also had the best precision of all MIL methods.
The segmentation results of the proposed MIL-CAM approach when taking the argmax of the softmax outputs (i.e., argmax MIL-CAM in the table) are shown in the third column in Fig. \[fig:4\_1c\]. The long roots are a challenging problem. Although our proposed method detects most of the root pixels, it expands the boundary of some roots. This expansion results in high recall (0.859) but low precision (0.186) as shown in table \[table:post\]. To mitigate this, we also applied a conditional random field (CRF) [@krahenbuhl2011efficient] postprocessing to the segmentation results of our approach. The default parameters of the CRF were used as 0.7 for the certainty of the label, 3 for the parameter of the smoothness kernel, 80 for the spatial parameter of the appearance kernel, 13 for the color parameter of the appearance kernel and 2 inference steps were run. Segmentation results after CRF postprocessing are shown in the fourth column in in Fig. \[fig:4\_1c\]. Postprocessing improved the precision of results from 0.186 to 0.667, and the mean Intersection-Over-Union (mIoU) from 0.180 to 0.513 as shown in Table \[table:post\]. The only approach with that outperformed the proposed MIL-CAM with CRF postprocessing on any metric was the U-Net method outlined in [@xu2019overcoming]. However, this U-Net was pre-trained using a large dataset consisting of 17567 MR images with full pixel-level annotation and, thus, did not have to overcome the weak label challenge.
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Conclusion
==========
In this work, we proposed MIL-CAM for weakly supervised MR image segmentation. The proposed MIL-CAM approach outperformed a variety of comparison attention map approaches as well as a variety of MIL segmentation methods, particularly when incorporating a CRF post-processing.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the U.S. Department of Energy, Office of Science, Office of Biological and Environmental Research award number DE-SC0014156 and by the Advanced Research Projects Agency - Energy award number DE-AR0000820.
|
---
abstract: 'Despite the success of reinforcement learning (RL) in various research fields, relatively few algorithms have been applied to industrial control applications. The reason for this unexplored potential is partly related to the significant required tuning effort, large numbers of required learning episodes, i.e. experiments, and the limited availability of RL methods that can address high dimensional and safety-critical dynamical systems with continuous state and action spaces. By building on model predictive control (MPC) concepts, we propose a cautious model-based reinforcement learning algorithm to mitigate these limitations. While the underlying policy of the approach can be efficiently implemented in the form of a standard MPC controller, data-efficient learning is achieved through posterior sampling techniques. We provide a rigorous performance analysis of the resulting ‘Bayesian MPC’ algorithm by establishing Lipschitz continuity of the corresponding future reward function and bound the expected number of unsafe learning episodes using an exact penalty soft-constrained MPC formulation. The efficiency and scalability of the method are illustrated using a 100-dimensional server cooling example and a nonlinear 10-dimensional drone example by comparing the performance against nominal posterior MPC, which is commonly used for data-driven control of constrained dynamical systems.'
author:
- |
Kim P. Wabersich\
Institute for Dynamic Systems and Control\
ETH Zurich\
`wkim@ethz.ch` Melanie N. Zeilinger\
Institute for Dynamic Systems and Control\
ETH Zurich\
`mzeilinger@ethz.ch`
bibliography:
- 'bibliography.bib'
title: 'Performance and safety of Bayesian model predictive control: Scalable model-based RL with guarantees '
---
Introduction
============
Driven by a constantly increasing research and development effort in the field of autonomous systems, including e.g. autonomous driving, service robotics, or various production processes in chemical or biological industry branches, the number of challenging control problems is growing steadily. Together with the ever increasing complexity of such systems, including physical constraints and safety specifications, this motivates research efforts towards automated and efficient synthesis procedures of high-performance control algorithms.
While significant progress in this context has been made with learning-based control for systems with continuous state and action spaces in the areas of machine learning and in particular reinforcement learning (RL), see e.g. [@Lillicrap2016], only few methods support data-efficient learning of control policies that can satisfy system constraints. In addition, when compared to classical control strategies such as simple PID or state-feedback control [@Sontag1998], the implementation effort and the required expert knowledge for tuning high performance RL algorithms based on, e.g. deep learning techniques, is potentially limiting and can hinder wide-spread adoption in industrial control applications.
Control design using machine learning tools has also been approached from control theoretic perspectives, see for example [@Recht2019; @hewing2020] for an overview. Particularly in the case of general, complex, and safety-critical control problems, model predictive control (MPC) techniques [@Morari1999; @Qin2000; @Darby2012] have shown significant impact on both, industrial and research-driven applications, see also Figure \[fig:mpc\_impact\].
[r]{}[0.3]{} {width="\linewidth"}
Due to its principled controller synthesis procedures and professional software tools [@Houska2011a; @forcesnlp2017; @Verschueren2019], MPC offers an important framework for learning-based control, see, e.g., the recent reviews [@Mesbah2018; @Rosolia2018a; @hewing2020].
MPC can be seen as an approximate solution to an optimal control problem, which is intractable to solve exactly. The central mechanism is based on solving an open-loop optimal control problem in the form of an optimization problem, the MPC problem, at discrete time instances. More precisely, based on the currently measured system state, the sequence of future control actions is optimized at every time step in real time using a model to predict the evolution of the underlying system. To compensate for uncertainties in the prediction model and external disturbances, only the first element of the computed optimal action sequence is applied to the system and the procedure is repeated at every time step.
By construction, this mechanism heavily relies on a sufficiently accurate prediction and reward model of the system, which typically results in time-consuming system modeling and identification procedures. As a result, research in learning-based MPC mainly focuses on automatically improving the model quality, either by relying on available system data [@hewing2020 Section 3] or through active data collection using exploration-exploitation mechanisms similar to RL see e.g. [@Mesbah2018] for an overview. While passive approaches often allow efficient implementation [@McKinnon2019; @Carron2019], they rely on sufficiently informative data with respect to the optimal system behavior, which is usually unknown beforehand. Passive methods therefore run into the risk of converging to suboptimal operation regimes.
This limitation is addressed in so-called dual MPC schemes that provide effective exploration-exploitation strategies by approximating the information gain of future data. By relying on approximate stochastic dynamic programming [@Sutton2018], these MPC-based techniques are closely related to reinforcement learning concepts, see e.g. [@Heirung2012; @Arcari2019]. In case of episodic tasks, an alternative strategy is to optimize model or cost function parameters of an MPC using automatic differentiation [@NIPS2018_8050], sensitivity analysis [@Gros2019], or Bayesian optimization [@Neumann-Brosig2019; @Piga2019]. While the underlying concepts are promising, their theoretical properties still need to be investigated and the techniques often have limited scalability.
By relying on a posterior sampling framework for model-based reinforcement learning, a Bayesian MPC scheme was proposed in [@Wabersich2020] with the goal of enabling practical and scalable reinforcement learning for industrial applications using concepts from model predictive control. We extend this basic idea to a theoretical framework for a variety of learning-based MPC controllers by analyzing the theoretical properties of Bayesian MPC and propose a modification that introduces cautiousness w.r.t. the constraints leading to the following main contributions.
*Performance:* In [@Wabersich2020], a fundamental regularity assumption of the future reward is adopted [@Osband2014 Section 6.1], which is a central ingredient to transfer well-known Thompson sampling analysis in a Bayesian optimization setting to RL and characterizes the resulting regret bound. In Section \[subsec:regularity\_linear\_concave\] we show that the regularity assumption is always satisfied in the important special case of linear mean transitions, concave rewards, and polytopic state and action constraints. For more general nonlinear mean transition functions and rewards, we provide relatively weak sufficient conditions in Section \[subsec:regularity\_nonlinear\] that ensure the required regularity and thereby provide an intuition for cases in which Bayesian MPC works and potentially fails in terms of cumulative regret bounds.
*Safety:* The cautious Bayesian MPC formulation uses a simple state constraint tightening that allows to rigorously relate the expected number of unsafe learning episodes to the cumulative performance regret bound in Section \[subsec:unsafe\_learning\_episodes\].
Preliminaries
-------------
*Notation:* We denote the $i$-th element of a vector $c\in\RR^n$ as $c_i$. A vector with $n$ elements equal to $1$ is denoted by $\mOnes{n}$ and if it is clear from the context we write $\mOnes{}$.
We consider systems that can be modeled as a random Markov Decision Process (MDP) of the form $$\begin{aligned}
\label{eq:general_mdp}
\mMDP \mDef (\XX, \UU, F, R, S^0)
\end{aligned}$$ with a compact set of admissible states of the form $\XX\mDef\{s\in\RR^{n} | g_s(s) \leq \mOnes{}\}$ and compact set of admissible actions $\UU\mDef\{a\in\RR^m | g_a(a) \leq \mOnes{}\}$ where $\mDefFunction{g_s}{\RR^n}{\RR^{n_s}}$ and $\mDefFunction{g_a}{\RR^m}{\RR^{n_a}}$. The state transition probability at time $t$ is described by $s(t+1)\sim F(s(t), a(t))$ over the state space $\XX$ starting from a random initial condition $s(0)\sim S^0$. We restrict our attention to time-invariant transition models of the form $$\begin{aligned}
\label{eq:transition}
s(t+1) &= f(s(t),a(t);\mPa_{F})+\epsilon_{F}(t),
\qquad t\in\mathbb N
\end{aligned}$$ that are parametric w.r.t. $\mPa_F$ and subject to $\sigma_F$-sub-Gaussian zero mean i.i.d. noise $\epsilon_{F}(t)$. The reward signal at time $t\in\mathbb N$ is distributed according to $r(t) \sim R(t, s(t), a(t))$ over $\mathbb R$ with $s(t)\in\XX$ and $a(t)\in\UU$, i.e. its distribution may change over time. Similarly as for the state transitions, we focus on reward models of the form $$\begin{aligned}
r(t, s(t), a(t);\mPa_\mR) &= \ell(t, s(t), a(t);\mPa_\mR) + \epsilon_\mR(t)
\label{eq:reward_model}
\end{aligned}$$ where $\epsilon_\mR(t)$ is $\sigma_R$-sub-Gaussian zero mean i.i.d. noise and $\mPa_\mR$ parameterizes the mean function.
In case of perfectly known transition and reward parameters $\mPa_\mR$ and $\mPa_{F}$, the goal is to find a control policy $\mDefFunction{\pi}{\mathbb N\times\XX}{\UU}$ such that application of $a(t) = \pi(t,s(t))$ maximizes the time-varying sum of reward signals starting from a given initial condition $s(0)\sim S^0$ over a finite horizon of $T$ time steps: $$\begin{aligned}
\label{eq:nominal_objective}
\max_\pi\mExpectationExp{E}{\sum_{t=0}^{T-1} \ell(t,s(t),a(t);\mPa_\mR)}
\text{ subject to~\eqref{eq:transition} with }
E \mDef [\epsilon_{F}(0), .., \epsilon_{F}(T-2)].
\end{aligned}$$
Importantly, maximization of needs to be performed while taking into account state and action constraints, i.e. $s(t)\in\XX$ and $a(t)\in\UU$ for all $t=0,...,T-1$. It should be noted that this yields a challenging control problem, even for small-scale systems with state dimension $n<5$, perfectly known parameters $\mPa_{F}$, $\mPa_R$, and noise-free rewards and transitions.
Soft-constrained model predictive control as an approximate optimal control policy {#sec:soft_mpc}
----------------------------------------------------------------------------------
In the idealized case of perfect system parameter knowledge, an approximate policy $\pi$ to maximize can be obtained by repeatedly solving a simplified open-loop optimal control problem, a so-called model predictive control (MPC) problem, initialized at the currently measured state $s(t)$. While MPC formulations vary greatly in their complexity, a simple formulation as originally proposed by [@Mitter1966] provides sufficient practical properties in terms of performance and constraint satisfaction for many applications. Thereby, we optimize over an action sequence $\{\mUpred_{k|t}\}$ subject to the system constraints while neglecting zero mean additive disturbances. The resulting MPC problem is given by
\[eq:mpc\_problem\] $$\begin{aligned}
\mMpcCost{\mPa}{t}(s) \mDef
\max_{\{\mUpred_{k|t}\}} &~~ \sum_{k=t}^{T-1} \ell(k, \mXpred_{k|t}, \mUpred_{k|t};\mPa_\mR)-I(\rho_{k|t})\label{eq:mpc_problem_reward} \\
\text{s.t.} &~~ \mXpred_{t|t} = s, ~~ \rho_{k|t}\geq 0, \label{eq:mpc_problem_initial_condition}\\
&~~ \mXpred_{k+1|t} = f(\mXpred_{k|t}, \mUpred_{k|t}; \mPa_F), ~ k = t,..,T-2, \label{eq:mpc_problem_transition}\\
&~~ \mXpred_{k|t} \in \bar\XX_\delta(\rho_{k|t}), ~ k = t,..,T-1,\label{eq:mpc_problem_state_constraint} \\
&~~ \mUpred_{k|t} \in \UU, ~ k = t,..,T-1, \label{eq:mpc_problem_action_constraint}
\end{aligned}$$
and can be efficiently solved online based on the current system state $s(t)$ using tailored MPC solvers [@Houska2011a; @forcesnlp2017; @Verschueren2019]. Ideally, the prediction horizon $T$ equals the task length, yielding a shrinking horizon MPC. For long task horizons $T$, another common approximation in MPC is to select a smaller prediction horizon and to operate in a receding horizon fashion, see e.g. [@hewing2020 Section 2.2].
Different from the original formulation as proposed by [@Mitter1966] and different from [@Wabersich2020], we enforce a modified state constraint . First, we use a tightened state constraint common in MPC for uncertain systems to foster closed-loop constraint satisfaction, see e.g. [@marruedo2002input] and [@hewing2020 Section 3] for an overview. By optimizing state trajectories subject to a tightened state constraint set, i.e. $g_s(s) \leq (1-c)\mOnes{}$ with $0<c<1$, we gain a safety margin to compensate for uncertain model parameters $\mPa_F$ and unknown external disturbances $\epsilon_F$ before state constraint violation occurs at some time step $t$ in the future, i.e. $g_s(s(t))\nleq\mOnes{}$. As a second modification, we soften the tightened state constraint in and include the extra negative reward term $-I(\rho)$ on the constraint relaxation in to ensure feasibility of problem as similarly done in [@Zeilinger2014]. The penalty is selected to realize a so-called exact penalty function as proposed by [@Kerrigan2000]. The resulting cautious soft-constraint formulation is given as $\bar\XX_\delta(\rho) = \{s\in\RR^n | g(s) \leq (1-\delta)\mOnes{} + \rho\}$ with parameter $\delta\in\RR$, $\delta>0$ defining the degree of cautiousness, slack variable $\rho \in \RR^{n_s}$, $\rho\geq 0$, and the exact penalty $I(\rho) = c_{1}^\top \rho + c_{2}\rho^\top \rho$ for sufficiently large linear penalty weights $c_{1}\in\RR^{n_s}$, $c_1>0$ [@Kerrigan2000]. Importantly, the tightening of the constraints using $\delta >0$ will be the main mechanism allowing to upper bound the expected number of unsafe learning episodes in Section \[subsec:unsafe\_learning\_episodes\].
Using an imperfect estimate $\tilde \mPa=(\tilde \mPa_\mR, \tilde \mPa_F)$ of the true system parameters $\mPa \mDef (\mPa_\mR,\mPa_F)$ in the MPC problem , we denote the expected closed-loop future reward, including a weighted constraint violation penalty, at time $t$ and state $s$ as $$\begin{aligned}
\label{eq:reward}
\mV{\mPa}{\tilde\mPa}{t}(s) \mDef
\mExpectationExp{E}{
\sum_{j=t}^{T-1} r(j, x(j), u(j);\mPa_\mR)
-I(\rho(j))
~\middle|~
\begin{matrix*}[l]
s(t) = s, \\
a(j) = \mPo(j, s(j);\tilde \mPa), \\
s(j+1) = f(s(j), a(j); \mPa_F) + \epsilon_F(j),\\
\rho(j) = \min_{\rho\geq 0}{\rho} \text{ s.t. } s(j) \in \bar\XX_\delta(\rho)
\end{matrix*}
}
\end{aligned}$$ with $E \mDef [\epsilon_{\mR}(t), .., \epsilon_{\mR}(T-1),\epsilon_{F}(t), .., \epsilon_{F}(T-2)]$ and $\mPo(j, s(j);\tilde\mPa) \mDef \mUpredOpt_{j|j}(s(j);\tilde\mPa)$, being the first element of the optimal input sequence of the MPC problem at time step $j$ with parameters $\tilde\mPa$ and $\rho(j)$ the required softening of state constraints.
Reinforcement learning problem
------------------------------
For MDPs of the form , we consider the case of unknown transition and reward distributions that are parametric according to , . The corresponding reinforcement learning problem is to improve the MPC policy, which is based on solving at every time step, through learning episodes that lead to reward maximization in a data-efficient manner. During each learning episode $e=0,1,..,N-1$ we therefore need to provide a control policy that trades-off information extraction and knowledge exploitation when applied to the MDP at each time step $t=0,1,..,T-1$ starting from $s(0)~\sim S^0$. We assume access to prior information about the MDP parameterization, such as production tolerances, to be given as $(\mPa_F, \mPa_\mR)\sim\mDistribution{\mPa}$. Collected data up to $N$ episodes is denoted by $$\begin{aligned}
\label{eq:data}
\mData_{N}\mDef
\left\{
\left(
t,
s_{t,e},
a_{t,e},
s_{t+1,e},
r_{t,e}
\right)_{t=0}^{T-1}
\right\}_{e=0}^{N-1}.
\end{aligned}$$ Conditioned on collected data , the corresponding posterior distribution up to episode $e$ is denoted by $\mPa_e \sim \mDistribution{\mPa|\mData_e}$.
Based on the acquired data over $N$ episodes, the performance of the RL algorithm is measured in terms of the expected Bayesian cumulative regret $$\begin{aligned}
\label{eq:cumulative_regret}
\mCre(N)\mDef
\mExpectationExp{\mPa, \mPa_e, \mData_e}{ \sum_{e=0}^{N-1} \mRe{e} }
\text{ with episodic regret }
\mRe{e} \mDef \mExpectationExp{s}{
\mV{\mPa}{\mPa}{0}(s)
-
\mV{\mPa}{\mPa_e}{0}(s)
}.
\end{aligned}$$ Using the notation of the expected future reward in , the cumulative regret quantifies the expected performance deviation between the MPC-based RL algorithm using episodically updated model parameters $\mPa_e$ and the optimal MPC-based policy with access to the true parameters $\mPa$ of the underlying MDP .
Scalable model-based RL: The Bayesian MPC algorithm
===================================================
Following the concept introduced in [@Wabersich2020], we propose to combine model-based RL using posterior sampling as introduced in [@Strens2000] and investigated by [@Russo2014; @Osband2013; @Osband2014] with a cautious model predictive control policy parametrization as described in Section \[sec:soft\_mpc\] to obtain a new class of model-based RL policies with safety guarantees, called Bayesian MPC. At the beginning of each learning episode $e$ we sample transition and reward parameters $\mPa_e$ according to their posterior distribution that results from the prior distribution $\mDistribution{\mPa}$ together with observed data $\mData_e$.
**Bayesian MPC algorithm $\qquad\qquad\quad$**
Initialize $\mData_0 = \emptyset$
The sampled parameters yield an MPC problem parametrization that would correspond to an MDP with parameters $\mPa_e$. However, since $\mPa_e \neq \mPa$, such an MPC policy might be inconsistent with the underlying system to be controlled, in particular if the posterior parameter variance is large. In this case, the sampled policy is likely to cause explorative closed-loop behavior, producing information-rich data. Compared to using, e.g., the current maximum a-posteriori estimate of the parameters as done in most learning-based MPC approaches, which we refer to as nominal posterior MPC [@hewing2020], the algorithm therefore generates explorative behavior in case of large posterior parameter uncertainties. As soon as the task-relevant parameter distributions begin to cumulate around the corresponding true process parameters, the parameter samples will start to cumulate as well, implying convergence of the sampled MPC performance to the MPC performance with perfectly known parameters. We provide a rigorous analysis of this effect in Sections \[subsec:regularity\_linear\_concave\] and \[subsec:regularity\_nonlinear\], which is one of the key ingredients to obtain a regret bound.
In addition to performance, if the MPC policy using the true MDP parameters is capable of ensuring cautious constraint satisfaction in expectation, i.e. if $\mExpectationExp{E}{I(\rho(t))}=0$ for $\delta >0$ under $a(j) = \mUpredOpt(j,s(j),\mPa)$ in , then we can additionally bound the expected number of unsafe learning episodes, i.e. the number of episodes in which the state constraints are violated. As formalized in Section \[subsec:unsafe\_learning\_episodes\], this can be achieved through a sufficiently large $c_1$ in the exact penalty $I(.)$ in together with a bound on the instant regret.
Besides the availability of efficient software tools to solve the MPC problem , see e.g. [@Houska2011a; @forcesnlp2017; @Verschueren2019], large-scale MPC problems often exhibit a distributed structure such as autonomous mobility-on-demand systems [@carron2019scalable] or common systems in process manufacturing [@Maestre2014]. The corresponding MPC problem can often be implemented in a distributed fashion in such cases, see e.g. [@Nedic2018], and is therefore scalable to arbitrary dimensions.
Analysis {#sec:analysis}
========
Since the Bayesian MPC algorithm can conceptually be used to enhance any existing MPC application for a wide variety of MDPs, we briefly recap general sufficient conditions from [@Wabersich2020] in this section to obtain finite-time regret bounds that are based on the framework proposed by [@Osband2014]. The main contribution of this paper is to establish that these conditions hold for the relevant case of linear systems in Section \[subsec:regularity\_linear\_concave\] and to provide sufficient conditions for the more general nonlinear case in Section \[subsec:regularity\_nonlinear\]. These results enable us together with cautious soft constraints from Section \[sec:soft\_mpc\] to derive a bound on the expected number of unsafe learning episodes under application of the Bayesian MPC algorithm in Section \[subsec:unsafe\_learning\_episodes\].
We start by reviewing the main steps of model-based RL based on posterior sampling arguments as presented in [@Osband2013; @Osband2014] to reformulate the regret in terms of the expected learning progress of the transition and reward function. By using a regularity assumption on the expected future reward under the sampled MPC controllers this then allows us to bound the cumulative regret using the so-called Eluder dimension, which expresses the learning complexity for different mean and reward function classes. Instead of the instant regret $\mRe{e}$ in , which includes the unknown optimal future reward $\mV{\mPa}{\mPa}{0}(s)$, we formulate the regret in terms of the sampled MPC controller applied to the corresponding sampled system, for which it is optimal, i.e. $$\begin{aligned}
\label{eq:rotated_regret}
\mExpectationExp{\mPa, \mPa_{e}, s, \mData_e }{\mtRe{e}}
&= \mExpectationExp{\mPa, s, \mData_e}{
\mExpectationExpMed{\mPa_{e}}{
\mV{\mPa_e}{\mPa_e}{0}(s)
-
\mV{\mPa}{\mPa_e}{0}(s)
|
\mPa, s, \mData_e
}
},
\end{aligned}$$ where $\mV{\mPa_e}{\mPa_e}{0}(s)$ is known based on the sample $\mPa_e$ and $\mV{\mPa}{\mPa_e}{0}(s)$ can be observed. Using posterior sampling arguments we can verify that $
\mExpectationExpSmall{\mPa, \mPa_{e}, s, \mData_e}{\mRe{e} - \mtRe{e}} =0
\Rightarrow
\mExpectationExpSmall{\mPa, \mPa_{e}, s, \mData_e}{\mRe{e}} = \mExpectationExpSmall{\mPa, \mPa_{e}, s, \mData_e }{\mtRe{e}}.
$ The reformulated regret allows another reformulation based on the Bellman operator as originally proposed by [@Osband2013] for discrete states and actions and sketched in [@Osband2014; @Wabersich2020] for the continuous case to end up with a regret bound of the form $$\begin{aligned}
\nonumber
\mExpectationSmall{\mtRe{e}} \leq
&\mExpectationExp{ }{
\sum_{t=0}^{T-1}
\mExpectationExp{\epsilon(t)}{
| \mV{\mPa_e}{\mPa_e}{t+1}(f(s(t),a(t);\mPa_e)+\epsilon_F(t))
- \mV{\mPa_e}{\mPa_e}{t+1}(f(s(t),a(t);\mPa)+\epsilon_F(t))|
}
}
+\\\label{eq:two_rotated regret terms}
&\mExpectationExp{}{
\sum_{t=0}^{T-1}
| r(t,s(t), a(t);\mPa_e)-r(t,s(t), a(t);\mPa) |},
\end{aligned}$$ with expectation over $\mPa, \mPa_e, s, \mData_e$. The second term in can be bounded via the conditional posterior through $
\mExpectationExpSmall{\mPa, s, \mData_e }{
\mExpectationExpSmall{\mPa_{e}}{
\sum_{t=0}^{T-1}
| r(t, s(t), a(t);\mPa_e) - r(t, s(t), a(t);\mPa)|
~ | ~ \mPa, s, \mData_e
}
}.
$ The first term, however, requires a regularity assumption that quantifies how errors in the expected one-step-ahead prediction $f(s(t),a(t);\mPa_e) - f(s(t),a(t);\mPa)$ cause deviations w.r.t. the one-step-ahead expected future reward $\mV{\mPa_e}{\mPa_e}{t+1}(.)$ as follows.
\[ass:continuity\_v\] For all $\mPa_e\in \RR^{n_\mPa}$ and $s^+,\tilde s^+\in\XX$ there exists a constant $L_V>0$ such that $$\begin{aligned}
\label{eq:ass_continuity_v}
\mExpectationExp{\epsilon_F(t)}{
| \mV{\mPa_e}{\mPa_e}{t+1}(s^+ + \epsilon_F(t))
- \mV{\mPa_e}{\mPa_e}{t+1}(\tilde s^+ + \epsilon_F(t))|
}
\leq
L_V \mNorm{2}{s^+ - \tilde s^+}.
\end{aligned}$$
Note that if the expected future reward could vary arbitrarily, even for very similar states $s^+$ and $\tilde s^+$, it might be impossible to provide any kind of regret bound. While previous literature only required this assumption, we will theoretically investigate its justification in the case of MPC-based policies in Section \[subsec:regularity\_linear\_concave\] and Section \[subsec:regularity\_nonlinear\].
The relationship between the regret and the mean deviation between the true and sampled reward and transition function as described previously allows us to derive a Bayesian regret bound using statistical measures. The first bound relates to the complexity of the respective mean function also known as the Kolmogorov dimension $\dim_K$, see also [@Russo2014]. In an online learning setup it is additionally necessary to quantify the difficulty of extracting information and accurate predictions based on observed data, which is measured in terms of the Eluder dimension $\dim_E$ [@Russo2014]. These measures further require boundedness of the mean reward and transition function as follows.
\[ass:bounded\_expected\_cost\_transition\] There exist constants $c_\mR$ and $c_F$ such that for all admissible $s\in\RR^n$, $a\in\RR^m$, $\mPa\in\RR^{n_{\mPa}}$, and $t=0,1,..,T-1$ it holds $|\ell(t,s,a;\mPa_\mR)|\leq c_\mR$, and $||f(s,a;\mPa_F)||\leq c_F$.
Following [@Osband2014; @Wabersich2020], we can combine these measures to obtain the following regret bound for the Bayesian MPC algorithm as an immediate consequence from Theorem 1 in [@Osband2014] with $\tilde \OO$ neglecting terms that are logarithmic in $N$.
\[thm:general\_regret\_bound\] If Assumptions \[ass:continuity\_v\] and \[ass:bounded\_expected\_cost\_transition\] hold then it follows that $$\begin{aligned}
\label{eq:cor_general_regret_bound}
\mCre(N)
\leq
\tilde \OO\left(
\sigma_R\sqrt{\mKdim(\ell)\mEdim(\ell)T N}
+L_V \sigma_F \sqrt{\mKdim(f)\mEdim(f)T N}
\right).
\end{aligned}$$
Specific bounds for different parametric function classes can be found, e.g., in [@Russo2014; @Osband2014].
Regularity of the value function for large-scale linear transitions and concave rewards {#subsec:regularity_linear_concave}
---------------------------------------------------------------------------------------
Regularity of the future reward as required by Assumption \[ass:continuity\_v\] is a central ingredient for the performance analysis and essentially determines the shape of the regret bound in Theorem \[thm:general\_regret\_bound\]. While explicit bounds on the Kolmogorov- and Eluder dimensions are available for relevant parametric function classes [@Russo2014; @Osband2014], we provide a bound on $L_V$ that holds under application of the Bayesian MPC algorithm. In this section we begin by focusing on the control relevant case of linear time-invariant transitions of the form $$\begin{aligned}
\label{eq:linear_transition}
s(t+1) = A(\theta_F)s(t) + B(\theta_F)a(t) + \epsilon_F(t)
\end{aligned}$$ and reward models that are either affine or quadratic and concave in the states and actions for each time step $t=0,1,..,T-1$. Furthermore, we restrict our attention to state and action spaces that are polytopic of the form $\XX\mDef\{s\in\RR^n | A_s s \leq b_s\}$ and $\UU\mDef\{a\in\RR^m | A_a a \leq b_a\}$. Based on these assumptions we establish Lipschitz continuity of the optimizer of the MPC Problem . Combining Lipschitz continuity of $\mUpredOpt_{t|t}$ with Lipschitz continuity of the mean transition and reward model allows us to establish Assumption \[ass:continuity\_v\].
\[thm:lipschitz\_continuity\_linear\] Consider MPC problem . If the mean transition is linear, the state and action constraints are polytopic, and the mean reward function is linear or quadratic and strictly concave for all time steps, then under application of the Bayesian MPC algorithm it follows that Assumption \[ass:continuity\_v\] holds.
The proof together with a detailed construction of $L_V$ according to Asssumption \[ass:continuity\_v\] can be found in Appendix \[app:proof\_lipschitz\_continuity\_linear\]. Combining this result with Corrollary \[thm:general\_regret\_bound\] and the specific bounds on the Eluder- and Kolmogorov dimensions from [@Russo2014; @Osband2014] provides the following performance bound without explicit need of Assumption \[ass:continuity\_v\].
\[cor:specific\_regret\_bound\] Under the same assumptions of Theorem \[thm:lipschitz\_continuity\_linear\], the cumulative Bayesian regret of the Bayesian MPC algorithm is bounded by $$\begin{aligned}
\mCre(N)
&\leq
\tilde \OO\left(
\sigma_{R}n_\ell\sqrt{2 T N}
+L_V \sigma_{F} n\sqrt{n(n+m) T N}
\right)
\end{aligned}$$ with $n_\ell$ mean reward parameters and $L_V$ according to in Appendix \[app:proof\_lipschitz\_continuity\_linear\].
The statement is an immediate consequence of Theorem \[thm:general\_regret\_bound\], Theorem \[thm:lipschitz\_continuity\_linear\], and [@Osband2014 Proposition 2,3].
Extension of regularity towards nonlinear transitions and rewards {#subsec:regularity_nonlinear}
-----------------------------------------------------------------
While the linear case as considered in the previous section covers a large portion of control applications, the increasing availability and performance of nonlinear MPC solvers motivates the extension to nonlinear reward and transition models. We therefore extend the analysis from Section \[subsec:regularity\_linear\_concave\] to the more general case of transition and reward functions $f$ and $\ell$ that are nonlinear and non-convex as well as more general state and action spaces of the form $\XX\mDef\{s\in\RR^n | g_s(s) \leq \mathbb 1\}$ and $\UU\mDef\{a\in\RR^m | g_a(a) \leq \mathbb 1\}$.
To this end, we use a similar line of reasoning as in the proof of Theorem \[thm:lipschitz\_continuity\_linear\] to provide sufficient conditions on the resulting nonlinear MPC problem that ensure Assumption \[ass:continuity\_v\]. In particular, we utilize results from [@Liu1995] to analyze local continuity properties of KKT-based solutions of the MPC problem as follows.
\[thm:lipschitz\_continuity\_nonlinear\] Let Assumption \[ass:bounded\_expected\_cost\_transition\] hold and consider the MPC problem with $f,\ell,I,g_s$, and $g_a$ continuously differentiable and Lipschitz continuous. If the linear independence (LI) and the strong second-order sufficient condition (SSOSC) according to [@Liu1995 3(a) and 3(d)] hold for all admissible $s\in\RR^n$, $\mPa\in\RR^{n_{\mPa}}$, and $t=0,1,..,T-1$, then it follows that Assumption \[ass:continuity\_v\] holds.
The main steps of the proof can be found in Appendix \[app:proof\_lipschitz\_continuity\_nonlinear\]. The linear independence (LI) condition refers to the linear independence of the gradients of the active constraints at an optimum of with respect to the decision variables and ensure necessity of the corresponding KKT conditions. In addition, the strong second-order sufficient condition (SSOSC) guarantees sufficiency of the KKT conditions and uniqueness of local solutions through a local positive definiteness condition of the Hessian of the Lagrangian w.r.t. the decision variables, also depending on the active constraints at the optimum. Consequently, if these conditions do not hold, situations where small deviations of $s$ cause a ‘jumping behavior’ between different local optimal solutions of the MPC problem can occur and potentially lead to a non-Lipschitz continuous future reward function.
While the imposed assumptions are difficult to verify, note that the LI condition is a common requirement for nonlinear solvers and that the SSOSC is the weakest condition to ensure existence and local uniqueness of local solutions of the MPC problem for small perturbations of the initial condition $s$, see [@Kojima1980]. Importantly, note that, e.g., a normally distributed $\epsilon_F$ helps to smooth the future reward through the expectation operator in and Assumption \[ass:continuity\_v\] may still be satisfied.
Bounding the expected number unsafe learning episodes {#subsec:unsafe_learning_episodes}
-----------------------------------------------------
While a tightened MPC formulation using the true parameters $\mPa$ typically provides state constraint satisfaction in expectation for many practical applications, the parameter samples $\mPa_e$ during application of the Bayesian MPC algorithm can vary significantly during initial learning episodes. It can therefore happen that the constraints are violated, even in expectation. In such learning episodes, however, the amount of constraint violation can partially be observed through the regret due to the exact penalty in the future reward function . As a consequence, if the MPC using the true system parameters $\mPa$ provides satisfaction of the tightened constraints in expectation, i.e. $s(t)\in\bar\XX_\delta(0)$, we can use regularity of the expected future reward to show that a converging parameter estimate yields a converging future reward and therefore converging constraint satisfaction. In other words, since the stage cost function is bounded and the constraints are tightened, a sufficiently large soft constraint penalty ensures observability and a bound on state constraint violations, which can be formalized as follows.
We first derive an upper bound on the instant regret $\Delta_e$ as defined in implying constraint satisfaction in Appendix \[app:safety\_analysis\]. By combining this intermediate result with the regret bound from Theorem \[thm:general\_regret\_bound\] we bound the cumulative expected number of unsafe learning episodes in Theorem \[thm:total\_number\_of\_unsafe\_episodes\]. To streamline notation we denote the state, action, and slack variable sequence in the expected future reward for a given initial state $s\sim S^0$ as $s_{\tilde\mPa}^\mPa(j)$, $a_{\tilde\mPa}^\mPa(j)$, and $\rho_{\tilde\mPa}^\mPa(j)$ for $j=0,..,T-1$ in the following.
\[thm:total\_number\_of\_unsafe\_episodes\] Let the conditions of Theorem \[thm:general\_regret\_bound\] hold and consider a weighting factor in the exact penalty term $I(.)$ in that satisfies $\min_i(c_{1,i}) \geq \frac{2Tc_R + c_\delta}{\delta}$ for some $c_\delta >0$. If $\mExpectationExp{E,s}{\rho_{\mPa}^\mPa(j)}=0$, then the total number of $N_{\mathrm{unsafe}}$ episodes, for which there exists a $j \in\mathbb N$, $0\leq j \leq T-1$ such that $\mExpectationExp{E,s}{s_{\mPa_e}^\mPa(j)\notin\XX}$ is bounded in terms of the cumulative regret by $N_{\mathrm{unsafe}}\leq\lceil CR(N)c_\delta^{-1}\rceil$.
A sublinear cumulative regret bound therefore ensures a decreasing ratio between the number of episodes with constraint violation and the total number of learning episodes, which vanishes at the rate of $c(1/N)$ for $N \rightarrow \infty$ and some positive constant $c$. This can be seen as a first step towards combined finite time safety and performance guarantees in case of model-based RL in continuous state and action spaces. Note that the upper bound $c_\delta$ and consequently also the lower bound on the exact penalty scaling could potentially be improved in the corresponding proof (Appendix \[app:safety\_analysis\]), e.g. by exploiting the concrete structure of $I$ including quadratic terms.
Numerical examples
==================
![Simulation results of numerical examples for 100 different experiments. Thin lines depict experiment samples and thick lines show the corresponding mean. **Left:** Cumulative regret of the large-scale thermal application detailed in Section \[app:cooling\_example\]. **Middle:** Maximum value of $\rho(j)$ as defined in over one episode. **Right:** Cumulative regret of exploration task as described in Section \[app:drone\_example\].[]{data-label="fig:example_plots"}](fig/matlab/server_regret-crop.pdf "fig:"){width="0.32\linewidth"} ![Simulation results of numerical examples for 100 different experiments. Thin lines depict experiment samples and thick lines show the corresponding mean. **Left:** Cumulative regret of the large-scale thermal application detailed in Section \[app:cooling\_example\]. **Middle:** Maximum value of $\rho(j)$ as defined in over one episode. **Right:** Cumulative regret of exploration task as described in Section \[app:drone\_example\].[]{data-label="fig:example_plots"}](fig/matlab/quad_cons-crop.pdf "fig:"){width="0.27\linewidth"} ![Simulation results of numerical examples for 100 different experiments. Thin lines depict experiment samples and thick lines show the corresponding mean. **Left:** Cumulative regret of the large-scale thermal application detailed in Section \[app:cooling\_example\]. **Middle:** Maximum value of $\rho(j)$ as defined in over one episode. **Right:** Cumulative regret of exploration task as described in Section \[app:drone\_example\].[]{data-label="fig:example_plots"}](fig/matlab/quad_regret-crop.pdf "fig:"){width="0.35\linewidth"}
We first consider the task of efficiently controlling a large-scale network with 100 cooling units, e.g. a server farm or production machines in manufacturing plants, that are arranged in a grid structure and have a strong thermal coupling with respect to locally neighboring units. The system dynamics and reward satisfy the assumptions of Section \[subsec:regularity\_linear\_concave\], see Appendix \[app:cooling\_example\] for further details. The goal is to find an energy efficient control policy that obeys maximal allowable temperatures despite parametric uncertainties in the system. In Figure \[fig:example\_plots\] (Left), we compare the proposed Bayesian MPC algorithm against commonly used nominal posterior MPC, i.e. selecting $\theta_e \mDef \mExpectation{\theta|\mData_e}$ [@hewing2020 Section 3], using 100 different system realizations. While both algorithms show reasonable learning performance and provide constraint satisfaction at all times, Bayesian MPC is able to significantly reduce the cumulative regret by almost $50\%$ compared to nominal posterior MPC.
In addition, we consider a generic drone search task falling into the problem class of Section \[subsec:regularity\_nonlinear\]. The goal is to collect information about an a-priori unknown position of interest using a quadrotor drone. While the prior of the 10-dimensional drone dynamics are selected according to [@bouffard2012board], we additionally simulate strong winds in different altitudes, which adds strong nonlinear effects to the dynamics. Once the target position is reached, the drone collects information before it returns to the base station for analysis and recharge. The overall goal therefore is to learn the drone dynamics, winds in different altitudes and the most informative search position. The safety-critical constraints are a maximum range of the drone together with a minimum altitude that need to be satisfied under physical actuator limitations, see Appendix \[app:drone\_example\] for further details. By comparing nominal posterior MPC against Bayesian MPC over 100 different experiments in terms of expected constraint satisfaction, we notice from Figure \[fig:example\_plots\] (Middle) that Bayesian MPC causes explorative behavior during initial episodes, which yields higher constraint violations compared to nominal posterior MPC. However, this behavior enables safety of future episodes and bounded cumulative regret Figure \[fig:example\_plots\] (Right) compared to posterior nominal MPC, which has unbounded cumulative regret.
Broader Impact {#broader-impact .unnumbered}
==============
The proposed RL algorithm is tailored to solve modern control engineering problems, i.e. problems where high-performance control under system constraints is essential. Nowadays, such applications are typically driven by model predictive control (MPC) techniques. Since the proposed RL algorithm provides an automated way of improving MPC controllers, any existing MPC application can potentially be enhanced through the presented method addressing a key challenge and development cost factor in industry. Prominent example systems and can be found in aerospace [@eren2017model], automotive [@del2010automotive], or process manufacturing [@morari1999model], where control methods help to reduce energy consumption by optimizing and coordinating processes. As for most control techniques, which act on a low-level planning instance, ethical and societal aspects mainly depend on the specific application in which they are used.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the Swiss National Science Foundation under grant no. PP00P2\_157601/1.
Additional information: Numerical examples
==========================================
All examples are implemented using the Casadi framework [@andersson2019casadi] together with the IPOPT solver [@wachter2006implementation].
Large scale thermal application {#app:cooling_example}
-------------------------------
![Cooling network structure: Each unit $i$ has a measured temperature state $s_i$ and cooling action $a_i$ and is affected by the temperature states of the top/down/left/right neighboring units arranged in a grid.[]{data-label="fig:experiment_cooling"}](fig/ppt/cooling-crop.pdf){width="0.5\linewidth"}
We consider the task of efficiently controlling a large-scale network of 100 cooling units, e.g. a server farm or production machines in manufacturing plants, that are arranged in a grid structure and have a strong thermal couplings with respect to locally neighboring units, see Figure \[fig:experiment\_cooling\].
Actions $a(t)\in\UU\subset\RR^{100}$ describe the applied cooling power to each cooling, which are subject to physical limitations $\UU\mDef\{a\in\RR^{100}|a\leq \bar a\}$. The system state is defined by the temperatures $s(t)\in\RR^{100}$ of each unit that needs to be below a given threshold $s(t)\in\XX\mDef\{s\in\RR^{100}|s\leq 100\}$ for all times. The thermodynamics of the plant are given as an MDP with linear mean dynamics such that each unit $i$ has unknown dynamics of the form $s_i(t+1)= A_{ii}s(t) + B_{i}a_i(t) + \sum_{j\in N_{s,i}}A_{ij}s_j(t) + C_i + \epsilon_F(t)$ with neighboring units indexed by $j$, known Gaussian parameter prior distribution $(A,B,C)\sim\mDistribution{\mPa_F}$ and Gaussian process noise $\epsilon_F(t)$. Note that the resulting overall dynamics can be stated in the form by extending the state space.
The goal is to minimize the overall expected energy consumption $\sum_i L_i a_i(t) + \epsilon_\mR$ by considering thermal couplings while keeping the temperature of each cooling unit below a specified maximum temperature, starting form a temperature level below $100$ degrees. The energy efficiency of each unit is described through parameters $L\in\RR^{100}$ that are sampled from a known Gaussian prior distribution $l\sim\mDistribution{\mPa_\mR}$ plus additive Gaussian measurement noise $\epsilon_\mR$. The overall plant consists partly of new cooling units with known efficiency and older cooling units with uncertain efficiency factors that are worse in expectation. Due to these different efficiency levels that are provided through the prior distribution, explorative behavior can be beneficial to exploit more efficient units. The exact numerical values and prior parametrisation can be found in the function $\texttt{server\_experiment.m}$ in the provided source code for the example.
Drone search application {#app:drone_example}
------------------------
![Illustration of the drone search application (Section \[app:drone\_example\]). **Left:** Conceptual drawing of the task including wind in different altitudes, the maximum range of the drone, and the unknown optimal position for surveillance. **Right:** Sample drone trajectory during one episode. Starting from the base station in the middle, the drone quickly approaches the position of maximum information gain.[]{data-label="fig:experiment_quadrotor"}](fig/ppt/quadrotor-crop.pdf "fig:"){width="0.57\linewidth"} ![Illustration of the drone search application (Section \[app:drone\_example\]). **Left:** Conceptual drawing of the task including wind in different altitudes, the maximum range of the drone, and the unknown optimal position for surveillance. **Right:** Sample drone trajectory during one episode. Starting from the base station in the middle, the drone quickly approaches the position of maximum information gain.[]{data-label="fig:experiment_quadrotor"}](fig/matlab/quad_traj.jpg "fig:"){width="0.4\linewidth"}
We consider the task depicted in Figure \[fig:experiment\_quadrotor\], with a quadrotor system as described in [@bouffard2012board] with a 10 dimensional state space.
The dynamics are of the form $s(t+1)=As(t) + Ba(t) + C\Phi(s(t)) + \epsilon_F$, where $A$ and $B$ matrices describe the unknown dynamics around the hovering state and and $C\Phi(s(t))$ models strong winds in different altitudes as $\Phi(s(t))=[k(s_3(t),\bar W_1),k(s_3(t),\bar W_2),k(s_3(t),\bar W_3)]^\top$ using radial basis functions $k(.,.)$ [@Friedman2001] with given hyper-parameters $W_i$ and unknown parameters $C$. The system has a three dimensional action space that allows to control the desired pitch and roll as well as vertical acceleration of the drone.
The system constraints are given by physical action constraints and a box constraint on the position states, which describes the minimal altitude and maximum range of the drone. Furthermore, the use of a linear model is only valid around the hovering state, yielding additional absolute pitch and roll constraints of $30~\mathrm{[deg]}$ to the system.
The reward signal corresponds to the information gained at the final position at the end of an episode and is modeled as the sum of equally spread radial-basis-functions at positions $p_i\in\RR^3$, $i=1,..,9$ , i.e. $r(T-1, s(T-1), a(T-1)) = \epsilon_\mR + \sum_{i=1}^9 \theta_{R,i} k(s_{1-3}(T-1), p_i)$ see Figure \[fig:experiment\_quadrotor\] (Right) for an example illustration. The unknown system dynamics parameters $\mPa_F\mDef C$, dynamics process noise $\epsilon_F$, reward parameters $\mPa_\mPa\mDef \{p_i\}$, and reward noise $\epsilon_\mR$ are normally distributed. The exact numerical values can be found in the function `quadrotor_example.m` in the provided source code for the example.
Proofs
======
Proof of Theorem \[thm:lipschitz\_continuity\_linear\] {#app:proof_lipschitz_continuity_linear}
------------------------------------------------------
It is sufficient to show global Lipschitz continuity of $\mV{\mPa_e}{\mPa_e}{t}(s)$ since existence of an $L_{V_t}>0$ such that $$\begin{aligned}
\mExpectationExp{\epsilon_F}{
| \mV{\mPa_e}{\mPa_e}{t+1}(s^+ + \epsilon_F)
- \mV{\mPa_e}{\mPa_e}{t+1}(\tilde s^+ + \epsilon_F)|
}
&\leq
\mExpectationExp{\epsilon_F}{
L_{V_t}||s^+ + \epsilon_F-\tilde s^+ + \epsilon_F||
}\\
&\leq
L_{V_t}||s^+ -\tilde s^+ ||
\end{aligned}$$ implies the desired result. Let $\tilde\ell(j,s,a)\mDef \ell(j,s,a;\mPa_e) - I(\rho(s))$ with $\rho(s)\mDef \min_{\rho\geq 0}\text{ s.t. }s\in\bar\XX_\delta(\rho)$ according to , which is Lipschitz continuous in $s$ and $a$ since $\XX$ and $\UU$ are polytopic. To streamline notation we denote the state and action sequence in the expected future reward with $\tilde \mPa = \mPa_e$ as $s(j, s, E)$ and $a(j, s)$ for $j=0,..,T-1$ with $s(0, s, E)=s$ and $E \mDef [\epsilon_{\mR}(t), .., \epsilon_{\mR}(T-1),\epsilon_{F}(t), .., \epsilon_{F}(T-2)]$ in the following. We have due to linearity of the expectation operator, Jensen’s inequality, the triangle inequality, and global Lipschitz continuity of $\tilde\ell$ that there exists $L_{\ell,j}$ such that $$\begin{aligned}
&|\mV{\mPa_e}{\mPa_e}{t+1}(s^+)
- \mV{\mPa_e}{\mPa_e}{t+1}(\tilde s^+)|\\
=&
\left|\mExpectationExp{E}{
\sum_{j=t+1}^{T-1} \tilde\ell(j, s(j,s^+,E), a(j,s(j,s^+,E)))
- \tilde\ell(j, s(j,\tilde s^+,E), a(j,s(j,\tilde s^+,E)))
}\right|\\
\leq&
\mExpectationExp{E}{
\sum_{j=t+1}^{T-1}
\left|\tilde\ell(j, s(j,s^+,E), a(j,s(j,s^+,E)))
- \tilde\ell(j, s(j,\tilde s^+,E), a(j,s(j,\tilde s^+,E)))
\right|
}\\
\leq&
\mExpectationExp{E}{
\sum_{j=t+1}^{T-1}
L_{\ell,j}||[s(j,s^+,E)^\top-s(j,\tilde s^+,E)^\top,
a(j,s(j,s^+,E))^\top-a(j,s(j,\tilde s^+,E))^\top]^\top||
}\\
\end{aligned}$$ It therefore remains to show that $s$ and $a$ are Lipschitz continuous in their second argument. Since $a(j,s)$ is the first element of the optimal action sequence according to and is guaranteed to be feasible due to the soft-constraint reformulation, it follows from [@borrelli2003constrained Thm. 1.8] for affine $\tilde\ell$ and from [@borrelli2003constrained Thm. 1.12] for strictly concave quadratic $\tilde\ell$ for any $j$ and $s,\tilde s\in\RR^n$ that there exists a $\bar K \in \RR^+$, $\bar K <\infty$ such that $$\begin{aligned}
||a(j,s)-a(j,\tilde s)|| \leq \bar K ||s-\tilde s||.
\end{aligned}$$
It remains to show that there exists an $L_s(j)$ such that $$\begin{aligned}
||s(j,s^+,E)-s(j,\tilde s^+,E)||\leq L_s(j) ||s^+-\tilde s^+||
\end{aligned}$$ We show that $$\begin{aligned}
&||s(j,s^+,E)-s(j,\tilde s^+,E)\leq L_s(j)||s^+ - \tilde s^+||\\
\Rightarrow & ||s(j+1,s^+,E)-s(j+1,\tilde s^+,E)||\leq L_s(j+1)||s^+ - \tilde s^+||
\end{aligned}$$ with $L_s(j)=L_s(j-1)(||A|| + ||B||\bar K)$ and $L_s(0)=1$ by induction.
Induction start: $$\begin{aligned}
j = 0:\quad ||s(0,s^+,E)-s(0,\tilde s^+,E)||=
||s^+ - \tilde s^+||\leq L(0)||s^+ - \tilde s^+||
\text{ with } L(0)=1
\end{aligned}$$ implying $$\begin{aligned}
||s(1,s^+,E)-s(1,\tilde s^+,E)||
&=||As^+ + B a(0, s^+) + \epsilon_F(0) -A\tilde s^+ - B a(0, \tilde s^+) - \epsilon_F(0)||\\
&\leq ||A(s^+ - \tilde s^+) + B (a(0, s^+)-a(0, \tilde s^+))||\\
&\leq \underbrace{(||A||+||B||\bar K)1}_{=L_s(1)}||s^+ - \tilde s^+||.
\end{aligned}$$
Induction step: $$\begin{aligned}
\text{For any }j>0:\quad ||s(j,s^+,E)-s(j,\tilde s^+,E)&\leq L_s(j)||s^+ - \tilde s^+||
\end{aligned}$$ we have $$\begin{aligned}
&||s(j+1,s^+,E)-s(j+1,\tilde s^+,E)||\\
=&||As(j,s^+,E) + B a(j, s(j,s^+,E)) + \epsilon_F(j)-As(j,\tilde s^+,E) - B a(j, s(j,\tilde s^+,E)) - \epsilon_F(j)||\\
=&||A(s(j,s^+,E)- s(j,\tilde s^+,E)) + B(a(j, s(j,s^+,E))-a(j, s(j,\tilde s^+,E)))|| \\
\leq&(||A||+||B||\bar K)||s(j,s^+,E)- s(j,\tilde s^+,E)||\\
\leq&\underbrace{(||A||+||B||\bar K)L(j)}_{L_s(j+1)}||s^+ - \tilde s^+||\text{ (induction step hypothesis)}
\end{aligned}$$ Combining these results yields $$\begin{aligned}
\label{eq:lipschitz_continuity_value_function}
L_V = \sum_{j=t+1}^{T-1} L_{\ell,j}(L_s(j)(1 + \bar K))<\infty.
\end{aligned}$$
Proof outline of Theorem \[thm:lipschitz\_continuity\_nonlinear\] {#app:proof_lipschitz_continuity_nonlinear}
-----------------------------------------------------------------
For any initial state $s_0$ there exists a corresponding optimal solution $\mUpredOpt_{t|t}$ to due to the soft-constraint formulation, Lipschitz continuity of the objective, Lipschitz continuity of the constraints in , and the compactness of the input constraints. Together with [@Liu1995 Theorem 3.7] it follows from the given assumptions that there exists a unique function $y(t,s) \mapsto [\mUpredOpt_{t|t},\mUpredOpt_{t+1|t},...,\mUpredOpt_{t+N-1|t}]^\top$ that is Lipschitz continuous w.r.t. all initial conditions $s\in \BB(s_0,r)$ with $\BB(s_0,r)\mDef\{s\in\RR^n |~||s-s_0||\leq r\}$ and $s_0$ fulfilling the KKT conditions corresponding to . Due to the SSOSC, the KKT conditions imply optimality of $y(t,s)$ and we conclude existence of a local Lipschitz constant $L_a(t,s_0)>0$ such that for all $s,~\tilde s\in\BB(r,s_0)$ it holds $||a(t,s) - a(t,\tilde s)||=||y_t(s) - y_t(\tilde s)||\leq L_a(t,s_0)||s - \tilde s||$. Since the action space is compact it also follows boundedness of $$\begin{aligned}
L_{\bar a} = \max_{s,\tilde s, ||s-\tilde s||>r} \frac{||a(t,s) - a(t,\tilde s)||}{||s-\tilde s||},
\end{aligned}$$ allowing us to select $\bar K\mDef\max\{L_{\bar a},L_a(t,s_0)\}$. From here we can proceed analogously to the proof of Theorem \[thm:lipschitz\_continuity\_linear\] using $||s(j+1,s^+,E) - s(j+1,\tilde s^+,E)||\leq
(L_{fs} + L_{fa}\bar K)||s(j,s^+,E) - s(j,\tilde s^+,E)||$ with $L_{fs}$ and $L_{fa}$ being the Lipschitz constants of $f$ w.r.t. the state $s$ and action $a$.
Bounding the expected number of unsafe learning episodes {#app:safety_analysis}
--------------------------------------------------------
\[lem:reward\_constraint\] Let Assumption \[ass:bounded\_expected\_cost\_transition\] hold. Consider the expected future reward in for a constraint tightening $\delta >0$ and $s\in\bar\XX_\delta(0)$. If $\mExpectationExp{E}{\rho_{\mPa}^\mPa(j)}=0$ for all $j=0,..,T-1$ and the weighting factor of the exact penalty term $I(.)$ in satisfies $\min_i(c_{1,i}) \geq \frac{2Tc_R + c_\delta}{\delta}$ for some $c_\delta >0$ then it holds $$\begin{aligned}
|\mV{\mPa}{\mPa}{0}(s) - \mV{\mPa}{\tilde\mPa}{0}(s)| < c_\delta
\Rightarrow \mExpectationExp{E}{s_{\tilde\mPa}^\mPa(j)\in\XX} \text{ for all } j=0,1,..,T-1.
\end{aligned}$$
For a proof by contradiction, consider the case $| \mV{\mPa}{\mPa}{0}(s) - \mV{\mPa}{\tilde\mPa}{0}(s)| < c_\delta$ and $\mExpectationExp{E}{s_{\tilde\mPa}^\mPa(\bar j)\notin\XX}$ for some $0\leq \bar j \leq T-1$. It holds $\max_i\mExpectationExp{E}{\rho_{i,\tilde\mPa}^\mPa(\bar j)}>\delta$ and $\mExpectationExp{E}{I(\rho_{\tilde\mPa}^\mPa(\bar j))}\geq
\min_i(c_{1,i})\max_i\mExpectationExp{E}{\rho_{i,\tilde\mPa}^\mPa(\bar j)}$ Next, we derive a lower bound on the absolute expected reward difference $$\begin{aligned}
|\mV{\mPa}{\mPa}{0}(s) - \mV{\mPa}{\tilde\mPa}{0}(s)|
&= \left|
\mExpectationExp{E}{
\sum_{j=0}^{T-1}\ell_{\mPa}^{\mPa}(j) - \ell_{\tilde\mPa}^{\mPa}(j)
+ I(\rho_{i,\tilde\mPa}^\mPa(j))
}
\right|
\end{aligned}$$ with $\ell_{\tilde \mPa}^{\mPa}(j)\mDef\ell(j, s_{\tilde\mPa}^\mPa(j), a_{\tilde\mPa}^\mPa(j);\theta_\mR)$ to show the contradiction. We distinguish two cases:
Case $\mExpectationExp{E}{\ell_{\mPa}^{\mPa}(j) - \ell_{\tilde\mPa}^{\mPa}(j)} \geq 0$ for all $j$: It follows directly that $|\mV{\mPa}{\mPa}{0}(s) - \mV{\mPa}{\tilde\mPa}{0}(s)| \geq \min_i(c_{1,i})
\max_i\mExpectationExp{E}{\rho_{i,\tilde\mPa}^\mPa(\bar j)}
\geq c_\delta$.
Case $\mExpectationExp{E}{\ell_{\mPa}^{\mPa}(j) - \ell_{\tilde\mPa}^{\mPa}(j)} < 0$ with $j\in\mathbb J$ for some index set $\mathbb J \subseteq\{0,..,T-1\}$: $$\begin{aligned}
& |\mV{\mPa}{\mPa}{0}(s) - \mV{\mPa}{\tilde\mPa}{0}(s)| \\
& \geq \left|\mExpectationExp{E}{\ell_{\mPa}^{\mPa}(\bar j) - \ell_{\tilde\mPa}^{\mPa}(\bar j) + \min_i(c_{1,i})\delta +
\sum_{j\in\{0,...,T-1\}\setminus\{\bar j\}} \ell_{\mPa}^{\mPa}(j) - \ell_{\tilde\mPa}^{\mPa}(j) + I(\rho_{\tilde \mPa}^{\mPa}(j))}\right|\\
& \geq \left|\mExpectationExp{E}{(T-1)2c_R + c_\delta + \sum_{j\in\{0,...,T-1\}\setminus\{\bar j\}} \ell_{\mPa}^{\mPa}(j) - \ell_{\tilde\mPa}^{\mPa}(j) + I(\rho_{\tilde \mPa}^{\mPa}(j))}\right| \\
& \geq \left|\mExpectationExp{E}{(T-1)2c_R + c_\delta + \sum_{j\in\mathbb J\setminus\{\bar j\}} \ell_{\mPa}^{\mPa}(j) - \ell_{\tilde\mPa}^{\mPa}(j) + I(\rho_{\tilde \mPa}^{\mPa}(j))
+ \sum_{j\notin\mathbb J\setminus\{\bar j\}} \ell_{\mPa}^{\mPa}(j) - \ell_{\tilde\mPa}^{\mPa}(j) + I(\rho_{\tilde \mPa}^{\mPa}(j))}\right|
\end{aligned}$$ where we use linearity of the expectation operator, $\min_i(c_{1,i}) \geq \frac{2Tc_R + c_\delta}{\delta}$, and the fact that $\ell_{\mPa}^{\mPa}(\bar j) - \ell_{\tilde\mPa}^{\mPa}(\bar j)> - 2c_R$. Similarly $\sum_{j\in\mathbb J\setminus\{\bar j\}} \ell_{\mPa}^{\mPa}(j) -
\ell_{\tilde\mPa}^{\mPa}(j) \geq -|\mathbb J|c_R \geq - (T-1)2c_R$ and $I(\rho_{\tilde \mPa}^{\mPa}(j)) \geq 0$ yielding $$\begin{aligned}
|\mV{\mPa}{\mPa}{0}(s) - \mV{\mPa}{\tilde\mPa}{0}(s)|
& \geq \left|c_\delta + \sum_{j\notin\mathbb J\setminus\{\bar j\}} \ell_{\mPa}^{\mPa}(j) - \ell_{\tilde\mPa}^{\mPa}(j) + I(\rho_{\tilde \mPa}^{\mPa}(j))\right|\\
&\geq c_\delta.
\end{aligned}$$ The lower bound implies $$\begin{aligned}
c_\delta > |\mV{\mPa}{\mPa}{0}(s) - \mV{\mPa}{\tilde\mPa}{0}(s)| &\geq c_\delta
\end{aligned}$$ yielding the contradiction.
Proof of Theorem \[thm:total\_number\_of\_unsafe\_episodes\] {#app:proof_total_number_of_unsafe_episodes}
------------------------------------------------------------
Let $\NN_{\mathrm{unsafe}}$ be the set of episode indices such that $\mExpectationExp{s}{|\mV{\mPa}{\mPa}{0}(s) - \mV{\mPa}{\mPa_e}{0}(s)|} > c_\delta$ for $e\in\NN_{\mathrm{unsafe}}$, i.e. potentially unsafe episodes for which there exists some $j$ s.t. $\mExpectationExp{E,s}{s_{\mPa_e}^\mPa(j)\notin\XX}$ out of a total of $N$ episodes. By summing up potentially unsafe episodes $e\in \NN_{\mathrm{unsafe}}$ we get $$\begin{aligned}
\label{eq:thm_total_number_of_unsafe_episodes}
\sum_{e\in\NN_{\mathrm{unsafe}}}\mExpectationExp{s}{|\mV{\mPa}{\mPa}{0}(s) - \mV{\mPa}{\tilde\mPa}{0}(s)|}
\geq N_{\mathrm{unsafe}} c_\delta
\end{aligned}$$ by Lemma \[lem:reward\_constraint\] with $N_{\mathrm{unsafe}}$ such that $|\NN_{\mathrm{unsafe}}| = N_{\mathrm{unsafe}}$. By definition, the cumulative regret provides a bound for the sum in and we therefore end up with $CR(N) \geq N_{\mathrm{unsafe}}c_\delta$, which proves the desired statement.
|
---
abstract: 'A cavity optomechanical magnetometer is demonstrated. The magnetic field induced expansion of a magnetostrictive material is resonantly transduced onto the physical structure of a highly compliant optical microresonator, and read-out optically with ultra-high sensitivity. A peak magnetic field sensitivity of 400 nT Hz$^{-1/2}$ is achieved, with theoretical modeling predicting the possibility of sensitivities below 1 pT Hz$^{-1/2}$. This chip-based magnetometer combines high-sensitivity and large dynamic range with small size and room temperature operation.'
author:
- 'S. Forstner'
- 'S. Prams'
- 'J. Knittel'
- 'E. D. van Ooijen'
- 'J. D. Swaim'
- 'G. I. Harris'
- 'A. Szorkovszky'
- 'W. P. Bowen'
- 'H. Rubinsztein-Dunlop'
title: Cavity Optomechanical Magnetometer
---
Ultra-low field magnetometers are essential components for a wide range of practical applications including geology, mineral exploration, archaeology, defence and medicine[@ref1]. The field is dominated by superconducting quantum interference devices (SQUIDs) operating at cryogenic temperatures[@ref11]. Magnetometers capable of room temperature operation offer significant advantages both in terms of operational costs and range of applications. The state-of-the-art are magnetostrictive magnetometers with sensitivities in the range of fT Hz$^{-1/2}$[@ref6; @ref6b], and atomic magnetometers which achieve impressive sensitivities as low as 160 aT Hz$^{-1/2}$[@ref8] but with limited dynamic range due to the nonlinear Zeeman effect[@ref11; @ref13]. Recently, significant effort has been made to miniaturize room temperature magnetometers. However both atomic and magnetostrictive magnetometers remain generally limited to millimeter or centimeter size scales. Smaller microscale magnetometers have many potential applications in biology, medicine, and condensed matter physics[@njpBouchardHall; @ref19]. A particularly important application is magnetic resonance imaging, where by placing the magnetometer in close proximity to the sample both sensitivity and resolution may be enhanced[@Stefan_ref22_29], potentially enabling detection of nuclear spin noise[@Meriles], imaging of neural networks[@ref19], and advances in areas of medicine such as magneto-cardiography[@ref1; @ref13] and magneto-encephalography[@Stefan_ref25].
In the past few years, rapid progress has been achieved on NV center based magnetometers. They combine sensitivities as low as 4 nT Hz$^{-1/2}$ with room temperature operation, optical readout and nanoscale size[@ref16] and are predicted theoretically to reach the fT Hz$^{-1/2}$ range[@ref17]. This has allowed three-dimensional magnetic field imaging at the micro scale using ensembles of NV-centers [@ref19], and magnetic resonance [@ref18] and field imaging[@ref17] at the nanoscale using single NV centers. In spite of these extraordinary achievements applications are hampered by fabrication issues and the intricacy of the read-out schemes[@ref20]. Furthermore miniaturization is limitied by the bulky read-out optics, the magnetic field coils for state preparation and the microwave excitation device[@ref19].
In this letter we present the concept of a cavity optomechanical field sensor which combines room temperature operation and high sensitivity with large dynamic range and small size. The sensor leverages results from the emergent field of cavity optomechanics where ultra-sensitive force and position sensing has been demonstrated[@ref21]. A cavity optomechanical system (COMS) is used to sense magnetic field induced deformations of a magnetostrictive material, which are detected with an all-in-fiber optical system suitable for the telecom wavelength range. The presence of mechanical and optical resonances greatly enhances both the response to the magnetic field and the measurement sensitivity.
Three implementations using different types of COMS are investigated, microscale Fabry-Perot resonators[@ref38], optomechanical zipper cavities[@ref39], and toroidal whispering gallery mode (TWGM) resonators[@ref22], all of which are approximately 50 $\mu$m in size. Theoretical modelling predicts ultimate Brownian noise limited sensitivities at the level of 200 pT Hz$^{-1/2}$, 10 pT Hz$^{-1/2}$, and 700 fT Hz$^{-1/2}$, respectively for each architecture. We experimentally demonstrate the concept with a TWGM resonator fabricated on a silicon wafer, achieving a peak sensitivity of 400 nT Hz$^{-1/2}$. The possibility to integrate TWGM resonators in a dense two-dimensional array with waveguides for optical coupling[@ref24] provides the potential for highly-integrated imaging magnetometers[@ref23].
A COMS based field sensor consists of a field-sensitive material, i.e. a magnetostrictive material coupled to the COMS’s mechanical oscillator. On application of a modulated field the field-sensitive material generates an oscillating mechanical stress field within the COMS structure exciting its mechanical eigenmodes. The vibrations modulate the length of the COMS’s optical resonator. By coupling light from a laser to the cavity at or close to the wavelength of a suitable optical resonance, the vibrations are imprinted on the transmission spectrum and thus can be detected all-optically. To estimate the field sensitivity the COMS based field sensor is modelled by a forced harmonic oscillator with a single mechanical eigenmode with eigenfrequency $\omega_m$, effective mass $m$, and quality factor $Q$. In this case the displacement of each volume element of the COMS $\vec{u}(\vec{r},t) = x(t) \vec{u}(\vec{r})$ is given by the product of the scalar displacement amplitude $x(t)$, and the spatial mode shape function $\vec{u}(\vec{r})$[@ref28].
![(color online). Maximum B-field sensitivity of three COMS based magnetometers. a) Cross-section of a TWGM resonator consisting of a Terfenol-D disc surrounded by an optical resonator (light blue). b) Cross-section of a Fabry-Perot resonator with Terfenol-D actuators that excite vibrational modes in a suspended micro-mirror[@ref38]. c) Top-view of an optomechanical zipper cavity attached to two Terfenol-D actuators[@ref39]. Red arrows represent movement of the Terfenol-D (in gray), black arrows mechanical vibrations.[]{data-label="fig1"}](OMmagnetometersPRL0501C.eps){width="8cm"}
In the frequency domain the motion of a forced harmonic oscillator driven at frequency $\omega$ is given by $x(\omega) = \chi(\omega) [F_{\rm sig}(\omega) + F_{\rm therm}(\omega) ]$, where $\chi(\omega)=[m (\omega_m^2 - \omega^2 + i \omega_m \omega/Q)]^{-1}$ is the mechanical susceptibility, $F_{\rm therm}(\omega) $ the Brownian noise force with the spectral density $\langle |F_{\rm therm}(\omega) |^2\rangle = 2 k T m \omega_m/Q$[@ref29], and $F_{\rm sig}(\omega)= F_{\rm sig} \delta(\omega-\omega_{\rm sig})$ an effective harmonic driving force at the signal frequency $\omega_{\rm sig}$. This force is generated by the time-dependent mechanical stress tensor ${\bf T}(\vec{r}) e^{i \omega_{\rm sig}t}$ induced by the external magnetic field, which produces the body force density $\vec{f}_{\rm sig}(\vec{r})=\nabla \cdot {\bf T}(\vec{r})$[@ref30]. The strength of the effective scalar driving force $F_{\rm sig}$ experienced by the mechanical mode depends on the overlap with the spatial mode shape function[@ref30; @ref31] and is given by $$F_{\rm sig} = \int \vec{f}_{\rm sig}(\vec{r}) \cdot \vec{u}(\vec{r}) dV. \label{2}$$ The displacement amplitude $x(\omega)$ shifts the resonance frequency of the optical resonator $\Omega_0$ by $\delta \Omega = g x (\omega)$, where $g$ is the opto-mechanical coupling constant[@ref28]. In an experiment this frequency shift is imprinted on the transmission spectrum of the optical cavity and measured with a spectrum analyzer. Including the spectral density of measurement noise $S_{\delta \Omega^2}^{\rm meas}(\omega)$ due to electronic noise, laser noise and thermorefractive noise, the resulting resonance frequency shift spectrum is $S_{\delta \Omega^2} (\omega)=g^2 \langle | x (\omega) |^2 \rangle + S_{\delta \Omega^2}^{\rm meas}(\omega)$. Expanding this expression we find $$\begin{aligned}
S_{\delta \Omega^2} (\omega) \! \!
% &=& \! \! g^2 \! \left | \chi (\omega)\right |^2 \left [ \left | F_{\rm sig}(\omega)\right |^2 \! \! \! + \! \left \langle \left | F_{\rm therm} \right |^2 \right \rangle \right ] \! + \! S_{\delta \Omega^2}^{\rm meas} (\omega) \nonumber\\
% \! \!
&=& \! \! g^2 \! \left | \chi (\omega)\right |^2 \left [ F_{\rm sig}^2 \delta(\omega-\omega_{\rm sig}) \! + \! \frac{2kT m \omega_m}{Q} \right ] \! + \! S_{\delta \Omega^2}^{\rm meas} (\omega) \nonumber,\end{aligned}$$ where the signal field is assumed to be single frequency at frequency $\omega_{\rm sig}$. The minimum detectable force $F_{\rm sig}^{\rm min}$ is obtained by integrating the signal and noise contributions over the bandwidth of the measuring system $\Delta \omega$ and determining the force at which the signal-to-noise ratio is unity with the result $$\frac{F_{\rm sig}^{\rm min}(\omega)}{\Delta \omega^{1/2}} = \left [\frac{2 k T m \omega_m}{Q} + \frac{S_{\delta \Omega^2}^{\rm meas} (\omega) }{g^2 \left | \chi(\omega)\right |^2} \right ]^{1/2}. \nonumber %\label{6}$$ Assuming a homogeneous magnetic field $B_x$ oriented in the $x$-direction and a suitably oriented magnetostrictive medium with only a single non-zero magnetostrictive coefficient $\alpha_{\rm mag}$ which causes the material to stretch in the same direction, the magnetostrictive induced stress tensor $\bf T$ has only a single component given by $T_{xx} = \alpha_{\rm mag} B_x$. Eq. (\[2\]) then yields the effective driving force that excites the mechanical eigenmode $$F_{\rm sig} = B_x \int \frac{\partial \alpha_{\rm mag}}{\partial x} u_x(\vec{r}) dV = B_x c_{\rm act}, \label{8}$$ where $u_x(\vec{r})$ is the x-component of $\vec{u}(\vec{r})$ and $c_{\rm act}$ characterizes how well the magnetic field is converted into an applied force on the oscillator and is referred to here as the magnetic actuation constant. The minimum detectable magnetic field can then finally be expressed as $$\frac{B_{x}^{\rm min}(\omega)}{\Delta \omega^{1/2}} = \frac{1}{c_{\rm act}} \left [\frac{2 k T \omega_m m}{Q} + \frac{S_{\delta \Omega^2}^{\rm meas} (\omega) }{g^2 \left | \chi(\omega)\right |^2} \right ]^{1/2}. \label{9}$$ The largest reduction is achieved at the resonance frequency $\omega_m$, where $\left | \chi(\omega)\right |$ is maximized. As expected, the isolation from the surrounding heat bath and the enhanced mechanical motion achieved by coupling the magnetostrictive material to an oscillator enhances the sensitivity. In the thermal noise dominated limit, the sensitivity scales as $m^{1/2}/c_{\rm act} Q^{1/2}$, while in the measurement noise limited regime it scales as $m/c_{\rm act} Q g$.
To estimate the achievable field sensitivity we consider three realistic scenarios based on different COMS, a TWGM resonator, an optomechanical zipper cavity[@ref39], and a miniature Fabry-Perot resonator with suspended micro-mirror[@ref38], as depicted in Fig. \[fig1\]a-c. In each case, the magnetostriction is provided by the rare earth alloy Terfenol-D which has a relatively large magnetostrictive coefficient of $\alpha_{\rm mag} = 5 \times 10^8$ N T$^{-1}$m$^{-2}$ at room temperature[@ref32; @ref33].The TWGM resonator based scenario is of most relevance to our experiments and consists of a central Terfenol-D cylinder of $24 \mu$m radius and $5 \mu$m height surrounded by a cylindrically shaped silica ring with $R = 30~\mu$m outer radius which constitutes the optical resonator. We consider a radial breathing mode with typical optomechanical coupling constant and mechanical quality factor of $g = \Omega_0/R = 6\times10^{19}$ m$^{-1}$s$^{-1}$[@ref28] and $Q=1000$ respectively, where the displacement amplitude $x$ is defined to correspond to the radial displacement at the outer surface of the ring. A finite element simulation (COMSOL) yielded the eigenfrequency $\omega_m = 2 \pi \times 5.6$ MHz and effective mass $m = 35$ ng[@ref28], and the magnetic actuation constant $c_{\rm act} = 0.19$ N/T was obtained by solving Eq. (\[8\])[@ref35]. A measurement noise level of $S_{\delta \Omega^2}^{\rm meas}(\omega)^{1/2} = 20$ Hz/Hz$^{1/2}$[@ref21] was chosen at an optical resonance frequency of $\Omega_0 = 1.8 \times10^{15}$ s$^{-1}$, as is typical for TWGM resonators. Fig. \[fig1\]a shows the resulting magnetic field sensitivity as predicted by Eq. (\[9\]), with a maximum predicted sensitivity of 700 fT Hz$^{-1/2}$ at resonance. By comparison, the peak sensitivities predicted for miniature Fabry-Perot resonators and optomechanical zipper cavities of similar spatial dimensions were 200 pT Hz$^{-1/2}$ and 10 pT Hz$^{-1/2}$, respectively (see Supplemental Information for calculations[@ref35]). The Fabry-Perot resonator based system was least sensitive, as the magnetostrictive material excites the mechanical modes of the suspended mirror indirectly and inefficiently by vibrating the entire mirror mount[@ref35]. The TWGM magnetometer shows the best sensitivity as the magnetostrictive material couples to the vibrating mode over its entire radius which results in the largest actuation constant $c_{\rm act}$ of the three systems.
![(color online). Experimental set-up showing a TWGM resonator with a Terfenol-D sample attached.[]{data-label="fig2"}](LayoutPRL0401B.eps){width="8cm"}
As a first proof of principal demonstration of a cavity optomechanical magnetometer, a piece of Terfenol-D (Etrema Products Inc.) with a size of roughly $50\times15\times10~\mu$m$^3$ was affixed to the top surface of a TWGM resonator using micromanipulators and two-component epoxy. The TWGM resonator had major and minor diameters of 60 and 6 $\mu$m, respectively, and a 10 $\mu$m undercut[@ref22]. Fig. \[fig2\] shows the experimental set-up. The TWGM resonator was placed between two 20 mm diameter coils that generated an HF magnetic field previously calibrated with a commercial Hall sensor. 980 nm light from a widely tunable, external cavity diode laser was passed through a fibre polarization controller and evanescently coupled to the resonator via a tapered optical fibre[@ref29]. The laser was thermally locked to the full width half maximum of an optical resonance[@Terry]. Mechanical vibrations of the TWGM resonator shift its optical resonances and thus modulate the transmitted light that is detected with an InGaAs photodiode. The transduction spectrum shown in Fig. \[fig3\]a was obtained by analyzing the transmitted light with a spectrum analyser. It shows multiple characteristic peaks corresponding to thermally excited mechanical modes. With a finite element solver the three vibrational modes around 10 MHz were identified as the lowest order flexural mode (centre) and the two lowest order crown modes. Application of a 250 $\mu$T magnetic field at $\omega_{\rm ref} = 2 \pi \times 10.38$ MHz resulted in a signal at that frequency above the Brownian noise confirming the ability of the sensor to detect magnetic fields. The magnetic field sensitivity was determined by $$B_x^{\rm min} (\omega_{\rm ref})/\Delta\omega^{1/2} = {\left | B_x \right|}/\! \!{\sqrt{{\rm SNR}(\omega_{\rm ref}) \, \Delta\omega_{\rm RBW}}}, \label{11}$$ where SNR is the signal to noise ratio, $B_x$ the applied magnetic field in direction $x$, and $\Delta \omega_{\rm RBW}$ the resolution bandwidth of the spectrum analyzer.
![(color online). (a) Brownian noise spectrum with magnetic excitation at 10.38 MHz. (b) System response as a function of applied B-field frequency. Inset: magnified system response centered around 10.385 MHz. Red curves: Lorentzian fits.[]{data-label="fig3"}](SAandNAPRL0401B.eps){width="7cm"}
To determine the sensitivity over a wide frequency range, a network analyzer was used to scan the frequency of the magnetic field from 7 to 13 MHz and simultaneously detect the system’s response $N(\omega)$. As expected, Fig. \[fig3\]b shows a Lorentzian line shaped peak distribution similar to the Brownian noise spectrum (Fig. \[fig3\]a), with an additional peak present around 11.5 MHz, due to the enhanced noise rejection of the phase sensitive network analyzer. In both cases the measured data agrees quite well with a model with five mechanical resonances[@ref29].
Importantly, the apparent large noise amplitudes visible in Fig. \[fig3\]b are stationary in time, and in fact corresponds to an ensemble of resonant features having $Q$-factors on the order of $10^4$. Two such features are shown in the inset. We believe that these resonances are due to ultrasonic waves in the Terfenol-D grains that are excited by a magnetostrictive mechanism. Similar resonances have been reported for other magnetostrictive materials with comparable $Q$-factors[@ref41; @ref42], though to our knowledge, this is the first such observation in Terfenol-D. Constructive and destructive interferences between neighboring resonances cause strong positive and negative variations in the spectrum. It is expected that the density of resonances could be greatly reduced by engineering the geometry of the Terfenol-D. For example, the simple geometry used in Ref. [@ref41] meant that only roughly two resonances per MHz were observed in agreement with theoretical predictions.
The sensitivity spectrum $B_{\rm sig}^{\rm min}(\omega)$ shown in Fig. \[fig5\] was obtained using the relation $B_{\rm sig}^{\rm min}(\omega) = \sqrt{S(\omega) N(\omega_{\rm ref})/N(\omega) S(\omega_{\rm ref})} B_{\rm sig}^{\rm min} (\omega_{\rm ref})$, with the reference sensitivity obtained from a measurement similar to that in Fig. \[fig3\]a using Eq. (\[11\]). Due to the Terfenol-D resonances, the sensitivity varies between $2 \mu$T Hz$^{-1/2}$ and 170 nT Hz$^{-1/2}$, depending on whether destructive or constructive interference occurs. At resonance we achieve a peak sensitivity of 400 nT Hz$^{-1/2}$ . As expected, this value is substantially above the 700 fT Hz$^{-1/2}$ predicted for an ideal system. In our first prototype the response of the Terfenol-D was transduced to the resonator through an epoxy layer and its geometry was not well controlled. As a result the overlap between the applied force and the material motion is expected to be quite poor. Furthermore, the mechanical modes best coupled were crown modes, which themselves do not couple with high efficiency to the optical field[@ref28]. These limitations could be overcome by controlled deposition of Terfenol-D via sputter coating. Further sensitivity enhancement may be achieved by using TWGM resonators with higher compliance such as is achieved with spoked structures[@ref34].
![(color online). B-field sensitivity as a function of frequency.[]{data-label="fig5"}](BminExpPRL0501B.eps){width="8cm"}
In conclusion, a cavity optomechanical magnetometer is reported where the magnetic field induced strain of a magnetostrictive material is coupled to the mechanical vibrations of an optomechanical resonator. A peak sensitivity of 400 nT Hz$^{-1/2}$ is achieved, with theoretical calculations predicting sensitivities in the high fT Hz$^{-1/2}$ with an optimized apparatus. This optomechanical magnetometer combines high-sensitivity, large dynamic range with small size and room temperature operation, providing an alternative to NV center based magnetometers for ultra-sensitive room temperature applications, low-field NMR and magnetic resonance imaging, and magnetic field mapping with high-density magnetometer arrays.
*Acknowledgements* This research was funded by the Australian Research Council Centre of Excellence CE110001013 and Discovery Project DP0987146. Device fabrication was undertaken within the Queensland Node of the Australian Nanofabrication Facility.
[99]{}
A. Edelstein, J. Phys.-Condes. Matter [**19**]{} 28 (2007). M. V. Romalis, and H. B. Dang, Materials Today [**14**]{} 258 (2011). F. Bucholtz, D. M. Dagenais, and K. P. Koo, Elec. Lett. [**25**]{} 1719 (1989). J. Zhai, Z. Xing, S. Dong, J. Li, and D. Viehland, App. Phys. Lett. [**88**]{}, 062510 (2006) H. B. Dang, A. C. Maloof, and M. V. Romalis, Appl. Phys. Lett. [**97**]{} 151110 (2010). D. Budker, and M. Romalis, Nat. Phys. [**3**]{} 227 (2007). L. M. Pham et al., New J. Phys. [**13**]{} (2011). L.-S. Bouchard, V. M. Acosta, E. Bauch and D. Budker New J. Phys. [**13**]{} 025017 (2011) ; L. T. Hall et al., PNAS [**107**]{} 18777 2010. S. Xu et al., Proc. Nat. Acad. Sci. [**103**]{} 12668 (2006)l; M. P. Ledbetter et al., Phys. Rev. Lett. [**107**]{} 107601 (2011). A. Laraoui, J. S. Hodges, C. Ryan, C. A. Meriles Phys. Rev. B [**84**]{} 104301 (2011). M. Hämäläinen, R. Hari, R. J. Ilmoniemi, J. Knuutila, and O. V. Lounasmaa, Rev. Mod. Phys. [**65**]{} 413 (1993). G. Balasubramanian et al., Nat. Mat. [**8**]{} 383 (2009). J. R. Maze et al., Nature [**455**]{} 644 (2008). M. S. Grinolds et al., Nat. Phys. [**7**]{} 687 (2011). R. S. Schoenfeld, and W. Harneit, Phys. Rev. Lett. [**106**]{} 030802 (2011). A. Schliesser et al., New J. Phys. [**10**]{} 19 (2008). T. Antoni et al., Opt. Lett. [**36**]{}, 3434 (2011). T. P. M. Alegre, R. Perahia, and O. Painter, Opt. Exp. [**18**]{} 7872 (2010). T. G. McRae et al., Phys. Rev. A [**82**]{} 023825 (2010). A. J. Maker, and A. M. Armani, Opt. Lett. [**36**]{} 3729 (2011). F. Xia, L. Sekaric, and Y. Vlasov, Nat. Phot. [**1**]{} 65 (2007). A. Schliesser, and T. J. Kippenberg, Adv. Atom. Mol. Opt. Phys. [**58**]{} 207 (2010). K. H. Lee et al., Phys. Rev. Lett. [**104**]{} 123604 (2010). A. N. Cleland, Foundations of nanomechanics: from solid-state theory to device applications (Springer, 2003). T. Briant et al., Phys. Rev. A [**68**]{} 033823 (2003). J. Verhoeven et al., Metal.and Mat. Trans. A [**21**]{} 2249 (1990). R. Kellogg, and A. Flatau, Smart Structures and Materials 1999: Smart Structures and Integrated Systems, Pts 1 and 2 3668, 184 (1999). See Supplemental Material http: . T. G. McRae, K. H. Lee, M. McGovern, D. Gwyther, and W. P. Bowen, Opt. Exp. **17** 21977 (2009). H. Ogi, J. App. Phys. [**82**]{} 3940 (1997). J.-I. Park et al., Sensors and Actuators a-Physical 140, [**84**]{} (2007).
G. Anetsberger et al., Nat. Phot. [**2**]{} 627 (2008).
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